# Two-Dimensional Full-Waveform Joint Inversion of Surface Waves Using Phases and Z/H Ratios
## 2 Calculation of 2D Rayleigh Wave Phase and Z/H-Ratio Kernel
### Sensitivity Calculation and Analysis  
To illustrate the procedure for obtaining the travel-time and Z/H-ratio-sensitivity kernels, we used the 2D spectral-element method (SPECFEM2D) to calculate the 2D sensitivity kernels via the adjoint-state method. The numerical parameters of the modeling in the computation are shown in Table 1. The model measures 800 km in the horizontal direction, with 320 uniform mesh nodes (quadrangles in 2D), and 100 km in the vertical direction, with 40 uniform nodes. A total of (4 \(\times\) 320 + 1) \(\ast\) (4 \(\times\) 40 + 1) = 206,241 unique grid points were used for the 2D spectral-element discretization with a four-degree polynomial ([PERSON] et al., 2005). The average grid spacing of the mesh was 2.5 km, which was sufficient to accurately simulate the surface waves at periods of greater than \(\sim\)8 s ([PERSON] and [PERSON], 2002). For simplicity and to minimize the possible interference of other arrivals on the Rayleigh waves, the P-wave velocity, the S-wave velocity, and the density were all held constant (Vp = 6000 m/s, vs. = 3500 m/s, and \(\rho\) = 2800 g/m\({}^{3}\)). The time interval of the simulation was set to 0.02 s to satisfy the numerical stability, and the time step was set to 12,000. We placed a Gaussian-type source-time function with a dominant frequency of 0.2 Hz at location (650 km, 0 km) to generate the 2D synthetic waveform of the surface wave, which was computed using SPECFEM2D. A station was set at location (150 km, 0 km). Figure 1 illustrates how to obtain the single-component phase sensitivity of the kernels (\(\mathrm{K}_{\beta}^{\mathrm{PZ}},\mathrm{K}_{\beta}^{\mathrm{PH}}\)) to \(\delta\)Vs on a vertical profile and horizontal profile in 2D media for surface waves with a period of 20 s. First, we obtained the seismograms of the Z-component (Figure 1a) and H-component (Figure 1b) recorded by the station, which were processed using a Gaussian filter with a central frequency of 0.5 Hz. Then, we obtained the adjoint source of the Z-component (Figure 1c) and H-component (Figure 1d) at the station using Equation (3). We performed one adjoint simulation by injecting the back adjoint source at the station into the model, which created adjoint wavefield \(s^{\mathrm{I}}\). Finally, the phase-sensitivity kernels of the Z-component (Figure 1e) and H-component (Figure 1f) at periods of 20 s were obtained using the time integration of the interaction of \(s^{\mathrm{I}}\) and the forward wavefield \(s\). To study the features of the phase kernel in the depth and lateral directions at different periods, the Z-component kernels at three other periods of 10 s (Figure 1g), 30 s (Figure 1i), and 40 s (Figure 1j) also were calculated using the same procedure. \begin{table}
\begin{tabular}{c c} \hline
**Model Size (Nx \(\times\) Nz)** & **x-Direction: 800 km; z-Direction: 100 km** \\  & **(300 \(\times\) 40 Elements)** \\ \hline Material properties & Vp = 6000 m/s, vs. = 3500 m/s, \(\rho\) = 2800 g/m\({}^{3}\) \\ Propagation time & 12,000 time steps; Time discretization: 0.02 s \\ Type of source-time function & Gaussian \\ Dominant frequency of source & 2 Hz \\ Source location & (650 km, 0 km) \\ Station location & (150 km, 0 km) \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical parameters for kernel calculation in the computation. Figure 2 illustrates how to obtain the single-component amplitude kernels (\(\mathrm{K}_{\beta}^{\mathrm{PZ}}\), \(\mathrm{K}_{\beta}^{\mathrm{PH}}\)) to \(\delta\)Vs on a vertical profile and horizontal profile in 2D media, as well as their differential kernels (\(\mathrm{K}_{\beta}^{\mathrm{PZ}}-\mathrm{K}_{\beta}^{\mathrm{PH}}\)), which is called the \(Z\)/H-ratio kernel (\(\mathrm{K}_{\beta}^{\mathrm{n}}\) ) in this paper. The procedure for obtaining the amplitude kernels was very similar to that for the phase kernels, except that the adjoint source was generated using different measurements followed by Equation (11).