## 3 Methodology
### Rayleigh Wave Ellipticity Curve Extraction and Inversion  
The shape of the obtained HVSR curves depends strongly of the local conditions of the ambient vibrations recordings: energy ratio of Rayleigh/Love waves, body wave contribution, lateral variation of soil properties, source pattern, acquisition device, etc. Under these circumstances, the Rayleigh wave ellipticity can be detectable between the fundamental frequency peak and the first minimum at higher frequencies (e.g., [30]).  
In recent years two different techniques have been mainly applied to reduce the contribution of body and Love waves in the calculation of the HVSR curve and provide a better estimation of the Rayleigh wave ellipticity: the H/V time-frequency analysis (HVTFA) [31] and the RayDec method [20]. The HVTFA technique uses the Continuous Wavelet Transform [32] to identify the P-SV waves and calculate the spectral ratio only with them. The RayDec method, based on the random decrement technique [33], uses the vertical component as a trigger and stacks a large number of horizontal and vertical signals in order to emphasize the Rayleigh waves with respect to the body and Love waves.  
Figure 2: Geological map of Oued-Fodda. Red line is the limit of the city. Thick black lines are the surface traces of the El-Asnam Fault (modified from [26]).  
In this study, we have used the RayDec technique to extract the Rayleigh wave ellipticity curve from ambient vibration recordings for each zone of the study area.  
The experimental curves are well reproduced with the RayDec and the HVSR techniques. However, particularly the RayDecellipticity curves better reproduce the right flank of the fundamental peak. To perform a reliable RWE inversion, we followed the strategy of [PERSON] et al. [21], for ellipticity curves with or without singularities. For curves with singularity (one clear peak), the right flank of f0 is inverted. For curves with two singularities, the part from the right flank of f0 to the end of f1 is inverted. For ellipticity curves without singularities, the whole broad peak of the RWE curve is inverted if such peak exists. For a better constraint, a small part of the left flank of f0 is added to the inversion process.  
The inversion of the RWE curves has been done using the Dinver software from the Geopsy package (www.geopsy.org) [34], which is based on the neighborhood algorithm [15]. The parameters that have to be set up before the inversion are: the number of sediments layers overlaying the bedrock, the interval of the variation of their thicknesses (hmin, hmax), densities (omin, \(\rho\)max), body wave velocities ranges (Vp and Vs), and Poisson ratio \(\upsilon\), ranging from 0.2 to 0.5 for soils and rocks.  
The inversion has a strong trade-off between depth and velocity, impossible to resolve with RWE curves alone. Therefore, such an inversion works only in combination with some prior knowledge of superficial Vs profile [35]. In this way, it is recommended to constrain the initial soil model using previous information (waves velocity, densities, layer thicknesses, etc.) from other geotechnical and geophysical techniques (borehole, seismic refraction, etc.).  
The inversion process produces a huge number of possible Vs profiles. For each generated soil model, the theoretical RWE curve is calculated, and then compared to the experimental one by a misfit parameter, which indicates how close the calculated curves are to the experimental ones.With all these considerations, the inversion of the HVSR curves is considered as a good tool to map the variation of Vs with depth.