Statistical analysis of backscatter coefficients in ERS-1 SAR images

Contents

1. Abstract and introductory remarks

2. Historical background

2.1 Some initial/practical problems

2.2 SAR and bathymetry

2.3 Start of statistical analysis

2.4 Contents of the thesis

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

3.1 Review

4. A statistical model for the scattering mechanism

4.1 The Rayleigh model

4.2 The K-distribution model

4.3 Problems with the K-distribution model

5. Digamma function

5.1 Basic properties of the digamma function

6. Generalized gamma distribution

6.1 Basic properties of the generalized gamma distribution

6.2 Maximum likelihood estimation

6.3 The ordinary gamma distribution

7. Exponential gamma distribution

7.1 Basic properties of the exponential gamma distribution

8. K-distribution

8.1 Basic properties of the K-distribution

8.2 Mode of the K-distribution

8.3 Maximum likelihood estimation in the K-distribution

9. Moment method

9.1 Brief introduction to the moment method

10. Method of moments

10.1 Moments and uniqueness of distributions

10.2 Moments and cumulants

11. Raghavan's method

11.1 General introduction to the method

11.2 The generalized gamma distribution

11.3 The K-distribution

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

12.1 Introduction to the chi-square test

12.2 Construction of categories

12.3 Number of observations and categories

12.4 Concluding remarks on the chi-square test

13. The runs test and the U test

13.1 The runs test

13.2 The U test

14. Kolmogorov's Dn test

14.1 Description of the Dn test

15. Which test should be chosen?

15.1 General problems with the category construction

15.2 The Dn test and the runs test

15.3 The Dn test and the chi-square test

15.4 Concluding remarks on the test comparisons

16. Data analysis, Parameter analysis

16.1 Parameter values in six homogeneous ROl's

16.2 Parameter values in two inhomogeneous ROl's

16.2.1 First consider 16014_1215, ROI

16.2.2 Second consider 16588_1215, ROI

16.2.3 Analysis of individual cells in 16014_1215, ROI

16.2.4 Analysis of individual cells in 16588_1215, ROI

16.2.5 Summary of analyses of the inhomogeneous ROl's

17. Data analysis, Statistical analysis

17.1 Test of statistical distributions

17.2 Tests of different test methods

17.3 Statistical analysis of 16481_2385, ROl2

17.4 Statistical analysis of 16014_1215,ROI and 16588_1215,ROI

17.5 Relationship between the K-distribution and the gamma distribution

18. Conclusions

19. Future recommendations

Objectives and Research Focus

This thesis investigates statistical methods for analyzing SAR image intensities, aiming to provide a foundation for future bathymetric detection. The primary research objectives include identifying and theoretically discussing relevant statistical distributions for SAR intensities, developing methods for statistical analysis over homogeneous areas, and evaluating the statistical behavior of these intensities. Key research areas include:

• Theoretical analysis of the K-distribution and generalized gamma distribution.
• Development and examination of parameter estimation techniques.
• Comparison of statistical goodness-of-fit test methods.
• Application of these methods to homogeneous and inhomogeneous sea areas.

Excerpt from the Book

Chapter Summary

Abstract and introductory remarks: Introduces the research goal of using SAR imagery for bathymetric purposes through statistical methods and outlines the thesis objectives.

Historical background: Reviews the pilot project initiated by the Royal Danish Administration of Navigation and Hydrography and discusses initial challenges in reading SAR image headers and calibration.

Contributions to the theory of K-distribution and generalized gamma distribution: Provides a literature review on K-distribution and generalized gamma distribution theory and their application to radar and ultrasound imaging.

A statistical model for the scattering mechanism: Explains the mathematical derivation of the Rayleigh model and introduces the K-distribution model for non-Gaussian scattering scenarios.

Digamma function: Introduces the digamma function, its mathematical properties, and its relevance to statistical distribution estimation.

