Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

Eq. (4.7) lnvolves integration over the coordinates of the receiver,
(4.10) requires integration over the coordinates of the source.
The kernel G and the inhomogeneous term Ell of the integral equation
(4.7) or (4.10) depend only on the normal. conductivity structure.
To determine the kernel 9 replace first the conductivity within
the anomalous domain by its normal val-ues. Then place at each point
of.the domain successively two n~utually perpendicular horizontal
and one verti-cal dipole and calculate the resulting vector fields
at each point of this domain. At a first glance the work involved
appears to be prohibitive, but it is sharply reduced by the reci-
prociky (4.9) and the isotropy of the normal conductor in horizon-
tal di.rection. Because of (4.9) , from the elements of Green's
tensor
G G G
xx xy xz
the three elements Gxz , G xy' Gyz need not to be calculated when
G G G is computed. From the remaining six elements G has
zy' yxr zy Y Y
the same structure as G xx' only rotated through 90'. The same re-
lation holds between G and GZx. Hence, there are only the four
z Y
independent elements G G G (say). The particular
XX' yx' GZxr zz
symmetry of the Gxxr Gyx, Gzz-conponent in connection with the re-
ciprocity (4.9) then shows that these components need to be eva1ua.-
ted only for points of observation above source points. Consider
for example a vertical dipole at xo=yo=O, zo. Then (4.9) yields
Because of the isotropy of the conductor in horizontal di.recti.on
(4.11 ) is alternatively written
GZZ (O,O,zOI~,y,~) = GZZ (O,O,Z~-X~-Y~Z~).
Now, GZZ has circular symmetry around the z-axis. Hence,