Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
IntechOpen is proud to announce that 191 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\n
Throughout the years, the list has named a total of 261 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
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\n\t\t\t
1. Introduction
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Radar monitoring of thick snow cover in polar regions with optical thickness (a product of the depth of the snow and the extinction coefficient) of order 1 or more from elevated above-ground and space platforms is of great importance for registration and for understanding glaciology processes caused by climatic change. Applicable to this issue, the volume component of the backscatter coefficient has a notable contribution to the total backscatter (Noveltis, 2005). Although the idea of the radar-cross section (RCS) term and its derivatives, like the backscatter coefficient, aims to separate as far as possible the sensor (radar) and target parameters, this distinction can rarely be fulfilled in the case of spatially extended geophysical targets (SEGT), such as atmospheric clouds and rain, as well as the thick snow cover that is the focus of this paper. Due to this feature, the analysis of backscatter from SEGT strongly depends on the relationship between the technical and physical-geometrical properties of the radar and target, respectively. The main parameters that govern the radar-target configuration for snow sounding are wavelength, antenna characteristics, pulse duration, sounding direction, extinction coefficient and the geometrical depth of the snow. A correct assessment of the volume component of the backscatter coefficient and an understanding of the realm of applicability of any backscattering model is possible only when the size of the scattering volume within a snow slab is known. This parameter depends on the factors mentioned above and should be distinguished for different situations in the practice of radar sounding of snow-covered terrain by Synthetic Aperture Radar (SAR), a scatterometer or an altimeter. This work attempts, at first, to determine the radar-target configurations inherent to volume scattering estimations of the thick snow cover under the different radar applications mentioned above. Next, we analyze the range of applications of the incoherent approach for backscatter magnitude estimation, currently one of the main techniques for snow characteristics assessment. The simplest incoherent approach, based on the so-called “particle” or “discrete” approximation, leads to the dependence of backscatter on the sixth moment of the particle size distribution function (PSDF) and the mean amount of particles in the scattering volume (Siegert & Goldstein, 1951, Battan, 1959, Ulaby et al., 1982). For a medium with losses (such as a thick snow slab) the modification of this approach is referred to as the semi-empirical model (Attema & Ulaby, 1978, , Ulaby et al.1982, Ulaby et al., 1996). There is some evidence of less backscatter occurring than expected by virtue of the conventional reflectivity factor and the backscatter coefficient for incoherent scatter from homogeneous thick snow slab (e.g., Rott et al., 1993). For dense media (mean distance between particles is less than the wavelength) there are many references in the literature that the backscatter is determined by media inhomogeneities (Naito & Atlas, 1967, Gossard & Srauch, 1983, Fung, 1994). To attempt to evaluate the contributions of these inhomogeneities, the radiative transfer (RT) and the dense media radiative transfer (DMRT) models were developed (e.g., Ulaby et al., 1982, Fung, 1994, Tsang et al., 2007). A review of these models and their modifications provided by Noveltis (2005) stated that in certain aspects, these approaches showed some successes. However, these models have not the close-form solutions that make difficult to use them and analyze results obtained. Kendra et al. (1998), based on experiments with artificial snow of varying depths, concluded that both conventional and dense-medium radiative transfer models fail to adequately explain the observed results.
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Finding an appropriate and relatively simple approach to calculate the volume component of backscatter to explain the observable deviations from the classical (incoherent) model is therefore a relevant task. The majority of previous studies, focusing on measurements of a dry snow at temperate latitudes, where the snow depth seldom exceeds 1 m and the corresponding optical thickness is much less than 1, have led to the conclusion that such an approach is not feasible because of the much weaker interaction of the electromagnetic radiation of radar wavebands with ice particles within the snow compared with the backscatter from the soil beneath. The current study focuses on the case of the Greenland ice sheet, however, where the depth of snow significantly exceeds that found in temperate latitudes. This favorable condition, in conjunction with the recent finding that the incoherent approach is only a specific case of a more comprehensive description of electromagnetic wave interactions with spatially-extended individual scatterers (Yurchak, 2009), provides a reason for more detailed investigations of the possibilities of the semi-empirical model for interpreting the observable features of radar backscatter from thick snow cover.
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2. Condition of complete burial of the probing pulse into snow medium
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To estimate the volume component of backscatter from a snow slab, it is first necessary to understand whether the probing pulse is completely buried within the snow slab or whether the illuminated volume is only part of the probing pulse volume. In weather radar meteorology, applicable to rain and clouds, this problem is known as “partial (or incomplete) beam filling,” with a corresponding factor included in the weather radar equation (e.g., Clift, 1985). For sounding thick snow cover, this issue is practically not discussed and is different compared with the sounding of meteorological targets. Complete burial depends on the sounding configuration and snow slab depth. The main criterion for complete burial is the location of the pulse scattering volume with angular size equal to the angular antenna beam width and with spatial length equal to one half of the actual spatial duration of the transmitted pulse into a snow layer. Thus, for complete burial, the angular and radial sizes of the backscatter volume should be matched with the snow slab depth, which is usually known only roughly for a particular geographical region. This condition poses problems for practical applications. Nevertheless, to better understand possible situations where complete burial is feasible, the requirements for the angular antenna beam size and the probing pulse duration will be analyzed separately for different major sounding configurations.
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2.1. Condition of complete burial of the angular (transverse) size of the pulse volume
The scheme of this configuration is shown in Figure 1 Here, and everywhere below, the snow slab is assumed to have limited depth and an unbounded horizontal extent.
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Figure 1.
Scheme of sounding for flat surface and plane wavefront
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Mathematically, the condition for complete burial in this case is:
where d\n\t\t\t\t\t\tb is the burial depth of the advance point (p) of pulse edge (leading or trailing) when the lagging point (c) reaches the surface and h is the snow depth. As follows from the geometry of Figure 2.1:
\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∠\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, and hence:\n\t\t\t\t\t\t
For φ0.5<<1 one can state\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, and condition (2.1) has the form:
If h~10 m and R\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t=800 km, then \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t1.25\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t5\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tand for θ~400, for example, the burial condition (2.1) is satisfied at roughly φ\n\t\t\t\t\t\t0.5<<0.0010. Obviously, this condition can not be fulfilled for any space-based radar system with a real aperture, but it is possible for the SAR if one assumes that the effective synthetic aperture radar beam illuminating an element of spatial resolution Δx on a flat surface can be equal to an extremely narrow, pencil-like beam with\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. For example, RadarSAT-1 has Δx ~12.5 m with an orbit height of approximately 800 km and thus, φ\n\t\t\t\t\t\t0.5~0.9*10-3 degrees. If one assumes that the typical size of the main lobe of the conventional antenna pattern is of order ~10, the complete burial of the transverse size of the pulse volume is possible for\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. This condition can definitely be fulfilled for airborne and above-surface elevated radars. The estimate obtained above should be considered only a rough approximation, because at nadir sounding (θ=0) and unbounded horizontal extent of snow slab, the wavefront can not be considered planar for the assessment provided. A more precise estimate is given below.
A real wavefront within a snow slab has a spherical shape. The impact of this shape on the estimation of d\n\t\t\t\t\t\tb as the look angle θ (Figure 1.) decreases begins at the moment when the line tangent to the spherical front at point p coincides with the horizontal line pg, Figure 2. This situation takes place when θ becomes equal to the angle between pc and the tangent:
As follows from geometrical sketch, the auxiliary angle β is equal to: \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tand, thus,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. Therefore, the impact of sphericity should be taken into account only for small look angles, when
Thus, the estimate (2.3), provided in the previous subsection, is valid for\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. An estimate for configurations close to nadir sounding will be carried out in the next subsection.
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2.1.3. Nadir sounding, flat surface spherical wavefront
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Figure 3.
Schematic of sounding in nadir direction
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This configuration is shown schematically in Figure 3 and the corresponding condition for complete burial is:
This estimate can be accepted for elevated platforms (for example, airborne radars and those mounted above the surface) but needs to be analyzed further for space platforms due to the sphericity of the surface, as the corresponding area illuminated on the Earth’s surface by conventional radar is large.
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2.1.4. Nadir sounding, spherical surface, spherical wavefront
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In the spherical surface approach, the depth of complete burial in altimeter mode is more than that for a flat surface described above by an increment Δz in the center of the beam. This effect is illustrated in Figure 4, and can be written as:
where\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, and y is the distance between the vertical axis and the beam periphery touching the spherical surface. Solving this system, the increment due to the sphericity is equal to:
where R\n\t\t\t\t\t\tE is the Earth’s radius. Taking into account that \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t<< 1, \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, and that the minus sign should be taken before the square root for physical reasons, the above formulae can be simplified significantly, yielding an increment equal to:
A numerical assessment under the parameters values listed above gives Δz~5 m. The estimation of the parameter \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t can be represented in the form:
This relationship coincides with the estimate obtained by Barrick (1972) for a spherical wave increment over the spherical mean sea surface. For R\n\t\t\t\t\t\t0=800 km, R\n\t\t\t\t\t\tE=6400 km and\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, one calculates \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t≈45 m. This result means that the Earth’s sphericity increases the complete burial parameter compared with the flat surface case (see 2.7) by a factor of ~(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t). For \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t8\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t the complete burial parameter equals to 1.125, i.e., a 12.5% increase.
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Based on known assessments of penetration depth of 2-5 m for Ku-band (e.g., Davis, 1996), the wave sphericity in practice does not allow for consideration of the complete burial of the transversal size of the probing pulse for a space-based altimeter and scatterometer. This sphericity causes the area illuminated on the surface to change as the probing pulse is buried into a snow slab. It can be easily shown that the radius of the illuminated area changes with the burial increment ΔR as\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
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2.1.5. Wavefront flattening
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The speed of an electromagnetic wave within a snow slab is lower than that in an air by\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, where ε’ is the real part of the dielectric permittivity of snow. For dry snow within a density range of 0.2-0.5 g/cm3, parameter ε’ changes roughly from 1.35 to 1.95 (Tiuri et al., 1984). Recently, a decrease in wave speed was also confirmed by direct measurements in snow (Scott et al., 2006). Due to the sphericities of the Earth and the wavefront, the paths of the wave for different rays into a snow slab for equal time are not the same that results from the distortion of the initial spherical wave front, as demonstrated in Figure 5.
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Figure 5.
Illustration of the flattening of a spherical wavefront within a snow slab
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The peripheral ray (ray Ob, for example) has a different path into a snow slab as compared with the central ray (Og), which that has a portion ag extending into the snow. This feature causes the flattening of the spherical wavefront. Thus, the snow slab works like dielectric lens antenna (e.g., Lo & Lee, 1993). Let us make a rough estimate of the impact of this phenomenon on the increment Δz. For a spherical wavefront (dashed curve in Figure 2.5), the following equation is valid: Ob=Oa+au+ug. Dividing both sides of this equation by speed of the wave propagation, c, yields the time of the wavefront arrival at point g:\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
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Due to delays in snow in part of ag=au+ug the equation for the wavefront arrival at point g is:\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, where v is the wave speed in snow. Because v<c, \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, providing proof of this flattening. To rate this phenomenon, let us suppose that the flattening front crosses the vertical at point u. This condition implies that the wave arrival times at points u and b are the same. This yields the equations: \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tand\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. Hence, the relative shortage (flattening) of the wavefront path in the center of the beam compared with propagation in the free space (air) is equal to:\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, and the corrected increment Δz (2.11) should be decreased by \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t~1.2…1.4 times.
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Moreover, due to the increase in snow density with depth, the dielectric permittivity also increases, additionally impacting the rays’ path configuration as takes place in a Luneburg lens. Thus, the actual meaning of the burial parameter lies between the two estimates carried out above:
Based on the above estimates, one can say that the surface sphericity practically contributes a relatively small amount to the estimate of the complete burial condition of the transverse dimension of the probing pulse, and consequently, the surface can be roughly considered as flat for any radar configurations regarding volume component assessment.
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2.1.6. Modes of volume scattering for spherical wave
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By analogy to spherical wave scattering from a surface (e.g., Moore & Williams, 1957) it is also reasonable to distinguish two modes of the volume component in the case of spherical wave scattering, as shown in Figure 6. For simplicity, the bounds of the snow slab are considered flat. Let us call the mode depth-limited (Figure 6a.) if at the moment when the central point of the spherical wavefront reaches the bottom of a snow slab or a penetration depth (D\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t), the peripheral point of the wavefront, crossing the surface, is still within the beamwidth footprint on the surface. This condition takes place when:
Modes of the volume scattering; a) depth-limited, b) beamwidth-limited
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Let us call the mode beamwidth-limited (Figure 6b.) if the peripheral points of the spherical wavefronts, crossing the surface, are beyond the beamwidth footprint on the surface for a significant part of the scattering volume. This situation takes place when:
Obviously, that the depth-limited mode is inherent to space based altimeter and scatterometr and the beamwidth-limited mode is realized for SAR. If the “significant part” in the definition of this mode is replaced with a more exact term, the “determined part,” the condition (2.15) can be modified and written in the form:
where \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tor \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t is the relative part of the scattering volume in which the beamwidth-limited mode is fulfilled. For example, if 90% of the scattering volume (\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t=0.9) is under the beamwidth-limited mode, the condition (2.15a) is \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0.2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tfor\n\t\t\t\t\t\t\t
2.2. Condition of complete burial of radial size of the pulse volume
\n\t\t\t\t
Suppose that some area of snow cover is illuminated by a radar located in orbit as shown in Figure 7. For simplicity, this scenario is depicted with a plane wavefront and flat surface.
\n\t\t\t\t
Figure 7.
Assessment of the complete burial of radial size of the probing pulse
\n\t\t\t\t
Let us denote the height of the orbit as R\n\t\t\t\t\t\n\t\t\t\t\t\t0\n\t\t\t\t\t, the distance from the radar to the center of the area as R, the anglular width of the main lobe of the antenna pattern as φ\n\t\t\t\t\t\n\t\t\t\t\t\t0.5\n\t\t\t\t\t and, the look angle as α. In addition, we assume that the cross-section of the main lobe is the circular. The illuminated area is assumed to be flat horizontally, and the bottom boundary surface of the snow slab is also flat and parallel to the top surface. Thus, the incidence angle, θ is equal to the look angle, α (θ=α).
\n\t\t\t\t
The main scaling parameters determining radial propagation are:
\n\t\t\t\t
1) The radial length of the pulse scattering volume, equal to half of its spatial extent in the snow medium:
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the wave propagation speed in snow, c is the wave propagation speed in the air, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the real part of the snow permittivity and τ\n\t\t\t\t\t\n\t\t\t\t\t\t0\n\t\t\t\t\t is the duration of the probing pulse;
\n\t\t\t\t
2) A one-way path in the snow, where the incidence power is decreased by “e” times, usually called the “penetration depth” in the literature
where k\n\t\t\t\t\t\n\t\t\t\t\t\te\n\t\t\t\t\t is the extinction coefficient, which characterizes the attenuation properties of the medium due to scattering and absorption. One important remark is necessary. Since radar sounding of snow is often performed in off-nadir mode, and the main lobe of the antenna pattern has a finite angular size, it should be
\n\t\t\t\t
Figure 8.
Relationships between the penetration depth (D\n\t\t\t\t\t\t\tp), penetration path (L\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t) and propagation depth (\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t) in a snow slab
\n\t\t\t\t
underlined that the extinction coefficient and the “penetration depth” are measured along the direction of the wave (ray) propagation and not only to the vertical. Therefore, calling the term \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∝\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t as the “…depth” should be considered a little bit confusing. A more appropriate name for this term, from a physical point of view, is the “penetration path” (L\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t), keeping in mind that the penetration depth (D\n\t\t\t\t\tp) is its value in the vertical direction. The depth reached by an electromagnetic wave propagating at an angle θ to the vertical and attenuating by “e” times represents the vertical component of the penetration path, and can be called the propagation depth\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, as illustrated in Figure 8. Thus, from this point onward in this paper, the following definitions and relationships are used:
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t↓\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the extinction coefficient in the vertical direction. In general, in inhomogeneous medium, the penetration path is a function of distance and direction\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where ψ is the azimuthal angle. For homogenous medium\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. We consider the probing pulse to be short if\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t; otherwise it is considered long. Obviously, the mode of sounding with current wave (CW) always belongs to the long probing pulse configuration. Also one should distinguish a case with fully scattering snow slabs, when the geometrical snow depth is equal to or less than the propagation depth (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t). Otherwise, one has a case of partially (because the wave does not penetrate to the bottom) scattering slow slab. The condition for complete burial, as it follows from Figure 7, is
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the slant snow depth and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
All of the above conditions are summarized in Table 1.
\n\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
#
\n\t\t\t\t\t\t\t
Condition
\n\t\t\t\t\t\t\t
Description
\n\t\t\t\t\t\t\t
Comments
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
1
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Short/long probing pulse (P.P.)
\n\t\t\t\t\t\t\t
Complete burial is satisfied for: short P.P. and fully scattering snow slab; long P.P. and partially scattering snow slab
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
2
\n\t\t\t\t\t\t\t
or \n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Fully/partially scattering snow slab
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
3
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Complete burial of the radial size of the probing pulse
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
Table 2.1. Conditions for complete burial of the radial size of the probing pulse
\n\t\t\t\t\t\t
\n\t\t\t\t\t
Table 1.
Conditions for complete burial of the radial size of the probing pulse
\n\t\t\t\t
Concluding remarks:
\n\t\t\t\t
If the conditions for complete burial of the radial size of the probing pulse are completed, it can at least be said that one sample of the return signal is formed by the scattering volume
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the illuminated base of the scattering volume. If, in addition, the conditions for complete burial of the transverse size of the probing pulse (see previous subsection) are also satisfied, that the scattering volume is determined by the entire pulse volume:
2.3. An assessment of the scattering volume under incomplete burial condition
\n\t\t\t\t
If the parameters of the probing pulse and of the snow slab do not satisfy the conditions presented in the table above, the radial size of the scattering volume is determined by the geometry of the snow slab. Several practically important cases for practical application are discussed below.
