Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 124

Here the mean number of the lightning discharges per unit time, , and the mean

magnitude of current moment, ǝ M Ǜ , are related to an individual thunderstorm with
number . Notice that Eq. ( 4.46 ) for the mean value of the magnetic field variations
has been derived for the stationary random process. Not surprisingly, this value is
independent of time.
In the measurements the magnetic variations observed on the ground have an
alternating-sign shape due to the presence of a variety of natural and man-made
noises. The mean value of such variations is normally close to zero and therefore is
of little importance. The correlation matrix at this point is a more suitable value for
adequate description of the magnetic variations. However, for the sake of generality,

in the analysis that follows we shall keep the mean values ǝ B j .t/ Ǜ and h B k .t 0 / i in
Eq. ( 4.44 ) despite these values are close to zero.
Far away from the lightning the perpendicular field b ' dominates over other
magnetic components. As the lightning far-field includes only the perpendicular
component, the spectral density given by Eqs. ( 4.90 ) and ( 4.91 ) is simplified to

D M 2 E sin 2 ' LJ LJ g ' r ;! LJ LJ

X

N

2

xx .!/ D 2 (4.47)

D 1

and

D M 2 E cos 2 ' LJ LJ g ' r ;! LJ LJ

N

X

2 ;

yy .!/ D 2 (4.48)

D1

where g ' r ;! is the Fourier transform of the propagation factor G ' r ;t , that is

Z

g ' r ;! D G ' r ;t exp . i!t/d!:

(4.49)

1

More usually we measure the so-called power spectrum of magnetic/electric
variations, which is proportional to j B .!/ j

2 . This spectrum determines
the magnetic/electric energy distribution over frequency. In a theory the power
spectrum can be correlated with the matrix of spectral densities of the stochastic
process, which are defined through Fourier integral of the correlation matrix ( 4.44 ).
More exactly, the power spectrum can be expressed through the sum D xx C yy
or D xx C yy C zz depending on whether the total magnetic field is
measured or only its horizontal components. As is seen from these equations, the
sum D xx C yy is independent of azimuthal angles ' . This means that this
value, as well as the power spectrum, depends only on the parameters which
in turn are the functions of distances from the ground-based station to the sites of
thunderstorms.

2 or j E .!/ j