IEEE Geoscience and Remote Sensing Magazine - March 2021 - 116

The spatial correlation models of SAR images also have
different analysis methods through the development of
single-pixel probability distribution models. The correlation analysis of SAR images was first discussed by Jakeman
and Pusey [3] when they proposed K-distributed sea clutter. They gave the expressions of the high-order moment
and the autocorrelation function of the radar cross-section
(RCS). They also provided the relationship between the autocorrelation function of the RCS and the number of scatterers, with which they defined the concept of the number
of equivalent scatterers. Subsequently, [4] and [5] derived
the influence of the number of scatterers that obey the negative binomial distribution on the spatial correlation.
A large number of previous models of non-Gaussian
clutter focus on the statistical analysis of the correlated Kdistributed clutter, and the correlation analysis is mainly
based on the development of a coherent scatterer model
described by a random walk process. Oliver et al. considered the influence of the imaging system and proposed
multiple surface cross-section models for describing the
K-distributed correlated clutter [6]-[9]. The K-distributed
correlated clutter is modeled as the output of the correlated
gamma-distributed RCS after the imaging system. It mainly
includes two models for simulating the RCS.
One is an accurate representation based on gamma-
Lorentzian intensity fluctuations [6], [7], which describe
the gamma-distributed RCS as the solution of the Fokker-
Planck equation (a continuous form of the birth-death-migration ratio equation [10], [11]) that can be obtained by numerical simulation using stochastic differential equations.
This model can accurately express the higher-order statistical characteristics of the correlated K-distributed clutter.
Its limitation is that it can represent only the exponential
correlation function and is suitable for 1D situations alone.
The other representation is an approximate model based
on a linear filter [8]. It uses an uncorrelated gamma-distributed RCS to obtain a correlated gamma-distributed RCS
through a linear filter. The uncorrelated gamma distribution can be regarded as a Gaussian random walk process.
This model is widely employed since it can describe arbitrary correlation functions and represent 2D clutter images.
The disadvantage of this method is that it can accurately
represent only second-order correlation moments, with
possible deviations for higher-order moments. Also, it is
applicable only to the case where the order of the gamma
distribution is a semi-integer. The author of [12] discussed
a method for simulating correlated clutter images with an
arbitrary power spectrum based on an approximate model.
To overcome the shortcoming that the approximate model
can describe only the semi-integer-order gamma distribution, more theoretical models and simulation methods of
correlated gamma distribution have gradually been proposed [13], [14], such as the memoryless nonlinear transform (MNLT) technique [13].
To characterize non-Gaussian clutter with more general
models, Conte et al. [15]-[18] proposed a complex spherical
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invariant random process (SIRP) to model SAR complex
clutter images. The advantage of the SIRP process is that,
as with the Gaussian distribution, it is completely determined by the mean and the covariance function. This
process has linear transformation properties that are similar to Gaussian variables, which facilitates the selection
of correlated structures. This model ensures that the joint
probability density function (PDF) of the real and imaginary components of complex clutter is circularly symmetric; that is, the phase obeys a uniform distribution. The
autocorrelation structure and cross-correlation structure
of the real and imaginary components are flexible and
can be stipulated according to a specific physical scene.
The closed-form, higher-order probability distribution expression of this model is available, and it can reduce to
the simplest case, where the amplitude obeys the Rayleigh
distribution. The SIRP is able to represent the K distribution, which is expressed as the product of a Rayleigh-distributed variable and a gamma-distributed variable, and
the Weibull distribution by constraining specific parameters. The disadvantage of this model is that it can represent only amplitude images that obey the distribution of
mixed Rayleigh laws. As an example, it cannot express the
lognormal distribution.
Based on the preceding research, Conte et al. [17] further
proposed a composite process to model radar clutter as a
product of the complex SIRP process and independent nonnegative random variables. The approach can be considered
as the modulation of the SIRP process, and its correlation
characteristics are introduced by the complex SIRP process. Correlation functions of clutter can also be expressed
as the product of the correlation functions of the complex
Gaussian process and a nonnegative random process. The
nature of the model is consistent with the product model
proposed by Ward [19] and the correlation analysis by Ulaby
et al. [20], both of whom model the speckle as multiplicative noise. The product model has also been one of the most
popular ones in the statistical representation of SAR images.
The authors of [17] provided simulation methods for correlated K-distributed clutter and correlated Weibull-distributed clutter based on the exogenous model [15].
This article analyzes the existing spatial correlation
models from two perspectives: the product model [19] and
the coherent scatterer model [6]. The product model describes the SAR image observations as the product of the
RCS component and the speckle component. The correlation is the product of the correlation function of the RCS
and the speckle. The product model is simple and easy to
implement and analyze, but it is difficult to establish a
corresponding relationship with the physical scattering
process. The coherent scatterer model, also known as the
discrete scatterer model, is a relatively complex statistical representation based on physical scattering processes. It analyzes the relevant characteristics of an SAR image based on
factors such as the number of scatterers, the RCS, and the
imaging impulse response. This method has strong physical
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