Evaluation of the discrete complex-image method for a NEC-like moment-method solution Page: 6 of 12
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are the free space Green's functions for the source and its image, G22 = exp(-jk2R2)/R2 and
G21 = exp(-jk2R1)/R1 with Ri = [p2 + (z Z/)2]1/2. The final terms, containing the remaining
Sommerfeld integrals, are
Sv _ -jwlpo 82 kv22 (6a)
Sz - (2 + k2) kv22 (6b)
- cos 4 +u22 + G21 -k2k G21 (6c)
S - = i 1 8v2+22 +G1- 2ky 2G21. (6d)
The final G21 terms are included in (6c) and (6d) to complete the field of the quasistatic image
in (5) as the field of a source in free space. After subtracting the quasistatic terms, the remaining
integrals u22 and v22 remain finite as R1 goes to zero, while the field components in equations (6)
have 1/R1 singularities from the derivatives of v22. The advantage of extracting the quasistatic
components for the discrete image approximation is that the functions Ru(kx2) and R(kz2)
in equation (3) decay as kz2 for large kx2, and these decaying functions are better suited to
approximation by a sum of exponentials than are RU and Rv which become constant. The
quasistatic terms are also subtracted in the interpolation method used in NEC, where the singular
remainders in equation (6) are multiplied by R1 so that they can easily be approximated with linear
or quadratic interpolation. Alternatively, the singularities in equations (6) can be approximated
 and also the next constant terms can be obtained. For example, in S,
82V22 k3 (k2 - ki) sin a 1
,9p2 ' (k2+k2)2 1+sina) R
jk2 (k, + k3k2 - kik2 - 2k1k2 - 214) jk4k (7
3(k1 + k2)(k2 + k2)2 2(k2 - k )(kl+ k2)5/2
for small R1, where
2(ki - k2)(k - kik2 + k2) kl + k\
C =tan-1 2k1- 4k1k2 + 5k2ki - 4k1ki 2k )
with Im(Ck) < 0 and sin a = (z + z')/Ri.
For the discrete image approximation Ru(kz2) and R(k-2) are approximated with a sum of
exponential functions in kx2 by means of the Prony [i4] or Matrix Pencil  methods. Since these
methods require equally spaced samples in a real variable, the substitution kz2 = k2[ta +t(tb - ta)]
is made. Applying the Matrix Pencil method with equally spaced samples over 0 < t < 1 yields
the exponential approximation over the range kx2 E (k2ta, k2tb) as
Rv(t) x( Aiek, Rv(k2) ~ 1 azebik2
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Burke, G.J. Evaluation of the discrete complex-image method for a NEC-like moment-method solution, report, January 5, 1996; California. (https://digital.library.unt.edu/ark:/67531/metadc671246/m1/6/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.