Statistical Properties of Surface Slopes via Remote Sensing

Via remote sensing, the use of radar images and optical processing of aerial photographs has been used. The interest in wave data is manifold; one element is the inherent interest in the directional spectra of waves and how they influence the marine environment and the coastline. These wave data can be readily and accurately collected by aerial photographs of the wave sun glint patterns which show reflections of the Sun and sky light from the water and thus offer high-contrast wave images.


Introduction
The complexity of wave motion in deep waters, which can damage marine platforms and vessels, and in shallow waters, same that can afflict human settlements and recreational areas, has given origin to a long-term development in laboratory and field studies, the conclusions of which are used to design methodology and set bases to understand wave motion behavior.
Via remote sensing, the use of radar images and optical processing of aerial photographs has been used. The interest in wave data is manifold; one element is the inherent interest in the directional spectra of waves and how they influence the marine environment and the coastline. These wave data can be readily and accurately collected by aerial photographs of the wave sun glint patterns which show reflections of the Sun and sky light from the water and thus offer high-contrast wave images.
In a series of articles, Cox and Munk (1954a, 1954b, 1955 studied the distribution of intensity or glitter pattern in aerial photographs of the sea. One of their conclusions was that for constant and moderate wind speed, the probability density function of the slopes is approximately Gaussian. This could be taken as an indication that in certain circumstances, the ocean surface could be modeled as a Gaussian random process. Similar observations by Longuet-Higgins et al. (1963) (cited by Longuet-Higgins (1962)) with a floating buoy, which filters out the high-frequency components, come considerably closer to the Gaussian distribution.
Other authors (Stilwell, 1969;Stilwell & Pilon, 1974) have studied the same problem considering a sea surface illuminated by a continuous sky light with no azimuthal variations in sky radiance. Different models of sky light have been used emphasizing the existence of a nonlinear relationship between the slope spectrum and the corresponding wave image spectrum (Peppers & Ostrem, 1978;Chapman & Irani, 1981).
Simulated sea surfaces have been analyzed by optical systems to understand the optical technique in order to obtain best qualitative information of the spectrum (Álvarez-Borrego, 1987;Álvarez-Borrego & Machado, 1985). (1) Because the source has a finite size, there are several incidence directions which are specular reflected to the camera. The directions, os  (where this angle is the angular dimension of the where rect(.) represents the rectangle function (Gaskill, 1978).
So, the projection of this source on the detector, after reflection, is given by where equation (1) We find then the "glitter function", given by This expression (eq. 12) tell us that the geometry of the problem selects a surface slope region and encodes like bright points in the image (glitter pattern).

Relationship among the variances of the intensities in the image, surface slopes and surface heights
The mean of the image, I  , may be written (Papoulis, 1981)   () , where  B  is defined by equation (12) and  p  is the probability density function in one dimension, where in a first approximation a Gaussian function is considered. Substituting in equation (13)  In the figure we can observe the dependence of this relationship with the angular position of the source, s  . In figure 2 we also can observe that for small incidence angles (0-10 degrees) and small values of variance of the surface slopes, it is possible to obtain bigger values in the variance of the intensities in the image. From equation (18), we can see that this behavior is independent of any surface height power spectrum that we are analyzing, because this relation depends on the probability density function of the surface slopes and the geometry of the experiment only. To solve this problem, it is necessary to analyze images which correspond at two or more incidence angles and to select a slope variance value which is consistent with all these data.
The relationship between 2   and 2   can be derived from (Papoulis, 1981) if we know the correlation function of the surface heights (this will be shown in next section of this chapter). Here,

 
C   is the correlation function of the surface heights and   C   is the correlation function of the surface slopes.

Relationship between the correlation function of the intensities in the image and of the surface heights
Our analysis involves three random processes: the surface profile,  , Ix Each process has a correlation function and it was shown (Álvarez-Borrego, 1993) that these three functions hold a relationship.
In order to achieve the inverse process, using equation (19) and equation (20), these two equations must meet certain conditions. For example, it is required that there exists one to one correspondence among the amount involved.
Using equation (19) the processed data can be numerically integrated twice, such that we obtain information of the correlation function of the surface heights,   C   , from the correlation function of the surface slopes, (20) is a more complicated expression, we cannot obtain an analytical result from it. A first integral can be analytically solved and for the second it is possible to obtain the solution by numerical integration. Resolving the first integral analytically, equation (20)   Also, from equation (19), it is possible to obtain the correlation function of the surface heights,  C   , from   C   and the require inverse process to determine the correlation function of the surface heights is completed.
A theoretical variance 2 I  can be calculated from equation (21). We wrote in Table 1

