IEEE Geoscience and Remote Sensing Magazine - March 2021 - 133

correlation in all directions, and the preceding formula
can be simplified as [36], [48]
1
R ^ r h = 2r

#unit circle R 1 ^hv $ ut hdut ,

(85)

where hv = vr2 - vr1 . Therefore, it is necessary to obtain
R 1 ^hv $ ut h based on the known R ^ r h by inverting (85),
which can be solved by the spectral representation
method. If it is assumed that the 2D random field Z is
isotropic, the radial spectral density function of the 2D
random field is [48]
f^~h =

~
s2

#0 ∞ R^ r hJ 0 ^~r hrdr,

(86)

where s 2 is the variance of the 2D random field and
J 0 ^ $ h is the zero-order Bessel function of the first kind.
The relationship between the radial spectral density
function f ^~ h of the 2D random field and the spectral
density function S 1 ^~ h of the 1D line process is provided in [48]:
s2
S 1 ^~h = 2 f^~h.
(87)
For any given correlation function R ^ r h , S 1 ^~ h can be
obtained by substituting the radial spectral density func-

Randoml
Positioned Scatterers

Turning Band Met od

Corrected aussian
istributed Field

MNLT Met od

tion f ^~ h of the random field into (87). The method for
generating 1D line processes is given in [49]:
Yi ^vrk $ ut ih = 2 cos ^ A i $ ^vrk $ ut ih + U ih
= 2 cos ^ A i ^ x k cos i i + y k sin i ih + U ih, (88)
where U i obeys a uniform distribution U i ~U 60, r@ and
A i is a random frequency following the probability density distribution f ^~ h . Finally, a 2D Gaussian random
field with a correlation function of R ^ r h can be obtained
using (83). In addition, [50] points out that anisotropic
2D random fields can be approximated by the product
of two mutually independent isotropic random fields.
3) Transform the correlated Gaussian random field into
a correlated gamma-distributed random field Z using the MNLT method. The MNLT method [2], [13],
[51] is employed to transform the correlated Gaussian
random field into a gamma-distributed random field.
It essentially equates variates of the Gaussian and
gamma distributions at equivalent cumulative values,
which brings
1
2r

o-1

2

#-∞ exp ;- y20 Edy 0 / #0 z o o Cz 0^oh exp 6- oz 0@dz 0 . (89)
y

The preceding equation defines the forward mapping
function y = F ^ z h for y " z and the inverse mapping
function z = F -1 ^ y h for y 0 ! z . Tough and Ward [52] gave
the relationship between the autocorrelation coefficient
t y ^ r h of the Gaussian random field and the autocorrelation coefficient t z ^ r h of the gamma distribution as
n
1 ∞ ty ^ r h
tz ^r h = r / n
2 n!
n=0

# c # exp ^- x 2 h H n ^ x h Q z c
∞

-∞

2
erfc ^ x h
m dx m ,
2

(90)

where H n ^ $ h is the Hermite polynomial defined by

Corrected amma
istributed Field
P(

G

LSP

k

k)

ξ i)

i(

ξi

RCS

ξ ik
→

rk
Imaging Function

Corrected
istributed Field

Ο

θi

ui

x

FIGURE 12. A correlated clutter simulation based on the coherent

scatterer model.
MARCH 2021

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

FIGURE 13. The TBM.

133
IEEE Geoscience and Remote Sensing Magazine - March 2021

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - March 2021