\n\t\t\t\t
\n\t\t\t\t\t
2.3.1. Flat surface, plane wavefront, long probing pulse and fully scattering snow slab
\n\t\t\t\t\t
In this scenario, the illuminated area, A\n\t\t\t\t\t\t\n\t\t\t\t\t\t\till\n\t\t\t\t\t\t on the snow cover (Figure 1.) is an ellipse, with the minor semi-axis equal to the radar beam cross-section radius \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and major semi-axis the same divided by the cosine of the incidence angle:\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. Thus:
If one uses the height of the satellite orbit\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, (2.23) transforms to:
The pattern of the scattering volume for this case is depicted in Figure 9, where the scattering volume is bounded by the surface and bottom planes.
\n\t\t\t\t\t
Figure 9.
An assessment of the scattering volume for a fully scattering snow slab
\n\t\t\t\t\t
In this case, the size of the scattering volume is the sum of the volume of a circular cylinder V\n\t\t\t\t\t\t0 with cylindrical element d\n\t\t\t\t\t\t0 and two similar volumes of the truncated cylinders V\n\t\t\t\t\t\t1 that mutually add up to a completed circular cylinder with cylindrical element d\n\t\t\t\t\t\t1 (the bases of all volumes are the same and equal to\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tmin\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t):
Because\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, we have, taking expression (2.25) into account:
Thus, the equivalent scattering volume is an elliptic cylinder with a base equal to the illuminated area on a surface and with a height element equal to the slab depth (under the assumption that the slab depth is less than the half of the spatial duration of the probe radar pulse). This result was obtained by Matzler (1987).
\n\t\t\t\t\t
The duration of the SAR probing pulse duration is equal to several tens of microseconds (e.g., 42 μs for RadarSAT-1 and 37.1 μs for ERS-1). Due to frequency chirp, the compressed probing pulse duration decreases by many times, resulting in a volume radial size equal to only several meters (5-13 m for RadarSAT-1 and 9.7 m for ERS-1, for example). Numerical data are provided based on Alaska Satellite Facility documents (“RadarSAT-1 Standard Beam SAR Images”, 1999; and “ERS-1 and ERS-2 SAR Images”, 1996).
\n\t\t\t\t\t
Figure 10.
Cross-section of the scattering volume for a spherical surface and wavefront. The position of the flattened wavefront at the moment when its central point (a’) reaches the penetration depth is depicted with a convex dashed line
\n\t\t\t\t\t
Because SAR images represent the backscatter pulse train from the entire path of a wave into a snow slab for any look angle as a point on a surface of some mean brightness (return power), the radial size of the scattering volume can be considered equal to either the slant size of the snow layer or the penetration path (whichever is smaller). That is, although the SAR pulse has a finite compressed spatial duration; it works like a long pulse due to the absence of radial discrimination in the sense of conventional radar terminology.
For this case, the bottom of the scattering volume is the geometrical placement of points located at a distance L\n\t\t\t\t\t\tp from the snow surface along the family of rays within the solid angle of the main antenna lobe, as shown in Figure 10.. The defining feature of this configuration is that the central and peripheral points of the spherical wavefront pass the length L\n\t\t\t\t\t\tp at different times, resulting in different radii of curvature for the top and bottom bounded surfaces.
\n\t\t\t\t
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
3. Volume component of the backscatter coefficient
\n\t\t\t
\n\t\t\t\t
3.1. Semi-empirical model for flat surface, plane wave and long probing pulse
\n\t\t\t\t
Based on the considerations provided above, one can formulate an estimation of the volume component of the backscatter within the framework of the so-called semi-empirical model (Attema & Ulaby, 1978), as for the case of incomplete burial of the probing pulse with a flat surface, plane wave and a long probing pulse, which better fits the conditions for sounding of a thick snow slab with SAR.
\n\t\t\t\t
The total radar backscatter from the illuminated area is composed of four components (Fung, 1994):
where σ\n\t\t\t\t\t\n\t\t\t\t\t\tas\n\t\t\t\t\t is the radar cross-section (RCS) due to backscatter from the air/snow (top) interface, σ\n\t\t\t\t\t\n\t\t\t\t\t\ts\n\t\t\t\t\t is the RCS due to backscatter from the snow volume, σ\n\t\t\t\t\t\n\t\t\t\t\t\tg\n\t\t\t\t\t is the RCS due to backscatter from the snow/ground (bottom) interface and σ\n\t\t\t\t\t\n\t\t\t\t\t\tgv\n\t\t\t\t\t is the RCS due to rescattering between ground and snow volume irregularities. As was summarized by Koskinen (2001), only the volume component is essential for dry snow. Therefore the essential portion of the total backscatter consists of the volume component and the contribution of the ground beneath:
Obviously, the radar becomes sensitive to the properties of the snow only when\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. In accordance with Ulaby et al. (1982), the general form for the backscatter coefficient from a surface is:
This notation presupposes that the illuminated areas on the slab top and on the ground (bottom of the slab) are the same. This assumption applies for small angle divergences of the radar beam, as is valid for strong directed antennae, low refraction on the air-snow interface and a ground surface that is flat and parallel to the surface of the snow slab. Accordingly, the volume component of the backscatter coefficient is:
In the incoherent approach,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t; i.e., σ\n\t\t\t\t\t\n\t\t\t\t\t\ts\n\t\t\t\t\t represents the total radar cross-section (RCS) of N scatterers contributing backscatter from a volume \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t of a snow slab (see subsection 2.3.1). Due to attenuation of the electromagnetic wave upon propagation within a snow mass, the total RCS of the snow slab is equal to:
where the attenuation coefficient, inherent to the i-th particle with a distance of ξ\n\t\t\t\t\t\n\t\t\t\t\t\ti\n\t\t\t\t\t from the coordinate origin O, is equal to:
where k\n\t\t\t\t\t\n\t\t\t\t\t\te\n\t\t\t\t\t is the extinction coefficient. The factor α takes into account the two-way distance of forth and back wave propagation. The assessment of the scattering volume described above relates only to the absolute value of the volume. In the case of wave directed propagation and, consequently, directed attenuation, the summation should be performed along the propagation axis, ξ. Because N>>1, the summation in (3.5) can be replaced by integration. As was mentioned above, this integration should be carried out along the direction of wave propagation, i.e., along the ξ axes. Small changes in the propagation direction at the air/snow interface are ignored due to minor differences in the corresponding refraction coefficients. Due to the random spatial distribution of scatterers within the scattering volume, this discussion considers only the mean backscatter characteristics averaged over several illuminated areas, with particles having independent spatial locations and RCSs.
where σ(ξ) is the running RCS (by the unit of a distance along axis ξ, dimension is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t) of the scattering volume. Assuming that the attenuation properties of snow remain the same along the propagation path\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, one can write:
Taking into account that \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t≠\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the radial size of the scattering volume (along axis ξ ), \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the mean total RCS of snow (ice) particles within the scattering volume (an oblique cylinder) while ignoring the attenuation and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the corresponding mean volume specific backscatter coefficient, we can transform the ratio \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tto the form:
Taking into account (3.8) and assuming the homogeneity of snow slab\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, we can conduct the integration of (3.12) and finally arrive at:
This expression is the mathematical formulation of the semi-empirical model and has been obtained by Attema & Ulaby (1978) and Ulaby et al. (1982). In this paper, this model will be cited as the “A-U model.” For this case, the backscatter coefficient (3.13) depends only on the specific volume backscatter of the snow medium and not on the pulse volume sizes. This remarkable feature is due to (1) “overcomplete” burial of the pulse length into the snow medium, and (2) the backscattering normalization factor A\n\t\t\t\t\till (see 3.4) is the basis of the probing pulse.
\n\t\t\t\t
The obvious imperfection of the model, as applied to sounding of thick snow, is the assumption of a constant extinction coefficient within the snow slab. The problem can be solved by designing an appropriate stratification model for the selected study area and modeling the spatial distribution of the extinction coefficient (e.g., Drinkwater et al., 2001).
\n\t\t\t\t
The next limitation of the model is due to wave sphericity. Under depth-limited mode conditions (see section 2), the illuminated area changes, and the running RCS in (3.7) can not be assumed to be statistically homogeneous within the scattering volume. The same is true regarding the extinction coefficient in (3.8) as well. Thus, for this case, the A-U model in form (3.13) should be used with care. For the beamwidth-limited mode, the form in (3.13) can be used, taking in mind the “determined part” of the scattering volume (see comments to 2.15a).
\n\t\t\t\t
The ground component of the backscatter coefficient is localized by the bottom location and can thus be expressed by its backscatter coefficient, taking into account the attenuation:
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the backscatter coefficient of the ground bottom surface governed only by the surface properties and its orientation in regards to the incidence of radar illumination..
where H\n\t\t\t\t\t\n\t\t\t\t\t\teff\n\t\t\t\t\t is the effective depth of snow sounding (EDS); i.e., the depth such that the backscatter from which occurs as if without attenuation, with backscatter equal to that which would occur from a slab of larger real depth with attenuation due to absorption and scattering.
\n\t\t\t\t
Parameter \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the optical thickness of snow (SOT) along the wave propagation direction. It is useful to express the backscatter coefficient through the dimensionless EDS using the geometrical snow depth, h:\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Given that expression, equation (3.16) can be written in the form:
is the maximal value of the backscatter coefficient for a given snow depth, h, ignoring both energy losses due to attenuation and scattering and the angular dependence of the scattering volume. The normalized effective depth of snow sounding (nEDS≡\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t) plays the role of a correction factor, and is equal to:
It is useful to find the dependence of the correction factor on the SOT magnitude and the incidence angle. Let us consider two extreme cases:
\n\t\t\t\t
a) “shallow” snow: \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t
meaning that the EDS is approximately equal to the geometrical snow depth. Nevertheless, the small magnitude of the volume component compared with the backscatter from the bottom layer (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t) makes this dependence difficult to use in practice. For example, a snow layer with h<~1 m is practically transparent to electromagnetic irradiance of the C-band.
\n\t\t\t\t
b) “thick” snow: \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t
where\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Equation (3.24) shows that the backscatter depends only on the penetration path (depth) under a constant incidence angle. In the “thick” snow regime, no additional snow accumulation contributes to the total backscatter due to the saturation effect. On the other hand, in this regime, the backscatter coefficient demonstrates an angular dependence, as the angle of incidence affects the propagation depth (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t). The sensitivity of the backscatter coefficient to the changes in snow thickness takes place in the so-called “intermediate” regime, when \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t Field data gives 20-30 meters of the penetration depth for the C-band (e.g., Hoen & Zebker, 2000), and yield values that can be considered the “working” range for probable snow measurements in this wave band. Values of the nEDS (3.19) can be assessed from the plot of the nEDS as a function of SOT, as shown in Figure 11.
\n\t\t\t\t
Figure 11.
Plot of the normalized effective depth of snow sounding versus the snow optical thickness
\n\t\t\t\t
To evaluate the angular dependence of the nEDS, we consider the normalized Snow Depth (nSD), which is the depth of snow normalized to the penetration path:
A plot of (3.27) reduced to zero dB at θ=0 for different nSD is shown in Figure 12.. As follows from this plot, the angular dependence of the volume component of the backscatter coefficient in a practically appropriate range of angles, 0-500 is rather weak and equal to ~2dB for the thick snow regime. For the intermediate regime, with nSD~1, its range of variation is about 1 dB. Because this range is of the same order as the errors, it is difficult to expect a notable angular dependence in practical measurements inherent to the intermediate regime.
\n\t\t\t\t
Figure 12.
The angular dependence of the normalized effective depth of snow sounding for the “thick” (nSD>>1) and “intermediate” regimes
\n\t\t\t\t
On the other hand, a notable angular dependence of the backscatter coefficient indicates that snow depth is greater than the penetration path. The intermediate regime represents practical interest for snow measurements. Suppose, for instance, the linear dependence of the geometrical snow depth on a horizontal distance S over the flat surface:
where k is the proportionality factor. Given that relation and taking into account (3.18) - (3.20), the normalized backscatter coefficient as a function of a distance is:
To specify the proportionality factor, let us assume that the geometrical snow depth reaches the propagation depth value at a distance, for example, equal to S\n\t\t\t\t\t\n\t\t\t\t\t\t0\n\t\t\t\t\t, i.e.,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Thus,
Illustration of the expected relative spatial behavior of the backscatter coefficient under a linear dependence of the snow depth on a distance
\n\t\t\t\t
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the normalized distance. For the intermediate regime, we can ignore the angular dependence of the backscatter coefficient. Let us also assume that the specific volume scattering is approximately the same for all points along that distance. The corresponding plot of expression (3.27), additionally normalized by a factor of\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, is depicted in Figure 13.:
Thus, in accordance with the semi-empirical model, the expected profile of the backscatter coefficient should have an exponential pattern for a transect taken over terrain with a linear increase in snow depth within the intermediate regime condition.
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
4. Comparison the A-U model estimates and the experimental radar data on the Greenland ice sheet
\n\t\t\t
For accurate comparison of the model with any experimental data, the study area must satisfy the “thick” snow condition to exclude the impact of the ground. Second, the snow pack must contain primarily Rayleigh ice particles with sizes less than approximately one-tenth of the wavelength. The most appropriate area matching this condition is the dry snow zone of the Greenland ice sheet. This region is not impacted by melt-freezing processes due to permanent sub-zero temperatures year-round. Therefore, the auspicious conditions for creation the ice lenses, tubes and large size aggregates, which inherent, for instance, to the percolation zone, are absent here. Additionally, the snow (firn) is up to several tens of meters deep (e.g., Zwally & Brenner, 2001), corresponding to the intermediate or even the thick snow regime. Due to known difficulties regarding in situ measurements in Greenland, there are no examples of the comparison of backscatter with snow depth in the literature for the intermediate or thick snow regimes.
\n\t\t\t
Figure 14.
Transect across hill (straight line crossing the dark spot on the SAR signature) in the north-eastern region of Greenland (left), and the corresponding backscatter (solid line, convex downwards curve) and elevation (dotted line, convex upwards curve) profiles (right)
\n\t\t\t
Nevertheless, a certain qualitative example of the sensitivity of backscatter to the snow depth relative to the intermediate regime can be provided based on a topographic feature (a hill) in the northeast region of Greenland between 810 and 820 N, as shown in Figure 14..
\n\t\t\t
The counter-phase behavior of the backscatter coefficient and the elevation profiles plotted above can be explained supposing that the snow depth is minimal on the hilltop and increases proportional to the distance from the top of the hill down to its base. This assumption closely matches the field and modeled data provided by Jaedicke et al. (2000).
\n\t\t\t
For quantitative assessment of the semi-empirical model more regular terrain should be chosen. A typical dry snow area is presented on the RadarSAT-1 SAR image of eastern Greenland in Figure 15..
\n\t\t\t
As is clearly seen, the magnitude of the backscatter coefficient is about -20 dB in the dry snow zone. This assessment coincides with data in the known literature (e.g., Drinkwater et al., 2001, Partington, 1998; Baumgartner et al., 1999, Forster et al., 1999) which also gives values of -12…-20 dB. We can estimate the corresponding value using the A-U model. We assume that the thick snow regime is valid, and that the snow slab consists primary of particles of roughly the same sizes; i.e., it is a monodisperse medium.
\n\t\t\t
For these conditions (see 3.24), ignoring backscatter from the bottom surface, we get:
\n\t\t\t
Figure 15.
Illustration of the backscatter coefficient values (left) range for an arbitrary transect (straight line on RadarSAT-1 SAR signature, right) from the percolation to the dry snow areas within the east margin of the Greenland Ice Sheet (directions are from left to right for the plot of the backscatter profile and contra versa for the SAR signature; date of data acquisition: November 2, 2007; data granule ID: R1_62609_SWB_271)
Further calculations below aim to discover the specific volume backscatter, σ\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t and the extinction coefficient, k\n\t\t\t\t\n\t\t\t\t\te\n\t\t\t\t. The estimates listed below will be conducted with the Rayleigh approach. In accordance with Ulaby et al. (1986), this approach bounds the upper electric size of particle, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, with the inequality\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where m is the complex refractive index of the particle matter and r\n\t\t\t\tu is the corresponding upper size of a particle. For ice particles with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t observed with C-band radar (λ=5.6 cm), the maximal size satisfying the Rayleigh approach is 2.6 mm. Following known relationships (e.g., Ulaby et al, 1986), the specific volume scattering is:
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t˙\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t1.7776\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the refractive index of the particle matter and n and η are the real and imaginary parts of the refractive index, respectively. The value of these components can be derived from the value of the complex dielectric permittivity of a material\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where ε’ and ε” are the real and imaginary parts of the dielectric permittivity.
where n\n\t\t\t\t\n\t\t\t\t\t0\n\t\t\t\t is the particle concentration.
