Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

The case m = 1 can be treated in a quite analogue way. If the
d
anomalous slab extens till the surface, i.e. z = 0, the pertinent
1
kernel is easily derived from (3.36a):
(Because of (3.30a) there has been a change of sign. 1
If there are more normal layers (in addition to the air half-space),
the problem is treated as for m = 2, with the air half-space as
+
last (L-th) layer, we have to calculate 6L and hence B; separately.
We can't use C.
The kernels K (m) are nicely peaked functions. The halfwidth is
approximately 2[zm-zo[, i.e. ttrice the vertical grid width. For an
insulator the tails are comparatively long (-l/y2), for a conductor,
an exponential decrease is inferred from (3.36). In general. two
points to the left and the right of the central point will give a
,
satisfactory approximation:
where
h /2
CO
y: 3h /2
'm)=2 i K(~) (u,zo)du, p1 (m)= JY ~(~)(u,z~)du, pp)= K(~)(U,U~
Po
0 3h /2
J2 Y
-
PWi - Pi# C pi = 1, hy = horizontal griqwidth.
(3.38) expresses in any application of the finite difference formu1
the anoma1.ous part of the electric field outside the anomalous slab
in terms of anomalous .field values at the boundary. At the vertical
boundaries the impedance boundary condition (3.24) is applied.
So far only the TE-case has been considered. The TI$-mode can be
handled similarly, taking only the different boundary condition
into account. The approprate formulas can be worked out as an
exercise.
I