with shape parameter
α
and scale parameter
β
, that is, Z
g
x ∼ Gx;
α
,
β
Zhang et al., 2010. Its probability density function can be expressed as follows
pZ
g
x;α,β =
Z
g
x
α −1
Γαβ
α
exp −
Z
g
x
β ⎛
⎝ ⎜⎜
⎞ ⎠
⎟⎟
10 where Γ
α
is the Gamma function. The moment of the bivariate Gamma distribution can be
obtained by differentiating Eq. 10. The mean and variance can be obtained as
[ ]
[ ]
2 2
1 2
2 1
αβ θ
θ αβ
θ θ
= ∂
∂ =
= ∂
∂ =
∂ ∂
=
= =
=
θ θ
θ
θ x
θ θ
x
L Z
Var L
L Z
E
g g
11
In isotropic stochastic processes, the covariance Cov Z
g
x,
Z
g
x+h of the bivariate Gamma function can be expressed as
an exponential form
2
, 3
exp
g g
g g
Cov Z h Z
h C h
r αβ
⎡ ⎤
+ ⎣
⎦ ⎛
⎞ =
= −
⎜ ⎟
⎜ ⎟
⎝ ⎠
x x
12 In this paper, an exponential family of covariance function is
chosen for the bivariate Gamma model. The second-order variogram of Z
g
x, denoted
γ
2,g
h, is thus an exponential variogram defined by Garrigues et al., 2007
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
− −
=
g g
r h
h 3
exp 1
2 ,
2
αβ γ
13 where r
g
is the R of the variogram, it will be used to indicate the global texture structure of water in the real sea ice SAR images.
αβ
2
is the sill of the variogram. Let X and Y be Gamma random variables with shape
parameter
α
1
and
α
2
, scale parameter
β
1
and
β
2
, respectively. A random variable Z = X − Y has the following distribution
function Zhang, 1983.
F
Z
z =
β
1 α
1
e
β
1
z
z
α
1
− j
α
1
− j 1
β
1 j
− 1
j −1
⎡ ⎣
⎢ ⎢
j=1 α
1
∑
β
2 i
−1
j − 2 + i
β
1
+ β
2 j
−2+i
i −1
i =1
α
2
∑
⎤ ⎦
⎥ ⎥
⎥ , z 0
1 −
β
1 α
1
α
1
−1 β
2 i
−1
α
1
− 2 + i β
1
+ β
2 α
1
−1+i
i −1
, z = 0
i =1
α
2
∑
1 −
β
1 α
1
e
β
2
z
α
1
−1 −z
α
2
−i
α
2
− i
i =1
α
2
∑
β
2 α
2
−1−i+ j
α
1
− 2 + j β
1
+ β
2 α
1
−1+ j
j −1
, z 0
j=1 i
∑
⎧
⎨ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪
⎩ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ 14
For a stationary Gamma function Z
g
x, the difference Z
g
x −
Z
g
x + h is a random variable with mean and variance equal to
zero and 2
γ
g
h. After the calculation of the distribution function of Z
g
x − Z
g
x + h with Eq. 14, the expectation of |Z
g
x −
Z
g
x + h| can be figured out. Furthermore, from Eq. 7
γ
1,g
h can be obtained Sheldon, 2004. According to different values
of
α
,
γ
1,g
h have different expressions. Taking
α
= 2 corresponding to 2-look SAR imagery as an example,
γ
1,g
h can be expressed as follows
2 8
3
, 2
, 1
h h
g g
γ γ
=
15 2.2.2 Mosaic model Z
m
x: A mosaic model on a domain D
can be defined by partitioning the domain D into a tessellation and assigning each cell of the tessellation a value independently
drawn from a distribution. Poisson mosaic model is built based on the Poisson line tessellation, in which the image domain is
partitioned by Poisson lines see Fig. 2 b. Each line can be characterized by two parameters: its distance to the origin,
denoted d with d 0, and a random orientation θ with [0, 2π] Kotz et al., 2000 see Fig. 2 a.
The combination of Poisson random lines and independent Gamma random variables within each cell defines our mosaic
random function Z
m
x. Lantuéjoul 2002 shows that the