\n\t\t\t
The extinction coefficient can be calculated as a sum of the absorption (k\n\t\t\t\t\n\t\t\t\t\ta\n\t\t\t\t) and scattering (k\n\t\t\t\t\n\t\t\t\t\ts\n\t\t\t\t) coefficients (Ulaby et al., 1986):
where Q\n\t\t\t\t\n\t\t\t\t\ta\n\t\t\t\t and Q\n\t\t\t\t\n\t\t\t\t\ts\n\t\t\t\t are the absorption and scattering cross-sections respectively. These parameters, under the Rayleigh approach, can be calculated by the following formulas:
The real part of the dielectric permittivity for ice, contained in the formulas above, is equal to 3.15 and in practice does not depend on temperature or wavelength. The imaginary part of the dielectric permittivity for ice is not constant with changes in temperature or the illumination frequency. In accordance with Matzler (1987) it can be expressed through the following empirical formula:
where F is frequency in GHz and A, B and C are the empirical coefficients. For a temperature of -150C, which is more appropriate for the dry snow case, the coefficients are equal to: A=3.5*10-4, B=3.6*10-5, C=1.2. Now, the expressions obtained for the backscatter coefficient calculation may be combined. Substituting (4.2) in (4.1), we get:
An important feature of this expression is its independence with regards to the particle concentration, a reflection of the property inherent to the saturation regime mentioned above in section III. Since the scattering volume is less than the spatial duration of the probing pulse an increasing particle number is equivalent to an increasing particle concentration. Thus, no additional amount of the snow over the snow pack with a thickness greater than the penetration depth can cause a notable increase in the backscatter coefficient. This circumstance is also very useful from a simulation point of view, since there is no microstructure parameter that needs to be assumed. For the numerical calculation listed below we assumed that the snow within the dry snow zone consists of snow grains that are actually grains of ice that look like grains of rice, with a typical size of ~1 mm. Taking into account the fact for ice: ε’=3.15 and ε’>> ε” (Tiuri et al., 1984, Matzler, 1987),\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t″\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Given that information, equation (4.15) for an intermediate incidence angle of ~400 (cosθ=0.77) is equal to:
The plot of (4.17) is depicted in Figure 16. for C-band radar with F=5.3 GHz (λ≈ 5.6 cm).
\n\t\t\t
Figure 16.
The dependence of the backscatter coefficient on particle size in accordance with the A-U model for C-band radar and snow depth greater than the penetration depth (“thick” snow regime)
\n\t\t\t
It follows from this plot that the observable backscatter coefficient values of -12…-20 dB can be caused by particles with radii of less than 0.35-0.17 mm. The current estimates are slightly lower compared with the same made by Partington (1998), who found that the backscatter coefficient would range from -20 dB for a mono-distribution of grain sizes with a mean radius of 0.25 mm to -2 dB for a mean grain radius of 1 mm. Our results showed a 0.17 mm mean radius for -20 dB and a 1 mm mean radius for -3 dB. The small discrepancy is probably due to the temperature dependence of the imaginary part of the ice’s dielectric permittivity. The grain sizes values assessed do not coincide with the field data. In accordance with Table 2. the mean particle radius within a ~1 m depth surface layer of dry snow is 0.5…2.0 mm. As seen in Figure 16.., these values should produce a backscatter coefficient of approximately from -8 dB to 0 dB. Thus, there is a discrepancy of ~12 dB between the observable data and the values forecast by the semi-empirical model. Therefore, the A-U model overestimates the backscatter coefficient significantly. It should be noted, in addition, that the particle radius can only increases with the snow depth due to the depth-dependent grain radius model (Alley et al., 1982):
where r\n\t\t\t\t\n\t\t\t\t\t0\n\t\t\t\t is the mean radius at the surface, C is the crystal growth rate and H\n\t\t\t\t\n\t\t\t\t\ta\n\t\t\t\t is the mean annual layer thickness. Given this model, various simulation experiments (e.g., Forster et al., 1999, Drinkwater et al., 2001) and field measurements (e.g., Jezek et al., 1994, Woods, 1994; Lytle & Jezek, 1994), the mean particle radius for the entire scattering volume is even greater than the surface values and the aforementioned discrepancy becomes even more firmly grounded.
\n\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
#
\n\t\t\t\t\t\t
Mean ice particle radius, mm
\n\t\t\t\t\t\t
Source
\n\t\t\t\t\t\t
Location
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
1
\n\t\t\t\t\t\t
0.5-2.0
\n\t\t\t\t\t\t
Benson, 1996
\n\t\t\t\t\t\t
Greenland
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
2
\n\t\t\t\t\t\t
1.0-1.5
\n\t\t\t\t\t\t
Schytt, 1964
\n\t\t\t\t\t\t
Spitsbergen
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
3
\n\t\t\t\t\t\t
Less than 0.75 mm
\n\t\t\t\t\t\t
Higham and Craven, 1997
\n\t\t\t\t\t\t
Antarctic
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
4
\n\t\t\t\t\t\t
0.2-0.6
\n\t\t\t\t\t\t
Woods, 1994
\n\t\t\t\t\t\t
Greenland
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
5
\n\t\t\t\t\t\t
0.1-0.7
\n\t\t\t\t\t\t
Lytle and Jezek, 1994
\n\t\t\t\t\t\t
Greenland
\n\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t\t\t\t
Table 4.1. Literature data on the mean ice particle size near the surface of dry snow cover
\n\t\t\t\t\t
\n\t\t\t\t
Table 2.
Thus, although the A-U model gives a qualitative assessment of the main features of backscatter behavior from a snow slab, the result of quantitative comparisons with field data does not match the theoretical predictions.
\n\t\t\t\n\t\t
\n\t\t
\n\t\t\t
5. Enhanced semi-empirical model of the volume component of the backscatter coefficient
\n\t\t\t
The problem highlighted in the previous section can be resolved by considering the statistical properties of the small-scale fluctuations in the particle concentration and its scattering properties within the scattering volume. The appropriate method for doing so that is the so-called “slice” approach, which was primarily suggested in weather radar meteorology (Marshal & Hitchfeld, 1953; Smith, 1964). This approach exploits the known radar feature in accordance with which the particles of a “cloud” located close to the front of the incident radar wave are considered to be approximately at the same distance from the radar and reflect incident electromagnetic wave almost coherently. One can consider that these particles are embedded in a fictitious thin cylindrical volume (“slice”), whose base coincides with the surface of spherical wave front and side-bounded by the main lobe of the antenna pattern. Thus, the scattering volume into a snow slab can be represented as an adjoining series of these slices, as illustrated in Figure 17.. Each slice is much narrower than the radar wavelength in the wave propagation direction (Δs<<λ).
\n\t\t\t
Figure 17.
Simplified slice approach scheme. The arbitrary position of a single slice is shown on the cross section of the scattering volume
\n\t\t\t
A slice’s radial size, Δs can actually be considered as a minimal spatial scale of the backscatter property fluctuation, if this scale is much smaller than the wavelength. Using this approach, it was shown by the author (Yurchak, 2009), that the specific volume component of the backscatter from any spatial extended geophysical target (SEGT), included the snow as well, can be presented as the incoherent sum of the radar cross-sections of individual scatterers only if the number of particles in the slices (n) are distributed in accordance with the Poisson law; i.e., the variance of the number of particles (Var(n) ) is equal to the mean number (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0.16\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t). Otherwise, the “classical” specific volume component should be corrected by a so-called deviation factor Y(ξ\n\t\t\t\t\n\t\t\t\t\ta\n\t\t\t\t\n\t\t\t\t, χ):
\n\t\t\t\t\n\t\t\tand \n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t is the Poisson index, \n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\tis the variation coefficient of the particle radar equivalent length (PREL≡\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t, σ\n\t\t\t\n\t\t\t\tp\n\t\t\t is the radar cross-section of an individual particle). Formulas (5.1) and (5.2) reflect the fundamental physical principal that the fluctuations of the medium parameters (inhomogeneities) are the cause of the electromagnetic wave scattering (e.g., Atlas, 1964, Fabelinskii, 1968). The parameter ξ\n\t\t\t\n\t\t\t\ta\n\t\t\t might be expressed through the measurable parameters of the snow particle size distribution function (PSDF), such as the variation coefficient of particle radius, \n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t, and the skewness coefficient, Sk:\n\t\t\t\t
Following Shi et al. (1993), if the PSDF for snow can be presented by the lognormal distribution, the parameters of relationship (5.3) are:\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, and\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Given these relations,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Thus, the PREL variation coefficient (5.3) depends, via ξ\n\t\t\t\t\n\t\t\t\t\tr\n\t\t\t\t only on the geometric standard deviation, S\n\t\t\t\t\n\t\t\t\t\tr\n\t\t\t\t. The corresponding plot of the deviation factor (5.2) is depicted in Figure 18. for different values of the Poisson index χ.
\n\t\t\t
Figure 18.
Deviation factor versus particle radius variation coefficient for different deviations of Poisson index χ from the Poisson law
\n\t\t\t
As can be concluded from analysis of the plot, the deviation of ~12 dB emphasized in the previous section can be explained by the homogeneous microstructure of the dry snow zone, with approximately identical particle sizes (ξ\n\t\t\t\t\n\t\t\t\t\tr\n\t\t\t\t<0.1) and negligible variations of the particle concentration (χ<<1) within the scattering volume. As an example of data supporting the current considerations, the results of comparative measurements of C-band backscatter from firn and in situ measurements of statistical parameters of firn dielectric permittivity for a study area in Antarctic (Zahnen et al., 2002) can be taken into account. In this study, dielectric permittivity was considered to be a measure of the heterogeneity of the snow/firn pack. A notable positive correlation was observed between backscattering signal strength and the standard deviation of the dielectric constant collected along 4 and 15 meters depth interval of several drills within a dry firn area. At the same time, a negative correlation takes place for a mean dielectric constant. Because the dielectric constant is approximately linearly proportional to the density of dry snow (e.g., Tiuri et al., 1984), and therefore linearly proportional to the number of ice particles per unit volume as well, these correlations match the behavior of the deviation factor (5.2) on the Poisson index. The index and, hence, scattering increase with increasing variations in particle number and decrease with increasing\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Of course, more comprehensive field data are needed to confirm this hypothesis.
\n\t\t\t
The considerations provided above are related to the simplest case of complete burial of the probing pulse into a snow slab in the beamwidth-limited mode. Obviously, that the result is still the same for incomplete burial of the radial size of the scattering volume. The transverse size change of a slice within the scattering volume in the depth-limited mode results in an additional gradient in particle number among slices, and the contribution of this additional gradient to the backscatter should be analyzed in the future.
\n\t\t
\n\t\t
\n\t\t\t
6. Normalized snow depth
\n\t\t\t
\n\t\t\t\t
6.1. Derivation of the normalized snow depth from the backscatter coefficient based on the enhanced semi-empirical model
\n\t\t\t\t
After enhancing the semi-empirical model by the “slice” approach, one resume assessment of the snow depth, the key parameter in all mass-balance studies. To derive this parameter, let us to write down the enhanced semi-empirical model (ESEM) in a form more fitting for the Greenland ice sheet:
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the backscatter coefficient from the ice layer, L\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t is the penetration path (2.18), Y is the deviation factor (5.2), and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the slant snow depth. Let us also denote: \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tas the saturation backscatter coefficient. Given these expressions, the normalized Snow Depth (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, see (3.25)) as a function of distance S along snow cover with a variable snow depth, h, can be derived from (6.1) to take the form:
Here, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t to simplify the notation. If \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tln\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t then
The equation above indicates that the estimated normalized snow depth depends on number and kind of scatterers (σ\n\t\t\t\t\t\n\t\t\t\t\t\t∞\n\t\t\t\t\t) and their statistical characteristics (Y) at small (less then wavelength) scales. Thus, to extract the nSD from measured radar data, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, one needs to know the product of the deviation factor and the saturation coefficient. To find the physical snow depth, the penetration path must also be known. It should
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Figure 19.
Typical backscatter coefficient profile across a transect within the margin of the Greenland Ice Sheet (in the vicinity of the Swiss camp)
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be underlined, additionally, that the nSD \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is not quite the same as the snow optical thickness\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Obviously,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tln\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. nSD is a combination of the electromagnetic wave direction (L\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t) and vertical geometrical (h) scaling factors of the snow depth. These two terms coincide only for sounding in the nadir direction.
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From the point of view of practical measurements, detailed analysis recommends stopping the nSD estimate at a distance when\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. It should also be noted also that the deviation factor (Y) might be less or more than unit (Figure 18.). Because the product \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is generally unknown, it is reasonable to establish its value as ~2 dB higher than the value of the backscatter profile plateau shown, for instance, in Figure 19. The backscatter profile has the main features inherent to the theoretical profile in the linear approach, as shown in Figure 13..
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Figure 20.
The normalized snow depth profile derived from the backscatter coefficient profile in Figure 19.
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The profile has an exponential-like increase at the beginning of transect that transitions into a plateau-like, approximately constant level. Most probably, this level represents the saturation level caused by thick snow of depth greater than the propagation depth. The corresponding nSD is shown in Figure 20.
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The nSD profile plotted in Figure 20. is obtained under the assumption that the deviation factor and the extinction coefficient are constant across the entire profile distance. Although the approach demonstrated here does not give the absolute value of the snow depth without certain assumptions, its advantage is that the assumptions are clearly delineated and put forth. In turn, if one has a calibration point (spot), the technique allows extension of the remote sensing methodology on vast remote areas with similar electromagnetic properties. In addition, the nSD is a relative indicator of the snow environment stability and redistribution.
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The nSD in form (6.3) is derived from the ESEM, which in turn is based on consideration of a flat surface, plane wavefront and long probing pulse. Hence, the result obtained is most appropriate for SAR sounding of a thick snow slab.
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6.2. An example of time transformation of nSD profiles within a selected Greenland marginal area
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The normalized snow depth is sensitive to the geometrical snow depth and to the extinction coefficient. Assuming the snow microstructure to be homogeneous within a limited area, one can suppose that nSD depends only on the snow depth and thus apply nSD for snow mass-balance assessment. An illustrative example of an nSD profile change after a 10-year period is shown in Figure 21..
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Figure 21.
Study area #3 (box with center: 75.070N and 54.300W) within the Greenland ice sheet margin with several transects normal to the coastline (left). Normalized snow depth profiles (right) for transect 3a (lower line in the study area) derived from RadarSAT-1 data. Distance scale: 1 sample=50 m. Solid line is nSD profile on 11/16/1997, dotted line is nSD profile on 11/12/2007
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The nSD profiles were derived from the RadarSAT-1 data archive from the Alaska Satellite Facility in Fairbanks (granules R1_10617_SWB_261 from 1997 and R1_62753_SWB_261 from 2007 with coordinates of the center scene 75.230N/54.290W and 75.220N/54.370W respectively, descending modes for both). As follows from the center locations of the data, the SAR shots were performed from practically the same point in the orbit and thus can be analyzed in parallel without additional corrections.
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A comparative assessment of the nSD profiles indicates notable differences between the profiles. Generally, these changes can be caused snow mass redistribution within the 10-year period, variation of the extinction coefficient due to possible snow metamorphism, or both factors simultaneously. The detailed analysis of these factors and their “weights” in the deviations of these profiles is beyond the scope of this work but is planning for further consideration.
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7. Model spectral dependence of the backscatter
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7.1. Spectral dependence of the backscatter in accordance with the A-U model
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The spectral (frequency) dependence of the backscatter coefficient within the framework of the semi-empirical model (A-U and ESEM) is the topic of the current section. The aim of this discussion is to find out how spectral features depend on the thick snow properties. Particularly, Davis & Moore (1993), namely assumed that a two-frequency radar system can provide quantitative estimates of snow physical properties.
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The model consists of two components that are definitely frequency-dependent: the specific volume component (4.2) and the normalized effective depth of sounding (3.19). Transforming the wavelength (λ) into the frequency\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, the combination of these two components gives, for the “intermediate” snow regime:
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the slant snow depth;\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The dimension of function (7.2) is [L-3].
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7.2. Estimation of spectral differences based on field penetration depth data
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For further analysis, it is convenient to deal with the dimensionless spectral function, which can be obtained by multiplying the spectral core (7.2) by the cube of the wavelength:
To evaluate (7.3) the spectral dependence of the penetration path (depth) should be estimated. One can state that the high frequency irradiance interacts primarily with the surface layer of snow with the relative small particles. And the low frequency irradiance penetrates deeper and, thus, is affected by particles which sizes are larger in accordance with the grain growth model (e.g., 4.18). For example, the grain radius changes from 0.25 to 0.6 mm within the depth of 10 m is in model of Drinkwater et al., (2001). In model of Forster et al., (1999) the range of the particles’ changes is even more (up to 1 mm) and additionally depends on the accumulation rate. Thus, to get more realistic spectral dependence, we should take into account the depth of penetration of the electromagnetic wave into the snow and the corresponding grain size change within this depth. Therefore, there is an obvious paradox: to calculate frequency dependence of the penetration depth we should know the latter a priory. Moreover, due to chain: “frequency-penetration depth-particle size” the mean particle radius impacted the radar irradiance turns out to be dependent on the irradiance frequency. In additional, the A-U model was derived assuming the constant extinction coefficient, as it was mentioned in section 3. The solution of these questions should be evaluated in further research. Taking in mind the restrictions listed above, the penetration depth is assumed to be dependent only on frequency in the following estimation. Since the significance of the particles’ size change compared with the fixed mean size in the extinction coefficient calculation is currently unclear, it is reasonable to take the field data available in the literature (Table 3.) for rough assessment the spectral dependence of the penetration depth.
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#
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Frequency, GHz
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Penetration depth, m
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Literature source
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1
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5 10 20 35
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18 9 4.3 1.5
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Hofer & Matzler, 1980
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2
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5 10 20 35
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20 10 4.8 1.4
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Matzler, 1987
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\n\t\t\t\t\t\t
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3
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5.2 6.6 10.3 10.7 18.0 37
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21.7 17.8 8.1 10.4 3.2 0.85
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Rott et al., 1993
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4
\n\t\t\t\t\t\t\t
10
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3.4±1.3
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Devis & Poznyak, 1993
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\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
5
\n\t\t\t\t\t\t\t
13.5
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3.5±1.5
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Davis, 1996
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\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
6
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5.3
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27±4
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Hoen & Zebker, 2000
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Table 3.
