Geoscience Reference
In-Depth Information
return strokes can be approximated by Eqs. (
3.7
) and (
3.10
), respectively. As we
have noted above, the
C
CG lightning discharges which transfer the positive charge
to ground differ from the
CG lightning discharges in amplitude and duration
of current, especially their long-lasting continuing current. In the case of high
amplitude, the statistical distributions of the negative and positive charge moments
are quite different (Williams et al.
2007
). In this notation, we introduce two types of
deterministic functions F .t/, that is, F
n
.t/ and F
p
.t/ and two independent random
values M
n
and M
p
for the negative and positive lightning, respectively.
Considering either of these current moments, the magnetic field of the single
lightning discharge can be written as
b
r
;t
t
n
D
M
n
G
r
;t
t
n
;
(4.42)
where the propagation factors
G
is supposed to be equal to zero as t<t
n
. These
functions are derivable from Maxwell equations for the Earth-Ionosphere resonance
cavity that should be supplemented by proper boundary conditions at the ground
and the ionosphere. In the simple model of the Earth-Ionosphere cavity considered
in Sects.
4.1
and
4.2
, the components of the function
G
can be extracted from
Eq. (
4.36
).
The net magnetic perturbation at the ground-recording station is also a ran-
dom quantity,
B
, which equals the sum of the magnetic perturbations caused
by individual lightning discharges. In the Cartesian reference frame fixed to the
ground-recording station, the net magnetic field of all the lightning due to global
thunderstorm activity is then
X
N
b
r
;t
;
b
r
;t
D
X
n
b
r
;t
t
n
;
B
.t/
D
(4.43)
D1
where
b
r
;t
is the magnetic field due to a thunderstorm with number .
It is common practice to describe the stochastic processes via mean
value/mathematical expectation and correlation functions of the stochastic process.
In the analysis that follows, we first introduce the basic characteristic of a
random process and then extend them to estimate the so-called power spectrum
of electromagnetic variations. The detailed calculations are found in Appendix B.
4.3.2
Correlation Matrix of Random Field Variations
The comprehensive analysis of random vector fields is mainly based on correlation
methods. We next compute the correlation matrix of the magnetic variations
defined as
‰
jk
t;t
0
D
ǝ
B
j
.t/B
k
t
0
Ǜ
ǝ
B
j
.t/
Ǜǝ
B
k
t
0
Ǜ
;
(4.44)
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