**4. Shear-Wave (SH) RTM**

### *4.1. Implementation of SH-RTM*

The steps of shear-wave reverse-time migration, similar to those of traditional reversetime migration, are as follows:

1. Forward extrapolating of the source wavefield: starting from a given or estimated source wavefield, we solve the equation to forward propagate the source wavefield. Thus, for a source emitting at source positions *xs*, *zs*,

$$V\_y(\mathbf{x}\_s, z\_s, t) = S(\mathbf{x}\_s, z\_s, t) \tag{26}$$

2. Shear-wave reverse continuation: For receiver wavefield propagation, we reverse the R of the seismic receiver recorded in time and then set the initial receiver position as the initial boundary condition. We then use the selected finite difference scheme

to solve the shear-wave equation (Equation (5)) iteratively to obtain the receiver wavefield. As shown in the following equation, where *xr*, *zr* is the position of the source transmitter and receiver, and *T* is the total duration of forward propagation.

$$V\_y(\mathbf{x}\_{r\prime} z\_{r\prime} t) = R(\mathbf{x}\_{r\prime} z\_{r\prime} T - t). \tag{27}$$

3. Imaging conditions of shear-wave application: the last step is to use the crosscorrelation of source wavefield and receiver wavefield obtained in the previous two steps to obtain the image of the underground structure.

$$I(\mathbf{x}, z) = \int\_0^T \mathcal{S}(\mathbf{x}, z, t) \mathcal{R}(\mathbf{x}, z, t) dt. \tag{28}$$

### *4.2. Shear-Wave Reverse-Time Migration in Marmousi Models*

A set of two-dimensional Marmousi models are set up, as shown in Figures 8 and 9, where the S-wave velocity (Figure 9) is modified by the ratio of horizontal to vertical P-wave velocity (Figure 8). The model size is 121 × 401 grid points, the spatial grid size Δx = 10 m, the time step is 1 ms, the Ricker wavelet of 20 Hz is excited, and the sampling time is 4 s. With these initial conditions, the reverse-time migration imaging is then carried out.

**Figure 8.** Marmousi model: acoustic velocity model.

**Figure 9.** Marmousi model: shear-wave velocity model.

We use the acoustic-wave equation and P-wave source loading method for the reversetime migration of the P-wave velocity (Figure 10), and the shear-wave equation and shear-wave source loading method for the reverse-time migration of the S-wave velocity (Figure 11). The result of using shear-wave velocity for reverse-time migration (Figure 11)

has clearer structural definitions and higher resolution compared with the result of acoustic reverse-time migration using P-wave velocity in Figure 10. We then continue with the SH reverse-time migration imaging experiment, by increasing the spatial gird size, setting Δx = 15 m, get new S-wave velocity Marmousi model (Figure 12) and the sampling time to 4000 ms. The images for different differential methods are then compared. The following figures (Figures 13–15) show the reverse-time migration results of the Marmousi model in the shear-wave equation using different finite difference schemes.

**Figure 10.** The reverse-time migration result with Marmousi acoustic velocity model.

**Figure 11.** The reverse-time migration result with Marmousi shear-wave velocity model.

**Figure 12.** Marmousi model: shear-wave velocity model.

**Figure 13.** CFD is used for the reverse-time migration result with Marmousi shear-wave velocity model, Δx = 15 m.

**Figure 14.** CCD is used to the reverse-time migration result with Marmousi shear-wave velocity model, Δx = 15 m.

**Figure 15.** CSCD is used for the reverse-time migration result with Marmousi shear-wave velocity model, Δx = 15 m.

As shown in Figures 13 and 14, the CCD scheme yields better imaging than the CFD scheme, where the red boxes mark the area of improvement, due to higher accuracy in accounting for numerical dispersion caused by the spatial gird size. Compared with Figures 13 and 14, the CSCD scheme produces the best imaging results of the Marmousi model (Figure 15).
