governing equation is reduced at the outset. Associat-

ing the degrees of freedom with element centroids and

face elements, each being a flat triangular facet. To

locating observation points there cause all "self ele-

achieve the numerical formulation we assume further

ment" singularities to drop out of computations: in all

that the current **J **is approximately constant over each

self element integrations

such element, associating its value with the location of

the element's centroid, **r**i, i.e., **J**i = **J**(**r**i) for the *i*th ele-

(ni ⋅ **J **j ) = 0 =

(13)

ment. Locating **r **only at the centroid of each element

^

.

means that what follows α is always one-half. Alto-

gether we obtain

Once the currents are obtained from solution of the

above algebraic system (12) they can be substituted in

1 **J ** ∫∫ *dS*′ **n **⋅ **J **∇*g*(r , **r**′) **J**

(

)

^i j

the equivalent of (7), expressed numerically in a man-

(10)

2 i

i

∑ *S*j

ner similar to (12), to obtain scattered E and H fields at

any observation point. In line with the assumptions

(ri , **r **′) = **n**i **H ** i*inc*

outlined above, we calculate the (re)radiated fields,

^

ignoring the influence of the ground surface disconti-

nuity, i.e., as if the radiating currents were in an infinite

where Sj is the area of the *j*th element. Thus over each

soil medium. The computer program was tested against

detailed near-field solutions for cases where analytical

element we need only evaluate the integral

solutions are available (sphere), and against reasonable-

ness checks for signal loss as a function of distance

, p = *s*, *t*, *n*

within the soil.

(11)

where *s *and *t *are locally constructed tangential coordi-

nates, and *n *is the local outward normal coordinate.

INCIDENT FIELD AND POLARIZATION

We used four-point Gaussian quadrature to ensure

PARAMETERS

accuracy in this integration.

Locating **r**i at each element centroid in succession

The basic setup assumed for the antenna-target

provides N vector equations in the N unknown current

system is shown in Figure 14. The antenna is simulated

by a sheet of currents deemed to be a reasonable

face. Expressing the ultimate governing relations in

approximation of those on a metal surface driven

terms of tangential *s *and *t *components we obtain

as a dipole antenna. We adjusted this current distribu-

tion so that its subsurface radiation pattern resembles

[(

) ]

)

(

patterns considered representative of those from a di-

1 **J ** ∑ **n **⋅ **s ***Vs *+ **s **⋅ **s ***Vn *J

^ i ^ j ij ^i ^ j ij sj

2 si

pole antenna resting on a ground surface. To accom-

plish this, the antenna current distributions are expressed

[(

) ]

)

(

as

∑ **n**i ⋅ **t **j Vsij + **s**i ⋅ **t **j Vnij Jtj =

^^

^^

( z *z*a )

[(

)

π*w*z ,

^

(12)

= **t**i ⋅ **H**ii*nc *12 **J ** ti ∑ **n**i ⋅ **s **j Vtij +

^

^^

(14)

) ]

[(

, *y * *y*a ≤ *w*y , *z * *z*a ≤ *w*z

(

)

+ **t**i ⋅ **s **j Vnij Jsj ∑ **n**i ⋅ **t **j Vtij +

^^

^^

where wy = 0.8 m and wz = 0.4 m. In choosing unit

) ]

(

+

^^

.

value maximum current magnitude, we implicitly nor-

^

malize all simulations with respect to that dimension

of input. In their general features, i.e., 3-dB beamwidth,

This is the algebraic system to be solved. In effect, the

lobe patterns, and angular locations where the fields

method consists of applying point matching over a PEC,

decline rapidly towards minima, patterns from these

with pulse basis functions.

distributions fit our needs here.

This formulation was chosen in part for its simplic-

Our antenna model thus consists of a rigorous rep-

ity and hence programming convenience. Because tan-

resentation of a synthetic or idealized structure. The

gential electric field components drop out, the initial

resulting fields are strictly in accordance with Maxwell's

22