Digital Signal Processing Reference
In-Depth Information
8.6.1.3
Alternative Formulations of KA
An alternative formulation for the vector (polarimetric) response of electromagnetic
scattering using the Kirchhoff tangent-plane approximation, here depending only on
the surface topography and its local Fresnel reflection coefficients is developed in
Treuhaft et al.
(
2011
, Eq. 16)
Z
A
i
.
r
0
/e
ikj
r
r
0
j
j
r
r
0
j
e
ikj
r
r
T
j
j
r
r
T
j
C
i
4
.
R p
ref
p
s
/.
k
i
.
r
0
/
k
s
.
r
0
//
n
.
r
0
/d
2
r
0
(8.43)
where
p
are the polarimetric base vectors,
E
0
and A
i
relate to the incident field as
E
i
.
r
0
/
D
E
0
e
ikj
r
0
r
T
j
=
j
r
0
r
T
jD
A
i
.
r
0
/
p
i
;
r
,
r
T
, and
r
0
are the receiver, transmitter
and surface integration point positions respectively;
R p
ref
is the vector field at the
surface due to local specular reflection by a facet (see
Treuhaft et al. 2011
, Eq. 14);
and subscripts
s
, and
i
refer to scattering (or received), and incidence respectively.
Note that the first term in the right-hand side of the equation corresponds to
the signal reaching the receiver directly from the transmitter (no scattering). This
term should be removed when interested in the reflected component solely. This
formulation can be useful to evaluate the polarimetric response of GNSS signals
through realistic realizations of the surface (Montecarlo-like simulations). Note
that in such numerical evaluation of the integrals, the surface sampling length
must be shorter than the electromagnetic wavelength. Similarly,
Clarizia et al.
(
2012
) presents a facet-like approach to the KA, consistent with full KA, but less
computationally expensive.
E
.
r
/
p
s
D
E
0
p
s
S
8.6.1.4
Validity Limits of Kirchhoff Approximations
The Kirchhoff Approximation (Eq.
8.25
) can be applied when both the correlation
length of the surface, l , and its average radius of curvature are greater than the
electromagnetic wavelength. For the radius of curvature to be large enough, the
surface vertical-scale roughness (standard deviation of the surface heights, )must
be small compared to l
2
=. Note that this latter condition limits the surface's
vertical-scale roughness with respect to its horizontal-scale one, l , but it still
permits large (provided that l is long enough).
Its Geometric Optics limit (Eqs.
8.30
-
8.33
,orEqs.
8.40
and
8.41
) has validity
when the standard deviation of the surface height is large compared to the
electromagnetic wavelength, that is, the incident radiation has small wavelength
compared to the surface structure (also called high-frequency limit).
Finally, the Physical Optics limit (Eq.
8.42
) is also called low-frequency regime,
and it requires both small vertical-scale roughness and small slopes statistics.
Table
8.3
shows the validity of each limit of the Kirchhoff Approximation.
Note that in all the formulations above, the frequency term of the electromagnetic
wave has been removed for simplicity. The factor e
i2f t
should be added back in
all of them, where f is the resulting carrier frequency after taking into account
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