Generalized gamma distribution: Defines the generalized gamma distribution and its properties, demonstrating how it serves as a flexible candidate for modeling backscatter coefficients.

Exponential gamma distribution: Discusses the distribution of SAR intensity in decibels, which follows the exponential gamma distribution when intensities are generalized gamma distributed.

K-distribution: Presents the properties, mode behavior, and parameter estimation challenges associated with the intensity K-distribution.

Moment method: Introduces moment-based parameter estimation, highlighting its limitations and inaccuracies when applied to complex distributions like the K-distribution.

Method of moments: Explores the more robust method of moments as a parameter estimation technique and addresses the uniqueness of moments for specific distributions.

Raghavan's method: Discusses Raghavan's alternative parameter estimation approach and evaluates its utility in relation to the method of moments.

Chi-square test for goodness-of-fit: Details the procedure for conducting chi-square tests, emphasizing the importance of category construction for reliable testing.

The runs test and the U test: Introduces the runs test to detect systematic deviations and describes the U test as a combination of chi-square and runs tests.

Kolmogorov's Dn test: Describes the Kolmogorov's Dn test as a distribution-free goodness-of-fit test and discusses its limitations in composite hypothesis cases.

Which test should be chosen?: Compares the different statistical tests, focusing on category dependence and the reliability of each test in different scenarios.

Data analysis, Parameter analysis: Presents the analysis of parameters for various homogeneous and inhomogeneous regions of interest using the SAR images.

Data analysis, Statistical analysis: Provides a comprehensive summary of the statistical distribution testing results and compares the efficacy of different test methods.

Conclusions: Summarizes the major findings, emphasizing the validity of generalized gamma and K-distributions for SAR backscatter and the challenges of parameter estimation.

Future recommendations: Suggests avenues for future research, including the use of airborne SAR, along-track interferometry (ATI), and more advanced spatial detection algorithms.

Keywords

SAR, ERS-1, Backscatter coefficients, K-distribution, Generalized gamma distribution, Ordinary gamma distribution, Parameter estimation, Maximum likelihood estimation, Method of moments, Chi-square test, Runs test, U test, Kolmogorov Dn test, Bathymetry, Signal-to-noise ratio

Frequently Asked Questions

What is the primary focus of this thesis?

The thesis focuses on the statistical analysis of SAR (Synthetic Aperture Radar) image intensities, specifically investigating how these intensities are distributed over homogeneous and inhomogeneous sea areas to develop robust methods for future bathymetric detection.

What are the central statistical distributions analyzed?

The work primarily examines the ordinary gamma distribution, the three-parameter generalized gamma distribution, and the three-parameter K-distribution as candidates for modeling backscatter coefficients in SAR images.

What is the primary research goal?

The main objective is to develop and examine statistical parameter estimation and goodness-of-fit testing methods that can characterize SAR image intensities and identify departures from homogeneity, which is essential for detecting bathymetric features.

Which statistical methods are evaluated for parameter estimation?

The research evaluates the maximum likelihood (ML) estimation, the "moment method," the "method of moments," and "Raghavan's method" for estimating parameters within the analyzed distribution families.

Which tests for goodness-of-fit are discussed?

The thesis analyzes the chi-square test, the runs test, the combined "U test," and Kolmogorov's Dn test, discussing their respective sensitivities and limitations, particularly in the context of composite hypotheses.

What is the core conclusion regarding the statistical models?

The study concludes that backscatter coefficients in homogeneous sea areas are well-described by both the generalized gamma distribution and the K-distribution, though parameter estimation remains computationally challenging for both.

Why is the signal-to-noise ratio critical in this study?

Low signal-to-noise ratios, common in dark areas or low wind conditions, complicate parameter estimation and negatively impact the reliability of statistical goodness-of-fit tests, making reliable detection difficult.

What are the limitations of the K-distribution model?

The K-distribution model often faces problems related to the estimation of the "look" parameter (L), which may require non-physical values to achieve a good fit, suggesting the model may be fundamentally limited or require modification.