Literature data on the experimental penetration path (depth)These field data can be approximated by a simple empirical power function
with the coefficient of determination equals to 0.9. In (7.4) the penetration path is in meters and frequency is in GHz. It should be noted that field data, provided in the first three rows of Table 3., are obtained based on measurements within the surface layer of the snow (up to 3 m). It means that these data also do not take into account adequately the actual stratification of the thick dry snow cover of the Greenland ice sheet and, therefore, the further estimation should be considered only as a first approximation. A plot of (7.3) is shown in Figure 22. This plot demonstrates a notable discrimination in behavior of the spectral function B(F,h\n\t\t\t\t\t\n\t\t\t\t\t\tθ\n\t\t\t\t\t) for different snow depths. Particularly, the magnitudes of the curves at a frequency of 5 GHz are notably different. Differences of the backscatter magnitude for frequencies of 5 and 14 GHz as a function of the slant snow depth are depicted in Figure 23.
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Figure 22.
Dimensionless spectral characteristics of the backscatter for several values of the slant snow depth based on the empirical spectral dependence of the penetration depth
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Figure 23.
Slant snow depth as a function of the magnitude difference between the dimensionless spectral components of the backscatter coefficients at 5 and 14 GHz
Spectral pattern of the experimental data of (Baumgartner et al., 1999) for an incidence angle of 400 for two test sites in Greenland
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where the slant snow depth is in meters and the spectral difference \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is in dB. An illustrative example of the spectral difference observed in Greenland can be provided from the known literature. Baumgartner at al. (1999) conducted field measurements of the backscatter coefficient from the dry snow site GITS (77.10N, 610W) and from the site NASA-U (730N, 50.50W) located between the dry snow and percolation zone for four wavelengths, 5.3, 10, 13.5 and 17 GHz, at temperature from -210C to -150C, with the ground-based radar. The primary goal of this experiment was to study the angular dependence of the backscatter. For our purposes, the spectral dependence related to the incidence angle of 400 (where the backscatter is expected to be due mainly to the volume scattering) can be extracted for the four points and is shown in Figure 24. Taking into account that\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where μ is an unknown coefficient that we assume to be the same for all wavelengths, one can obtain the spectral ratio or the spectral difference in the logarithmic units for a pair of frequencies F1 and F2:
Differences in the backscatter coefficients of these frequencies for both test sites are approximately the same and equal roughly to -13 dB, as shown in Figure 24.. Assuming the backscatter error of ~±0.5 dB, the corresponding spectral difference (from 7.7) is approximately equal to 0.4±0.5 dB. In accordance with (7.5), the slant snow depth is ~6.7-20.8 m. Since these data are related to an incidence angle of 400, the estimate of actual snow depth is 5.1-15.9 m. This value is in accordance with a note in the paper that the depth of snow was ~7-8 meters for the NASA-U site, at least.
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Thus, a rough assessment of the snow depth within several selected dry snow areas, based on the spectral dependence of the backscatter in the frames of the SEM (ESEM), gives results comparable with in-situ measurements.
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7.3. Spectral characteristics of the penetration depth for model monodisperse cold snow
\n\t\t\t\t
It is interesting to compare the empirical (7.4) and theoretical frequency dependence of the penetration path (depth). Calculation of the extinction coefficient has been completed in the same manner as in section 3, with the additional estimation of particle concentration n\n\t\t\t\t\t\n\t\t\t\t\t\t0\n\t\t\t\t\t (see (4.8)):
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the fractional volume of ice in snow (0.2-0.4), ρ\n\t\t\t\t\t\n\t\t\t\t\t\ts\n\t\t\t\t\t and ρ\n\t\t\t\t\t\n\t\t\t\t\t\ti\n\t\t\t\t\t are density of snow and ice, respectively. We also should demarcate the frequency and particle size inherent to the Rayleigh approach (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t30\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tlg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\tlg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t). The penetration depth (path) in snow is calculated for several combinations of snow density and particle size for a temperature of -150C. The result of this calculation is shown in Figure 25. together with the empirical dependence (7.4). The gently sloping of the empirical curve compared with the theoretical ones indirectly support the idea, mentioned above, about the contribution of the particle size in spectral behavior of the penetration depth (see the corresponded comments in the previous subsection).
\n\t\t\t\t
In Figure 7.5, the results obtained from these calculations for T= -150C and the snow density of 0.3 g/m3 are compared with the known values (Ulaby et al., 1986, Fig. 19.64, p.1608) for T= -10C and density of 0.24 g/m3. The plot indicates the increasing of the penetration depth at low temperatures for the low frequency region, a trend most obvious for small particles of 0.5 mm radius. This behavior is explainable because the losses in snow caused by absorption decrease as the temperature decreases (Ulaby et al., 1986, Fig.Е.3, p. 2027), and the absorption mechanism prevails over scattering in the low frequency region. Pointed changes become more apparent for small particles due to more significant role absorption plays for them as compared with large particles.
\n\t\t\t\t
Figure 25.
Theoretical spectral dependence under the Rayleigh approach of the penetration depth for a temperature of -150C (our calculations) with comparison of the empirical relationship (7.4)
\n\t\t\t
\n\t\t
\n\t\t
Figure 26.
Comparison of the spectral dependence of the penetration depth calculated with Mie formulas for a temperature of -10С (Ulaby et al., 1986) with our calculations under the Rayleigh approach for a temperature of -150С
\n\t\t
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8. Backscatter depth profile model for snow sounding with broadband pulses of short effective (compressed) length
\n\t\t\t
\n\t\t\t\t
8.1. Phenomenological approach in the case of very short probing pulse
\n\t\t\t\t
The above considerations apply when the scattering volume was less then the probing pulse volume. In the case of altimeter sounding of thick snow cover with short broadband chirp pulses this condition is not satisfied, the backscatter coefficient is represented not by the point of a radar signature but in the form of a backscatter profile along the wave propagation path. When \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t5\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t14\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t5\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t14\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t13.41\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, the volume component of the total backscatter from a snow area with the center located at a slant distance of R beneath the surface should simply mirror the backscatter coefficient depth distribution taking attenuation into account:
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t˙\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the total radar cross-section (RCS) of N scatterers contributing backscatter from a scattering volume V\n\t\t\t\t\t\n\t\t\t\t\t\tsct\n\t\t\t\t\t from a distance R. Since the scattering volume is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, and taking the correction on the deviation factor into account (see section 5), the equation (8.1) can be presented in the form:
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the specific volume backscatter, and the return signal is proportional to the cosine of the incidence angle and has the envelope of the exponential form distorted by the variability of the statistical, scattering and absorption properties of the snow medium along the wave propagation path. The illuminated area A\n\t\t\t\t\till (transverse size of the scattering volume) changes during the burial of the probing pulse into the snow slab due (mainly) to the wave sphericity in the depth-limited mode (see section 2). However, this effect does not impact the accuracy of the specific volume backscatter coefficient presented here, due to its the mutual cancellation in the nominator (via V\n\t\t\t\t\tsct) and the denominator of equation (8.2). As a result of this cancellation, for very short probing pulses, the backscatter envelope depends only on changes in the specific volume backscatter with distance.
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
8.2. Analytical derivation for commensurate length of the scattering volume and snow depth
\n\t\t\t\t
In general case, when \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, the interactions between a probing pulse with a normalized form U(R) and an extended target such as snow cover is described by the convolution integral (Moore & Williams, 1957):
where P\n\t\t\t\t\t\n\t\t\t\t\t\tr\n\t\t\t\t\t\n\t\t\t\t\t(t) is the return signal, P\n\t\t\t\t\t\n\t\t\t\t\t\tT\n\t\t\t\t\t\n\t\t\t\t\t(t) is the transmitted pulse profile and P\n\t\t\t\t\t\n\t\t\t\t\t\ts\n\t\t\t\t\t\n\t\t\t\t\t(t) is a term that includes the distribution range of the scattering facets, their scattering properties and the effects of the antenna gain. As applied our consideration of volume backscatter, one can write
If, for example, the probing pulse has a Gaussian form, which is usually an adequate approximation for short pulse systems (e.g., Brown, 1977), the weighting function of the scattering volume is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where Δ\n\t\t\t\t\t\tq\n\t\t\t\t\t is the spatial duration of the scattering volume weight function at level\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. For level q=0.5 (-3 dB), β=-4ln0.5=2.8. If, in addition, the microstructural, statistical and attenuation properties of the snow slab do not change along the propagation path within the snow pack, one can write:
For short pulses, if their envelope can be represented by the delta-function, i.e., \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, equation (8.5) yields the corresponding result obtained earlier (8.3) for the very short pulses configuration. For the long pulses, where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, which yields to the standard form of the A-U model (3.13). For the general case, a closed-form solution of the equation (8.5) can be obtained.
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the error function, one can get the integral of equation (8.5) for a Gaussian probing pulse:
If one denotes\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, then (8.5) has the form:
Finally, if one denotes: \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t(see 3.24), \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, the closed-form solution of equation (8.6) using these dimensionless parameters can be written as:
Distance normalization in (8.9) is chosen through the parameter S for the purpose of comparing the backscatter profile calculated here with the modeled “clear” backscatter profile derived for a very short pulse (8.2) and governed only by attenuation. The equation obtained is close to the volume component of the SV model of Davis & Moore (1993). The distinctions of the current approach are that it uses a slightly different description of the pulse-snow interaction, the end result is written in terms of the backscatter coefficient depth profile, and the state of homogeneity of the snow mass is taken into account through the deviation factor, Y. It is interesting, that equation (8.9) coincides with the backscatter model from the sea surface, obtained by Barrick (1972), exactly by the form with, of course, other meaning of model parameters.
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
8.3. Comparison of the results obtained with the phenomenological approach
\n\t\t\t\t
\n\t\t\t\t\t
8.3.1. Very short probing pulse
\n\t\t\t\t\t
For the limit case when\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, and taking into account that \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t (Abramowitz & Stegun, 1972), the corresponding backscatter coefficient is:
This result confirms relationship (8.2) obtained above based on phenomenological considerations.
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8.3.2. Long probing pulse
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Taking into account the asymptotic value of the error function \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t (Abramowitz & Stegun, 1972), and taking only the first term of the extension, we get the value of (8.9) for γ>>1:
For long pulses, the available distance variation is much less than the pulse length, i.e.,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tlim\n\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. Hence,
This expression also confirms the result of the A-U model (enhanced) under thick snow condition (3.24). Thus, an important positive feature of the model (8.9) is that its results coincide with those obtained with phenomenological approaches for extreme values of the scattering volume (much less and much more than the penetration path respectively). Nevertheless, it should be noted that ignoring the change in the illumination area during the burial of the probing pulse into snow media is, strictly speaking, valid only for a short probing pulse or for the beamwidth-limited mode. For long and intermediate-length probing pulses with wavefront sphericity in the commonly-existing depth-limited configuration, this effect should be taken into account because there are slices (see sections 2 and 5) of different transverse sizes within\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
\n\t\t\t\t
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8.4. Numerical calculation of the return signal for intermediate length of the probing pulse
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A plot of the normalized backscatter coefficient \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t (S\n\t\t\t\t\tmax is the normalized distance where function (8.7) has the maximum value at a given γ) is shown in Figure 27.
\n\t\t\t\t
Figure 27.
Illustration of the reproducibility of the snow backscatter uniform profile taking into account only attenuation (bold dash line) and finite probing pulse length normalized to the penetration depth (parameter\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t).
This figure illustrates the contribution of the pulse length to the return signal envelope pattern. The shorter pulse clearly gives a better approximation of the real backscatter profile. Contemporary altimeters use a probing pulse with a physical duration of 20-50 microseconds and a corresponding effective duration (due to intrapulse frequency chirping) equal to 3.125 nanoseconds (e.g., Rosmorduc et al., 2006). In spatial units, these values are equivalent to \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tmax\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t∗\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t≈0.5 m. For L\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t~10-50 m, the expected values of the parameter γ range from 0.05 to 0.01. An important feature of this plot is that the value of the maximum of the return signal for any maximum position (S\n\t\t\t\t\tmax) and values of parameter γ are the same:
The semi-empirical model of the volumetric component of the backscatter coefficient, proposed by Attema & Ulaby (1978) (SEM or A-U model) is considered in more detail. The basic advantage of the SEM is its mathematical simplicity and clear physical sense but the SEM, derived using the incoherent approach, does not explain some remote sensing experimental data collected over the dry snow zone of the Greenland ice sheet. We discuss the main configurations of radar sounding of snow, including the relationships between the technical and positional parameters of radar on one hand and the geometrical and electromagnetic properties of snow cover on the other. Based on this consideration, the field of applicability of the SEM is figured out for the further assessments. Some qualitative assessments of the configuration of the scattering volume were carried out using plane and spherical surface approaches. More detailed study of the diffraction of space-bounded spherical electromagnetic wave on a large and strongly absorbing dielectric sphere with a rough surface is needed to evaluate this problem further. Detailed SEM estimates and experimental radar data over the dry snow zone of the Greenland ice sheet are provided for the negative temperatures inherent to the winter seasons. Data analysis shows a significant discrepancy between the expected and actually measured values of the backscatter coefficient. Possible reasons for the observed deviations of the SEM results are discussed based on the enhanced semi-empirical model (ESEM) and analysis of the dependence of the backscatter coefficient on the snow optical thickness and its spectral function. The ESEM is obtained on the basis of the so-called "slice" approach, taking into account partially-coherent backscattering due to finite wave front size and considering the contribution of the statistical characteristics of small-scale fluctuations of concentration and of sizes of the ice particles in a snow pack. The known semi-empirical model is only a special case of the ESEM. Using the ESEM, some observable radar signatures of the Greenland ice sheet, including the dry snow zone, can be interpreted in terms of the statistical characteristics of the snow microstructure. The model obtained retains the mathematical simplicity and physical clarity of the known A-U model and significantly expands its application for the interpretation of experimental data. Available field data, which confirm the consistency of the ESEM with observed values of the backscatter coefficient, are also provided. As a consequence of this analysis, the concept of the normalized snow depth (nSD) is formulated, and its practical application for monitoring snow mass displacement is described. The frequency-dependent core of the backscatter has been analyzed as well. Empirical and theoretical spectral characteristics of the penetration depth are also presented. The result allow for improvements in our understanding of opportunities for a two-frequency method of sounding thick snow covers and promote more accurate interpretations of the known experimental data. The extension of the ESEM for sounding thick snow cover with broadband pulses of short effective length is also discussed, resulting in an exact description of the return signal envelope as a function of the electromagnetic and microstructural characteristics of the snow cover. The validity of the model is supported by the coincidence of the model estimates with assessments obtained for several specific cases using phenomenological approach. In conclusion, this work demonstrates that the enhanced version of the semi-empirical model can be applied successfully in the analysis of radar backscatter from thick snow cover.
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Acknowledgments
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This work was supported by NASA’s Cryospheric Sciences Program. The author would like to thank Dr. W. Abdalati for his permanent attention to this work and useful discussions.
