Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 123

where j;k D x;y; z and the angular brackets denote the averaging over all
available/possible realizations of the random process. This matrix is defined in such
a way that the component ‰ jk may vanish in the case of statistically independent
functions B j and B k . Below we focus only on magnetic variations while the same
technique can be applied to electric components as well.
The field fluctuation is assumed to be a steady stochastic process. This means
that the correlation matrix depends on t and t 0 in such a way that it is a function of
only the time difference D t 0 t. In such a case the spectral density of this random
process must be delta correlated (Rytov et al. 1978 ; Weissman 1988 ), that is,
‰ jk !;! 0 D ı ! ! 0 jk .!/: (4.45)

The spectral matrix jk .!/ is of our prime interest in this section.
In what follows the lightning discharge occurrence 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 to 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. We refer
the reader to Appendix B for details about the Poisson probability law. In reality,
just after the end of the lightning discharge the probability for the next discharges
slightly decreases since it is necessary about 5 s to reconstruct the total charge of
thunderstorm cloud (Uman and Krider 1982 ; Uman 1987 ). We shall ignore this fact
assuming that the mean time 1 between two adjacent discharges is much greater
than the last value.
Lightning activity in each thunderstorm is considered as an independent random
process. Consider first the random process associated with the thunderstorm center
with number . The moments of lightning discharge appearance, t n , and the current
moment magnitudes, M n , are supposed to be statistically independent of each
other and their probability distributions are independent of the impulse number n .

Recall that the propagation factor G r ;t describing the shape of magnetic field
generated by individual lightning discharge is considered as deterministic functions.
In the simple model where the Earth-Ionosphere forms a spherically symmetric
resonance cavity, the functions G for the vertical CG lightning can be considered
as a universal function of the angular distance between the lightning and the

ground-based station, that is, G D G ;t . The influence of the solar wind and
radiation on the ionospheric plasma and the presence of the Earth's magnetic field
bring the actual ionosphere is nonuniform and asymmetric. This means that the
shape of the functions G can depend on the lightning/thunderstorm coordinates.

The mean value of ǝ b .t/ Ǜ under above requirements is obtained in Appendix B.
Assuming that the lightning discharge processes at each thunderstorm are indepen-
dent of each other and taking into account Eqs. ( 4.71 ) and ( 4.43 ), we obtain the
mean value of net magnetic field generated by all the thunderstorms

Z

N

X

ǝ M Ǜ G r ;t 0 dt 0 :

h B iD (4.46)