Technical report | Generating Correlated Gamma Sequences for Sea-Clutter

ABSTRACT

This report presents a hybrid method for simulating sequences of correlated Gamma random variables for modelling sea clutter, using a combination of linear and/or nonlinear transforms. Depending on the shape parameter, this method minimises the use of non-linear transformations. Mathematically the method is simpler than its counterpart methods which leads to a quicker simulation run time. Two memoryless non-linear transform (MNLT) approaches are also studied with comparative results showing that the hybrid approach is more computationally efficient and slightly more accurate for low shape parameters. The drawback of the proposed method is, however, that it can only handle positive correlations whilst the two MNLT methods are capable of handling both positive and negative correlations.

EXECUTIVE SUMMARY

To support the Generic Phased Array Radar Modelling (GPARM) simulation program of the Defence Science and Technology Organisation (DSTO), which in turn supports many Australian defence programs including SEA1448 (ANZAC ASMD), AIR7000 (future maritime patrol and response capability) and AIR5077 (Wedgetail), this report proposes a hybrid method and examines other existing methods for simulating sequences of correlated random variables with a Gamma distribution.

Radar sea-clutter is a dominant undesired signal seriously affecting radar performance over the sea surface. It is now widely accepted that in most scenarios, radar sea-clutter can be modelled as a compound non-Gaussian random process. One of the most popular is the compound Kdistribution which consists of two parts: a fast-varying component representing sea-clutter speckle which is modelled as a complex Gaussian with zero mean and unit variance and a slowly-varying component which is Gamma distributed and represents the underlying sea-clutter intensity. These two components are assumed to be mutually independent.

Depending on radar parameters and sea surface conditions, each component of the received sea-clutter may be correlated and appropriate correlation models should be included in the simulation. For K-distributed sea-clutter, methods for simulating both correlated Gaussian and Gamma processes are required. Simulation of the former process is straightforward and can be realised by a linear transform using either spherically invariant random processes (SIRP) or Fourier synthesis. The advantage of the linear transform is that the desired correlation properties are easily maintained. However, simulation of the correlated Gamma distribution is more difficult and may require application of the so-called memoryless non-linear transform (MNLT) to generate the desired correlation.

This report proposes a hybrid method for simulating sequences of correlated Gamma random variables. The approach depends on both the desired shape parameter and in some cases the correlation coefficient. For most distributions with a shape parameter greater than 0.5, the method only requires linear transforms to generate the desired Gamma correlation. However in other cases, such as when the shape parameter is less than 0.5, the method requires the MNLT to achieve the desired correlation. The MLNT technique proposed in this report is different from its counterparts. Compared to the other methods, the proposed one is mathematically simpler making its numerical implementation easier and more computationally efficient. The drawback is, however, that it can only handle positive correlation coefficients.

Two implementations of the MNLT are also examined and evaluated in this report. The first directly implements the numerical integration described by Tough and Ward, while the second is a polynomial auto-correlation method extended from the former by Weinberg and Gunn. Results for the former have a high level of accuracy and speed due to the optimised code. The latter also performs reasonably well except for small shape parameters less than 0.3. Whilst these methods are much more complex then the hybrid method, they are able to handle negative correlation coefficients. A comparison between methods shows that the hybrid method has a reduced simulation run time and provides slightly higher accuracy in generating the desired correlated sequences, particularly for small shape values.

Another contribution in this report is the provision of criteria for realisable correlation functions. It is shown that not every correlation function can be simulated but only those that have a positive semi-definite or positive definite covariance matrix.

This report serves as a detailed reference for computer programmers and research scientists who want to implement correlated Gamma processes to simulate distributions such as the Kdistribution.