Geoscience Reference
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where
Z
T
1
T
h
G
.t
t
m
/
i
D
G
.t
t
m
/dt
m
:
(4.68)
0
Assuming for the moment that the time interval T in Eq. (
4.68
) is much greater than
the duration of the lightning discharge and making allowance for
h
M
m
i D h
M
i
,
yields
Z
D
.n/
b
.t/
E
D
G
t
0
dt
0
:
n
h
M
i
T
(4.69)
1
It should be noted that the mean value (
4.69
) is independent of time. This is
due to the fact that the stationary random process has been assumed. Substituting
Eq. (
4.69
)for
ǝ
.n/
b
.t/
Ǜ
into Eq. (
4.65
), and taking into account that
X
nP .n/
Dh
n
iD
T;
(4.70)
nD0
gives the mean value of
h
b
.t/
i
Z
G
t
0
dt
0
:
h
b
.t/
iD
h
M
i
(4.71)
1
To treat a correlation matrix of the random process in Eq. (
4.44
), we consider
a steady stochastic process such as Poisson random process. This means that the
correlation matrix (
4.44
) depends only on the time difference
D
t
0
t, that is,
‰
jk
./
D
ǝ
B
j
.t/B
k
.t
C
/
Ǜ
ǝ
B
j
.t/
Ǜ
h
B
k
.t/
i
; (4.72)
where the mean values
ǝ
B
j
.t/
Ǜ
are given by Eq. (
4.46
) and j;k
D
x;y;
z
.
Substituting Eq. (
4.43
)for
B
.t/ into Eq. (
4.72
) leads to
X
N
X
N
ǝ
B
j
.t/B
k
.t
C
/
Ǜ
D
ǝ
b
j;
r
;t
b
k;
r
;t
C
Ǜ
;
(4.73)
D
1
D
1
where
r
and
r
denote the coordinates of the corresponding thunderstorms.
If
¤
, then
ǝ
b
j;
r
;t
b
k;
r
;t
C
Ǜ
D
ǝ
b
j;
r
;t
Ǜǝ
b
k;
r
;t
Ǜ
;
(4.74)
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