IEEE Geoscience and Remote Sensing Magazine - March 2021 - 96

◗◗ When the number of scatterers in a resolution cell obeys

r and
the negative-binomial distribution with mean N
r " ∞, then it can be deduced that the scattered ampliN
tude and intensity obey the K distribution regardless of
the distribution of scattered amplitude [10], [54]. The
following is the detailed derivation process.
Suppose that the number N of scatterers is a negativebinomial distributed statistical random variable independent of " a i , and " z i , with PDF:
	

r N
aN
k
N + a -1
a
r , ah = c
m
p ^ N; N
r N + a , (41)
N
N
a1+ a k

where N could be seen as the number of failures before the
a successes in a series of independent Bernoulli trials, and
r h . The
the probability of success for each trial is a /^a + N
corresponding mean and variance are
	

r , Var ^ N h = a -1 + N
r -1. (42)
Mean ^ N h = N

The characteristic function of the scattered field was derived in (29). Now assume that the scattered amplitude a i
of each scatterer is IID and N is a statistical random variable obeying the negative-binomial distribution in (41).
r ; then the folTake the expectation of N, and let a = a/ N
lowing characteristic function could be derived:
	

r /a h _ 1 - G J 0 _ au / N
r i H iA - a. (43)
M Nr ^u h = 71 + ^ N

It holds that
	

lim M Nr ^u h = 61 + G a 2 H u 2 /4a@ - a, (44)

N " ∞

which is exactly the characteristic function of the K distribution:
	

p^ Ah =

2b a bA ka
K a - 1 ^bAh , (45)
C^ah 2

where A is the scattered amplitude. Then, the intensity
distribution can be obtained using the transformation
formula p Z v ^ z h = p A ^ z h / ^2 z h, and it is equivalent to
p Kv ^ Z v = z; b, a, n h in Table 2 when it satisfies b = 2 b ,
n = 1.
◗◗ Assuming that the number of scatterers in a single-resolution cell obeys the Poisson distribution with expectation
r shown in (46) and X obeys the CD shown in (47),
X=N
then it can be derived that this is equivalent to the negativebinomial distribution in (41) according to the total probability formula in (48). Therefore, the K-distributed SAR
image in (45) can be obtained regardless of the distribution
of scattered amplitude according to (2) [23], [54]:
	
	
96

rN
r h ,
r h = N exp ^- N
p^N | X = N
N!
1 a a ka a - 1
~a k
p^X = ~h =
r ~ exp a - N
r ,
C^ah N

(46)
(47)

	

p^N h =

#0 ∞ p^N | X = ~hp X^~hd~.

(48)

◗◗ The K-distributed SAR image can also be modeled in

the perspective of a random walk of Brownian motion
[51]. The real R and imaginary T components of the
complex scattered field are modeled as two Wiener processes with a zero mean. When the stop time Z = z is
given, R and T are normal distributed variables with a
variance of z /2, then a Rayleigh distributed amplitude
A = R 2 + T 2 can be obtained as
2A A
p ^ A ; z h = z e - z . (49)
2

	

If the stop time Z obeys the CD ,
	

p^ z h =

^ b/2 h2a a - 1
zb 2
z exp a - 4 k. (50)
C^ah

Then, according to the Bayesian formulation,
	

p^ Ah =

#0 3 p^ A| zh p^ z hdz .(51)

Subst it ut i ng t he t ra n sfor m at ion for mu la p Z v ^ z h =
p A ^ z h / ^2 z h , it can be deduced that the scattered
f ield intensity v both obey the K distribution [see
p Kv ^ Z v = z; b, a, n h in Table 2].
RICE DISTRIBUTION
The Rice (or Rician) distribution is the probability distribution of the amplitude of a circularly symmetric
Gaussian random variable with a nonzero mean. In the
coherent scatterer model, the Rice distribution can be
obtained by assuming that the scattered field is a summation of a constant A 0 and an infinite number of random variables [25], [55]; that is, the scattered field can
be expressed as
	

N

E = A 0 + / a i e jz i = A 0 + A n e jin. (52)
i=1

The scattered intensity can be written as
	

v = E 2 = A 20 + A 2n + 2A 0 A n cos i n . (53)

Assuming that 1) the number of scatterers N " ∞, 2) the
amplitudes of each scatterer " a i , are IID, and 3) the phase
of the scattered field obeys a uniform distribution in ^0, r@,
then it can be deduced that the scattered field intensity
obeys the Rice distribution, which is
	

p Rice ^v h =

v A0
v + A 20
1
1 I 0d
2 exp ' 2o
2o 2
o 2 n,(54)

where I 0 is the modified zero-order Bessel function of the
first kind.
When the constant A 0 is zero, the Rice distribution becomes the Rayleigh distribution. Because the Rice distribution
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

MARCH 2021
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