Geoscience Reference
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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 /
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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