IEEE Geoscience and Remote Sensing Magazine - March 2021 - 101

distribution of the number of scatterers and outputs the
distribution of SAR image intensity. Under the ISS model,
when the number of scatterers N in a single-resolution cell
is infinite, the intensity distribution of the scattered field is
an impulse function [10]; when N is a variable that obeys
the negative-binomial distribution with an infinite mean,
the intensity of the scattered field obeys a CD [10], [63]. The
statistical distributions under the ISS model are given in the
next section.
IMPULSE FUNCTION
When the number N of scatterers in a resolution cell is
an infinite constant, the distribution of scattered intensity is an impulse function. Dividing a by N in the Laplace transform Q N ^ s h of the scattered intensity in (91), if
N " ∞, the Laplace transform Q N ^ s h and corresponding
PDF by inverse Laplace transform are [63]
lim Q N ^ s h = exp ^- s G b 2 Hh, (92)

	

N " ∞

	

p ^v h = d ^v - G v Hh, (93)

where d ^ $ h denotes impulse function. The scattered intensity is therefore a constant in this limit.

Gamma-distributed intensity is generated by the following rate equation, which is a continuum analog of the birthand-death immigration model:
	

22 P
2P
2P
mv
=
+ 62m - o + ^n - mh v@ 2v + ^n - mh P, (97)
2t
2v 2

where m, n, and o are the birth, death, and immigration
rates, respectively.
The solution of (97) is most readily accomplished in
terms of the generating function of P ^v, t h by the Laplace
transform
Q ^ s, t h = G exp ^- sv h H, (98)

	

which satisfies the partial differential equation
2Q
2D
2t = - s :o + ^n - m + ms h 2s Q. (99)

	

Assume that the initial intensity is v 0 , and the boundary
conditions on Q are
Q ^0, t h = 1; Q ^ s, 0 h = exp ^- sv 0 h; 

	

(100)

then, the solution can be obtained as
CD
The " Statistical Modeling of the Rayleigh Speckle " section
introduces the modeling of the CD based on the Rayleigh
speckle model. This section gives the following two other
modeling processes for the CD : 1) the CD under the ISS
model and 2) the CD by solving the rate equation, which
describes the continuous fluctuations of the birth-anddeath immigration model.
1)	 When the number of scatterers in a resolution cell obeys
the negative-binomial distribution with an infinite
r , as shown in (41), then the scattered intensimean N
ty obeys the CD [10], [63]. The following is derivation
process.
r for Q N ^ s h, as shown in (91), the averDividing a by N
age form of Q N ^ s h can be written as
	

r /a h^1 - G exp ^- sa 2 /N
r h H h@- a , (94)
Q Nr ^ s h = 61 + ^ N

where G$H denotes the expectation.
r " ∞, the limit form of (94) is
If N
	

lim
Q Nr ^ s h = ^1 + s GvH /a h- a; (95)
r

N " ∞

then, the gamma-distributed scattered intensity can be obtained using the inverse transform in (95)
	

p^vh = a

a ka v a - 1
av k
exp a .
G vH C ^ a h
G vH

(96)

2)	 The scattered intensity v is considered a continuous
fluctuation controlled by the process of the birth-anddeath immigration model. The solution of the rate equation will produce a scattered intensity of CD [63].
MARCH 2021

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE



Q ^ s, t h = d

o/m
m-n
v 0 ^ n - m h is
n exp <
F,
m - n + m^i - 1hs
m - n + m^i - 1hs
(101)

where
i ^ t h = exp 6^m - n h t@ . (102)

	

If n 2 m and t " ∞, then the equilibrium distribution is
Q^ s h = d

	

o/m
m-n
n . (103)
m - n - ms

The inverse Laplace transformation of (103) gives the following CD :
	

p^vh = ;

^n - mh

m

E

o/m

^n - mhv
v o/m - 1
E . (104)
exp ;C ^ o/ m h
m

Taking n = o/m , o = o/ ^n - mh , then (104) is equivalent
to p Gamma v in Table 2.

STATISTICAL MODELING BASED
ON EMPIRICAL MODELS
Different from statistical modeling based on the coherent scatterer model defined previously, statistical modeling based on the empirical model no longer considers
the scattering process but rather starts directly from the
image itself. The main target of this approach is to find
a tractable distribution that is consistent with the actual
SAR images.
As displayed in Figure 11, according to the presence of
analytical expressions and the complexity of the model,
empirical statistical modeling can be classified into three
101
IEEE Geoscience and Remote Sensing Magazine - March 2021

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