Geometry of the model (Gaussian case considering a variable detector angle)
A more real physical situation is shown in figure 4. The surface,   x  , is illuminated by a uniform incoherent source S of limited angular extent, with wavelength  . Its image is formed in D by an aberration-free optical system. The incidence angle s  is defined as the angle between the incidence angle direction and the normal to the mean surface and represents the mean angle subtended by the source S.   d i  corresponds to the angle subtended by the optical system of the detector with the normal to point i of the surface, i. e.  1 tan , where H is the height of the detector and x  is the interval between surface points. We can see that in this more realistic physical situation, angle d  is changing with respect to each point in the surface. It is worth noticing that a variable d  does not restrict the sensor field of view.
i  is the angle subtended between the normal to the mean surface and the normal to the slope for each i point in the surface The apparent diameter of the source is  . Light from the source is reflected on the surface for just one time, and, depending on the slope, the light reflected will or will not be part of the image. Thus, the image consists of bright and dark regions that we call a glitter pattern.
The interval characterized by equation (25)

Relationships among the variances of the intensities in the image and surface slopes
The mean of the image I  may be written as (Álvarez-Borrego & Martín-Atienza, 2010) www.intechopen.com


   , where   i B  is the glitter function defined be equation (24).   i p  is the probability density function, where a Gaussian function is considered in one dimension. Substituting in equation (29) Substituting the equation (31) in equation (33)   The explanation for this is very simple: if the camera stays at H=100 m, it will receive more reflection of light at large s  , because the geometry of reflection. When H increases, the camera will receive less light reflection of large incidence angles but will have more light reflection for small incidence angles. Therefore, when the camera is at a larger height, will have more reflection from light incidence angles smaller than light of larger incidence angles. Thus we can say that the results presented by Álvarez-Borrego in 1993, Cureton et al., 2007 and Álvarez-Borrego & Martín-Atienza in 2010 are correct for the Gaussian case.
In certain cases, if we have data corresponding to one s  value, it is not possible to obtain a single value for the variance of the surface slopes 2   . To solve this problem, it is necessary to analyze images which correspond at two or more incidence angles and to select a slope variance value which is consistent with all these data (Álvarez-Borrego, 1995). From equation (34), we can see that this relation depends on the probability density function of the surface slopes and the geometry of the experiment only.

Relationship between the correlation functions of the intensities in the image and of the surface slope
The relationship between the correlation function of the surface slopes   Although it is possible to obtain an analytical relationship for the first integral, for the second integral the process must be numeric.  (37). We wrote in Table 2 the values of the image variance in order to normalize the correlations in figure 7 for different values for s  and H (100, 500, 1000 and 5000 m).

Geometry of the model (Non-Gaussian case considering a variable detector angle)
The model, considering d  as variable, is shown in figure 4. We think this is a more realistic situation.

Relationships among the variances of the intensities in the image and surface slopes considering a non-Gaussian probability density function
The mean of the image I  may be written as ( The detector angle d  is a function of the position x, thus, the specular angle is a function of the distance x from the nadir point of the detector, n = 0, to the point n = i (see equation (22)).    Figures   8 and 9 show this relationship considering the skewness and the skewness and kurtosis in the non-Gaussian probability density function respectively. We can see that the behavior of the curves looks very similar to the Gaussian case ( figure 6). The values for skewness and kurtosis were taken from a Table showed by Plant (2003) from data given by Cox & Munk (1956), for a wind speed of 13.3 m/s with the wind sensor at 12.5 m on the sea surface level.

Relationship between the correlation functions of the intensities in the image and of the surface slope considering a non-Gaussian probability density function
As mentioned before, our analysis involves three random processes: the surface profile  x  , its surface slopes   x  and the image   Ix. Each process has a correlation function and it was shown in (Álvarez-Borrego, 1993) that these three functions are related.
The relationship between the correlation function of the surface slopes   Although it is possible to obtain an analytical relationship for the first integral, for the second integral the process must be numeric. Thus, equation (43)   A theoretical variance 2 I  can be calculated from equation (45). We wrote in Table 3 the values of the image variance in order to normalize the correlations in figure 10 for different values for s  and H (100, 500, 1000 and 5000 m).  Table 3. Values of the image variance in order to normalize the correlations in figure 10 for different values for s  and H.

Conclusions
We derive the variance of the surface heights from the variance of the intensities in the image via remote sensing considering a glitter function given by equation (12) when the geometry consider a detector angle of 0 o d   , and considering a glitter function given by the equation (24) considering a geometrically improved model with variable detector line of sight angle, given by figure 4. In this last case, we consider Gaussian statistics and non-Gaussian statistics. We derive the variance of the surface slopes from the variance of the intensities of remote sensed images for different H values. In addition, we discussed the determination of the correlation function of the surface slopes from the correlation function of the image intensities considering Gaussian and non-Gaussian statistics.
Analyzing the variances curves for Gaussian and non-Gaussian case it is possible to see the behavior of the curves for different incident angles when H increases. This behavior agrees with the results presented by Álvarez-Borrego (1993) and Geoff Cureton et al. 2007, andMartin-Atienza (2010) for the Gaussian case.
These new results solve the inverse problem when it is necessary to analyze the statistical of a real sea surface via remote sensing using the image of the glitter pattern of the marine surface.