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\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/9554.pdf",chapterXML:"https://mts.intechopen.com/source/xml/9554.xml",downloadPdfUrl:"/chapter/pdf-download/9554",previewPdfUrl:"/chapter/pdf-preview/9554",totalDownloads:3141,totalViews:140,totalCrossrefCites:5,totalDimensionsCites:6,totalAltmetricsMentions:0,impactScore:2,impactScorePercentile:77,impactScoreQuartile:4,hasAltmetrics:0,dateSubmitted:null,dateReviewed:null,datePrePublished:null,datePublished:"October 1st 2009",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/9554",risUrl:"/chapter/ris/9554",book:{id:"3343",slug:"advances-in-geoscience-and-remote-sensing"},signatures:"Boris Yurchak",authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Condition of complete burial of the probing pulse into snow medium",level:"1"},{id:"sec_2_2",title:"2.1. Condition of complete burial of the angular (transverse) size of the pulse volume",level:"2"},{id:"sec_2_3",title:"2.1.1. Oblique sounding, flat surface, plane wavefront",level:"3"},{id:"sec_3_3",title:"2.1.2. Oblique sounding, flat surface, spherical wavefront",level:"3"},{id:"sec_4_3",title:"2.1.3. Nadir sounding, flat surface spherical wavefront",level:"3"},{id:"sec_5_3",title:"2.1.4. Nadir sounding, spherical surface, spherical wavefront",level:"3"},{id:"sec_6_3",title:"2.1.5. Wavefront flattening",level:"3"},{id:"sec_7_3",title:"2.1.6. Modes of volume scattering for spherical wave",level:"3"},{id:"sec_9_2",title:"2.2. Condition of complete burial of radial size of the pulse volume",level:"2"},{id:"sec_10_2",title:"2.3. An assessment of the scattering volume under incomplete burial condition",level:"2"},{id:"sec_10_3",title:"2.3.1. Flat surface, plane wavefront, long probing pulse and fully scattering snow slab ",level:"3"},{id:"sec_11_3",title:"2.3.2. Long probing pulse, spherical wave, spherical surface, partially scattering slab",level:"3"},{id:"sec_14",title:"3. Volume component of the backscatter coefficient ",level:"1"},{id:"sec_14_2",title:"3.1. Semi-empirical model for flat surface, plane wave and long probing pulse ",level:"2"},{id:"sec_15_2",title:"3.2. Effective snow depth",level:"2"},{id:"sec_17",title:"4. Comparison the A-U model estimates and the experimental radar data on the Greenland ice sheet",level:"1"},{id:"sec_18",title:"5. Enhanced semi-empirical model of the volume component of the backscatter coefficient",level:"1"},{id:"sec_19",title:"6. Normalized snow depth",level:"1"},{id:"sec_19_2",title:"6.1. Derivation of the normalized snow depth from the backscatter coefficient based on the enhanced semi-empirical model",level:"2"},{id:"sec_20_2",title:"6.2. An example of time transformation of nSD profiles within a selected Greenland marginal area",level:"2"},{id:"sec_22",title:"7. Model spectral dependence of the backscatter ",level:"1"},{id:"sec_22_2",title:"7.1. Spectral dependence of the backscatter in accordance with the A-U model",level:"2"},{id:"sec_23_2",title:"7.2. Estimation of spectral differences based on field penetration depth data",level:"2"},{id:"sec_24_2",title:"7.3. Spectral characteristics of the penetration depth for model monodisperse cold snow",level:"2"},{id:"sec_26",title:"8. Backscatter depth profile model for snow sounding with broadband pulses of short effective (compressed) length ",level:"1"},{id:"sec_26_2",title:"8.1. Phenomenological approach in the case of very short probing pulse",level:"2"},{id:"sec_27_2",title:"8.2. Analytical derivation for commensurate length of the scattering volume and snow depth ",level:"2"},{id:"sec_28_2",title:"8.3. Comparison of the results obtained with the phenomenological approach ",level:"2"},{id:"sec_28_3",title:"8.3.1. Very short probing pulse",level:"3"},{id:"sec_29_3",title:"8.3.2. Long probing pulse ",level:"3"},{id:"sec_31_2",title:"8.4. Numerical calculation of the return signal for intermediate length of the probing pulse ",level:"2"},{id:"sec_33",title:"9. Summary ",level:"1"},{id:"sec_34",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAbramowitz\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tStegun\n\t\t\t\t\t\t\tI. A.\n\t\t\t\t\t\tEds\n\t\t\t\t\t1972 Hanbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Wiley.\n\t\t\t'},{id:"B2",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAlley\n\t\t\t\t\t\t\tR. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBolzan\n\t\t\t\t\t\t\tJ. F.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWhillans\n\t\t\t\t\t\t\tI. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1982 Polar firn densification and grain growth. 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IEEE Journal of Oceanic Engineering, OE-9 , 5\n\t\t\t\t\t377\n\t\t\t\t\t382 , December 1984.\n\t\t\t'},{id:"B45",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tTsang\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPan\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLiang\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLi\n\t\t\t\t\t\t\tZ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tCline\n\t\t\t\t\t\t\tD. W.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tTan\n\t\t\t\t\t\t\tY.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007 Modeling active microwave remote sensing of snow using dense media radiative transfer (DRMT) theory with multiple-scattering effects. IEEE Trans. Geosci. Remote Sens..\n\t\t\t\t\t45\n\t\t\t\t\t4 April 2007, 990\n\t\t\t\t\t1004 .\n\t\t\t'},{id:"B46",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tUlaby\n\t\t\t\t\t\t\tF. T.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMoore\n\t\t\t\t\t\t\tR. K.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFung\n\t\t\t\t\t\t\tA. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1982 Microwave Remote Sensing, Active and Passive. Vol. II. Radar Remote Sensing and Surface Scattering and Emission Theory. Artech House, Inc., Norwood.\n\t\t\t'},{id:"B47",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tUlaby\n\t\t\t\t\t\t\tF. T.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMoore\n\t\t\t\t\t\t\tR. K.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFung\n\t\t\t\t\t\t\tA. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1986 Microwave Remote Sensing, Active and Passive. Vol. III. From Theory to Applications. Artech House, Inc., Norwood.\n\t\t\t'},{id:"B48",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tUlaby\n\t\t\t\t\t\t\tF. T.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSiqueira\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNashashibi\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSarabandi\n\t\t\t\t\t\t\tK.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1996 Semi-empirical model for radar backscatter from snow at 35 and 95 GHz. IEEE Trans. Geosci. Remote Sens., 34\n\t\t\t\t\t5\n\t\t\t\t\t1059\n\t\t\t\t\t1065 , September 1996.\n\t\t\t'},{id:"B49",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWoods\n\t\t\t\t\t\t\tG. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1994 Grain growth behavior of the GISP2 ice core from central Greenland. Technical Report Series 94\n\t\t\t\t\t002 , Earth System Science Center, Penn State University, USA.\n\t\t\t'},{id:"B50",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tYurchak\n\t\t\t\t\t\t\tB. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2009 Radar volume backscatter from spatially extended geophysical targets in a “slice” approach. IEEE Trans. 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University of Maryland, Baltimore County, GEST
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1. Introduction
Idiopathic Pulmonary Fibrosis (IPF) is the most common type of Idiopathic Interstitial Pneumonia. It is more prevalent in men, and its incidence increases with age, especially beyond the fifth decade [1]. Its incidence is estimated to be 3–9 cases per 100,000 per year in the western hemisphere [2]. As per a systematic review, the prevalence is estimated to be 0.5–27.9/100,000 [3]. Although newer therapies such as Pirfenidone and Nintedanib are available to slow the progression of the disease, the mortality and prognosis remain dismal, comparable to that of certain malignancies [4]. A key consideration has been the lack of optimal understanding of the pathophysiological mechanisms underlying the disease process, as interventions can then be targeted to modify the disease process and achieve better outcomes for the patients [4]. In a genetically susceptible individual, many risk factors have been proposed [1]. One such factor that has been closely associated with IPF is Gastroesophageal reflux disease (GERD) [1]. Their association has been hypothesized, studied, and targeted therapeutically. However, its role as a causative and aggravating factor has not yet been crystallized. Our chapter aims to review the association of GERD with IPF, its alleged role in causing or promoting lung injury, the effect of GERD therapy on IPF, recommendations from clinical guidelines, and the direction for future research.
1.1 Gastroesophageal reflux disease (GERD) and its relevance to IPF
GERD is a disease caused by reflux of stomach contents into the esophagus and beyond, causing troublesome symptoms and complications [5]. It causes esophageal and stomach symptoms characterized by chest pain, nausea, bloating, heartburn, and regurgitation. It can also cause extraesophageal symptoms such as throat pain, burning, lump in the throat, the sensation of needing to clear the throat, hoarseness of voice, cough, wheezing, bronchospasm, etc. [5, 6]. Importantly not all reflux events are symptomatic as there could be non-acid reflux [5, 6, 7].
Prevalence of GERD is very common in the western world, with North American estimates being 18.1–27.8% [8]. Europe, similarly, has a prevalence of up to 25% [8]. In a United Kingdom general practice database, IPF was much more likely to be associated with a diagnosis of GERD (65%) or use of anti-reflux therapy (71%) when compared to controls [4, 7, 9]. The prevalence of erosive esophagitis and hiatal hernia, both of which are associated with increased reflux, is also much higher in pulmonary fibrosis patients when compared to the general population [8, 10]. Hence there is a strong epidemiological association between these two disease entities.
GERD occurs commonly as a result of increased frequency of transient lower esophageal sphincter relaxations (TLESRs), which are defined as brief moments of lower esophageal sphincter tone inhibition that are independent of a swallow [11]. Other pathophysiological mechanisms implicated in the causation of GERD are reduced lower esophageal sphincter (LES) pressure, reduced upper esophageal sphincter pressure, reduced esophageal motility, Hiatal hernias, which distorts the gastroesophageal junctional anatomy, impairment of esophageal clearance, and sluggish gastric emptying [4, 5, 7]. A combination of these factors leads to the reflux establishing contact with mucosa in the upper gastrointestinal tract, pharynx, tracheobronchial tree, and lungs, causing extra esophageal symptoms as previously described [4, 5, 7].
Evaluation of GERD can be made by direct visual examination by esophagogastroduodenoscopy (EGD). The chief advantage is that the mucosa can be visualized directly and is helpful in the diagnosis of possible complications of GERD, including Barret’s esophagitis, esophagitis, gastritis, gastric and esophageal stricture, and malignancy. However, pH monitoring better evaluates reflux, wherein a pH measuring probe is placed in the esophagus [5]. The primary measurement is the amount of time spent with a pH less than 4.0 [5]. However, it has its inherent limitations, as non-acid reflux cannot be measured and can remain totally asymptomatic. This limitation has been overcome by the placement of channels that measure impedance. Liquid reflux has low impedance and high conductance, while gaseous reflux, such as belching, has high impedance with low conductance [12]. Combined 24-hr multichannel intraluminal impedance-pH monitoring (MII-pH) are available to determine the amount of refluxate, its proximal extent, and/or the presence of both acid and weakly acidic reflux [7, 13]. The chief metric when using MII-pH is the “Total number of refluxes” (Pathological when more than 80 and normal if less than 40 in a 24-hour period) and esophageal “Acid exposure time (AET)” as the percentage of time with pH less than 4.0 in the distal esophagus [14]. The use of MII-pH in GERD associated with extra esophageal disease, particularly in IPF, is rather novel and promising to help illuminate the pathophysiological mechanisms between the two diseases [15]. It is noteworthy that IPF belongs to a group of diseases that are only possibly or likely associated with GERD, and its role is only speculated [7]. The use and application of MII-pH for the study of extraesophageal diseases and symptoms has not been as productive as for typical GERD [7].
1.2 The pathophysiological relationship and co-existence between GERD and IPF
The relationship between IPF and GERD is quite intriguing. The epidemiological association suggests that there appear to be plausible biological and mechanical factors underlying this pathophysiology.
It is suggested that GERD is associated with decreased upper and lower esophageal sphincter tone (hypotensive esophagogastric junction) with or without increased frequency of transient lower esophageal sphincter relaxations (TLESRs), leading to increased refluxate with an associated micro-aspiration of the gastric contents into the trachea and lungs [16, 17, 18]. Contrary to this proposed theory, it has been proposed that lung fibrosis causes decreased lung compliance along with lower lung elasticity, resulting in increased negative intrathoracic pressure during inspiration that is transmitted to the mediastinal structures, including the esophagus and its sphincters [17]. This causes increased transient lower esophageal sphincter relaxations (TLESRs) with lower and upper esophageal sphincters [17]. There is also a pressure gradient across the diaphragm in respiratory diseases like IPF, which may promote these favorable refluxate mechanisms, especially during coughing, increased respiratory excursions during exacerbations, and may potentially be further aggravated by hypoxia/hypercapnia, medications like antacids, glucocorticoids, and obstructive sleep apnea/hypopnea syndromes [19]. Hiatal hernia alters the physical and physiologic function of the lower esophageal sphincter, thereby promoting reflux [20, 21, 22]. Furthermore, it has been proposed that esophageal dysmotility may contribute to reflux [23, 24]. Ultimately, the result of these phenomena is that the gastric refluxate, which contains both acidic and non-acidic contents, leads to delayed esophageal clearance and micro-aspiration in the tracheobronchial tree injure the pulmonary parenchyma consisting of both alveolar and interstitial components [4, 7, 19]. The healing of this injury eventually occurs by fibrosis, and the pulmonary remodeling that ensues culminates in a distorted fibrotic architecture [4, 7].
Many studies have been performed to provide evidence and study the relationship between GERD and IPF. Most of these studies have limitations and often conclude with contradictory results. Therefore, evidence has shown a co-existence and/or association between IPF and GERD. However, causality has yet to be determined [4, 7, 25].
Gao et al. conducted a study involving 69 IPF patients, 62 healthy volunteers, and 88 IPF negative GERD patients. The prevalence of GERD was high in patients with IPF, and in relation to their comparator group showed the variable presence of esophageal dysmotility and decreased lower and upper esophageal sphincter pressure. IPF patients also had increased reflux events proximally and impaired bolus transit time [16]. Raghu et al. studied 65 patients with IPF who were subjected to 24-h pH monitoring and esophageal manometry with a comparison group of 133 asthmatic patients and symptoms of GERD. The prevalence of abnormal gastroesophageal reflux in IPF patients was high at 87%, with 76% and 63% demonstrating abnormal distal and proximal esophageal acid exposures, respectively; a finding higher than within the comparison group [18]. The study also showed that the presence of GERD was not always symptomatic, and there was no correlation with IPF severity [18]. This was further confirmed in a study involving 28 patients with histologically confirmed IPF using hypopharyngeal multichannel intraluminal impedance (HMII) [26]. HMII used a specialized impedance catheter to directly measure laryngopharyngeal reflux (LPR) and full column reflux (reflux 2 cm distal to the upper esophageal sphincter). The study included 16 males and 12 females with a mean age of 60.4 years (range, 41–78) and a BMI of 28.4 (range, 21.1–38.1), respectively. Abnormal proximal exposure was present in 54% (15/28) of patients. This latter group was more likely to have a defective lower esophageal sphincter (LES) compared with those without (93% vs. 75%). Fourteen patients (56%) had abnormal esophageal motility, including aperistaltic esophagus (n = 9), suggesting that this may be common in this patient population [26].GERD was noted to be highly prevalent at more than 70% in patients with IPF; abnormal proximal reflux events such as LPR and full column reflux were also quite common despite a frequently negative DeMeester score (It is a composite of six different parameters which measures acid exposure giving a pH score used to diagnose GERD), suggesting that nonacid reflux (25% of patients) is prevalent in this patient population [26]. A high rate of esophageal mucosal injury and a longer acid clearance time was also noted [26].67–76% of the systematic review demonstrated abnormal esophageal acid exposure off PPI treatment [27].
In another study conducted by Savarino et al. [28], 40 IPF patients were studied alongside 40 non-IPF ILD patients and 50 healthy volunteers, who served as controls. Patients were off reflux therapy and underwent a High-resolution Lung CT scan (HRCT) and pH-impedance monitoring. Patients with IPF had significantly increased esophageal acid exposure, the number of acidic, weakly acidic, and proximal reflux events relative to the comparison groups. Pulmonary fibrosis HRCT scores correlated well with reflux episodes in both the distal and proximal esophagus. Patients with IPF had more bile acids and pepsin (p < 0.03) in bronchoalveolar lavage fluid (62% and 67%, respectively) and saliva (61% and 68%, respectively) relative to the comparison groups [28]. Gavini et al. conducted an elegant study involving 45 pre-transplants patients with IPF who had received pulmonary function tests within the last 3 months. Patients were off reflux therapy and had no reflux surgery. They measured GER on multichannel intraluminal impedance and pH study (MII-pH). Six pH/acid reflux parameters with corresponding MII/bolus reflux measures were prespecified. Multivariate analyses were applied using forward stepwise logistic regression. Severe pulmonary dysfunction was defined using diffusion capacity for carbon monoxide (DLCO) ≤40%. Abnormal total reflux episodes and prolonged bolus clearance time (OR = 1.21 p = 0.05), but not the refluxate pH values, were significantly associated with pulmonary dysfunction severity on univariate and multivariate analyses [29]. Overall, it appears that esophageal dysmotility, the total number of acidic, weakly acidic, and non-acidic refluxes with prolonged bolus clearance time, appear to impact the underlying lung pathology.
Animal and human studies have shown that the presence of gastric contents (pepsin, bile acids, gastric acid) via microaspiration in bronchoalveolar lavage (BAL) fluid can cause tissue damage and inflammatory infiltrate [28, 30, 31, 32, 33, 34, 35]. Histologically presence of thickened alveolar walls, collagen deposition in the interstitium, epithelial-mesenchymal transition, and presence of various fibrogenic factors has been found [28, 34, 35]. The latter consists of TGF-beta, NFκB, Farnesoid X receptor, and others. TGF-beta can be induced by gastric contents, leading to fibroblast proliferation and fibroproliferative changes [4, 7].
1.3 The role of proton pump inhibitor (PPI)/histamine-2 receptor blockers (H2RA) and anti-reflux surgery in IPF
There has been a long-standing interest in the use of anti-secretory therapy/anti-reflux surgery in IPF patients, given that GERD has been thought of as having a relationship with IPF [4, 7]. While it is not unreasonable to give anti-secretory therapy to patients with symptomatic GERD patients, it has certainly been hard to objectively justify the use in all patients with IPF, some of whom may not have any reflux or reflux with non-acidic gastric contents [36]. This has indeed been a recommendation from international guidelines, albeit it was a week level of recommendation [37]. As per literature, PPIs are the most frequently used medications, and further discussion will relate henceforth to PPI.
PPIs are known for increasing the pH of gastric acid; a mechanism thought to prevent microaspiration of acidic contents into the lung and hence potentially protect against acid-induced pneumonitis [37]. In vitro studies show that PPIs like Esomeprazole have pleiotropic effects, can inhibit expression of pro-inflammatory molecules like vascular cell adhesion molecule-1, inducible nitric oxide synthase, tumor necrosis factor-alpha (TNF-α), and interleukins (IL-1β and IL-6), and exhibit antioxidant and anti-fibrotic properties by downregulation of profibrotic proteins including receptors for transforming growth factor β (TGFβ), fibronectin and matrix metalloproteinases (MMPs) [38, 39]. They may also inhibit apoptosis of pneumocytes expressing Surfactant (SP-C) [38, 39]. Retrospective studies have also demonstrated that PPIs may prolong transplant-free survival of IPF patients [38, 39].
However, PPIs are not without risks. They have been shown to alter the microbiome of the respiratory tract and increase the risk of pneumonia [17]. Furthermore, they increase the risk of micronutrient deficiencies like Vitamin B12, cause dementia, Clostridium difficile infection, decrease bone density and increase the risk of fractures. They may increase the risk of chronic kidney disease progressing to end-stage renal disease [40]. However, it is to be noted that most of the evidence for this comes from observational data and meta-analyses, which have their own inherent limitations [40].
Anti-reflux surgery is an important therapeutic option in patients with GERD. Nissen fundoplication and Laparoscopic anti-reflux surgery (LARS) are the two most performed surgeries, both of which are generally safe in IPF [4, 14]. Lee JS et al. reported a retrospective cohort of 204 IPF patients consisting of individuals with symptoms of GERD (34%), a history of GERD (45%), reported use of GERD medications (47%), and Nissen fundoplication (5%). After the multivariate adjustment, the use of GERD medication was associated with a lower radiologic fibrotic score. It was also an independent predictor of longer survival time in patients with IPF [41]. Lee JS et al. also reported the combined results of 3 prospectively collected randomized controlled trial data, including 242 patients only from the placebo arm. Although the data came from RCTs, this was not an RCT. Of the total 242 patients, 124 patients were taking PPI/H2RA, and 118 patients were not taking any antisecretory therapy. In IPF, a slower decline in Forced vital capacity (FVC) has shown a correlation with improved survival time in IPF [42]. The study showed that there was a slower decline in FVC in the PPI/H2RA group, which was statistically significant. Also, there were fewer acute exacerbations in the PPI/H2RA group, and this result did not contribute to the slower decline in FVC. However, there was no change in mortality, presumably due to the follow-up period not being sufficient. This study result generated an interesting hypothesis that the use of PPI/H2RA could slow disease progression [43].
