Estimation of $K$ -Distribution Shape Parameter From Sonar Data: Sample Size Limitations

The K-distribution has been found to provide a good description of the probability density function (pdf) of the matched filter envelope of sonar signals backscattered from the seafloor. Furthermore, the scale and shape parameters that specify the K-distribution, under certain assumptions, have been related to physical descriptors of the seafloor. Specifically, the shape parameter has been related to the number of contributing scatterers and the scale parameter has been related to the average scatterer size. There exists, therefore, the possibility of estimating seafloor scattering properties by determining the shape and scale parameters of the K-distribution that best fits the signal returned from the seafloor. This paper investigates the relationship between data sample size and the accuracy to which K-distribution shape parameter can be determined. Computer-generated random data are considered and two types of test are used to study the ease with which the data distribution can be described. The Kolmogorov-Smirnov (KS) test and a parameter estimation technique based on the method of moments are used and it is shown that, for these tests, the sample size necessary to distinguish reliably between Rayleigh- and K-distributed data is proportional to the square of the shape parameter of the K-distribution. It is also shown that the sample size necessary to determine the shape parameter to the nearest integer value is proportional to the shape parameter raised to the fourth power. This proportionality is shown to be consistent with consideration of Cramer-Rao lower bound (CRLB). The implications for practical sonar scenarios are discussed.