## 4 Synthetic Data Example  
In this section, a synthetic example is performed to investigate the capability of 2D pseudo-viscoelastic FWI in the time domain to image shallow complex structures using IFOS2D. The true model used to simulate the observed data for three parameters (P- wave and S- wave velocity model and density model) is generated inspired by a real model located in Iran, which contains large synclinal shape of evaporite layers with very high velocity, faulted in the left side and the thickness of the high-velocity layers varies through the section. The seismic velocity of this evaporite layers is between 3840 and 5420 m/s, according to the percent of the containing salt compare to anhydrite, depth and thickness of the layer, which is in the range investigated in different studies [41; 42]. The surrounding carbonate and shale layers show velocities around 2800-3420 m/s. The main problem in the seismic data with the abovementioned problem is to image target layers below the high-velocity layer, which is supposed to be resolved by FWI method. Therefore, in our study, in the first step, we tried to build a synthetic model with same geometry and shape of the high-velocity layer. In the next step, we tried to select velocities for each layer according to the real velocity of the media. In this step, since the provided forward and reverse codes for FWI in this study are mainly used for near-surface data, rather than deep seismic; so to prevent instability in analysis, we scaled down all the velocities of the layers in the model with a constant value. Therefore, we modeled the high-velocity layer near the surface with velocity close to 600 m/s. It should not be considered as the real high-velocity layers in deep earth, but a downscaled version of that.  
Due to high velocity, propagation of the surface and body waves through the complex near-surface layers would be more complicated. This example can test the capability of the FWI to image a complex velocity structure. The model space has a size of 400 grid points in the horizontal direction and 160 grid points in the vertical direction. Therefore, the actual dimension would be 50 \(\times\) 20 m considering a grid spacing of 0.125 m. A total of 19 shots and a total of 73 receivers located at the constant depth of 0.2 m that record both horizontal and vertical components are used. A cubed sine wavelet with a center frequency of 31.25 Hz generated by a hammer source is used as the source signal. The CPML frame is marked by the black dashed line. A viscoelastic medium is considered in this example and approximated a constant quality factor of \(Q_{s}=Q_{p}=20\) in the analyzed frequency band up to 60 Hz (a high-cut frequency filter of 10, 20, 30, 40, 50, and 60 Hz is used in stages) with three relaxation mechanisms of a generalized standard linear solid. A minimum of five iterations are taken into account at each stage. A 1D linear gradient is used to build the background of the true model and the background is considered as the initial model for each parameter in inversion. All models are updated simultaneously during inversion. It is worth mentioning that the true and initial velocity models are built with a \(\frac{v_{p}}{v_{s}}\) ratio of 1.5 and a total propagation time of 0.6 s is considered. Initially, we tried the \(V_{p}/V_{s}\) ratio of 1.5 because it is the minimum ratio which can be used as a reasonable value for sediments or soft rocks near the surface. In the following, we have selected the \(V_{p}/V_{s}\) ratio of 2.5 which is more realistic for our example. The PCG is carried out to solve the inverse problem.