Excerpt

## Contents

Summary in Danish

Acknowledgement

1. Abstract and introductory remarks

2. Historical background

3. Contributions to the theory of К-distribution and generalized gamma distribution

4. A statistical model for the scattering mechanism

5. Digamma function

6. Generalized gamma distribution

7. Exponential gamma distribution

8. K-distribution

9. Moment method

10. Method of moments

11. Raghavan’s method

12. Chi-square test for goodness-of-fit

13. The runs test and the и test

14. Kolmogorov’s Dn test

15. Which test should be chosen?

16. Data analysis, Parameter analysis

17. Data analysis, Statistical analysis

18. Conclusions

19. Future recommendations

References

Appendix

20. Appendix, Text material

20.1 Appendix, A statistical model for the scattering mechanism

20.2 Appendix, Calibration of ERS-1 SAR images

20.3 Appendix, Exponential gamma distribution

20.4 Appendix, K-distribution

20.5 Appendix, Saddlepoint method for approximating the K-distribution

20.6 Appendix, Raghavan’s method

20.7 Appendix, Chi-square test for goodness-of-fit

20.8 Appendix, The runs test and the и test

20.9 Appendix, Kolmogorov’s Dn test

20.10 Appendix, Numerical and iterative calculations

20.11 Appendix, Probability theory

20.12 Appendix, Statistical definitions and properties

21. Appendix, Data material

21.1 Data results for image: 16481_2385, ROM

21.2 Data results for image: 16481_2385, ROI2

21.3 Data results for image: 17055_2385, ROI1

21.4 Data results for image: 17055_2385,

21.5 Data results for image: 17055J2385,

21.6 Data results for image: 17299_2385,

21.7 Data results for image: 16014_1215,

21.8 Data results for image: 16588_1215,

22. Appendix, Images

## Summary in Danish

Denne Ph.D. rapport omfatter udviklingen af metoder til statistisk analyse af SAR billeder. I den videnskabelige litteratur diskuteres det meget, hvordan amplituder og/eller intensiteter i SAR billeder fordeler sig statistisk i havområder. Ofte antages det at signalet (backscatter coefficienterne) I et SAR billede følger en gamma fordeling, og dette testes ved hjælp af et chi-i-anden test. Det har også været fremført at man i stedet for gamma fordelingen skal bruge en K-fordeling. Vi har undersøgt 6 homogene og 2 inhomogene havområder i fem ERS-1 SAR.PRI tre-looks billeder fra Grønland 1994.

Denne rapport indeholder diskussion om følgende emner; 1) En række resultater vedrørende den generaliserede gamma fordeling og K- fordelingen nævnes eller bevises. Bl.a. vises en del resultater om modus i K-fordelingen, såsom at K-fordelingen altid vil have mindst ét modus.

2) Hverken for den generaliserede gamma fordeling eller for K-fordelingen kan maximum likelihood (ML) ligningerne løses direkte. Derfor er det nødvendigt at estimere nogle startværdier som kan bruges i en iterativ løsning af ML ligningerne. Forskellige metoder er her testet: i) en moment metode hvor man bruger forholdet mellem forskellige momenter i fordelingen, ii) en moment metode hvor man bruger middelværdi, varians og evt. tredje centrale moment af logaritmen til observationerne, og iii) Raghavan’s metode hvor man bruger forholdet mellem det aritmetriske og det geometriske gennemsnit til estimation. For den generaliserede gamma fordeling kan reelt set kun ii) anvendes med succes. Kun i stærkt støjende eller inhomogene havområder kan estimation med ii) ikke gennemføres. For K-fordelingen kan alle tre metoder bruges, men i) giver ofte dårlige resultater eller bryder ned, og det sker at iii) bryder ned. Der er ingen eksempler på at ii) er brudt ned, men det er teoretisk set muligt. Det skal her siges, at estimation i K-fordelingen kun forsøges hvis estimationen i den generaliserede gamma fordeling er lykkedes, hvorfor det ikke kan udelukkes at ii) også ville bryde ned for K-fordelingen hvis ii) bryder ned for den generaliserede gamma fordeling. Grunden til at estimationerne i) og iii) bryder sammen skønnes at være at den anvendte værdi for antallet af looks (tre) bør sættes til en værdi højere end 3 eller alternativt at K-fordelingsmodellen i visse tilfælde bør justeres.

3) Chi-i-anden testet er meget vanskeligt at bruge korrekt og det bør ofte suppleres med andre tests. Chi-i-anden testet måler kun absolutte afvigelser mellem det observerede og det forventede, mens testet ikke er i stand til måle om afvigelserne er systematiske eller tilfældige. Et random runs test er i stand til måle hvorvidt afvigelserne er tilfældige eller systematiske, og dette test kan derfor supplere et chi-i-anden test. De to tests kan kombineres til et såkaldt и test. Disse tre tests er imidlertid meget følsomme overfor hvordan de indgående kategorier er konstrueret, så man skal være meget omhyggelig med at konstruere kategorierne “korrekt”. Der bør hverken være for få kategorier (da dette mindsker testets styrke) eller for mange kategorier (da dette øger støjens indflydelse på resultatet).

En anden type test er Kolmogorov’s Dn test, som udelukkende måler den maksimale afvigelse mellem den observerede og den forventede fordelingsfunktion. Dette test involvereringen kategori-opdeling, mentii gengæld er testet (ligesom runs testet) bygget

4) Ud fra en antagelse om at den generaliserede gamma fordeling er den korrekte fordeling, kan man teste hvorvidt de ovennævnte tests opfører sig “korrekt”.

Det viser sig at Kolmogorov’s test er alt for optimistisk overfor de testede hypoteser (dvs. testsandsynlighederne er for høje). Runs testet giver generelt også ret høje testsandsynligheder (men langt fra så høje som Dn testet). Chi-i-anden testet og и testet giver som regel nogenlunde ligefordelte testsandsynligheder, omend der kan være variationer der sikkert bunder i hvordan kategorierne er konstrueret i det konkrete tilfælde. Hvis der er “for mange” kategorier vil chi-i-anden testet typisk give meget lave test sandsynligheder og runs testet vil give meget høje test sandsynligheder (et problem der ofte optræder ved lave signal-støj forhold). Dette problem kan delvis udjævnes ved brug af u testet, hvorfor и testet må betegnes som mindre sensitivt overfor en forkert kategori konstruktion end chi-i-anden testet. Hvis antallet af kategorier sættes ned (i den hensigt at skabe en mere ligelig fordeling af testsandsynlighederne), sættes samtidigt testets styrke ned.

Generelt må man derfor sige at и testet er et fornuftigt test at bruge når man vil teste om observationerne (backscatter coefficienterne) følger en given fordeling, men at man skal være meget opmærksom på afvigelser som skyldes konstruktionen af kategorier. Dn testet kan her supplere til at afgøre, om hypotesen reelt er forkert eller om konstruktionen af kategorier er forkert.

