Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 136

since the lightning discharges processes associated with different thunderstorms are
supposed to be statistically independent of each other.
Consider now the case of D . This implies that both magnetic variations,

b j; r ;t and b k; r ;t C are taken at the same thunderstorm. For the reason
of convenience, we can therefore omit the inferior index and argument r .
Since we assume that the lightning discharges form a random Poisson process, the
corresponding terms on the right-hand side of Eq. ( 4.73 ) can be written analogous
to Eq. ( 4.65 ), that is

D . n / b j .t/b k .t C / E P .n/;

X

ǝ b j .t/b k .t C / Ǜ D

(4.75)

nD0

where the angular brackets denote a conditional mean of the function

X

n

X

n

b j .t/b k .t C / D M m 1 M m 2 G j .t t m 1 /G k .t C t m 2 /; (4.76)

m 1 D 1 m 2 D 1

that is, the mean value of this function under the condition that there were n flashes
during the interval .0;T/. Here we made use of Eq. ( 4.42 ).
As the components of magnetic field are represented in the spherical coordinate
system in which the source is located on the polar axis z 0 shown in Fig. 4.3 from
the right of the main image, the components of the magnetic field in the local
Cartesian coordinate system fixed at a ground-based station can be found through
the transformation matrix O A given by Eq. ( 4.41 ). For example, if we choose first to
study the component ‰ xx ./ the function G x is given by

G x D G ' sin ' G cos ':

(4.77)

For the sake of generality, we shall consider all the components b r , b , and b '
though only the component b ' given by Eq. ( 4.36 ) is nonzero in the simple model
of the waveguide treated above. As we shall see subsequently the TE mode, which
contains the components b r and b , can be excited as well due to the generation of
Hall currents in the E-layer of the ionosphere.
Taking into account the definition of a conditional mean ( 4.66 ) and assuming
that the moment T is much greater than the duration of the lightning discharge, we
obtain

1

T 2 h k 1 h k 2 ; if m 1 ¤ m 2
1

h G k 1 .t t m 1 /G k 2 .t C t m 2 / iD

T k 1 k 2 ./; if m 1 D m 2 ;

(4.78)

where the subscripts k 1 and k 2 denote or '. Here the cross-correlation function
k 1 k 2 ./ takes the following form