Regularity Analysis of Airborne Natural Gamma Ray Data Measured in the Hoggar Area (Algeria)

With growing attention on global environmental and climate change, geoscience has experienced rapid change and development in the last three decades. Many new data, methods and modeling techniques have been developed and applied in various aspects of geoscience. The chapters collected in this book present an excellent profile of the current state of various data, analysis methods and modeling techniques, and demonstrate their applications from hydrology, geology and paleogeomorphology, to geophysics, environmental and climate change. The wide range methods and techniques covered in the book include information systems and technology, global position system (GPS), digital sediment core image analysis, fuzzy set theory for hydrology, spatial interpolation, spectral analysis of geophysical data, GIS-based hydrological models, high resolution geological models, 3D sedimentology, change detection from remote sensing, etc. most chapters focus on studies of a particular method or technique.


Introduction
The airborne Gamma Ray (GR) measurements have been used since decades in geophysical research. The airborne measurement of gamma radiation emitted by naturally occurring elements finds applications in: geological mapping (Graham and Bonham-Carter, 1993;Jaques et al., 1997;Doll et al., 2000, Aydin et al., 2006Sulekha Rao et al., 2009), regolith and soil mapping (Cook et al., 1996;Wilford et al., 1997;Bierwirth and Welsh, 2000), mineral exploration (Brown et al., 2000), and hydrocarbon research (Matolín and Stráník, 2006). Potassium (K), Uranium (U) and Thorium (Th) are the three most abundant, naturally occurring radioactive elements. The K element is the main component of mineral deposits, while Uranium and Thorium are present in trace amounts, as mobile and immobile elements, respectively. The concentration of these different radioelements varies between different rock types, thus the information provided by a gamma-ray spectrometer can be exploited for needs of the rocks cartography. The obtained maps allow to localize radioelement anomalies corresponding to zones disrupted by a mineralizing system. The approach presented in this chapter deepens the results derived from the conventional study. It consists on a mono(two)-dimensional fractal analysis of natural radioactivity measurements recorded over the Hoggar area (Algeria). The natural radioactivity measurements, like other geophysical signals, contain a deterministic and a stochastic components. The former part holds information related to the regional aspect, while the latter reflects the local heterogeneities. As the raw spectrometric data need to be processed before any exploitation, the stochastic component can be altered and some information about heterogeneities is lost. Here, we show first the fractal behavior of the analyzed GR measurements. In addition, it is demonstrated that this behavior is not affected by all the pre-processing operations (spectrometric corrections and 2D-interpolations). The corrections are not then necessary. Since the analyzed data exhibit a fractal exponent varying with the spatial position, they are modeled as paths of multifractional Brownian motions (mBms) (Peltier and Lévy-Véhel, 1995).  (Caby et al., 1981, modified)

Overview on the analyzed GR measurements
The analyzed GR measurements are recorded during a magneto-spectrometric survey accomplished, between 1971 and 1974 over the Hoggar, for the purpose of the mining research and the regional geological mapping. The average of the flight height is fixed at 500 feet (approximately 150 m).  The direction of the profiles: perpendicular to the geological structures.  The distance between lines varies from 2 to 5 kilometers according to the areas, but on average it is about two kilometers.  The distance between the observation points is approximately 46.2 m (152 feet).

Corrections of the airborne natural activity measurements
The measurements acquired during an airborne spectrometric survey can not be exploited in a raw state, but need to be corrected mainly from aircraft background, stripping (or Compton) effect and height effect (IAEA, 2003).

Background corrections
There are three components of the background correction:  The instrument background (called ''aircraft background'' in airborne gamma spectrometry),  The cosmic background arisen from the reaction of primary cosmic radiation with atoms and molecules in the upper atmosphere.  The effect of atmospheric radon. In portable or car-borne gamma ray surveys, the background component is usually small relative to the signal from the ground. The observed count rates in the four channels: Total Count (TC), Potassium (K), Uranium (U) and Thorium (Th), are corrected for the background effects using the following formulae:

Stripping correction
This correction, also known as the channel interaction correction, consists of removing ('strips') count rates from each of the K, U and Th for gamma rays not originating from the radioelement or decay series being monitored. For example, Th series gamma rays appear in both the U and K channels, and U series gamma rays appear in the K channel. The corrections are given by:

Height correction
This correction is applied only on airborne gamma spectrometric measurements. The gamma radiation decreases exponentially with the elevation. Since the height of the aircraft changes continuously, the airborne Gamma Ray spectrometric data need to be corrected to a nominal survey height above the ground.
TC corr , K corr , U corr and Th corr : Values of the corrected count rates in the four channels, TC obs , K obs , U obs and Th obs : Values of the observed count rates in the four channels, h : Real survey height , h 0 : Nominal survey height (h 0 =150 m), TC , K , U and Th : Linear attenuation coefficients in the four channels. The estimated values of these coefficients are (Groune, 2009) : K = 6.8617 10 -3 m -1 , U = 6.3726 10 -3 m -1 , and Th = 5.2247 10 -3 m -1 . The TC is calculated as the approximate average of the three coefficients: TC = 6.56 10 -3 m -1 . www.intechopen.com

