# 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
The amplitude-sensitivity kernels of the \(Z\)-component and H-component with periods of 20 s obtained using the adjoint-state method are shown in Figure 2e,f. Thus, the \(Z\)/H-ratio-sensitivity kernel at a period of 20 s (Figure 2g) could be obtained by subtracting the H-component amplitude-sensitivity kernel from the \(Z\)-component amplitude-sensitivity kernel. Similarity, the \(Z\)/H-ratio kernel at periods of 10 s (Figure 2h), 30 s (Figure 2j), and 40 s (Figure 2k) are also displayed and will be discussed later in this section. Figure 1: Procedure for obtaining the phase-sensitivity kernels for the surface waves with a period of 20 s, for a source (star) and station (triangle) pair. Seismograms were filtered using a Gaussian filter (central frequency: 0.5 Hz) recorded by the station for the (**a**) \(Z\)-component, (**b**) H-component, (**c**) adjoint source of the \(Z\)-component, and (**d**) adjoint source of the H-component. The shaded area represents the time window over which the measurement of the phase delay was made. (**e**) The phase-sensitivity kernels of the \(Z\)-component, and (**f**) the phase-sensitivity kernels of the H-component. The \(Z\)-component phase-sensitivity kernels at other periods of (**g**) 10 s, (**h**) 20 s, (**i**) 30 s, and (**j**) 40 s were also calculated to study the features of the phase kernel in the depth and lateral directions at different periods. The experiment described above shows that both the 2D-phase and Z/H-ratio kernels provide more information than the 1D sensitivity kernels. The 1D kernels only had sensitivities in the vertical direction because they did not consider the source direction. In contrast, the 2D kernels varied with distance from the station and with depth, which indicated that they had strong resolving powers in both the depth and lateral directions. Both kernels had different features in the vertical and lateral directions. The phase kernel was concentrated along the propagation path, with a much broader sensitivity range than the Z/H-ratio kernels, while the Z/H ratio sensitivity was only concentrated near the station. The reason for this is that the amplitude-sensitivity kernels of the Z-component and the H-component had the same pattern on the source side and a common propagation path, and the use of the component-differential calculation could remove the observational errors in the source and propagation path. This feature was consistent with previous  
Figure 2: Procedure for obtaining the amplitude- and Z/H-ratio-sensitivity kernels for surface waves with a period of 20 s, for a source (star) and station (triangle) pair. Seismograms were filtered using a Gaussian filter (central frequency: 0.5 Hz) recorded by the station for the (**a**) Z-component, (**b**) H-component, (**c**) adjoint source of the Z-component, and (**d**) adjoint source of the H-component. The shaded area represents the time window over which the measurement of the amplitude reduction was made. (**e**) The amplitude-sensitivity kernels of the Z-component, and (**f**) the amplitude-sensitivity kernels of the H-component. (**g**) The Z/H-ratio-sensitivity kernels obtained by subtracting the H-component amplitude-sensitivity kernel from that of the Z-component. The Z-component Z/H-ratio kernels at other periods of (**h**) 10 s, (**i**) 20 s, (**j**) 30 s, and (**k**) 40 s were also calculated to study the features of the Z/H kernel in the depth and lateral directions at different periods. Note that only the kernels near the station (the region outlined by the dashed line in (**g**)) were plotted. studies of the Z/H ratio, especially for 1D cases; that is, the Z/H ratio was only sensitive to the structures beneath the station. To compare the sensitivity to the S-wave velocity in depth for these two data sets, we reduced the spatial dimension of the phase and the Z/H-ratio kernels shown in Figures 1g-j and 2g-j from 2D to 1D by laterally summing the values on the horizontal plane in each vertical grid and plotting them as a function of depth.