IEEE Geoscience and Remote Sensing Magazine - March 2021 - 93

STATISTICAL MODELING BASED ON
THE NR SPECKLE MODEL
The Rayleigh speckle model is used to describe low-resolution radar images. The model assumes that there is a large
number of independent scatterers in a resolution cell, the
scattered amplitude of the scatterers are IID, and the scattering phase is uniformly distributed. However, such assumptions are seldom valid in high-resolution radar imagery. The NR speckle model is developed based on the
coherent scatterer model. In this model, the radar echo
appears to contain a " target signal " with fluctuations, and
the following are the main explanations for this phenomenon [9]:
◗◗ The scattered amplitude in a resolution cell is no longer a constant but has some fluctuating characteristics.
As shown in Figure 7, a i ^ r h represents the -fluctuating
characteristic of the scattered amplitude of the i th scatterer at different observation positions r. At a specific
time t 0, the number of scatterers at the specified observation position r0 is N (N = 4 in Figure 7), and the
spatial-fluctuation characteristics of the scatterer a i ^ r h
cause the scattered field at the position r0 to be determined by a few strongly contributed scatterers. Therefore, even when the number N of scatterers is large,
such fluctuations cause NR scattered echoes. This phenomenon gave birth to the concept of an " equivalent
number of scatterers, " which is introduced in the " NR
Speckle Model " section.
◗◗ The number N of scatterers in a resolution cell is small,
and the central limit theorem is no longer satisfied.
The scattering echo at this time is no longer a Rayleigh
echo, and the fluctuation characteristics of the number
of scatterers influence the probability distribution of a
single pixel. The number of scatterers can be modeled
as a random variable obeying a Poisson distribution
whose expectation is also a random variable obeying
some distribution.
◗◗ The distribution of the phase of scattered field in a resolution cell is no longer uniformly distributed. The discussion of phase distribution is relatively rare because
it is generally assumed that the scatterers are randomly
positioned and the maximum distance between the
scatterers is much larger than the wavelength; therefore,
the scattered phase is usually sufficient to satisfy the
uniform distribution.
The next section first theoretically deduces and analyzes
the probabilistic statistical model under the NR speckle
model and then introduces various statistical distributions
based on the NR speckle model.
NR SPECKLE MODEL
Rewrite the expression of the scattered field E ^ r h in the coherent scatterer model (4) as follows:
E^ r h = / a i e
N

i=1

jz i

= / ^a i cos z i + ja i sin z ih = R + jT = Ae .
N

ji

M Ri ^~h = M Ti ^~h / E 6exp ^ j~a i cos z ih@
=

	

#- rr p z ^z ih exp ^ j~a i cos z ihdz i

1 r
= 2r # exp ^ j~a i cos z ih dz i
-r
= J 0^a i ~h ,


(26)

where J 0 is the first type of zero-order Bessel function.
Assuming that the number of scatterers N is a constant,
then the characteristic function of the sum of N independent random variables is the product of the characteristic
function of each variable, and the characteristic function
of the real and imaginary components of the scattered field
can be written as [9], [25]
M R ^~h = M T ^~h = % M Ri ^~h = % M Ti ^~ h = % J 0 ^a i ~h. (27)
N

N

N

i=1

i=1

i=1

Because the phase angles of the summation of the individual components are uniformly distributed within
^- r, r@, the joint probability density and joint characteristic function of the real and imaginary components of the
summed phase vector is circularly symmetric. The 2D characteristic function M R, T ^~ R, ~ Th is [9], [23]
M R, T ^~ R, ~ Th = M R, T _ ~ 2R + ~ T2 i = % J 0 _ a i ~ R2 + ~ T2 i . (28)
N

i=1

Denoting u = ~ 2R + ~ 2T as the characteristic function of
the amplitude A of the scattered field, E ^ r h [9], [25] can be
written as
M N ^u h = % J 0 ^a i uh . (29)
N

	

i=1

a(r)

a1(r)
a2(r)
a3(r)
a4(r)
r

r0
N=4

a (r 0) =

∑

ai (r 0)

i=1

i=1

(25)
MARCH 2021

Take the following assumptions:
◗◗ a i, a j, z i, z j, ^ i ! j h are collectively independent random variables
◗◗ " z i , is uniformly distributed at 60, 2r h .
The characteristic functions M Ri ^~ h and M Ti ^~ h of the
real component a i cos z i and the imaginary part a i sin z i of
the ith scatterer can be written as

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

FIGURE 7. The amplitude fluctuation of an NR speckle model.

93
IEEE Geoscience and Remote Sensing Magazine - March 2021

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