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z
z
1
2
N
*
y
*
2
ʸ μ
1
O
x
z
˕ μ
W
E
b ˕
b
ʸ
S
Fig. 4.3 Coordinate systems used in this study. 1 —ground-based recording station.
2 —thunderstorm. A base coordinate system is fixed to the center O of the Earth in such a
way that the recording station is situated on the polar axis z . The same picture from a viewpoint
looking down on the Earth from above is shown from the right of the main image. Presented here
are the station, the local Cartesian coordinate system, and horizontal components, b and b
system of coordinates with polar axes z 0 , which coincides with the lightning
current, as shown in Fig. 4.3 . It is usually the case that the far-field/wave zone
includes only the perpendicular component, b ' , whereas the near-field consists of
all the components due to nonuniformity and asymmetry of the actual ionosphere
(Surkov et al. 2005 , 2006 ). More usually we measure the field components in
the Cartesian system of coordinates fixed to the ground recording station. In this
case the components of the magnetic field can be expressed through b r , b , and b '
as follows:
0
1
A D
0
@
1
0
1
A :
b x
b y
b z
0 cos ' sin '
0 sin ' cos '
10 0
b r
b
b '
@
A
@
(4.41)
In the text we shall use the abbreviation O A for the transformation matrix in
Eq. ( 4.41 ).
Far from the site of lightning discharge occurrence, the electromagnetic field
generated by the lightning discharge can be expressed via only lightning parameter,
that is, via the current moment of the stroke m.t/. As before we approximate the
actual current moments with the function m.t/ D MF .t/, where the magnitude, M
of the current moment is assumed to be a random value whereas the function F .t/
describing the current moment profile is a deterministic function of time (Surkov
et al. 2005 , 2006 ). For example, the current moment of individual and multiple
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