## 3.2 - A simple model for the radar signature of dunes and yardangs

After presenting a qualitative description of radar scattering profiles across linear dunes and mega-yardangs, we tried to reproduce the variations of the observed radar backscattered power using simple surface scattering models, whose parameters were estimated from various sources. We considered two surface scattering models, widely used by the radar remote sensing community: the Geometric Optics Model – GOM – suited to rough surfaces (surface roughness being defined with respect to the radar wavelength, most natural surfaces are rough at X-band) (Fung and Eom, 1981), and the Integral Equation Model – IEM – valid for medium-rough to smooth surfaces (Fung et al., 1992). The latter model was previously used to reproduce the radar scattering of linear dunes of the Great Sand Sea in Egypt, using SIR-C/X-SAR scenes (Paillou et al., 2014).

Besides the radar wavelength and polarisation, input parameters for these models are local incidence angle $\theta$ (computed from the radar look angle and local slope at the observed point), surface roughness described by rms-height $\sigma$ and correlation length $L$ (assuming a Gaussian autocorrelation function), and the dielectric constant $\epsilon$ of the material constituting the surface. The dielectric constant of dry sediments and sedimentary rocks in desert regions does not vary much and is the 3–6 range for its real part, with an imaginary part close to zero (Ulaby et al., 1990). We fixed the dielectric constant to that of silicate ($\epsilon$ = 3.5) for all surfaces. Note that small changes in this parameters are not important when computing radar backscattered power, as compared to the effects of variations in the incidence angle and surface roughness. The topography of the various selected areas was obtained using SRTM and GDEM data, GDEM being mainly used to fill in holes in the SRTM coverage, especially in dune areas. Taking into account the look angle and orbit inclination of the TerraSAR-X radar, we computed slope maps in the radar range direction, from which we derived the local incidence angle $\theta$ at each pixel. The resolution of digital elevation models provided by SRTM (90 m) is coarser than the resolution of the TerraSAR-X images, so it was not possible to reproduce scattering effects due to small-scale features, in particular natural corner reflectors. As regards surface roughness parameters, only the ones for the surface of Egyptian dunes were estimated by previous studies (Paillou et al., 2014). We then considered a Bayesian inversion approach to estimate the surface roughness of dunes and interdunes for both Egyptian and Namibian cases, and of crests and valleys for both Iranian and Chadian yardangs. We computed the probability $P$ of similarity between the actual radar backscattered power in TerraSAR-X images $\sigma_{TX}^0$ and the computed radar backscatter power $\sigma_{th}^0$, for roughness parameters $\sigma$ varying between 0.02 and 0.5 cm and $L$ varying between 0.2 and 3.5 cm, assuming a tolerance criteria $\tau$= 0.03:

$P(\sigma,L) = \frac{1}{\tau} \exp{\left(-\left(\frac{\sigma_{TX} ^0 - \sigma_{th}^0\left(\sigma,L\right)}{\tau^2}\right)^2\right)}$

The GOM model was used when $\sigma \gt$ 0.3 cm and the IEM model was used when $\sigma \lt$ 0.3 cm. After exploring the roughness parameter space, we kept the ($\sigma$, $L$) couple which corresponds to a maximum probability of similarity between the actual TerraSAR-X and the computed backscattered power. Fig. 5 shows an example of this Bayesian approach for dunes and interdunes of the Great Sand Sea in Egypt.

Figure 5: Bayesian inversion for the surface roughness of an Egyptian dune. The probability $P$ increases from blue to red, and the black dot is the location of the maximum probability. Top: side of the dune facing the radar illumination (maximum at $\sigma$= 0.07 cm and $L$ = 1.39 cm). Middle: side of the dune opposite to the radar illumination (maximum at $\sigma$= 0.09 cm and $L$ = 1.95 cm). Bottom: rough interdune (maximum at $\sigma$= 0.48 cm and $L$ = 0.63 cm). The discontinuity in probability at $\sigma$= 0.3 cm is due to the transition from GOM to IEM model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Once incidence angle and surface roughness parameters were estimated, we computed theoretical radar scattering profiles using GOM and IEM models, and compared them to observations for both linear dunes and mega-yardangs. Fig. 6 shows examples of comparison between computed and observed radar profiles across linear dunes in Egypt and Namibia, and mega-yardangs in Iran and Chad. One can see that the computed radar profiles and the actual TerraSAR-X ones are quite similar on the average. The high-frequency variations in TerraSAR-X profiles could not be reproduced, due to the coarse resolution of the SRTM topography. Nevertheless, simple surface scattering models such as GOM and IEM allow us to explain fairly well the variations of the actual radar scattering, as previously shown in Paillou et al. (2014) for dunes only. These results confirm the qualitative interpretation of the radar signature of linear dunes and mega-yardangs presented in the previous sub-section.

Figure 6: Comparison between radar scattering profiles computed using our surface scattering model (bold black line) and TerraSAR-X actual data (grey line). From top to bottom: Egyptian dune, Namibian dune, Iranian yardang, and Chadian yardang, corresponding to the sites shown in Fig. 3 and Fig. 4. Vertical axis is the backscattered power (in dB) and x-axis is a normalised distance across the studied structure. The dash-line represents the TerraSAR-X noise level (around −28 dB).