Evaluation of the discrete complex-image method for a NEC-like moment-method solution Page: 7 of 12
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with aj = Aie-Bit/Ab-t) and b= = Bi/[k2(tb - ta)]. When the exponential approximation of
R2(k42) is substituted into (2) the integral for each term can be evaluated using the Sommerfeld
identity (4) to get a sum of discrete images
22 '' at (8)
with Rj = [p2 + (z + z' + jb2)2J1/2. Thus v22, and similarly u22 are approximated by a sum of
free-space Green's functions for images in complex space.
In applying the discrete image method a contour similar to C2 in figure 2 is usually chosen
for the approximation . Hence, ta = 1 and tb is a negative imaginary value. This deformation
from Co yields a linear path in the kz2 plane and also avoids surface-wave poles that would occur
for a stratified medium. It also moves the path further from the Zenneck pole in Rv, making the
3. NUMERICAL RESULTS
The accuracy of the discrete image approximation was tested against the numerical evaluation
routines in SOMNEC  after reducing the error limits in the Romberg adaptive integration by
two orders of magnitude and increasing the accuracy of the Bessel and Hankel functions. With
these changes the relative error in the integration seems to be around 10-6 to 10-7.
The error from the discrete image approximation for v22 is shown in figure 3a for a lossy
ground with the number of images Nt varied. The matrix pencil method was used with N = 300
samples and tb = -j10. Nt = 12 was the maximum number of terms that could be obtained from
the matrix pencil method with the tolerance in the singular-value decomposition set to 10-10.
The approximation is seen to converge rapidly in the region of 0.1 < R/Ao < 1. The error with
varying tb is shown in figure 3b, where Nt was always the maximum returned by matrix pencil.
The increased error for small R1 is due to the truncation of the approximation contour at k2tb as
the integrand converges more slowly, so a larger tb reduces the error. The use of the Sommerfeld
identity for equation (8) implies an integration contour to infinity, but the error is uncontrolled
beyond k2tb and increases as the integrand decays. The increased error in figure 3b for large R1
and large tb is due to insufficient sampling, and would be reduced with a larger N at the cost
of increased time for determining the image parameters. Figure 3c shows the error with varying
tb for grazing incidence along the ground. The error due to truncation at tb occurs sooner as R1
is decreased than for points off the interface, since the integrand decays more slowly without the
e-ik2(z+z') term. In numerical integration the Hankel function form of equation (1) could be used
in this case, with the integration contour deformed downward to a steepest descent path. The
contour C2 used in the discrete image approximation is more nearly optimum for large (z + z')/p.
For large R1 the error increases rapidly, apparently from difficulty in approximating R~ near the
Zenneck pole near kz2 = k2. A small tb reduces the error for large R1 with increased error for
small R1. Figure 3d shows the same result as 3c but for dielectric ground. In this case the error
increases more rapidly for large R1, perhaps due to the difficulty in approximating the lateral
wave. Although the lateral wave, with wavenumber k1, must be synthesized from exponentials in
k2 the approximation is successful for a number of cycles determined by the number of discrete
image terms. For example, the lateral wave was approximated to about R1/A = 0.7 with 5
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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/m1/7/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.