Geoscience Reference
In-Depth Information
Appendix B: Mean Value and Correlation Function
of Random Process
In this appendix the process of lightning discharges is treated as a sequence of
independent random events, which obeys the Poisson random distribution. This
implies that the elementary probability, dP , of the lightning origin from the moment
t till t
C
dt is proportional to dt and does not depend on t, that is dP
D
dt, where
stands for the mean number of the lightning discharges per unit time. Hence the
probability, P .n/, of appearance of n lightning discharges during a time interval
.0;T/ is given by a Poisson distribution
n
exp .
h
n
i
/
nŠ
P .n/
D
h
n
i
;
(4.64)
where
h
n
iD
T stands for the mean number of the lightning discharges per time T .
We first consider a single thunderstorm as a source of lightning activity. For
simplicity, we shall omit the subscript for the number of this thunderstorm.
Let
b
.
r
;t/ be the net magnetic field at the point
r
originated from the lightning
discharges happened at random moments. Here we ignore the spatial distribution of
the lightning discharges in a thunderstorm area since the magnetic field is measured
far away form the recording station. For reasons of convenience, we shall therefore
omit the argument
r
of the function. The mean value of the random value
b
.t/ can
thus be written as
D
.n/
b
.t/
E
P .n/;
X
h
b
.t/
iD
(4.65)
nD0
where P .n/ is the Poisson distribution (
4.64
). The angular brackets on the right-
hand side of (
4.65
) denote a conditional mean of the function
b
.t/, which is the
mean value of
b
.t/ under the condition that there were n lightning flashes during
the interval .0;T/ (Rytov et al.
1978
):
Z
T
D
.
n
/
b
.t/
E
n
X
1
T
D
b
.t
t
m
/dt:
(4.66)
mD1
0
Suppose now that the random moment of the lightning discharge/impulse occur-
rence, t
m
, and the magnitude of current moment M
m
, are statistically independent
and their probability distributions are independent of the impulse number n.On
account of Eq. (
4.42
) we get
D
.
n
/
b
.t/
E
X
n
D
1
h
M
m
ih
G
.t
t
m
/
i
;
(4.67)
m
D
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