LASNE et al.: PHASE SIGNATURE FOR DETECTING WET SUBSURFACE STRUCTURES
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B. Two-Layer IEM Model
Because we consider smooth to medium rough surfaces, the
IEM model can be used. The two-layer scattering problem con-
siders a smooth homogeneous sand layer covering a rougher
layer (paleosoil) as represented in Fig. 8. We shall consider both
surface autocorrelation functions to be Gaussian. For the sand
cm and
cm. These values
layer, we take
were derived from laser profiler measurements [14]. For the wet
cm and
cm. These
paleosoil layer, we take
Fig. 9. Total backscattered power
is represented as the vectorial sum of
values were estimated from hand-dug pits showing the sandpa-
and paleosoil surface
contributions.
the dune's surface
leosoil interface.
The IEM model validity range is expressed by the following
and is transmitted to the paleosoil's surface with a transmission
conditions:
angle .
to the airborne
The total complex backscattered signal
(13)
SAR is the coherent sum of the contribution of the dune sur-
where is the wavenumber of the incident wave
.
and paleosoil surface
(we neglect volume scattering
face
Only the single-scattering term for each layer will be
terms). It can be written as
considered here, that is the cross-polarized returns are zero.
Considering the first-order radiative transfer solution, the total
(6)
can be written as [27]
backscattered power
with each term being described by its amplitude
and phase
(14)
. Equation (6) can be represented as the sum of two vectors
where pp HH or VV, and
is the surface scattering
as shown in Fig. 9. We can set a phase origin so that the phase
term from the upper sand layer given by
is zero. We can then write for
associated with the sand layer
HH and VV polarizations
(7)
(15)
, and
with
(8)
As a first approximation, we can consider that the phase of
HH and VV signals backscattered by the paleosoil is the same
(16)
(which is not true, as it will be shown in the following using
and
FDTD simulations):
(17)
(9)
The phase
is only a function of the distance traveled by
represents the Kirchhoff field coefficients, and
In (16),
the incident wave through the sand layer of thickness
corresponds to the complementary field coefficients [27].
Equation (17) is the Fourier transform of the th power of the
repre-
Gaussian surface correlation function. In (14),
sents the noncoherent scattering from the paleosoil layer atten-
(10)
uated by the sand layer. It can be approximated by
(18)
being the remainder of divided by .
with
The phase of the total backscattered signal
in Fig. 9 can
is given by (15) applied to the paleosoil
where
be expressed by
characteristics.
As the covering sand layer is very homogeneous, the volume
and the interaction
scattering term of the first layer
(11)
between volume inhomogeneities and the lower
term
medium (surfacevolume interaction) are neglected here.
It leads to a phase difference between HH and VV signals
Finally, we obtain the copolarized phase difference from (12)
(19)
(12)