Furthermore, Fidler et al. conducted a systematic review and meta-analysis, studying the effect of pharmacological therapy of GERD in IPF patients, which showed a significant improvement in IPF related survival (adjusted risk: HR 0.45) but no effect on all-cause mortality. There was a change in progression-free survival, FVC, acute exacerbation, and other Pulmonary function test parameters. In patients with FVC less than 70% of predicted, there was an increase in pulmonary infection, which was significant as this is a known side effect of PPI affecting patients with more advanced disease [44]. It follows from this discussion that the studies once again have small numbers, mostly observational, and hence have limitations providing poor or limited quality of evidence [44].
In a randomized controlled trial, Raghu et al. analyzed data from 27 patients who underwent Laparoscopic anti-reflux surgery (LARS) and 20 patients who did not undergo surgery with FVC measurement at 48 weeks as the endpoint in an intention to treat analysis. All patients had abnormal acid GER with a confirmed DeMeester score of ≥14·7; measured by 24-h pH monitoring and preserved forced vital capacity (FVC) of more than 50%. Patients were allowed to use Nintedanib and Pirfenidone. Patients in the surgery group had a slower decline in FVC, which was not statistically significant at 48 weeks in the non-surgery group (p = 0·28)}. Acute exacerbation of IPF, hospitalization for respiratory etiology, and mortality were also less in the surgical group, however not to statistical significance [45].
2. Discussion
GERD has been known to be co-existent with many Pulmonary disorders such as Systemic Sclerosis, Chronic obstructive pulmonary disease (COPD), Bronchial Asthma, IPF, Bronchiectasis, Aspiration Pneumonia, Lung transplant complications such as Bronchiolitis obliterans (BOS), etc. [6]. These plethora’s of lung conditions being associated with GERD are likely due to the shared common genetic embryological and developmental origin of the two organ systems from the foregut [6, 60]. In addition, they share the intrathoracic cavity and also have the same vagal innervation [6, 60]. As such, two predominant theories are in vogue, the “Refluxate theory” and the “Reflux theory,” which attempt to explain the disease mechanisms with their common origin and development as background. The “Refluxate theory,” as previously described, implicates acid reflux from the GI tract and its micro-aspiration into the Respiratory tree, causing physicochemical damage to the latter culminating in fibrosis [6]. The “Reflex theory” pertains to the reflex increase in bronchoconstriction and airway resistance in response to the presence of acid in the esophagus and respiratory tree [6]. Furthermore, as discussed previously, the presence of pulmonary fibrosis may aggravate the gastroesophageal reflux due to decreased compliance, elasticity, and need for increased negative intrathoracic pressure generated during inspiration, causing increased gradient across thoracic and abdominal compartments [6]. Hence there is possibly a bidirectional relationship between the two organ systems, as depicted in Figure 1. A Summary of the studies that evaluated the role of antireflux therapy and surgery in the management of IPF is available in Table 1.
Figure 1.
The bidirectional relationship between IPF and GERD.
Fourteen patients with early fundoplication had better survival when compared to those with reflux and no intervention.
As compared to patients with reflux who did not have early fundoplication, those who had early fundoplication had less incidence of BOS at 1 and 3 years.
In all four patients, PFTs stabilized or improved, and their status was maintained with proper PPI therapy.
At the latest follow-up, all of the patients were still alive, and none of them had an acute exacerbation of IPF or required therapy for respiratory difficulties during this time.
During the average 15-month follow-up, there were no perioperative complications or a reduction in lung function.
Patients with idiopathic pulmonary fibrosis treated with fundoplication had better oxygen levels, but the oxygen requirements of control patients with idiopathic pulmonary fibrosis who did not have the surgery increased significantly
Compared to matched controls, IPF patients with hiatal hernia who used antacid medicines had substantially higher DLCO and better composite physiologic index scores.
The use of anti-acid medications at the start of the study had no effect on the therapeutic effect of Nintedanib on slowing FVC decrease in IPF patients.
Antacid therapy was not associated with disease progression, all-cause mortality, IPF-related mortality, absolute FVC decrease of 10% or more, mean observed change in FVC and FVC percent of predicted, hospital admission rate, 6 Minute walk distance(MWD) stratified by baseline FVC, and adverse events at 52 weeks.
There were no significant differences in disease progression, all-cause mortality, IPF-related mortality, all-cause hospitalization rate, or mean change in % FVC at 52 weeks between the two groups (with or without antacid therapy).
PPI usage was linked to fewer high attenuation regions in CT scans of a large group of asymptomatic community-dwelling middle-aged and older people, suggesting a possible benefit in ILD.
In both antisecretory therapy treated and nontreated individuals, the yearly decline rate of FVC was identical in both Nintedanib/placebo-treated patients.
Antisecretory medicine did not influence the therapeutic effect of Nintedanib and was not related to a better course of illness
Studies designed to test the relationships between the two diseases entities have several limitations. They are mostly retrospective, have small sample sizes, with poorly defined inclusion and exclusion criteria, resulting in many confounders. While these limitations can be addressed partially by conducting prospective studies, randomized controlled data with a large sample size will remain elusive due to the prolonged time required for a disease process like IPF takes to evolve and manifest [25]. Besides, diseases like IPF are not clearly recognizable early, and GERD with non-acid reflux or poorly acidic reflux may not manifest with classic symptoms [6, 25], hence denying the opportunity for early recognition and follow up. Hence, our reliance on smaller case-controlled studies with a few well-conducted meta-analyses has only revealed an association between GERD and IPF, far from the nine causality criteria propounded by “Hill” [61, 62]. Although not ruling out causality, a weak association between the two diseases still needs to be viewed with an abundance of caution as the effects of residual confounding generate sufficient bias to prevent a robust causal inference from these types of studies [62]. Although such challenges will limit future studies, investigating therapeutic interventions like LARS and PPIs along with disease-modifying therapies like Nintedanib and Pirfenidone may improve outcomes for our IPF patients [25].
The large database-based clinical studies with robust timestamping of initiation of each disease entity will be helpful in establishing a temporal relationship. A machine learning model development is the need of the hour to answer this clinical question.
3. Conclusion
The co-existence of IPF and GERD is very common. There is likely a bidirectional pathophysiological relationship between the two disease entities. Although there is no causality established, current guidelines do recommend therapy with PPI in all patients with IPF. There remain many important challenges to the study of these coexisting conditions, and it may not be possible to obtain robust data establishing causality. Nevertheless, an attempt can be made to further conduct well-designed interventional studies to benefit patients in need.
Acknowledgments
Kristina Kardum Cvitan, Author Service Manager, IntechOpen.
Conflict of interest
The authors declare no conflict of interest.
\n',keywords:"idiopathic pulmonary fibrosis (IPF), gastro-esophageal reflux disease (GERD), microaspiration, proton pump inhibitor, risk factors",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/80364.pdf",chapterXML:"https://mts.intechopen.com/source/xml/80364.xml",downloadPdfUrl:"/chapter/pdf-download/80364",previewPdfUrl:"/chapter/pdf-preview/80364",totalDownloads:62,totalViews:0,totalCrossrefCites:1,dateSubmitted:"December 15th 2021",dateReviewed:"January 3rd 2022",datePrePublished:"February 4th 2022",datePublished:null,dateFinished:"February 4th 2022",readingETA:"0",abstract:"Idiopathic pulmonary fibrosis (IPF) and Gastroesophageal reflux disease (GERD) commonly co-exist. Pathophysiological mechanisms causing IPF are still not well understood, and GERD has been implicated in both as a probable causative and disease-promoting entity. Although not conclusively proven, this relationship has been the subject of several studies, including therapeutic interventions aimed at treating GERD and its resultant effect on IPF and related outcomes. Our review aims to present the current concepts and understanding of these two disease processes, which are multifaceted. Their complex interaction includes epidemiology, pathophysiology, diagnosis, treatment, review of research studies conducted to date, and future directions for research.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/80364",risUrl:"/chapter/ris/80364",signatures:"Nitesh Kumar Jain, Anwar Khedr, Hisham Ahmed Mushtaq, Brian Bartlett, April Lanz, Greta Zoesch, Stephanie Welle, Sumeet Yadav, Thoyaja Koritala, Shikha Jain, Aysun Tekin, Ramesh Adhikari, Aishwarya Reddy Korsapati, Mool Chand, Vishwanath Pattan, Vikas Bansal, Ali Rabaan, Amos Lal, Hasnain Saifee Bawaadam, Aman Sethi, Lavanya Dondapati, Raghavendra Tirupathi, Mack Sheraton, Maureen Muigai, David Rokser, Chetna Dengri, Kovid Trivedi, Samir Chandra Gautam, Simon Zec, Ibtisam Rauf, Mantravadi Srinivasa Chandramouli, Rahul Kashyap and Syed Anjum Khan",book:{id:"9816",type:"book",title:"Idiopathic Pulmonary Fibrosis",subtitle:null,fullTitle:"Idiopathic Pulmonary Fibrosis",slug:null,publishedDate:null,bookSignature:"Dr. Salim Surani and Dr. Venkat Rajasurya",coverURL:"https://cdn.intechopen.com/books/images_new/9816.jpg",licenceType:"CC BY 3.0",editedByType:null,isbn:"978-1-83969-240-6",printIsbn:"978-1-83969-239-0",pdfIsbn:"978-1-83969-241-3",isAvailableForWebshopOrdering:!0,editors:[{id:"15654",title:null,name:"Salim",middleName:null,surname:"Surani",slug:"salim-surani",fullName:"Salim Surani"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_1_2",title:"1.1 Gastroesophageal reflux disease (GERD) and its relevance to IPF",level:"2"},{id:"sec_2_2",title:"1.2 The pathophysiological relationship and co-existence between GERD and IPF",level:"2"},{id:"sec_3_2",title:"1.3 The role of proton pump inhibitor (PPI)/histamine-2 receptor blockers (H2RA) and anti-reflux surgery in IPF",level:"2"},{id:"sec_5",title:"2. Discussion",level:"1"},{id:"sec_6",title:"3. Conclusion",level:"1"},{id:"sec_7",title:"Acknowledgments",level:"1"},{id:"sec_10",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'Raghu G, Collard HR, Egan JJ, Martinez FJ, Behr J, Brown KK, et al. An official ATS/ERS/JRS/ALAT statement: Idiopathic pulmonary fibrosis: evidence-based guidelines for diagnosis and management. American Journal of Respiratory and Critical Care Medicine. 2011;183:788-824. DOI: 10.1164/rccm.2009-040GL'},{id:"B2",body:'Hutchinson J, Fogarty A, Hubbard R, McKeever T. Global incidence and mortality of idiopathic pulmonary fibrosis: A systematic review. The European Respiratory Journal. 2015;46:795-806. DOI: 10.1183/09031936.00185114'},{id:"B3",body:'Kaunisto J, Salomaa ER, Hodgson U, Kaarteenaho R, Myllärniemi M. Idiopathic pulmonary fibrosis--A systematic review on methodology for the collection of epidemiological data. BMC Pulmonary Medicine. 2013;13:53. DOI: 10.1186/1471-2466-13-53'},{id:"B4",body:'Wang Z, Bonella F, Li W, Boerner EB, Guo Q, Kong X, et al. Gastroesophageal reflux disease in idiopathic pulmonary fibrosis: uncertainties and controversies. Respiration. 2018;96:571-587. DOI: 10.1159/000492336'},{id:"B5",body:'Clarrett DM, Hachem C. Gastroesophageal reflux disease (GERD). Missouri Medicine. 2018;115:214-218'},{id:"B6",body:'Okwara NC, Chan WW. Sorting out the relationship between esophageal and pulmonary disease. Gastroenterology Clinics of North America. 2021;50:919-934. 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Laparoscopic fundoplication in patients with end-stage lung disease awaiting transplantation. The Journal of Thoracic and Cardiovascular Surgery. 2006;131:438-446. DOI: 10.1016/j.jtcvs.2005.10.014'},{id:"B48",body:'Fisichella PM, Davis CS, Lundberg PW, Lowery E, Burnham EL, Alex CG, et al. The protective role of laparoscopic antireflux surgery against aspiration of pepsin after lung transplantation. Surgery. 2011;150:598-606. DOI: 10.1016/j.surg.2011.07.053'},{id:"B49",body:'Raghu G, Morrow E, Collins BF, Ho LA, Hinojosa MW, Hayes JM, et al. Laparoscopic anti-reflux surgery for idiopathic pulmonary fibrosis at a single center. European Respiratory Journal. 2016 Sep;48(3):826-832. DOI: 10.1183/13993003.00488-2016. Epub 2016 Aug 4. PMID: 27492835'},{id:"B50",body:'Kreuter M, Wuyts W, Renzoni E, Koschel D, Maher TM, Kolb M, et al. Antacid therapy and disease outcomes in idiopathic pulmonary fibrosis: A pooled analysis. The Lancet Respiratory Medicine. 2016;4:381-389. 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PMID: 28526800'},{id:"B58",body:'Costabel U, Behr J, Crestani B, Stansen W, Schlenker-Herceg R, Stowasser S, et al. Anti-acid therapy in idiopathic pulmonary fibrosis: Insights from the INPULSIS® trials. Respiratory Research. 2018;19:167. DOI: 10.1186/s12931-018-0866-0'},{id:"B59",body:'Jo HE, Corte TJ, Glaspole I, Grainge C, Hopkins PMA, Moodley Y, et al. Gastroesophageal reflux and antacid therapy in IPF: Analysis from the Australia IPF Registry. BMC Pulmonary Medicine. 2019;19:84. DOI: 10.1186/s12890-019-0846-2'},{id:"B60",body:'Mansfield LE. Embryonic origins of the relation of gastroesophageal reflux disease and airway disease. The American Journal of Medicine. 2001;111(Suppl 8A):3S-7S. DOI: 10.1016/s0002-9343(01)00846-4'},{id:"B61",body:'Hill AB. The environment and disease: association or causation? Proceedings of the Royal Society of Medicine. 1965;58:295-300'},{id:"B62",body:'Vaezi MF, Yang YX, Howden CW. Complications of proton pump inhibitor therapy. Gastroenterology. 2017;153:35-48. DOI: 10.1053/j.gastro.2017.04.047'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Nitesh Kumar Jain",address:"jain.nitesh@mayo.edu",affiliation:'
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Our business values are based on those any scientist applies to their research. The values of our business are based on the same ones that all good scientists apply to their research. We have created a culture of respect and collaboration within a relaxed, friendly, and progressive atmosphere, while maintaining academic rigour.
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IntechOpen is a dynamic, vibrant company, where exceptional people are achieving great things. We offer a creative, dedicated, committed, and passionate environment but never lose sight of the fact that science and discovery is exciting and rewarding. We constantly strive to ensure that members of our community can work, travel, meet world-renowned researchers and grow their own career and develop their own experiences.
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If this sounds like a place that you would like to work, whether you are at the beginning of your career or are an experienced professional, we invite you to drop us a line and tell us why you could be the right person for IntechOpen.
Integrity - We are consistent and dependable, always striving for precision and accuracy in the true spirit of science.
\n\n
Openness - We communicate honestly and transparently. We are open to constructive criticism and committed to learning from it.
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Disruptiveness - We are eager for discovery, for new ideas and for progression. We approach our work with creativity and determination, with a clear vision that drives us forward. We look beyond today and strive for a better tomorrow.
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What makes IntechOpen a great place to work?
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IntechOpen is a dynamic, vibrant company, where exceptional people are achieving great things. We offer a creative, dedicated, committed, and passionate environment but never lose sight of the fact that science and discovery is exciting and rewarding. We constantly strive to ensure that members of our community can work, travel, meet world-renowned researchers and grow their own career and develop their own experiences.
\n\n
If this sounds like a place that you would like to work, whether you are at the beginning of your career or are an experienced professional, we invite you to drop us a line and tell us why you could be the right person for IntechOpen.
