IEEE Geoscience and Remote Sensing Magazine - March 2021 - 95

other modeling methods of K distribution. The scattered
field of the K distribution can also be obtained when the
number of scatterers obeys the negative-binomial distribution with infinite expectations and the amplitude obeys
arbitrary distributions. The K-distributed amplitude and
finite constant of the number of scatterers also lead to Kdistributed scattered intensity. Similarly, the scattered field
obeying the Rayleigh distribution can also be obtained by
the Rayleigh distributed amplitude and finite constant of
the number of scatterers.
In addition, when the real and imaginary components
of the scattered field are IID variables with infinite variance
and the number of scatterers is an infinite constant, the NR
speckle model can be degenerated into the GCL theorem
model [26], which is introduced in the " Statistical Modeling Based on the GCL Theorem " section.

determined by the fluctuations of the number of scatterers
and their scattered amplitude [9], [10], [23], [51], [53], [54].
◗ When the scattered amplitude of each scatterer is independently and identically K-distributed with probability distribution p ^a; r h and N is an arbitrary finite constant, then the scattered intensity v is K-distributed [9].
The following is the corresponding derivation process.
Assume that p ^a; r h is K-distributed, which is

K DISTRIBUTION
The amplitude and intensity of the scattering field of the
K distribution have been derived from the product model
in the " Statistical Distributions of the Product Model " section. Here, the other four ways of modeling K distributions
are introduced based on the NR speckle model, which is

This is equivalent to the K-distributed intensity equation
p Kv ^ Z v = z; b, a, n h in Table 2 when it satisfies

Infinite Constant

Pv ^v; r h =

Negative
Binomial
istribution

Finite Constant

(39)

(40)

where Z v = z = v.

Γ

NR

Inverse Γ

NR

Beta istribution of
t e Second ind

NR

Beta istribution of
t e First ind

NR

I

istribution
istribution

G

istribution
W

istribution

istribution
t ers

Infinite Constant
mplitude

Constant
rbitrar

ariable

istribution

istribution
Ra leig

Finite ariance II
istribution
Infinite ariance II
istribution

istribution

CL T eorem Model

Sum of a Constant and an
Infinite Number of ariables
Real and Imaginar
Components

Scattered Field of
a Single Scatterer

NR Spec le Model

M

b/ v b v
c
m K M - 1^b v h.
C^Mh 2

b = b 2 /4, a = M, n = 1,

pectation

Number of
Scatterers per
Resolution

Infinite ariable

Poisson
istribution

(38)

Substituting p ^a; r h into (33), the scattered-intensity distribution Pv ^v; r h can be derived as

pectation

t ers
Finite ariable

2b a ba kv + 1 ^ h
K v ba , v 2 -1.
C^1 + vh 2

p ^ a; r h =

NR

istribution

NR

istribution

NR

Ra leig

istribution

NR

RiI

istribution

NR

Rice

istribution

FIGURE 8. The statistical distributions of scattered intensity based on an NR speckle model.
MARCH 2021

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

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IEEE Geoscience and Remote Sensing Magazine - March 2021

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