5) Ofte kan man bruge en generaliseret gamma fordeling eller en K-fordeling til at beskrive backscatter coefficienters fordeling i homogene havområder, men i en del tilfælde er det også tilstrækkeligt at bruge den almindelige gamma fordeling. I områder med meget lave backscatter coefficienter (lavt signal-støj forhold) vil de statistiske test generelt forkaste at intensiteterne følger nogen af de tre fordelinger (hvis ikke estimationen er mislykkedes som nævnt ovenfor).

i inhomogene havområder sker det oftest at alle tre fordelinger bliver forkastet. Det kan dog ske at en eller flere af fordelingerne accepteres hvis fordelingen af lave og høje værdier i området er sådan sammensat, at observationerne kunne have kommet fra en af fordelingerne, hvis de ellers havde været tilfældigt fordelt indenfor området. De anvendte statistiske tests undersøger nemlig ikke for systematik i den rumlige fordeling af data.

De udviklede metoder og resultater skønnes at være generelt anvendbare på SAR billeder fra homogene havområder, ønsker man derimod at detektere teksturer (afvigelser fra homogenitet) må testmetoderne suppleres med andre metoder, herunder todimensionale detektionsmetoder.

## Acknowledgement

I wish to thank Farvandsvæsenet (The Royal Danish Administration of Navigation and Hydrography) for the very large financial support to this thesis Without this support, this work could certainly not have been completed.

I am in great dept to my manager Arne Nielsen and my former manager Erik Buch, who have had confidence in the perspectives of the project and in my abilities to solve the problems involved. Their support has been of an extreme importance and value to me and the significance of this can hardly be overestimated.

I also want to thank my other colleagues in Oceanographic Department for their moral and encouraging support to my work. Especially, I want to thank Jørgen Eeg for lots of good and inspiring advises.

I want to thank Henning Skriver at the Electromagnetics Institute, Technical University of Denmark, for his valuable and patient support to my understanding of SAR images. His practical and theoretical support has been of large importance to me.

I want to thank professor Fernando Sansó from the Politechnical Institute of Milan. At two occasions, he has taken the time to read my material, and his comments and corrections have been extremely valuable to me. His support eliminated several misunderstandings and significantly raised the standard of several parts of the thesis.

I want to thank professor Søren Nørvang Madsen at the Electromagnetics Institute, Technical University of Denmark, for a very fruitful discussion of the derivation of the K- distribution.

I want to thank professor Preben Gudmandsen at the Electromagnetics Institute, Technical University of Denmark. He was of great importance right from the beginning when I started looking at SAR images, and during my work he has helped me a lot at several occasions.

I want to thank Dr Dale Winebrenner, Applied Physics Laboratory, University of Washington, USA, and professor Eric Jakeman, Department of Electrical and Electronic Engineering, The University of Nottingham, UK, for a very fruitful explanation of several problems in the derivation of the K-distribution.

I want to thank my supervisor, professor Carl Christian Tscherning at the Geophysical Institute, University of Copenhagen, for being willing to support this work as a Ph.D. thesis. He has done a great effort to help me and he has given me many good advises. I also thank him for bringing me into contact with professor Fernando ¿ansò.

I want to thank my family and friends whose moral and encouraging support has been very important to me.

Finally, but certainly not least, I want to thank the European Space Agency, ESA, for having supported my work as a pilot project. This support actually forms the basis of the whole thesis since all the SAR images which I have got have been delivered as part of this pilot project. Without ESA’s support, there would have been no images and this thesis would never have been written.

## Chapter 1 Abstract and introductory remarks

The perspective of this thesis is to be able to use spaceborne or airborne measurements for bathymetric purposes, preferable at Greenland. Such methods are thought of as alternative or supplementary to traditional hydrographic ship measurements. The basic hope is to be able to detect bathymetric changes by statistical methods with as few in situ measurements as possible; in situ measurements at Greenland can namely be very expensive and difficult to get. The use of statistical methods is an alternative approach to the solution of the bathymetric problem since hydrodynamical and physical modelling is normally used for this purpose.

In the thesis we have analysed SAR (Synthetic Aperture Radar) images from the ERS-1 satellite as possible candidates for space- and airborne methods. The SAR images have been acquired during the summer and autumn 1994 at the coast of Greenland (near Qaqortoq).

It has proved difficult to compare the acquired SAR images with depth measurements. Therefore, the contents of the thesis is more modest than a bathymetric analysis would have been. The intention is to develop methods which can be used for bathymetric (and probably other) purposes in the future. The focus is laid on the analysis of homogeneous sea areas, i.e. areas with no bathymetric or other signatures. However, a few inhomogeneous areas have been analysed too. Information about what characterizes a “normal” situation may be used to detect “non-normal” phenomena.

Consequently, the primary objectives of this thesis are

1) to find and theoretically discuss the relevant statistical distributions for intensities in SAR images,

2) to develop and examine methods which can be applied for a statistical analysis of intensities in SAR images over homogeneous open sea areas - an analysis which hopefully can be used to detect departures from homogeneity,

3) to use these methods for an examination of the statistical behaviour of the intensities in SAR images.

Objective 2) is subdivided into two parts:

2a) development of parameter estimation methods for statistical distributions,

2b) examination of different types of statistical test methods.

Ad 1)

Several statistical distributions have been discussed in the literature as candidate to description of the distribution of radar intensities. We have considered the following distributions:

i) the two-parameter gamma distribution

ii) the three-parameter generalized gamma distribution

iii) the three-parameter K-distribution

The two-parameter (ordinary) gamma distribution is a special case of the generalized gamma distribution and is supposed to be a good candidate when the so-called Rayleigh model holds.

Our primary interest has been to examine the distribution of intensities (calibrated backscatter coefficients, o° values), but with a simple transformation of the statistical distributions the test methods can equally well be used on amplitudes. Actually, for the generalized gamma distribution it turns out that amplitudes are generalized gamma distributed if and only if intensities are generalized gamma distributed. It is seen that if intensities are generalized gamma distributed then intensities in dB values are exponentially gamma distributed.

Ad 2a)

The most widely used estimation method is maximum likelihood estimation. If maximum likelihood estimation is possible, this method is preferred because of its many important properties. Unfortunately, in many cases there is no exact analytical solution to the maximum likelihood equations and consequently other estimation methods must be used. In this thesis we consider three alternative methods: the “moment method”, the “method of moments” and “Raghavan’s method”. Parameter estimates obtained by these alternative methods can be considered as the “final estimates” or they can be used as start values for an iterative approach to a maximum likelihood solution. We have chosen the latter solution (the iterative approach) as we consider maximum likelihood estimation to be preferable to other estimation methods.

