IEEE Geoscience and Remote Sensing Magazine - March 2021 - 86

and a widely applicable product model can be obtained
[2], [14] (see the " Statistical Modeling Based on the Product
Model " section).
As opposed to the Rayleigh speckle model, the original,
and more general, coherent scatterer model gives rise to the
NR speckle, i.e., without taking the Rayleigh assumptions
[9]. The NR speckle covers the statistical modeling developed from the coherent scatterer model, whose amplitude
does not follow the Rayleigh distribution, which includes
the product, NR speckle, GCL theorem and ISS models. The
" Statistical Modeling Based on the NR Speckle Model " section introduces the NR speckle model, which can be applied to heterogeneous regions. The NR speckle model can
be degenerated into the GCL theorem model under the assumption that 1) the real and imaginary components of the
scattered field obey distributions with infinite variance and
2) the number of scatterers is an infinite constant [20]-[22],
[26] (see the " Statistical Modeling Based on the GCL Theorem " section). In the coherent scatterer model, when the
phase is uniformly distributed, the coupling term caused by
the scattering phase can be neglected; then, an approximate
model, i.e., the ISS model, can be obtained [11]-[13], [27],
[28] (see the " Statistical Modeling Based on an ISS Model "
section). The ISS model can be used for a correlated clutter
simulation. It was once referred to as the surface model [11],
[12]. Here, we do not use this convention so as to avoid confusion with the " rough surface model " found in EM scattering theory.
The " Statistical Modeling Based on an Empirical Model "
section introduces the statistical approach based on empirical models. These models no longer consider the scattering
process but rather fit only the statistics of the pixel. The main
idea is to establish a statistical distribution that can better
fit observed SAR image data. Depending on whether there
exist analytical expressions and whether the model is elaborate, empirical models can be divided into single empirical
distribution, finite mixture, and nonparametric. The single
empirical distribution model uses one suitable probability
distribution [1]. The finite-mixture statistical model [29],
[30] is a semiparametric model that employs multiple probability distributions. It involves more parameters than does
the single-distribution model but has a wider range of applications. Both models have analytical expressions. The nonparametric approach uses general mathematical models to
fit real SAR images, such as the Parzen-window method [31],
neural networks [18], [32], [33], and support vector machine
(SVM) [19]. The nonparametric approach is usually elaborate
and has no analytical expressions.
STATISTICAL MODELING BASED ON
THE COHERENT SCATTERER MODEL
COHERENT SCATTERER MODEL
Assuming N discrete ideal point scatterers in a resolution
cell, the backscattering field contribution of the ith scatterer ( i = 1, 2, f, N ) is [34]
86

	

E i = K i a i cos ^~t - 2kri + i ih , (1)

where K i is the constant related to the radar system, such as
free-space propagation loss, antenna gain, and so on; a i denotes the ith scatterer amplitude; i i denotes the ith scatterer phase; ri denotes the distance between the antenna and
the scatterer; k = 2r/m is the wavenumber; and ~ denotes
the angular frequency. In free space, k = ~ n 0 f 0 = ~/c,
where c denotes the speed of light.
The phasor-domain counterpart of (1) is
	

E i = K i a i e jz i,

z i = i i - 2kri, (2)

where z i denotes the phase angle of E i .
Consider the following assumptions [34]:
Assumption 1: The scatterers per resolution are statistically
independent, and the interaction between the adjacent scatterers can be ignored. Then, the total instantaneous field E
can be expressed as a coherent superposition:
	

N

E = / K i a i e jz i. (3)
i=1

Assumption 2: The maximum distance between targets
Tr = ri - r j max is far less than the average distance between
the antenna and targets; the antenna gain is uniform across
the illuminated area, e.g., K i = K, i = 1, 2, f, N. For convenience, we shall set K = 1; then
	

N

E = / a i e jz i = Ae ji, (4)
i=1

where A and i are t he amplitude and phase of E,
respectively.
As shown in (4), if assumptions 1 and 2 are satisfied,
the complex scattered field E can be modeled as a random
walk model in the complex plane, and it can be expressed
as the summation of the scattered contributions of N independent scatterers in a resolution cell. This is called coherent
scatterer model, also referred to as the discrete scatterer model
in [1].
According to the coherent scatterer model in (4), the
probability distribution of the observation in a resolution
cell is mainly determined by four factors:
1)	 The spatial-fluctuation distribution of scatterer amplitude a i
in a resolution cell [9]. For simplicity, it is generally assumed that such amplitudes are independent and identically distributed (IID) as 1) constant, 2) variable, or 3)
a sum of a known constant and an infinite number of
variables. For complex scenes, it needs to be modeled as
a finite mixture of multiple scatterers [35].
2)	 The number N of scatterers in a resolution cell. This quantity can either be fixed across resolution cells, or it can
be a random variable. In either case, it can be finite or
infinite.
3)	 The distribution of scatterer phase z i in a resolution cell. It
can be modeled as a uniform distribution or nonuniform distribution.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

MARCH 2021
IEEE Geoscience and Remote Sensing Magazine - March 2021

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