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Saleh and Amal I. Hassan",coverURL:"https://cdn.intechopen.com/books/images_new/11120.jpg",editedByType:"Edited by",publishedDate:"June 23rd 2022",editors:[{id:"144691",title:"Prof.",name:"Hosam M.",middleName:null,surname:"Saleh",slug:"hosam-m.-saleh",fullName:"Hosam M. Saleh"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"10696",title:"Applications of Calorimetry",subtitle:null,isOpenForSubmission:!1,hash:"8c87f7e2199db33b5dd7181f56973a97",slug:"applications-of-calorimetry",bookSignature:"José Luis Rivera Armenta and Cynthia Graciela Flores Hernández",coverURL:"https://cdn.intechopen.com/books/images_new/10696.jpg",editedByType:"Edited by",publishedDate:"June 23rd 2022",editors:[{id:"107855",title:"Dr.",name:"Jose Luis",middleName:null,surname:"Rivera Armenta",slug:"jose-luis-rivera-armenta",fullName:"Jose Luis Rivera Armenta"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},subject:{topic:{id:"680",title:"Mathematical Modeling",slug:"engineering-acoustical-engineering-mathematical-modeling",parent:{id:"110",title:"Acoustical Engineering",slug:"engineering-acoustical-engineering"},numberOfBooks:1,numberOfSeries:0,numberOfAuthorsAndEditors:27,numberOfWosCitations:8,numberOfCrossrefCitations:8,numberOfDimensionsCitations:19,videoUrl:null,fallbackUrl:null,description:null},booksByTopicFilter:{topicId:"680",sort:"-publishedDate",limit:12,offset:0},booksByTopicCollection:[{type:"book",id:"5708",title:"Computational and Experimental Studies of Acoustic Waves",subtitle:null,isOpenForSubmission:!1,hash:"518d2ac3c49f5c4c48d4f3f3b0729232",slug:"computational-and-experimental-studies-of-acoustic-waves",bookSignature:"Mahmut Reyhanoglu",coverURL:"https://cdn.intechopen.com/books/images_new/5708.jpg",editedByType:"Edited by",editors:[{id:"15068",title:"Dr.",name:"Mahmut",middleName:null,surname:"Reyhanoglu",slug:"mahmut-reyhanoglu",fullName:"Mahmut Reyhanoglu"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}],booksByTopicTotal:1,seriesByTopicCollection:[],seriesByTopicTotal:0,mostCitedChapters:[{id:"56872",doi:"10.5772/intechopen.70590",title:"Acoustic Wave Monitoring of Fluid Dynamics in the Rock Massif with Anomaly Density, Stressed and Plastic Hierarchic Inclusions",slug:"acoustic-wave-monitoring-of-fluid-dynamics-in-the-rock-massif-with-anomaly-density-stressed-and-plas",totalDownloads:1110,totalCrossrefCites:1,totalDimensionsCites:5,abstract:"The geological environment is an open system, on which external and internal factors act. They lead it to an unstable state, which, as a rule, manifests itself locally in the form of zones, called dynamically active elements, which are indicators of potential catastrophic sources. These objects differ from the host geological environment by structural forms, which are often forming of a hierarchical type. The process of their activation can be observed using monitoring with wave fields, for mathematical support of which new modeling algorithms have been developed using the method of integral and integral-differential equations. A new approach to the interpretation of wave fields has been developed, to determine contours or surfaces of locally stressed hierarchical objects. An iterative process of solving the theoretical inverse problem for the case of determining configurations of 2D hierarchical inclusions of the k-th rank is developed. When interpreting monitoring results, it is necessary to use data from such monitoring systems that are tuned to study the hierarchical structure of the environment.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Olga Hachay and Andrey Khachay",authors:[{id:"150801",title:"Prof.",name:"Olga",middleName:"Alexandrovna",surname:"Hachay",slug:"olga-hachay",fullName:"Olga Hachay"},{id:"219182",title:"MSc.",name:"Andrey",middleName:null,surname:"Khachay",slug:"andrey-khachay",fullName:"Andrey Khachay"}]},{id:"57258",doi:"10.5772/intechopen.71203",title:"Sound Waves in Complex (Dusty) Plasmas",slug:"sound-waves-in-complex-dusty-plasmas",totalDownloads:1398,totalCrossrefCites:5,totalDimensionsCites:5,abstract:"Wave properties of strongly coupled complex dusty (SCCD) plasmas evaluated using the equilibrium molecular dynamics (EMD) simulation technique. In this work, the plasma normalized longitudinal current correlation function CL(k,t) and transverse current CT(k,t) are calculated for a large range of plasma parameters of Coulomb coupling parameter (Γ) and screening strength (κ) with varying wave’s number (k). In EMD simulations, we have analysed different modes of wave propagation in SCCD plasmas with increasing and decreasing sequences of different combinations of plasmas parameters (κ, Γ) at varying simulation time step (Δt). Our simulation results show that the fluctuation of waves increases with an increase of Γ and decreases with increasing κ. Additional test shows that the presented results for waves are slightly dependent on number of particles (N). The amplitude and time period of CL(k,t) and CT(k,t) also depend on different influenced parameters of κ, Γ, k and N. The new results obtained through the presented EMD method for complex dusty plasma discussed and compared with earlier simulation results based on different numerical methods. It is demonstrated that the presented model is the best tool for estimating the behaviour of waves in strongly coupled complex system (dusty plasmas) over a suitable range of parameters.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Aamir Shahzad, Muhammad Asif Shakoori, Maogang He and Sajid\nBashir",authors:[{id:"288354",title:"Dr.",name:"Aamir",middleName:null,surname:"Shahzad",slug:"aamir-shahzad",fullName:"Aamir Shahzad"}]},{id:"58101",doi:"10.5772/intechopen.72215",title:"Wave Propagation in Porous Materials",slug:"wave-propagation-in-porous-materials",totalDownloads:1554,totalCrossrefCites:1,totalDimensionsCites:5,abstract:"This chapter provides different models for the acoustic wave propagation in porous materials having a rigid and an elastic frames. The direct problem of reflection and transmission of acoustic waves by a slab of porous material is studied. The inverse problem is solved using experimental reflected and transmitted signals. Both high- and low-frequency domains are studied. Different acoustic methods are proposed for measuring physical parameters describing the acoustic propagation as porosity, tortuosity, viscous and thermal characteristic length, and flow resistivity. Some advantages and perspectives of this method are discussed.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Zine El Abiddine Fellah, Mohamed Fellah, Claude Depollier, Erick\nOgam and Farid G. Mitri",authors:[{id:"143693",title:"Dr.",name:"Zine El Abiddine",middleName:null,surname:"Fellah",slug:"zine-el-abiddine-fellah",fullName:"Zine El Abiddine Fellah"},{id:"144519",title:"Prof.",name:"Claude",middleName:null,surname:"Depollier",slug:"claude-depollier",fullName:"Claude Depollier"},{id:"178778",title:"Prof.",name:"Mohamed",middleName:null,surname:"Fellah",slug:"mohamed-fellah",fullName:"Mohamed Fellah"},{id:"209074",title:"Dr.",name:"Erick",middleName:null,surname:"Ogam",slug:"erick-ogam",fullName:"Erick Ogam"},{id:"227468",title:"Dr.",name:"Farid G",middleName:null,surname:"Mitri",slug:"farid-g-mitri",fullName:"Farid G Mitri"}]},{id:"57674",doi:"10.5772/intechopen.71647",title:"Optimized Finite Difference Methods for Seismic Acoustic Wave Modeling",slug:"optimized-finite-difference-methods-for-seismic-acoustic-wave-modeling",totalDownloads:1526,totalCrossrefCites:1,totalDimensionsCites:3,abstract:"The finite difference (FD) methods are widely used for approximating the partial derivatives in the acoustic/elastic wave equation. Grid dispersion is one of the key numerical problems and will directly influence the accuracy of the result because of the discretization of the partial derivatives in the wave equation. Therefore, it is of great importance to suppress the grid dispersion by optimizing the FD coefficient. Various optimized methods are introduced in this chapter to determine the FD coefficient. Usually, the identical staggered grid finite difference operator is used for all of the first-order spatial derivatives in the first-order wave equation. In this chapter, we introduce a new staggered grid FD scheme which can improve the efficiency while still preserving high accuracy for the first-order acoustic/elastic wave equation modeling. It uses different staggered grid FD operators for different spatial derivatives in the first-order wave equation. The staggered grid FD coefficients of the new FD scheme can be obtained with a linear method. At last, numerical experiments were done to demonstrate the effectiveness of the introduced method.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Yanfei Wang and Wenquan Liang",authors:[{id:"218676",title:"Prof.",name:"Yanfei",middleName:null,surname:"Wang",slug:"yanfei-wang",fullName:"Yanfei Wang"}]},{id:"57603",doi:"10.5772/intechopen.71411",title:"In-Fiber Acousto-Optic Interaction Based on Flexural Acoustic Waves and Its Application to Fiber Modulators",slug:"in-fiber-acousto-optic-interaction-based-on-flexural-acoustic-waves-and-its-application-to-fiber-mod",totalDownloads:1320,totalCrossrefCites:0,totalDimensionsCites:1,abstract:"The design and implementation of in-fiber acousto-optic (AO) devices based on acoustic flexural waves are presented. The AO interaction is demonstrated to be an efficient mechanism for the development of AO tunable filters and modulators. The implementation of tapered optical fibers is proposed to shape the spectral response of in-fiber AO devices. Experimental results demonstrate that the geometry of the tapered fiber can be regarded as an extra degree of freedom for the design of AO tunable attenuation filters (AOTAFs). In addition, with the objective of expanding the application of AOTAFs to operate as an amplitude modulator, acoustic reflection was intentionally induced. Hence, a standing acoustic wave is generated which produces an amplitude modulation at twice the acoustic frequency. As a particular case, an in-fiber AO modulator composed of a double-ended tapered fiber was reported. The fiber taper was prepared using a standard fusion and pulling technique, and it was tapered down to a fiber diameter of 70 μm. The device exhibits an amplitude modulation at 2.313 MHz, which is two times the acoustic frequency used (1.1565 MHz); a maximum modulation depth of 60%, 1.3 dB of insertion loss, and 40 nm of modulation bandwidth were obtained. These results are within the best results reported in the framework of in-fiber AO modulators.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Miguel Ángel Bello Jiménez, Gustavo Ramírez-Meléndez, Erika\nHernández-Escobar, Andrés Camarillo-Avilés, Rosa López-Estopier,\nOlivier Pottiez, Cristian Cuadrado-Laborde, Antonio Díez, José L.\nCruz and Miguel V. Andrés",authors:[{id:"46578",title:"Dr.",name:"Miguel V.",middleName:null,surname:"Andrés",slug:"miguel-v.-andres",fullName:"Miguel V. Andrés"},{id:"46579",title:"Dr.",name:"Antonio",middleName:null,surname:"Diez",slug:"antonio-diez",fullName:"Antonio Diez"},{id:"46580",title:"Dr.",name:"José L.",middleName:null,surname:"Cruz",slug:"jose-l.-cruz",fullName:"José L. Cruz"},{id:"160262",title:"Dr.",name:"Olivier Jean Michel",middleName:null,surname:"Pottiez",slug:"olivier-jean-michel-pottiez",fullName:"Olivier Jean Michel Pottiez"},{id:"160283",title:"Dr.",name:"Miguel",middleName:null,surname:"Bello-Jiménez",slug:"miguel-bello-jimenez",fullName:"Miguel Bello-Jiménez"},{id:"182010",title:"Dr.",name:"R.",middleName:null,surname:"López-Estopier",slug:"r.-lopez-estopier",fullName:"R. López-Estopier"},{id:"220895",title:"MSc.",name:"Gustavo",middleName:null,surname:"Ramírez-Meléndez",slug:"gustavo-ramirez-melendez",fullName:"Gustavo Ramírez-Meléndez"},{id:"220896",title:"MSc.",name:"Erika",middleName:null,surname:"Hernández-Escobar",slug:"erika-hernandez-escobar",fullName:"Erika Hernández-Escobar"},{id:"220897",title:"BSc.",name:"Andrés",middleName:null,surname:"Camarillo-Avilés",slug:"andres-camarillo-aviles",fullName:"Andrés Camarillo-Avilés"},{id:"220902",title:"Dr.",name:"Christian",middleName:null,surname:"Cuadrado-Laborde",slug:"christian-cuadrado-laborde",fullName:"Christian Cuadrado-Laborde"}]}],mostDownloadedChaptersLast30Days:[{id:"58101",title:"Wave Propagation in Porous Materials",slug:"wave-propagation-in-porous-materials",totalDownloads:1554,totalCrossrefCites:1,totalDimensionsCites:5,abstract:"This chapter provides different models for the acoustic wave propagation in porous materials having a rigid and an elastic frames. The direct problem of reflection and transmission of acoustic waves by a slab of porous material is studied. The inverse problem is solved using experimental reflected and transmitted signals. Both high- and low-frequency domains are studied. Different acoustic methods are proposed for measuring physical parameters describing the acoustic propagation as porosity, tortuosity, viscous and thermal characteristic length, and flow resistivity. Some advantages and perspectives of this method are discussed.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Zine El Abiddine Fellah, Mohamed Fellah, Claude Depollier, Erick\nOgam and Farid G. Mitri",authors:[{id:"143693",title:"Dr.",name:"Zine El Abiddine",middleName:null,surname:"Fellah",slug:"zine-el-abiddine-fellah",fullName:"Zine El Abiddine Fellah"},{id:"144519",title:"Prof.",name:"Claude",middleName:null,surname:"Depollier",slug:"claude-depollier",fullName:"Claude Depollier"},{id:"178778",title:"Prof.",name:"Mohamed",middleName:null,surname:"Fellah",slug:"mohamed-fellah",fullName:"Mohamed Fellah"},{id:"209074",title:"Dr.",name:"Erick",middleName:null,surname:"Ogam",slug:"erick-ogam",fullName:"Erick Ogam"},{id:"227468",title:"Dr.",name:"Farid G",middleName:null,surname:"Mitri",slug:"farid-g-mitri",fullName:"Farid G Mitri"}]},{id:"57258",title:"Sound Waves in Complex (Dusty) Plasmas",slug:"sound-waves-in-complex-dusty-plasmas",totalDownloads:1398,totalCrossrefCites:5,totalDimensionsCites:5,abstract:"Wave properties of strongly coupled complex dusty (SCCD) plasmas evaluated using the equilibrium molecular dynamics (EMD) simulation technique. In this work, the plasma normalized longitudinal current correlation function CL(k,t) and transverse current CT(k,t) are calculated for a large range of plasma parameters of Coulomb coupling parameter (Γ) and screening strength (κ) with varying wave’s number (k). In EMD simulations, we have analysed different modes of wave propagation in SCCD plasmas with increasing and decreasing sequences of different combinations of plasmas parameters (κ, Γ) at varying simulation time step (Δt). Our simulation results show that the fluctuation of waves increases with an increase of Γ and decreases with increasing κ. Additional test shows that the presented results for waves are slightly dependent on number of particles (N). The amplitude and time period of CL(k,t) and CT(k,t) also depend on different influenced parameters of κ, Γ, k and N. The new results obtained through the presented EMD method for complex dusty plasma discussed and compared with earlier simulation results based on different numerical methods. It is demonstrated that the presented model is the best tool for estimating the behaviour of waves in strongly coupled complex system (dusty plasmas) over a suitable range of parameters.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Aamir Shahzad, Muhammad Asif Shakoori, Maogang He and Sajid\nBashir",authors:[{id:"288354",title:"Dr.",name:"Aamir",middleName:null,surname:"Shahzad",slug:"aamir-shahzad",fullName:"Aamir Shahzad"}]},{id:"56289",title:"Acoustic Analysis of Enclosed Sound Space as well as Its Coupling with Flexible Boundary Structure",slug:"acoustic-analysis-of-enclosed-sound-space-as-well-as-its-coupling-with-flexible-boundary-structure",totalDownloads:1293,totalCrossrefCites:0,totalDimensionsCites:0,abstract:"Combustion instability is often encountered in various power systems, a good understanding on the sound field in acoustic cavity as well as its coupling with boundary flexible structure will be of great help for the reliability design of such combustion system. An improved Fourier series method is presented for the acoustic/vibro-acoustic modelling of acoustic cavity as well as the panel-cavity coupling system. The structural-acoustic coupling system is described in a unified pattern using the energy principle. With the aim to construct the admissible functions sufficiently smooth for the enclosed sound space as well as the flexible boundary structure, the boundary-smoothed auxiliary functions are introduced to the standard multi-dimensional Fourier series. All the unknown coefficients and higher order variables are determined in conjunction with Rayleigh-Ritz procedure and differential operation term by term. Numerical examples are then presented to show the correctness and effectiveness of the current model. The model is verified through the comparison with those from analytic solution and other approaches. Based on the model established, the influence of boundary conditions on the acoustic and/or vibro-acoustic characteristics of the structural-acoustic coupling system is addressed and investigated.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Jingtao Du, Yang Liu and Long Liu",authors:[{id:"203133",title:"Prof.",name:"Jingtao",middleName:null,surname:"Du",slug:"jingtao-du",fullName:"Jingtao Du"},{id:"203657",title:"Dr.",name:"Yang",middleName:null,surname:"Liu",slug:"yang-liu",fullName:"Yang Liu"},{id:"203658",title:"Dr.",name:"Long",middleName:null,surname:"Liu",slug:"long-liu",fullName:"Long Liu"}]},{id:"57214",title:"A Novel Idea of Coherent Acoustic Wave-Induced Atmospheric Refractivity Fluctuation and Its Applications",slug:"a-novel-idea-of-coherent-acoustic-wave-induced-atmospheric-refractivity-fluctuation-and-its-applicat",totalDownloads:1435,totalCrossrefCites:0,totalDimensionsCites:0,abstract:"The physical mechanism of generating the lasting tropospheric refractivity fluctuation with a stable array-distributed structure by coherent acoustic waves is investigated. An example of the quantitative calculation of atmospheric refractive index is given and analyzed. Based on the theory of electromagnetic wave propagation and scattering in the troposphere, the feasibility to purposefully affect radio wave propagation is qualitatively demonstrated by the experiment of the coherent acoustic source-induced laser interference fringe change. The potential application aspects of synthetically controlling the radio wave propagation by the artificial refractivity fluctuation structure are preliminarily proposed. This chapter will promote the development of the coherent acoustic wave-induced tropospheric refractivity fluctuation, and it has the important theoretical significance and potential application value to purposely apply the positive or negative effects on radio wave propagation.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Shuhong Gong, Yu Liu, Muyu Hou and Lixin Guo",authors:[{id:"218965",title:"Dr.",name:"Shuhong",middleName:null,surname:"Gong",slug:"shuhong-gong",fullName:"Shuhong Gong"},{id:"220994",title:"BSc.",name:"Yu",middleName:null,surname:"Liu",slug:"yu-liu",fullName:"Yu Liu"},{id:"220995",title:"BSc.",name:"Muyu",middleName:null,surname:"Hou",slug:"muyu-hou",fullName:"Muyu Hou"},{id:"220996",title:"Dr.",name:"Lixin",middleName:null,surname:"Guo",slug:"lixin-guo",fullName:"Lixin Guo"}]},{id:"57603",title:"In-Fiber Acousto-Optic Interaction Based on Flexural Acoustic Waves and Its Application to Fiber Modulators",slug:"in-fiber-acousto-optic-interaction-based-on-flexural-acoustic-waves-and-its-application-to-fiber-mod",totalDownloads:1320,totalCrossrefCites:0,totalDimensionsCites:1,abstract:"The design and implementation of in-fiber acousto-optic (AO) devices based on acoustic flexural waves are presented. The AO interaction is demonstrated to be an efficient mechanism for the development of AO tunable filters and modulators. The implementation of tapered optical fibers is proposed to shape the spectral response of in-fiber AO devices. Experimental results demonstrate that the geometry of the tapered fiber can be regarded as an extra degree of freedom for the design of AO tunable attenuation filters (AOTAFs). In addition, with the objective of expanding the application of AOTAFs to operate as an amplitude modulator, acoustic reflection was intentionally induced. Hence, a standing acoustic wave is generated which produces an amplitude modulation at twice the acoustic frequency. As a particular case, an in-fiber AO modulator composed of a double-ended tapered fiber was reported. The fiber taper was prepared using a standard fusion and pulling technique, and it was tapered down to a fiber diameter of 70 μm. The device exhibits an amplitude modulation at 2.313 MHz, which is two times the acoustic frequency used (1.1565 MHz); a maximum modulation depth of 60%, 1.3 dB of insertion loss, and 40 nm of modulation bandwidth were obtained. These results are within the best results reported in the framework of in-fiber AO modulators.",book:{id:"5708",slug:"computational-and-experimental-studies-of-acoustic-waves",title:"Computational and Experimental Studies of Acoustic Waves",fullTitle:"Computational and Experimental Studies of Acoustic Waves"},signatures:"Miguel Ángel Bello Jiménez, Gustavo Ramírez-Meléndez, Erika\nHernández-Escobar, Andrés Camarillo-Avilés, Rosa López-Estopier,\nOlivier Pottiez, Cristian Cuadrado-Laborde, Antonio Díez, José L.\nCruz and Miguel V. Andrés",authors:[{id:"46578",title:"Dr.",name:"Miguel V.",middleName:null,surname:"Andrés",slug:"miguel-v.-andres",fullName:"Miguel V. 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\r\n\tIn order to scientifically address significant issues such as climate change, which puts into question our very survival as a species, the current pandemic with its massive physical, socio-economical, and psychological consequences, and the rise of AI which challenges our established economic structures, we need to ask insightful questions: What is truly human? How can humans develop further? The answers to these questions are necessary not only to find new solutions to the current challenges, but also to shape new visions of what can come next.