For the generalized gamma distribution only the method of moments can be used for parameter estimation (or more precisely: the moment method can only be used iteratively and with large instability and Raghavan’s method can only be used in combination with other methods). Only in extreme cases the method of moments does not work. For the К-distribution all three estimation methods can be used. The moment method generally works poorly and often there is no solution at all. In some cases Raghavan’s method fails. When Raghavan’s method and the method of moments both work, they give almost identical results.

Ad 2b)

The most widely used test for goodness-of־fit is the chi-square test. The chi-square test is, however, based on a grouping of data in categories, and it turns out that the test is very sensitive to how these categories are constructed. The chi-square test also has the drawback that it only measures absolute deviations between the observed and the expected number of observations, while systematic deviations/tendencies are not registered. Therefore, other test methods have been examined too:

i) The runs test, which is a test for systematic deviations in data, il) The u test, which is a combination of the chi-square test and the runs test into one test. The и test therefore tests for both absolute and systematic deviations between the observed and the expected observations.

iii) Kolmogorov’s Dn test. Contrary to the other tests, this test does not depend on any category construction. The Dn test is sensitive to the largest difference between the observed and the expected distribution functions. The Dn test can be considered as a combination of a test for absolute and systematic deviations like the и test.

It turns out that generally the и test behaves much better than the Dn test.

Ad 3) It turns out that backscatter coefficients in the examined homogeneous sea areas can be assumed to follow a generalized gamma distribution as well as a K-distribution.

## Chapter 2 Historical background

ln 1993, ESA accepted a pilot project proposal with the title “Use of SAR images for detection and measuring sea bottom topography near the coast of Greenland”. The pilot project was proposed by Farvandsvæsenet. As a part of this project, 67 ERS-1 SAR.GEC images were received from three locations in Greenland: Sisimiut, Attu and Qaqortoq (the suffix “.GEC” means that the images are “geocoded” i.e. geometrically corrected). The images were acquired during the period where hydrographers in Farvandsvæsenet measured depths (bathymetry) in the same areas. There already were some depth measurements in the three areas, and the idea was to compare signals in SAR images with bathymetric measurements in order to investigate possible relationships /correlations. During satellite passage the hydrographers were supposed to measure current and wind velocity.

The reason for choosing the geocoded SAR.GEC product instead of the precision image SAR.PRI product (from which the GEC image is produced) was that the GEC product was claimed by ESA to be geometrically correct to within 150 meters. Therefore, it was supposed that an image warping (geometrical correction) could be avoided by choosing the GEC product. (At that time we used the image processing software EMImage, developed by the Electromagnetics Institute at the Technical University of Denmark, and large image warpings seemed to be rather difficult to perform with that software. Later on, after the development of EMImage had been stopped, IDL/ENVI was introduced. In ENVI image warping is a relatively easy task.)

### 2.1 Some initial/practical problems

There were several problems of practical (but very time consuming) nature, which were necessary to solve before the research could begin:

1) It took some time to be able to read the header informations (leader files) in the images. No one else in Denmark used the geocoded product, so this work had to be done from scratch (as it was not done by ESA). A complication in this header reading work was that there were inconsistencies between the leader file documentation ((ESA, 1994) and (ESA, 1995)) and the real leader files.

2) Comparison between the measured coastline (delivered by Kort- og Matrikelstyrelsen) and the coastline in the images disclosed discrepancies of far more than the claimed 150 meters. In some images the difference was more than 1 km. It took a long time to detect the origin of this error. The error turned out to be a consequence of the way geocoded images are processed. The assumptions which form the basis of this processing have the consequence that GEC images are useless in mountanous areas (like Greenland). A documentation of these problems has been given in (Sølvsteen, 1996).

3) A calibration of the image signals is necessary (see chapter 20.2). As such calibration routines were not delivered by ESA, it had to be developed from scratch. There was no official ESA program which was able to calibrate ESA’s images. After some time, an undocumented program from The German Processing Facility (D-PAF) was received.

This program was able to read the leader file information and calibrate the images. The program was developed only for internal use in D-PAF. The D-PAF program showed up to have several errors:

i) The most severe error was the power loss correction. The routine for power loss correction compared each pixel value with the power loss table and adjusted the pixel value according to this. The consequence of this was that all high pixel values became even higher and all low pixel values became even smaller. And this is certainly not the idea of power loss correction.

ii) The factor c¡ in (20.2.1) depends on the look angle. But in the program incidence angles were used. The difference between those angles are about 3-4 .

iii) In order to handle the image warping, the prograrn used the satellite heading angle (i.e. the clockwise angle between geographic North and the flight direction calculated at the satellite position in space). However, the corresponding angle projected to the ground should be used. The difference between these two angles becomes relatively large at high latitudes.

iv) There seemed to be a difference between the leader file North coordinates and the real North coordinates (a difference equal to the length of the image). This error is an error in the leader file but was not compensated for in the program.

### 2.2 SAR and bathymetry

The theory of “SAR and bathymetry” has been covered by several authors, for instance: (Alpers and Hennings, 1984), (Valenzuela et al., 1985), (Shuchman et al., 1985), (Hennings et al., 1988), (Hennings, 1988) and (Calcoen and van der Kooij, 1993) Generally, this theory has been developed under the assumption that a strong tidal flow passes over a linear large-scale topographic feature (typically sandbanks). The shape of the topographic feature must be relatively constant over a large area, and the tidal flow must have a strong component perpendicular to the feature (i.e. parallel to the direction of bathymetric changes).

The relationship between depth changes, intensity changes and current changes is (to first order) (Alpers and Hennings, 1984):

Abbildung in dieser eseprobe nicht enthalten

Here 5I/I0 is the relative change in intensity over the feature (l0 is a sort of background level). The x-axis is perpendicular to the bathymetric feature (parallel to the direction of bathymetric change), и is the current speed, dll/dx is the current change in the direction of bathymetric change, ď is the slope of the feature, d is the water depth.

Since the slope ď is part of (2.2.1) it implies that the bathymetric feature can only change in one direction - i.e. the feature has a linear shape. Furthermore, the expression dll/dx has only meaning if the current has to pass over the feature (and not around it).

Due to the presence of rocks and underwater mountains and due to the lack of linear shaped sandbanks near the coast of Greenland, the conditions that lead to (2.2.1) are not fulfilled and the bathymetric theory given above is not applicable to Greenland. Also the presence of different water layers at Greenland (with different salinity and temperature) made it problematic to rely too much on the validity of (2.2.1). Consequently, it was not the aim of the pilot project to implement (2.2,1) on SAR images from Greenland. Such an implementation would be an impossible task to perform. Instead, the aim was to find some statistical relationships between signatures in the images and measured sea depths.

