The Generation of Correlated Gamma Distributed Random Fields for the Simulation of Synthetic Aperture Radar Images

Abstract

Simulation plays an important role in both the development and analysis of new radar imaging and processing systems and in the analysis of their data. We have developed a synthetic aperture radar (SAR) simulation system, cSAR, that simulates the raw signal data. An important component of this system is the simulation of clutter, or the random spatial fluctuations of backscatter. The prevailing statistical model for clutter is the K distribution. This model is founded on a gamma distributed scattering coefficient. In this paper we present a system for simulating correlated gamma distributed fields of scattering coefficients. Our system starts with the requirement that in radar and other coherent imaging scenarios the locations of elemental scene scatterers must be random to achieve fully developed speckle in the image. We start with a set of scatterers whose spatial location follows a uniform distribution. To generate a correlated random field with this random distribution of scatterers we use the turning bands method. We show that the autocorrelation function (ACF) of the simulated random fields matches the analytical ACF. To transform the correlated Gaussian random field into a correlated gamma random field we use the memoryless nonlinear transform (MNLT). It has been shown that the MNLT skews the ACF of the simulated field. We present a method to preserve the form of the ACF of the field using a mapping function developed by Tough and Ward [J. Phys. D: Appl. Phys., 32 (1999), pp. 3075--3084]. We demonstrate that the ACF of the simulated field is preserved. Once the random field of scatterers is created, the scattering coefficient $s$ can be approximated with a local summing process, which emulates the incoherent imaging process of a SAR sensor. There is evidence in the literature that the sum of correlated gamma variates on a regular grid is also gamma distributed. We develop a new theory for the first and second order linear sum process (LSP) statistics for a random sum of random variables. We use this theory to derive two estimators for the gamma order parameter. Through a Monte Carlo experiment, it is shown that $s$, as represented by the LSP, does generally follow a gamma distribution for scenarios with at least nine scatterers per resolution cell. The order parameter and correlation properties are modified from the original random field according to patterns predicted by our theory, though some parameters must be calculated through simulation.