Empirical Bayesian Estimation of the Interferometric SAR Coherence Magnitude

SAR interferometry has developed rapidly in recent years and now allows measurements of subtle deformation of the Earth's surface with millimeter accuracy. All state-of-the-art processing methods require a precise coherence estimate. However, this estimate is a random variable and biased toward higher values. Up to now, little is published on the Bayesian estimation of the degree of coherence. The objective of the article is to develop empirical Bayesian estimators and to assess their characteristics by simulations. Bayesian estimation is understood as a regularization of the maximum likelihood estimation. The more information is used and the stricter the general prior, the more accurate the estimate will be. Three levels of prior information are developed: (1) an uninformative prior and an informative prior which can be implemented as (2a) less strict prior and (2b) strict prior. The informative priors are described by a single parameter only i.e., the maximum underlaying coherence. The article reports on the bias, the standard deviation and the root-mean-square error of the developed estimators. It was found that all empirical Bayes estimators improve the coherence estimation from small samples and for small underlaying coherences compared to the conventional sample estimator, e.g., a zero underlaying coherence is estimated by the expected
a posteriori
estimator without additional information with a 33.3% reduced bias using three samples only. Assuming the maximum underlaying coherence is 0.6, the bias is reduced by 51.3% for the strict prior and by 36.6% for the less strict prior. In addition, it was found that the methods work very well even for the extremely small sample size of only two values.