The idea of using statistical methods for bathymetry detection is inspired by the methods given in (Sølvsteen, 1995). Sølvsteen has used statistical methods in NOAA/AVHRR images for cloud-detection and for an examination of correlations/relationships between thermal infrared channels, in situ sea surface temperature (SST) measurements, and satellite viewing angle. The cloud-detection algorithm method is based on i) exclusion of extreme values, li) measurements of variations in matrices in the visible channel 2, and iii) measurements of the correlation coefficients between the two thermal infrared channels 4 and 5 and between channels 2 and 4. Furthermore, the correlation coefficient between channel 4 and the difference between channels 4 and 5 has been used to separate cloud contaminated areas from cold water areas - this is actually a very crucial point in a cloud-detection procedure since it may be difficult to determine whether a low temperature is due to clouds or to cold water. The SST analysis shows that the usual split-window algorithm for SST calculations is unreliable. This documentation is again based on comparisons of and correlation coefficients between some of the data which are included in the SST algorithm, namely i) channel 4 and the difference between channels 4 and 5, ii) satellite viewing angle, and iii) in situ bulk temperature measurements. Consequently, the whole analysis in (Sølvsteen, 1995) is based on statistical analysis of variations in the pixel values and on correlations between satellite and in situ measurements.

A similar idea was searched for in the analysis of SAR images for bathymetric purposes: would it be possible to use variations of backscatter coefficients to detect bathymetric changes, and would it be possible to measure any correlation between signals in SAR images and in situ depth measurements? Obviously, the presence of speckle in SAR images would complicate such a statistical method compared with the analysis of NOAA/AVHRR images.

In the search for statistical correlations between depth measurements and SAR signals, many major problems turned up - problems which lead to the abandonment of the originally planned project:

1) In areas where something “interesting” (= signatures that could be of bathymetric nature) showed up in a SAR image there were no depth measurements.

2) There were very dense depth measurements in areas with no bathymetry-like SAR signals.

3) “Interesting” phenomena usually only appear in one SAR image. In a few cases a phenomenon appears at the same location in two images, but then they look different. The reason for this could be that an (eventual) appearance of a bathymetric feature depends strongly on the way the current and the wind hit the feature.

4) There are different currents and winds from bay to bay because of curvatures of bays and mountains sheltering from winds. Therefore, in situ current/wind measurements are needed in every bay which is to be analysed. 2/3 of the satellite passages were during nighttime, and the hydrographers did not measure any currents and wind velocities during these passages. For some of the other images measurements lacked for other reasons, and for the rest of the images measurements were only performed at one location. Therefore, in practice these measurements were of no use.

5) Due to the lack of sandbanks and the presence of (spiky) rocks, it is more probable that water masses will pass around the bathymetric features than over them Therefore, turbulent currents are likely to turn up near bathymetric changes - where (2.2.1) assumed currents to be linearly shaped.

Because of these problems, it became apparent that it would be extremely difficult to complete the work. Actually, point 1) meant that a further analysis had to be postponed until more depth measurements became available.

Furthermore, it was realised that in order to be able to detect statistical changes of bathymetrical nature, it would be necessary to know what characterizes a “normal” situation. It was necessary to know how SAR signals appear in homogeneous sea areas.

Therefore, the work was changed into another direction, which could form a basis for a future detection of bathymetrical changes. The new topic of the work was to examine the statistical behaviour of backscatter coefficients in ERS-1 SAR images acquired from homogeneous open sea areas. This approach may be compared with the cloud- detection method mentioned above, where deviations from the normal situation are used for detection of contaminated areas. The idea is to use the knowledge of a normal sea state situation to detect when a non-normal situation occurs - such a non-normal situation could namely be due to a bathymetric feature.

This thesis discusses these examinations of the statistical distribution of backscatter coefficients in homogeneous sea areas.

### 2.3 Start of statistical analysis

The most basic theory concerning the distribution of backscatter coefficients ( 0° values) states that backscatter coefficients follow an ordinary gamma distribution (Skriver, 1990). Therefore, the initial statistical tests were done for this distribution. The tests rejected the hypothesis that σ° values follow a gamma distribution. Results of these tests were presented in a poster at an ERIM conference in 1997 (Sølvsteen, 1997). The conclusion of these tests were, however, based on the parameter estimation method given by Skriver (1990) which is not a maximum likelihood (ML) estimation. (Skriver (1990) uses the fact that for the Γ(α,β) distribution we have ΕΧ=αβ and VX = aß2. Then he estimates a = (EX)2/VX and ß = VX/EX where EX and VX are the observed meanvalue and variance. Such a method in which fractions of moments are used for parameter estimations is often referred to in the literature as the “moment method” (see chapter 9).) Since the estimated parameters were not ML parameters, the test probabilities were generally too small and the conclusions in (Sølvsteen, 1997) judged the gamma distribution too severely.

### 2.4 Contents of the thesis

At the 1997 ERIM conference where the poster was presented, J. Campbell at Canada Centre for Remote Sensing suggested to look closer at the «־distribution, and he also briefly mentioned the generalized gamma distribution as a possibility. Since then the main focus of the work has been to examine how well the «-distribution and the generalized gamma distribution fit the actual observations (backscatter coefficients). The results of these studies are presented in this thesis.

During the work with statistical tests it was discovered that the result of a chi-square test very much depends on the construction of and sizes of categories. This discovery lead to an intensive study of and experimentation with chi-square tests. These considerations and analyses are also presented in this thesis. Furthermore, the author became aware of other types of tests, the runs test, the и test and Kolmogorov’s Dn test. Due to the huge amount of tests being performed, there has been an exceptional opportunity to test different test methods and to compare results from different ways of testing (these test comparisons are described in the data analysis).

Another problem with statistical tests is how to estimate the parameters. Apart from the maximum likelihood estimation (which is not always possible to perform directly), other estimation schemes have been proposed (the “moment method”, the “method of moments” and “Raghavan’s method”). These estimation schemes have been examined in this thesis.

The statistical methods developed above are used on six homogeneous sea areas, and their ability to detect deviations from the normal situation has been tested on two inhomogeneous sea areas. Some of the results from this analysis were presented in a poster at a 1998 ERIM conference (Sølvsteen, 1998).

Consequently, this thesis consists of four topics of equal importance:

1) A discussion of some fundamental properties of the generalized gamma distribution and the «-distribution. These distributions are of interest as candidates for the statistical distribution of backscatter coefficients in 3-look SAR.PRI images.

2) A discussion of methods to estimate parameters in a distribution. Statistical tests for goodness-of-fit need an estimated distribution from the theoretical distribution family to compare with the observed distribution. It is not without importance how this estimated distribution is constructed, and therefore the result of the goodness-of-fit test also depends on the parameter estimation method.