\r\n
\r\n\tNeuroscientific research linking brain functions has produced a perspective on human development that includes normal, impaired, and enhanced neurophysiological, emotional and cognitive functioning. Human development has been considered the very aim of education and of educative processes. Indeed, the capabilities built through educational training are included in the UN’s human development index, according to which such capabilities are the ultimate criteria to assess the development of a country, rather than economic growth alone. Yet a full understanding of what Human Development truly constitutes, remains open. For example, tackling the question of what distinguishes human beings from other animals, and what humans’ possible development trajectory might look like, calls for a multidisciplinary approach. Consequently, contributions to such an inquiry might come from very different scientific fields, ranging from cognitive neuroscience to socioeconomics. For instance, in the field of neuroscience, self-awareness—the most specific characteristic of human beings—has been investigated in connection with its neural correlates. Recent research points to self-awareness as the particular ability of our species, directly connecting it to our abstract thinking which in turn enables envisioning new possible futures and self-development
\r\n
\r\n\tTo achieve a broad, multidisciplinary perspective on possible human development, subjects will be considered through varied— yet related—approaches. We will provide a complex yet consistent framework through which we will explore a substantial amount and variety of theories and case studies. Our ultimate goal will be to produce useful indications for policy making in diverse contexts, assist teachers and parents with child development in an optimal way, and enhance theoretical and practical knowledge.
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\r\n\tIn general, the harsher the environmental conditions in an ecosystem, the lower the biodiversity. Changes in the environment caused by human activity accelerate the impoverishment of biodiversity.
\r\n
\r\n\tBiodiversity refers to “the variability of living organisms from any source, including terrestrial, marine and other aquatic ecosystems and the ecological complexes of which they are part; it includes diversity within each species, between species, and that of ecosystems”.
\r\n
\r\n\tBiodiversity provides food security and constitutes a gene pool for biotechnology, especially in the field of agriculture and medicine, and promotes the development of ecotourism.
\r\n
\r\n\tCurrently, biologists admit that we are witnessing the first phases of the seventh mass extinction caused by human intervention. It is estimated that the current rate of extinction is between a hundred and a thousand times faster than it was when man first appeared. The disappearance of species is caused not only by an accelerated rate of extinction, but also by a decrease in the rate of emergence of new species as human activities degrade the natural environment. The conservation of biological diversity is "a common concern of humanity" and an integral part of the development process. Its objectives are “the conservation of biological diversity, the sustainable use of its components, and the fair and equitable sharing of the benefits resulting from the use of genetic resources”.
\r\n
\r\n\tThe following are the main causes of biodiversity loss:
\r\n
\r\n\t• The destruction of natural habitats to expand urban and agricultural areas and to obtain timber, minerals and other natural resources.
\r\n
\r\n\t• The introduction of alien species into a habitat, whether intentionally or unintentionally which has an impact on the fauna and flora of the area, and as a result, they are reduced or become extinct.
\r\n
\r\n\t• Pollution from industrial and agricultural products, which devastate the fauna and flora, especially those in fresh water.
\r\n
\r\n\t• Global warming, which is seen as a threat to biological diversity, and will become increasingly important in the future.
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\r\n\tThe environment is subject to severe anthropic effects. Among them are those associated with pollution, resource extraction and overexploitation, loss of biodiversity, soil degradation, disorderly land occupation and planning, and many others. These anthropic effects could potentially be caused by any inadequate management of the environment. However, ecosystems have a resilience that makes them react to disturbances which mitigate the negative effects. It is critical to understand how ecosystems, natural and anthropized, including urban environments, respond to actions that have a negative influence and how they are managed. It is also important to establish when the limits marked by the resilience and the breaking point are achieved and when no return is possible. The main focus for the chapters is to cover the subjects such as understanding how the environment resilience works, the mechanisms involved, and how to manage them in order to improve our interactions with the environment and promote the use of adequate management practices such as those outlined in the United Nations’ Sustainable Development Goals.
\r\n\tPollution is caused by a wide variety of human activities and occurs in diverse forms, for example biological, chemical, et cetera. In recent years, significant efforts have been made to ensure that the environment is clean, that rigorous rules are implemented, and old laws are updated to reduce the risks towards humans and ecosystems. However, rapid industrialization and the need for more cultivable sources or habitable lands, for an increasing population, as well as fewer alternatives for waste disposal, make the pollution control tasks more challenging. Therefore, this topic will focus on assessing and managing environmental pollution. It will cover various subjects, including risk assessment due to the pollution of ecosystems, transport and fate of pollutants, restoration or remediation of polluted matrices, and efforts towards sustainable solutions to minimize environmental pollution.
\r\n\tWater is not only a crucial substance needed for biological life on Earth, but it is also a basic requirement for the existence and development of the human society. Owing to the importance of water to life on Earth, early researchers conducted numerous studies and analyses on the liquid form of water from the perspectives of chemistry, physics, earth science, and biology, and concluded that Earth is a "water polo". Water covers approximately 71% of Earth's surface. However, 97.2% of this water is seawater, 21.5% is icebergs and glaciers, and only 0.65% is freshwater that can be used directly by humans. As a result, the amount of water reserves available for human consumption is limited. The development, utilization, and protection of freshwater resources has become the focus of water science research for the continued improvement of human livelihoods and society.
\r\n
\r\n\tWater exists as solid, liquid, and gas within Earth’s atmosphere, lithosphere, and biosphere. Liquid water is used for a variety of purposes besides drinking, including power generation, ecology, landscaping, and shipping. Because water is involved in various environmental hydrological processes as well as numerous aspects of the economy and human society, the study of various phenomena in the hydrosphere, the laws governing their occurrence and development, the relationship between the hydrosphere and other spheres of Earth, and the relationship between water and social development, are all part of water science. Knowledge systems for water science are improving continuously. Water science has become a specialized field concerned with the identification of its physical, chemical, and biological properties. In addition, it reveals the laws of water distribution, movement, and circulation, and proposes methods and tools for water development, utilization, planning, management, and protection. Currently, the field of water science covers research related to topics such as hydrology, water resources and water environment. It also includes research on water related issues such as safety, engineering, economy, law, culture, information, and education.
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Biochemistry also studies small signaling molecules, coenzymes, inhibitors, vitamins, and hormones, which play roles in life processes. Biochemical experimentation, besides coopting classical chemistry methods, e.g., chromatography, adopted new techniques, e.g., X-ray diffraction, electron microscopy, NMR, radioisotopes, and developed sophisticated microbial genetic tools, e.g., auxotroph mutants and their revertants, fermentation, etc. More recently, biochemistry embraced the ‘big data’ omics systems. Initial biochemical studies have been exclusively analytic: dissecting, purifying, and examining individual components of a biological system; in the apt words of Efraim Racker (1913 –1991), “Don’t waste clean thinking on dirty enzymes.” Today, however, biochemistry is becoming more agglomerative and comprehensive, setting out to integrate and describe entirely particular biological systems. The ‘big data’ metabolomics can define the complement of small molecules, e.g., in a soil or biofilm sample; proteomics can distinguish all the comprising proteins, e.g., serum; metagenomics can identify all the genes in a complex environment, e.g., the bovine rumen. This Biochemistry Series will address the current research on biomolecules and the emerging trends with great promise.",coverUrl:"https://cdn.intechopen.com/series/covers/11.jpg",latestPublicationDate:"June 24th, 2022",hasOnlineFirst:!0,numberOfOpenTopics:4,numberOfPublishedChapters:314,numberOfPublishedBooks:31,editor:{id:"31610",title:"Dr.",name:"Miroslav",middleName:null,surname:"Blumenberg",fullName:"Miroslav Blumenberg",profilePictureURL:"https://mts.intechopen.com/storage/users/31610/images/system/31610.jpg",biography:"Miroslav Blumenberg, Ph.D., was born in Subotica and received his BSc in Belgrade, Yugoslavia. He completed his Ph.D. at MIT in Organic Chemistry; he followed up his Ph.D. with two postdoctoral study periods at Stanford University. Since 1983, he has been a faculty member of the RO Perelman Department of Dermatology, NYU School of Medicine, where he is codirector of a training grant in cutaneous biology. Dr. Blumenberg’s research is focused on the epidermis, expression of keratin genes, transcription profiling, keratinocyte differentiation, inflammatory diseases and cancers, and most recently the effects of the microbiome on the skin. He has published more than 100 peer-reviewed research articles and graduated numerous Ph.D. and postdoctoral students.",institutionString:null,institution:{name:"New York University Langone Medical Center",institutionURL:null,country:{name:"United States of America"}}},subseries:[{id:"14",title:"Cell and Molecular Biology",keywords:"Omics (Transcriptomics; Proteomics; Metabolomics), Molecular Biology, Cell Biology, Signal Transduction and Regulation, Cell Growth and Differentiation, Apoptosis, Necroptosis, Ferroptosis, Autophagy, Cell Cycle, Macromolecules and Complexes, Gene Expression",scope:"The Cell and Molecular Biology topic within the IntechOpen Biochemistry Series aims to rapidly publish contributions on all aspects of cell and molecular biology, including aspects related to biochemical and genetic research (not only in humans but all living beings). We encourage the submission of manuscripts that provide novel and mechanistic insights that report significant advances in the fields. Topics include, but are not limited to: Advanced techniques of cellular and molecular biology (Molecular methodologies, imaging techniques, and bioinformatics); Biological activities at the molecular level; Biological processes of cell functions, cell division, senescence, maintenance, and cell death; Biomolecules interactions; Cancer; Cell biology; Chemical biology; Computational biology; Cytochemistry; Developmental biology; Disease mechanisms and therapeutics; DNA, and RNA metabolism; Gene functions, genetics, and genomics; Genetics; Immunology; Medical microbiology; Molecular biology; Molecular genetics; Molecular processes of cell and organelle dynamics; Neuroscience; Protein biosynthesis, degradation, and functions; Regulation of molecular interactions in a cell; Signalling networks and system biology; Structural biology; Virology and microbiology.",annualVolume:11410,isOpenForSubmission:!0,coverUrl:"https://cdn.intechopen.com/series_topics/covers/14.jpg",editor:{id:"165627",title:"Dr.",name:"Rosa María",middleName:null,surname:"Martínez-Espinosa",fullName:"Rosa María Martínez-Espinosa",profilePictureURL:"https://mts.intechopen.com/storage/users/165627/images/system/165627.jpeg",institutionString:null,institution:{name:"University of Alicante",institutionURL:null,country:{name:"Spain"}}},editorTwo:null,editorThree:null,editorialBoard:[{id:"79367",title:"Dr.",name:"Ana Isabel",middleName:null,surname:"Flores",fullName:"Ana Isabel Flores",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRpIOQA0/Profile_Picture_1632418099564",institutionString:null,institution:{name:"Hospital Universitario 12 De Octubre",institutionURL:null,country:{name:"Spain"}}},{id:"328234",title:"Ph.D.",name:"Christian",middleName:null,surname:"Palavecino",fullName:"Christian Palavecino",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0033Y000030DhEhQAK/Profile_Picture_1628835318625",institutionString:null,institution:{name:"Central University of Chile",institutionURL:null,country:{name:"Chile"}}},{id:"186585",title:"Dr.",name:"Francisco Javier",middleName:null,surname:"Martin-Romero",fullName:"Francisco Javier Martin-Romero",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bSB3HQAW/Profile_Picture_1631258137641",institutionString:null,institution:{name:"University of Extremadura",institutionURL:null,country:{name:"Spain"}}}]},{id:"15",title:"Chemical Biology",keywords:"Phenolic Compounds, Essential Oils, Modification of Biomolecules, Glycobiology, Combinatorial Chemistry, Therapeutic peptides, Enzyme Inhibitors",scope:"Chemical biology spans the fields of chemistry and biology involving the application of biological and chemical molecules and techniques. In recent years, the application of chemistry to biological molecules has gained significant interest in medicinal and pharmacological studies. This topic will be devoted to understanding the interplay between biomolecules and chemical compounds, their structure and function, and their potential applications in related fields. Being a part of the biochemistry discipline, the ideas and concepts that have emerged from Chemical Biology have affected other related areas. 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Behind these definitions are hidden all the aspects of normal and pathological functioning of all processes that the topic ‘Metabolism’ will cover within the Biochemistry Series. 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Thus proteomics, an area of research that detects all protein forms expressed in an organism, including splice isoforms and post-translational modifications, is more suitable than genomics for a comprehensive understanding of the biochemical processes that govern life. The most common proteomics applications are currently in the clinical field for the identification, in a variety of biological matrices, of biomarkers for diagnosis and therapeutic intervention of disorders. From the comparison of proteomic profiles of control and disease or different physiological states, which may emerge, changes in protein expression can provide new insights into the roles played by some proteins in human pathologies. Understanding how proteins function and interact with each other is another goal of proteomics that makes this approach even more intriguing. Specialized technology and expertise are required to assess the proteome of any biological sample. Currently, proteomics relies mainly on mass spectrometry (MS) combined with electrophoretic (1 or 2-DE-MS) and/or chromatographic techniques (LC-MS/MS). MS is an excellent tool that has gained popularity in proteomics because of its ability to gather a complex body of information such as cataloging protein expression, identifying protein modification sites, and defining protein interactions. The Proteomics topic aims to attract contributions on all aspects of MS-based proteomics that, by pushing the boundaries of MS capabilities, may address biological problems that have not been resolved yet.",annualVolume:11414,isOpenForSubmission:!0,coverUrl:"https://cdn.intechopen.com/series_topics/covers/18.jpg",editor:{id:"200689",title:"Prof.",name:"Paolo",middleName:null,surname:"Iadarola",fullName:"Paolo Iadarola",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bSCl8QAG/Profile_Picture_1623568118342",institutionString:null,institution:{name:"University of Pavia",institutionURL:null,country:{name:"Italy"}}},editorTwo:{id:"201414",title:"Dr.",name:"Simona",middleName:null,surname:"Viglio",fullName:"Simona Viglio",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRKDHQA4/Profile_Picture_1630402531487",institutionString:null,institution:{name:"University of Pavia",institutionURL:null,country:{name:"Italy"}}},editorThree:null,editorialBoard:[{id:"72288",title:"Dr.",name:"Arli Aditya",middleName:null,surname:"Parikesit",fullName:"Arli Aditya Parikesit",profilePictureURL:"https://mts.intechopen.com/storage/users/72288/images/system/72288.jpg",institutionString:null,institution:{name:"Indonesia International Institute for Life Sciences",institutionURL:null,country:{name:"Indonesia"}}},{id:"40928",title:"Dr.",name:"Cesar",middleName:null,surname:"Lopez-Camarillo",fullName:"Cesar Lopez-Camarillo",profilePictureURL:"https://mts.intechopen.com/storage/users/40928/images/3884_n.png",institutionString:null,institution:{name:"Universidad Autónoma de la Ciudad de México",institutionURL:null,country:{name:"Mexico"}}},{id:"81926",title:"Dr.",name:"Shymaa",middleName:null,surname:"Enany",fullName:"Shymaa Enany",profilePictureURL:"https://mts.intechopen.com/storage/users/81926/images/system/81926.png",institutionString:"Suez Canal University",institution:{name:"Suez Canal University",institutionURL:null,country:{name:"Egypt"}}}]}]}},libraryRecommendation:{success:null,errors:{},institutions:[]},route:{name:"profile.detail",path:"/profiles/82658",hash:"",query:{},params:{id:"82658"},fullPath:"/profiles/82658",meta:{},from:{name:null,path:"/",hash:"",query:{},params:{},fullPath:"/",meta:{}}}},function(){var t;(t=document.currentScript||document.scripts[document.scripts.length-1]).parentNode.removeChild(t)}()