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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 42, NO. 8, AUGUST 2004

Fig. 13.  Phase difference

at the sand­paleosoil interface obtained from the FDTD simulations for

(dashed lines) and

(plain lines) for

various incidence angles and rms heights.

The one-sided spectral density

of the surface rough-

We computed the scattered electric field for the two copo-

, can be written as

ness which is the Fourier transform of

larized modes HH and VV at each point of our geometry as a

function of time. Fig. 12 shows the simulated radar signal for

(21)

both copolarizations. We can clearly identify the radar echoes

arising from both the dune's surface and the subsurface pale-

This roughness function was used to determine the permit-

osoil as indicated by the arrows. The first echo corresponds to

tivity distribution in the cells of our geometrical model (cf.

the air­dune interface response and the second peak is related

Fig. 11).

to the sand­paleosoil interface response.

Simulations were performed for the geoelectrical model

B. Numerical Simulations

of Fig. 11 at a frequency of 1.6 GHz corresponding to the

RAMSES L-band SAR, under a variety of incidence angles

In order to simulate the backscattered radar echo for the pre-

ranging from 20 to 50 with 10 steps, and for various values

viously described stratigraphy and to compute the phase dif-

of the rms height of the paleosoil roughness. The emitted wave-

ference between the HH and VV polarizations, we used the

form used is a modulated Gaussian with a central frequency of

FDTD algorithm, which provides an exact electromagnetic so-

1.6 GHz and a 500-MHz bandwidth, and a maximum amplitude

lution [35]. In the FDTD technique, the geoelectrical model is

of 150 V/m. Several receivers were placed along the surface of

built by means of elementary cubic cells (cf. Fig. 11) called Yee

the sand layer to measure the backscattered signal in both H

cells. Each cell is characterized by three electromagnetic param-

and V polarizations.

eters: conductivity, permittivity, and relative permeability (for

FDTD numerical simulations allowed us to observe a phase

magnetic materials). The FDTD algorithm solves the Maxwell's

difference between H and V polarizations for the contribution

equations and computes the backscattered electric field for each

backscattered by the sand­paleosoil interface. This means

cell in the geoelectrical model. Such an algorithm allows in par-

in (10) should be described as

and

values,

ticular to take into account multiple-scattering effects, which

corresponding to the simulated phase difference. In order to

was not the case in our previous IEM simulations. The FDTD

evaluate this phase difference at the sand­paleosoil interface,

method is iterative and time is divided into small discrete steps

, we first had to compute the correlation func-

that we note

by using the Yee algorithm [36], which converts Maxwell's dif-

, which is the inverse

tion of the HH and VV signals

ferential equations into finite-difference equations. After exci-

Fourier transform FFT

of the energy spectral density func-

tation by a radar pulse, the three-dimensional components of

(Wiener­Khintchine relation)

tion

the electric and magnetic fields are computed at each time step

for each geometry cell, allowing to follow the wave propagation

FFT

(22)

across the geoelectrical model. We set the elementary cell size

to 5 mm in order to respect the stability criterion (a typical value

with

given by

of ten cells per wavelength is required to ensure temporal accu-

(23)

racy and numerical stability).