3) A discussion of methods to perform statistical tests for goodness-of-fit. If it shall be examined whether backscatter coefficients follow a given distribution or not, a statistical test must be performed. The result of such a test depends very much on how the test is performed.

4) The results of the tests for the distributions mentioned in 1 ) using the methods given in 2) and 3).

Topic 1) is covered by the following chapters: 3, 4, 5, 6, 7, 8, 20.1, 20.3Γ 20.4, 20.5

Topic 2) is covered by the following chapters: 9, 10, 11, 20.6, 20.10, 20.11, 20.12

Topic 3) is covered by the following chapters: 12, 13, 14, 15, 20.7, 20.8, 20.9, 20.10, 20.11, 20.12

Topic 4) is covered by the following chapters: 16, 17, 20.2, 21, 22

## 3. Contributions to the theory of K-distribution and generalized gamma distribution

In this chapter we shall give a brief review of some contributions given in the literature to the theory of (especially) the К-distribution and the generalized gamma distribution. The mathematical derivation of the К-distribution model will be given in chapter 4.

### 3.1 Review

Jakeman and Tough (1988) mention that the К-distribution was first considered in 1943 and that since then it has occasionally been used for fitting purposes. They mention that the work with the К-distribution has been intensified since Jakeman and Pusey (1976) showed that the К-distribution could be derived from a random walk model. The theoretical derivation of the К-distribution is specifically covered in chapter 4 and will not be repeated here.

It has been shown that the К-distribution fits very well to strong-scattering phenomena in a wide variety of scattering situations. Jakeman and Pusey (1978) mention the scattering of laser light from i) a turbulent layer of nematic liquid crystal, ii) a turbulent layer of air, iii) a turbulent layer of water, and iv) an extended region of atmospheric turbulence, and furthermore v) the scattering of starlight by the upper atmosphere, and vi) the scattering of microwave radiation by a small area of rough sea. So the applicability of the К-distribution is certainly not restricted to SAR images. Jakeman and Pusey (1978) document how well the К-distribution works for case i). Eltoft and Høgda (1998) show that the К-distribution works significantly better than the Rayleigh distribution in selected ocean SAR scenes.

Joughin et al. (1993) analyse first-year and old ice amplitudes in a four-look SAR image. They make chi-square tests for different numbers of samples (N) and categories (k). In an area with N=100 and к = 7 they find that most tests accept the K-distribution but that there are some rejections (especially for old ice areas). For N=1000 and k=15 there are no rejections. (In chapter 12 we shall see that their category sizes may be too large, especially for N=1000, and that this may explain their very high test probabilities for N=1000.) Though the order parameter V is higher for first-year ice than for old ice the variation in V values is so large that V cannot be used to detect first-year ice from old ice. They find - not surprising - that the variation in estimated V values is much larger for N=50 and N=100 than for N=500 and N=1000. Joughin et al. are the only ones in the referenced material who perform statistical tests, and they also seem to have the largest data material.

Eltoft and Høgda (1998) compare Rayleigh and K-distribution models for different airborne and satellite borne c band SAR scenes over the ocean. They examine results for different wind and wave conditions and for different incidence angles. They find that the Rayleigh model fits data better for high incidence angles than for low incidence angles (and in this context the ERS-1 SAR incidence angle of 23° is considered as a low incidence angle); at low incidence angles specular effects are supposed to be more important than at high incidence angles. Furthermore, they observe that the non- Rayleigh signals are more dominant in areas with low wind speed than in areas with high wind speed and that the strongest deviations from the Rayleigh model occur when the wind is blowing in the opposite direction of the waves. Finally, they observe a correlation between the degree of deviation from the Rayleigh model and the azimuth smearing.

Eltoft and Høgda observe that the order parameter in the K־distribution increases when the Rayleigh model becomes more plausible. Consequently, the order parameter can be used as a measure of the deviation from the Rayleigh model.

The observation that areas of low wind (low backscatter coefficients) have higher deviation from the Rayleigh model than high wind speed areas is in accordance with our own observations. This is also the case for the observation that the order parameter V in the K-distribution may be used to characterize the level of “Rayleigh-ness”. However, their conclusion, that for V larger than six the two distributions are equal, is not in accordance with our observations: here we find that V should be higher than 20-25 for the two distributions to be equal.

Jao (1984) discusses the radar signals from areas with different types of scatterers (terrain clutter) and presents a few data analyses from highly cluttered areas in order to argue for the K־distribution. He derives the К-distribution for intensities by the same methods as given by Jakeman et al.: he uses a negative binomial distribution to generate the К-distribution and he derives the negative binomial distribution from a compound Poisson model and from a birth-death-immigration model.

Weng et al. (1991) are interested in ultrasound В-scan images and they use the fractions of different moments in order to estimate V in the К-distribution. They use v to characterize the clustering (the number of scatterers) in the medium.

Shankar et al. (1993) are also interested in ultrasound В-scan images which they use to detect breast tumors. They use fractions of different moments and the signal-to-noise ratio in order to separate Rayleigh and non-Rayleigh areas from each otherľ Tumors are typically non-Rayleigh.

Oliver (1993) discusses maximum likelihood (ML) estimators for the intensity K- distribution and the bias which occurs from using other estimation methods. For instance he uses the method of moments (without calling it that). He is primarily interested in the limits of large values of L (the number of looks), and he is interested in texture analysis. Oliver also derives the intensity K־distribution by the same methods as Joughin et ai. (1993) use for the amplitude K-distribution.

Blacknell (1994) works in the same direction as Oliver (1993). He examines the errors (compared to ML estimates) for three different kinds of K-distribution estimators, namely estimates based on 1) the mean and second moment of the observations, 2) the mean and the mean of the logarithm of the observations, and 3) the mean and the variance of the logarithm of the observations. These methods are equivalent with 1 ) the moment method, 2) Raghavan’s method, and 3) the method of moments. All these methods will be discussed in this thesis. Blacknell examines the errors of the these estimators in relation to the ML estimates as 1) a function of L (1 < L < 100) for fixed order V=1, and

2) a function of V (0.1 < V < 10.0) with M = 256/L independent samples. He finds that the moment method works poorly while the method of moments often works a bit worse than Raghavan’s method.

It should be noted that Blacknell’s calculations are based on Fisher’s information matrix and not on real observations and that consequently practical problems with the performance of the estimators are not discovered. Furthermore, the number of samples is rather small and our case with L=3 and many V values higher than 10 is not covered; but obviously, such calculations can be performed from Blacknell’s equations.

Barakat (1986) derives the К-distribution for the weak-scatterer case. In this case the phases are not uniformly distributed as in the strong-scatterer case (which we are interested in). Barakat assumes that the phases follow a von Mises distribution. The derivation is based on the assumption that the number of scatterers follows a negative binomial distribution.