Impact of the pre-processings on the fractal properties of the airborne gamma ray measurements
Once all the corrections are applied, the corrected measurements grid is regrided using twodimensional interpolation algorithms to get a regular sampled grid which is processed by a local regularity analysis. The set of operations (corrections and interpolations) affects the stochastic component of the raw airborne spectrometric measurements, which holds information about heterogeneities. Therefore the fractal properties of the raw data may be changed. In the first stage, we have obtained the ''corrected'' and the ''corrected and interpolated'' data grids from the ''raw'' grid data corresponding to the measurements of the three channels: K, Th and U. The 2D-interpolation algorithms used in this study are: the trianglebased linear, the triangle-based cubic and the nearest neighbor interpolation algorithms. Since the results obtained by the different interpolation methods are close, only those related to the triangle-based linear algorithm are presented. First, five vertical profiles are extracted from the three considered grids (''raw'', ''corrected'' and ''corrected and interpolated '' grids) from the measurements of the three channels (Fig.2). The Fourier amplitude spectrum and the local Hölder exponent H(x) are computed for each data profile. (The geological map of the Hoggar, from Caby et al., 1981).
Regarding the computation of H(x), we need a sequence S k,n (i) defined by the local growth of the increment process: where n is the signal X length, k is a fixed window size, and m is the largest integer not exceeding n/k. The local Hölder function H(x) at point is given by Lévy-Véhél, 1994, 1995;Muniandy et al., 2001 ;Li et al., 2007Li et al., , 2008Gaci et al., 2010): From figure 3, it can be seen that all the calculated amplitude spectra, represented in a loglog plan, decay algebraically, the analyzed data exhibit then a fractal behavior. Moreover, the latter is described by a Hölder exponent varying with the latitude of the measure. Hence the data can be considered as paths of multifractional Brownian motions (mBms) (Peltier and Lévy-Véhel, 1995;Gaci et al., 2011). A significant result deserves to be noted is the fact that the spectra obtained from the ''raw'', ''corrected'' and ''corrected and interpolated'' measurements display a similar form. That is the applied operations (corrections and interpolations) do not affect the fractal aspect of the raw data. On the left, the measurements profile, in the middle the module of the amplitude spectrum of the measurements profile versus the wavenumber (rad/degree) in the log-log scale, and on the right, the local Hölder function. The raw data (blue), the corrected data (red) and the corrected and interpolated data (green). Moreover, the estimated Hölder functions obtained from the three types of measurements present very close values. Again, we confirm that the fractal properties of the raw data are not modified by both pre-processing operations. The implementation of the different 2Dinterpolation algorithms illustrates that the choice of the interpolation algorithm has a very slight effect on the estimated H value. An important result to be noted: the spectrometric corrections are not necessary for a fractal analysis which can be carried out directly on the raw measurements. By doing so, the stochastic component of the measurements is kept intact.

Local regularity analysis of airborne spectrometric data
In this section, we establish local two-dimensional regularity maps, from the interpolated raw GR data measured in the three channels: K, Th and U, using a wavelet-based algorithm via the two-dimensional Multiple Filter Technique (2D MFT). We obtained the latter technique by generalizing the mono-dimensional version (Dziewonski et al., 1969;Li, 1997) to the 2D-case (Gaci, 2011).

Spectrometric data interpolation
Considering the limitations of the computer's processing capacity, we consider the GR measurements recorded, in the K, Th and U channels, over the zone whose geographical coordinates are defined by: longitude: 3° 13' 58''-6°59' 26'' E, and latitude: 20° 27' 35''-25° 06' 37'' N. The 2D-interpolation of the raw spectrometric data is performed owing to the kriging algorithm. The interpolated GR grids data related to the K, Th and U channels are illustrated respectively by figures 4, 5 and 6.

