point matching of the field are reported. The accuracy of the discrete image approximation is
investigated, and the speed is compared with the interpolation method in NEC.
2. THE DISCRETE IMAGE APPROXIMATION FOR A HALF-SPACE

The discrete image approximation is obtained by numerical
processing of the Sommerfeld integrals for the potentials due to
a half-space, after first extracting a quasistatic term to improve
convergence. The method is outlined here to obtain the field
components needed in NEC. The solution for the field of a source
in the presence of a half-space can be found in many references,
for example [1, 2]. The geometry of the problem is shown in
figure 1, using the convention that the lower medium is medium
1 and the upper is medium 2. The wavenumbers are ki = ko(ei-
joi/weo)1/2 and k2 = ko(e2 - ja2/weO)1/2 with ko = wjf'5 oe
and eiwt time variation assumed. The form of the solution used
here is from [2] in terms of the potentials U22 and V22 which,
together with the free-space Green's functions for the source and
its image, yield all of the components of electric field in the upper

z
source i
E2, Q2, IA

iz'
V

y

E1, 01, /10
Fig. 1. Geometry for source
and evaluation points above a
half-space
medium due to vertical and

horizontal electric dipoles in the upper medium. The potentials involve infinite integrals over the
radial component of wavenumber kp which can be written in terms of either Bessel or Hankel
functions in kp as

[,          e - jk (z+ +Z)
V2.= 2     Rv(kz2)           Jo(kp)k dkp
o            kz2
o         e-jkz(z+z')
]  Rv(kz2)           H2 (kpp)kp dk,
l' e-jks(z+z')
U22 = 2     Ru(kz2)           Jo(kpp)kp dkp
0              (kz2)
Ru(kz2) H2 (kpp)kpdkp
_00            kz2

(la)
(ib)
(lc)
(ld)

where

Rv(kx2) =   k+2    ,
kykz1 +kk2

Ru(k2)=  kx +  k2

and kzl = (kl - k2)1/2 and kx2 = (k2 - kp)1/2 with Im(kzi, kx2) < 0. The basic contours for
evaluating these integrals on the real axis are shown in figure 2 as Co for the Bessel function form
and C1 for the Hankel function form.
As R1 = [p2 + (z + z')2]1/2 becomes small the integrals in (1) converge more slowly, leading
to a Ri1 singularity in the integrals. The quasistatic term containing this singularity can be
extracted by subtracting the constant limits that Rv and Ru approach as kp or kx2 become large.

2

?z