To the best of the author’s knowledge the generalized gamma distribution has not been discussed very much in relation to the scattering mechanism. In the referenced material only Gran (1992) discusses this topic but without mentioning any relevant test results. Some fundaments to the theory of the generalized gamma distribution have been laid by Stacy (1962) and Stacy and Mihram (1965).

From this review of some of the contributions to the analysis of SAR images it can be seen that this thesis may contribute significantly in the following areas:

1) The analyses are based on statistical tests and not solely on graphs.

2) Many important experiences have been done with statistical tests, and alternative tests to the chi-square test have been examined.

3) Many important experiences have been done with parameter estimation methods.

4) A relatively large data material has been analysed (compared to what has been referred to above).

5) The generalized gamma distribution has been included in the analysis.

## Chapter 4 A statistical model for the scattering mechanism

The aim of this chapter is to explain how the К-distribution can be derived for the distribution of amplitudes or intensities in radar images. It is assumed that the phases are uniformly distributed from 0 to- a so-called strong-scatterer situation. The chapter falls into three parts: first we discuss the so-called Rayleigh model, second we discuss the “extension” of the Rayleigh model to the К-distribution model, and third we discuss some situations where the К-distribution model may fail.

### 4.1 The Rayleigh model

The backscattered amplitude from a resolution area may be considered as a sum of contributions from several elements (scatterers) within the area. Each scatterer returns a signal (amplitude and phase) which is represented as a two-dimensional vector, referred to as an amplitude vector. The amplitude vector consists of an amplitude length and an angle (a phase) in relation to the radar. The resultant amplitude measured by the radar is a vector sum of these amplitude vectors from the scatterers.

At this stage it can already be seen that the mean value and the variance of the amplitude length must depend on each other (in contrast, for instance, to the normal distribution where there is no relationship between the mean value and the variance). The smallest possible amplitude vector length is of course zero (all the contributing vectors may cancel each other out to a resultant zero vector). The largest possible amplitude vector length is the sum of the lengths of all contributing amplitude vectors. Therefore, we must have that the longer the individual contributing amplitude vectors are (i.e. the higher the mean value of the resultant amplitude length), the higher will the variance also be.

We shall now prove that (for a one look image) the amplitude length follows a Rayleigh distribution and the intensity follows an exponential distribution. In this proof we closely follow the methods given in (Skriver, 1990, pp.14-17).

Let the resultant amplitude vector A be given in the two dimensional coordinates (X,Y), and let the angle (in relation to the X axis) and the length of the j’th contributing amplitude vector be Φ¡ and Ц , respectively. Furthermore, let the number of scatterers (= the number of contributing amplitude vectors) be N. Then we have:

Abbildung in dieser eseprobe nicht enthalten

(4.1.1)

Now suppose that the following assumptions hold:

Assumptions (4.1.2)

1. The angles cþj in (4.1.1) are independent and uniformly distributed on the interval [0,2π] (strong-scattering).

2. The lengths a-s and the angles cj)j are independent.

3. N is large.

4. The a/s are identically distributed, or at least: none of the a/s dominate the others.

Then it can be proven that the sums X and Y are asymptotically normally distributed (the central limit theorem, (20.11.10)), identically distributed and independent:

X ~ N(0, o2)

Y ~ N(0, a2)

where the variance σ2 depends on the distribution of the a/s. Consequently, X and Y follow the distribution

Abbildung in dieser eseprobe nicht enthalten

(4.1.3)

As X and Y are independent, the joint density function is the product of the marginal density functions:

Abbildung in dieser eseprobe nicht enthalten

(4.1.4)

Now convert to polar coordinates:

Abbildung in dieser eseprobe nicht enthalten

(4.1.5)

Abbildung in dieser eseprobe nicht enthalten

So we may consider the following transformation

(4.1.6)

The derivative of T(x,y) with respect to the vector A is a 2x2 matrix:

Abbildung in dieser eseprobe nicht enthalten

(4.1.7)

The determinant of this derivative is

Abbildung in dieser eseprobe nicht enthalten

(4.1.8)

Now using the transformation theorem for density functions (20.11.12) to get from (4.1.4)

Abbildung in dieser eseprobe nicht enthalten

(4.1.9)

From this we get the marginal distributions for A and Θ:

Abbildung in dieser eseprobe nicht enthalten

(4.1.10)

where f(A) is a Rayleigh distribution. We can find the density function for the intensity I = A2 by using a density transformation on f(A) (20.11.12):

Abbildung in dieser eseprobe nicht enthalten

(4.1.11)

where h(l) is an exponential distribution. The mean intensity is then E(l) = 2σ2. In chapter 6 we shall see that the Rayleigh distribution and the exponential distribution are both special cases of the generalized gamma distribution. Furthermore, the exponential distribution is a special case of the ordinary gamma distribution with parameters (1,202).

Until now we have only considered the one look case. Now we shall consider the multilook case. Suppose we have L looks. Then the intensity I in the L-look image will be

Abbildung in dieser eseprobe nicht enthalten

(4.1.12)

Now assume that the intensities 11י l2,...lL are statistically independent of each other. (Typically, in the case of an actual radar system neighbouring looks are slightly correlated, e.g. 10% orso.) Then the distribution of I in (4.1.12) is a gamma distribution with parameters (L, 202). (This follows from the fact that a sum of L independent gamma distributions with identical second parameters 202 will be a gamma distribution where the first parameter is the sum of each of the first parameters in the individual gamma distributions and where the second parameter is 202.)

In order to find the distribution of the amplitude in a multilook image, where the amplitude is the square root of the intensity I in (4.1.12), we can either do a straight forward transformation using (20.11.12) again or we can use a transformation result which will be given in chapter 6. In (6.1.10) we see that if l~g(L,1,2a2)

Abbildung in dieser eseprobe nicht enthalten

(4.1.13)

This latter distribution is referred to as the generalized Rayleigh distribution with 2L degrees of freedom and parameter 2'/2 · σ (Joughin et al., 1993)7

So we have proven that if the assumptions in (4.1.2) holds (that is if the central limit theorem can be applied) then the intensities in an L־look image follow a gamma distribution and the amplitudes follow a generalized Rayleigh distribution. Because of the involved Rayleigh distribution, this situation is often referred to as the “Rayleigh model”.

### 4.2 The К-distribution model

Abbildung in dieser eseprobe nicht enthalten

Now suppose that some of the assumptions in (4.1.2) do not hold in such a way that the central limit theorem (20.11.10) cannot be applied. In this case the Rayleigh model does not apply, and amplitudes and intensities are not any longer expected to follow a generalized Rayleigh distribution and a gamma distribution, respectively. Experiments have shown that in this situation the so-called K-distribution fits the amplitude distribution well. The amplitude К-distribution is defined as

(4.2.2)

Here the function Κη(χ) is the modified Bessel function of the second kind of order η. In chapter 8 we shall give a detailed description of the intensity K-distribution, while in this section we shall discuss how the К-distribution model can be derived. Since the model derivations are primarily based on amplitudes, we shall use the amplitude K-distribution in this section. But it should be noted that similar derivations apply for the intensity K- distribution.