Establishment of local regularity maps from interpolated spectrometric data
Using a wavelet-based algorithm, we estimate Hölder exponent maps, from the interpolated GR measurements recorded in the three channels (K, Th and U).
Recall that the two-dimensional continuous wavelet transform (2D-CWT) is given by a convolution product of a signal   sx , y and an analyzing wavelet   g x,y (Chui, 1992;Holschneider, 1995): where "a" is the scale parameter, "b x " and "b y " are the respective translations according to Xaxis and Y-axis (the symbol " -" denotes the complex conjugate). Alternatively, it can be computed via the Fast Fourier Transform: Here, we compute the wavelet coefficients via FFT using the two-dimensional multiple filter technique (2D MFT). The latter technique is obtained by generalizing the one-dimensional version (1D MFT), suggested by Dziewonski et al. (1969) and improved by Li (1997), to the two-dimensional case. It consists of filtering a two-dimensional signal using a Gaussian filter nm G( k , , )  given by (Gaci, 2011): Where  is a constant, ( 1 , 2 ) and ( 1 , 2 ) are respectively the -3 dB points of the Gaussian filters G 1 and G 2 , respectively.
A fractal surface   sx , y verifies the self-affinity property (Mandelbrot, 1977(Mandelbrot, , 1982Feder, 1988) : Where H is the Hurst exponent (or the self-affinity parameter). The symbol  means the equality of all its finite-dimensional probability distributions. For sufficiently large values of k, the scalogram, defined as the square of the amplitude spectrum:   2 Pk , x , y S( k ,x , y)  , can be expressed as: is the local spectral exponent which is related to the local Hurst (or Hölder) exponent,   Hx , y . The spectral exponent   x,y  in each point   x,y is computed as the slope of the scalogram versus the wavenumber in the log-log plan, the   Hx , y value is then derived using the equation (11). The implementation of the wavelet-based algorithm, using the generalized multiple filter technique, allows to establish regularity maps from the interpolated GR measurements recorded in the three channels (K, Th and U) (Fig. 7). In order to interpret the resulting maps in terms of geology, a geological map of the studied zone is considered. The results show that the H maps, derived from the measurements of all the channels, exhibit almost an identical image of the local regularity. By reporting the faults affecting the studied zone on the obtained regularity maps, we remark that the faults locations correspond to local minima of H values. The main accident (the 4°50' fault) is noticeable on almost all the regularity maps. However, the regularity maps present local minima of H values in some places, probably due to less important faults which have to be checked on updated detailed geological maps.
(a) A geological map of the studied zone 1 -Archaean granulites; 2 -Gneiss and metasediments, series of Arechchoum (Pr1); 3 -Gneiss with facies amphibole, series of Aleskod (Pr2); 4 -Indif. gneiss (Pr3); 5 -Pharusian Greywackes; 6 -Arkoses and conglomerates, series of Tiririne (Pr4); 7 -Volcano-sediments of Tafassasset (Pr4); 8 -Molasses (purple series) of Cambrian; 9 -Pan-African syn-orogenic granites; 10 -Pan-African Granites; 11 -Pan-African post-orogenic granites; 12 -Granites of Eastern Hoggar; 13 -Late pan-African Granites; 14 -Basalts and recent volcanism; 15 -Paleozoic cover; 16 -Fault. Now, we try to establish a correspondence between the obtained regularity maps and the geological map of the area. Since the obtained regularity maps are similar, we choose that estimated from the measurements recorded in the Th channel. Then, on the considered geological map and H map, we delimit in dotted lines the geological formations; the same color corresponds to the same geological facies (Fig. 8). These two maps show that a considered lithology is not characterized by the same value of the H coefficient. These obtained preliminary results reveal that the H value can not be used as an attribute to characterize lithology, while it could be used for the recognition and the establishment of the network faults.

Conclusion
This study presents a regularity analysis undertaken on the airborne spectrometric natural radioactivity measured, in three channels: K, Th and U, over the Hoggar area (Algeria). It reveals that the investigated data exhibit fractal properties depending on the spatial measurement location, thus can be modeled using multifractional Brownian motions. As the spectrometric corrections do not affect these properties, the regularity analysis can be carried out directly on the interpolated raw measurements. The Hölder exponent maps, obtained from the Gamma Ray measurements recorded in the three channels, show a similar local regularity. Besides, a strong correlation is derived between the H exponent values and the faults locations. Indeed, a fault corresponds to local minima H values, the H exponent value can then be used to identify the faults. However, it does not allow to characterize the lithological facies.

Acknowledgements
This work was supported by the Algerian -French program CMEP-PHC Tassili N°09 MDU 787. With growing attention on global environmental and climate change, geoscience has experienced rapid change and development in the last three decades. Many new data, methods and modeling techniques have been developed and applied in various aspects of geoscience. The chapters collected in this book present an excellent profile of the current state of various data, analysis methods and modeling techniques, and demonstrate their applications from hydrology, geology and paleogeomorphology, to geophysics, environmental and climate change. The wide range methods and techniques covered in the book include information systems and technology, global position system (GPS), digital sediment core image analysis, fuzzy set theory for hydrology, spatial interpolation, spectral analysis of geophysical data, GIS-based hydrological models, high resolution geological models, 3D sedimentology, change detection from remote sensing, etc. Besides two comprehensive review articles, most chapters focus on in-depth studies of a particular method or technique.