Before going into details about this derivation it should be emphazised that there does not seem to be a physically rigorous derivation of the K-distribution model (Winebrenner, 1998, personal communication). The most common derivations of the K-distribution model are based on assumptions which have some intuitive and maybe even experimental justification (Winebrenner, 1998, personal communication).

If we look at the assumptions (4.1.2) and assume that they do not hold, a natural question will be: “in what way do they not hold?”. An inspection of the derivations of the ^-distribution model, given for instance in (Jakeman and Tough, 1987), shows that the amplitude vectors are assumed to be independent and the phases are assumed to be uniformly distributed. Therefore, conditions 1. and 2. in (4.1.2) are fulfilled. Consequently, the problems seem to occur in conditions 3. and 4.

It turns out that there are two different derivations of the К-distribution model, where the first (represented by Jakeman and Tough) assumes that condition 3. is not fulfilled, while the second (represented by Joughin, Percival and Winebrenner) assumes that condition 4. is not fulfilled. Jakeman and Tough examine the amplitude distribution for an n-dimensional random walk of amplitude vectors, whereas Joughin, Percival and Winebrenner examine the amplitude distribution for an L־look SAR image.

In both derivations the aim is to describe the so-called bunching or clustering effect of the scatterers, i.e. the situation where the scatterers are not uniformly distributed over the illuminated area and/or where the backscattered signal (amplitude) is statistically correlated with the location in the illuminated area.

First we shall consider the derivation given in (Jakeman and Tough, 1987). They consider an amplitude vector which is a sum of N independent, identically distributed ndimensional amplitude vectors. As the amplitude vectors are identically distributed, it is seen that condition 4. in (4.1.2) is fulfilled. The sum of the n-dimensional amplitude vectors is considered as a random walk in n dimensions. This approach is possible because all vectors are independent and identically distributed, and because a random walk is actually a sum of vectors.

Now they let the number of scatterers N be a random variable. This assumption is in contrast to the situation in (20.11.10) where the number n of random variables in the sums is a constant. So in spite of the fact that <N> goes to infinity in the finai step of the derivation, condition 3. in (4.1.2) is not fulfilled.

Abbildung in dieser eseprobe nicht enthalten

Jakeman (1980) and Jakeman and Tough (1987) let N be negative binomial distributed:

(4.2.3)

Jakeman (1980) deduces the negative binomial distribution by letting the number of scatterers arise from a birth-death-immigration situation. That is, in the area under consideration discrete (bunched) scatterers arise, disappear or migrate. The parameter a in (4.2.3) characterizes the bunching/clustering effect of the scatterers.

Another way of arguing for the negative binomial distribution could be the following:

If we assume that

i) within a given small patch in the illuminated area a scatterer has a certain (small) probability p to occur, the patch being so small that two or more scatterers cannot occur,

ii) the probability p is the same for all such small patches in the illuminated area, and

iii) each scatterer occurs independent of the others, then we are let to a Poisson distribution for the number of scatterers in the illuminated area.

This Poisson model breaks down if the scatterers are not uniformly distributed over the illuminated area. At a sea element one may expect, for instance, that the number of scatterers depends on where on a wave crest the radar looks: there are expected to be more facets (mirroring elements) at that side of the wave which turns towards the satellite than on that side which turns away from the satellite. Many other local effects within the illuminated area may have an influence on the local number of scatterers. So what is expected is a clustering (bunching) of scatterers in the illuminated area.

If therefore we assume that the scatterers have different probabilities to occur depending on where in the illuminated area the patch is located, then we may be let to a negative binomial distribution. The negative binomial distribution can, namely, be considered as derived from an “extended” Poisson model where different scatterers may have different probabilities to occur (this type of argument is also used by Jakeman and Pusey (1978)).

More precisely, the negative binomial distribution can be generated as a compound distribution where the probability for scatterers in a small area is Poisson distributed with parameter λ and where λ is gamma distributed over the whole illuminated area (Stuart and Ord, 1994, pp.182-183). Let namely λ be gamma distributed with parameters a and 1/β:

Abbildung in dieser eseprobe nicht enthalten

(4.2.4)

(This approach to the negative binomial distribution is also used by Jao (1984) who also mentions that the choice of a gamma distribution is purely hypothetical.)

The parameters in the gamma distribution express the level of bunching/clustering in the illuminated area and the different probabilities to observe scatterers in different areas. These considerations justify in an intuitively way the use of a negative binomial distribution for the number of scatterers, and they also show how the assumption of a clustering effect is built into the negative binomial model.

From the assumption of N being negative binomial distributed and without assuming anything about the distribution of the individual amplitude vectors apart from conditions

Abbildung in dieser eseprobe nicht enthalten

1 .,2. and 4. in (4.1.2), Jakeman and Tough (1987) derive the К-distribution for the total amplitude length for an n-dimensional random walk when <N> approaches infinity:

(4.2.5)

The '๙ that appears in the constant b in (4.2.5) is, as can be seen in chapter 20.1, the step length of each of the N amplitude vectors scaled with <Ы>'А in such a way that E(a2) is a constant when <N> approaches infinity. Comparison with formula (2.14) in (Jakeman and Tough, 1987) for the intensity moments shows that we must have E(a2) = E(A2), where E(A2) is calculated by the probability function P(A).

In chapter 20.1, (4.2.5) is derived explicitely for the case where n=2, and some other results for the two-dimensional random walk are presented too.

Though (4.2.5) is actually the distribution of the length of an n-dimensional vector, it can be related to the case of an n/2- look image (where each look is the length of a twodimensional vector) in the following way:

Abbildung in dieser eseprobe nicht enthalten

Recall from (4.1.12) that the L-look amplitude is the square root of the sum of the L intensities. Each intensity is the squared length of a two-dimensional vector, i.e.

(4.2.6)

where each of the akj’s are supposed to be independent of each other. The length

(4.2.6) is exactly the length of a 2L-dimensionaI vector. Now look at the case where amplitude coordinates (ak,, a^ ), k=1 ,...,L have been received from each of the L looks. Instead of squaring and summing these coordinates two and two in order to get the sum of the L intensities, we may as well construct a 2L-dimensional vector A =

(a^ , a12, a21, a22,..., aL1, and then find the length of A as in (4.2.6). The result will be the same. Since all the akj’s are supposed to be independent of each other, the vector A may be considered as the result of a random walk in 2L dimensions. So for n = 2L we are in the case (4.2.5).

It should be noted that we have assumed that it is the same (negative binomial distributed) N scatterers which contribute to each of the 2L coordinates in A. This may be considered as being the case if the multi-looking is a so-called frequency domain multi-looking which is the way that ESA has processed the images used for this study. In this case each look in an L-look image is produced from the same illuminated area by using approximately 1/L of the total azimuth bandwith for each look. If instead one averages over L neighbouring pixels in order to get an L-look image, this assumption will not hold.

Second we consider the derivation of the amplitude К-distribution given by Joughin et

Abbildung in dieser eseprobe nicht enthalten

al. (1993). They use what they call a “mixture model”. In condition 4. in (4.1.2) it was assumed that all amplitudes were identically distributed. Now assume that this is only locally true. This means that the radar cross section varies from one L-look pixel to the next (for instance because of wave effects). (The radar cross section is a measure of the strength by which a radar target reflects the received signal.) In (4.1.13) it was seen that in this case A locally follows a generalized Rayleigh distribution with 2L degrees of freedom and parameter 2й · σ. Because of the varying radar cross section, however, the parameter ๐ will vary from pixel to pixel. Previously, we have seen that the mean intensity in a one-look pixel is 2๙. Now replace the mean intensity 2๙ with 2ü);.Then for given ω the amplitude follows the generalized Rayleigh distribution

(4.2.7)

Abbildung in dieser eseprobe nicht enthalten

(4.2.8)

then Ρ(Α) becomes

Abbildung in dieser eseprobe nicht enthalten

(4.2.9)

where the last equality sign comes from an application of (Gradshteyn and Ryzhik, 1994, formula 3.478, 4.). (4.2.9) is easily seen to be identical with (4.2.5) if we let L = 2n and recall that <l> = E(l) = E(A2) = E(a2).

A similar derivation of the intensity К-distribution can be found in (Oliver, 1993).

Some notes should be made about the calculations in (4.2.9), or more precisely about the gamma distribution (4.2.8):

i) First of all: why a gamma distribution? There does not seem to be a physical answer to this question. On the other hand, modelling the cross section fluctuations with a gamma distribution does not seem to be unreasonable from an experimental point of view. So the the level of the backscattered signal being related to the level of clustering/bunching is supposed to be governed by the gamma distribution. Furthermore, choosing Ρ(ω) in (4.2.9) to be a gamma distribution makes it possible to calculate P(A) in an analytical way. And finally, the result, the К-distribution, seems to describe the observations well and since a gamma distribution yields this result, it seems reasonable to choose it.

So the choice of the gamma distribution is based on a mixture of heuristical, empirical/experimental and mathematical/analytical grounds ((Jakeman and Tough, 1987), (Winebrenner, 1998, personal communication) and (Jakeman, 1998, personal communication)).

As a connection to the previously mentioned derivation of the redistribution, Jakeman and Tough (1987) and Joughin et al. (1993) claim that the gamma distribution is a continuous analogue to the negative binomial distribution. Here it is not quite clear from the context how the limiting procedure from the negative binomial to the gamma distribution should be performed. For instance, according to Andersson and Tolver Jensen (1983, p. 196) we must have for a negative binomial distributed variable p that V(P) E(P), and it is not obvious that we must necessarily have that <l>/(2La) > 1 in (4.2.8) (actually, this is never the case in our examinations). In chapter 20.1 we have a discussion of this topic.

ii) Second, note that E(l) = E(A2) in (4.2.8) is the mean intensity (= mean squared amplitude) calculated from P(A) in (4.2.9).

iii) Third, how are the parameters in the gamma distribution found? Since 2ω = 2σ2 is the local mean intensity in a one-look image, we have that CD is the half of the local mean intensity. The intensity in an L-look image is I = Iļ + ... + lL, so the average of the intensities in the L looks is l/L. If ω varies according to a gamma distribution over a large area, it is reasonable to demand that the mean value of ω is equal to half the mean intensity per look over the area. Therefore, if we call the first parameter in the gamma distribution a, then we must demand that the second parameter β is equal to <l>/(2La) so that E(co) = aß = <I>/(2L),

We have now finished the derivations of the K־distribution model. It has been seen that we can derive a К-distribution for an n-dimensional random walk of amplitude vectors and a К-distribution for an L-look SAR image which are identical for n = 2L. The random walk derivation is based on the assumption that the number of scattereres in a resolution cell is negative binomial distributed, while the L-look derivation is based on the assumption that the mean intensity of the pixels varies according to a gamma distribution over a large area. As mentioned above, the negative binomial distribution may be obtained if observations are Poisson distributed with a parameter λ which varies according to a gamma distribution. So there is a common basic assumption in both derivations, namely the introduction of a gamma distribution.

In this way the underlying physical assumptions for the К-distribution model and the generalized gamma distribution model (see chapter 6) are not very different: in the K- distribution model a gamma distribution is introduced to modulate the Rayleigh situation, while in the generalized gamma distribution model an extra parameter is introduced in the generalized Rayleigh distribution (for amplitudes) or in the gamma distribution (for intensities). This extra parameter in the generalized gamma distribution expresses the level of bunching in the same way as the modulating gamma distribution does in the K- distribution model.

Finally, one interesting observation concerning the K־distribution should be mentioned. First recall that the characteristic function (c.f.) for a probability function f(x) is defined as (see also section 10.2):

Abbildung in dieser eseprobe nicht enthalten

Jakeman and Tough (1987) prove that when <N> tends to infinity, the distribution of the amplitude vector A is given by the following characteristic function:

Abbildung in dieser eseprobe nicht enthalten

(4.2.10)

where a and и are the lengths of a and u, respectively. It is seen that the c.f. is independent of the directions (phases) of и and A (recall that <a2> = <A2>) . (4.2.10) shows that the distribution of the amplitude vector is infinitely divisible (see definition of infinitely divisibility in chapter 20.11). The infinitely divisibility can be seen in the following way:

Abbildung in dieser eseprobe nicht enthalten

Let X have distribution as in (4.2.10). Let к be an arbitrary integer and let xv ... xk be independent, identically distributed random variables with distribution as in (4.2.10) but with parameters (a/k , <a2>/k). Then it is seen that the distribution of X is that of the sum X-I + ... + Xk:

Abbildung in dieser eseprobe nicht enthalten

(4.2.11)

(4.2.11) not only shows that the distribution of A is infinitely divisible; it also shows the sum of к independent, identically distributed amplitude vectors has a distribution of the same type (only the two parameters are multiplied by k).

**[...]**

- Quote paper
- Claus Sölvsteen (Author), 1999, Statistical analysis of backscatter coefficients in ERS-1 SAR images, Munich, GRIN Verlag, https://www.grin.com/document/437304

Publish now - it's free

Comments