# 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
### D Full-Wave Sensitivity Kernels of the Z/H Ratio Equation  
For the amplitude reduction measurements, the natural logarithm of the amplitude reduction (\(\delta ln\,\,A\)) can be expressed as follows:  
\[\delta\ln A=-\frac{\int_{t_{1}}^{t_{2}}\overline{s}(t)\delta u(t)dt}{\int_{t_ {1}}^{t_{2}}|\overline{s}(t)|^{2}dt}=\int\Big{[}K_{a}^{q}(\mathbf{r})\delta ln \alpha+K_{\beta}^{q}(\mathbf{r})\delta ln\beta+K_{\rho}^{q}(\mathbf{r})\delta ln\rho \Big{]}d^{2}x, \tag{4}\]  
Here, A is defined as the ratio between (1) the amplitude of the cross-correlogram of the observed and synthetic waveforms at lag time \(\delta t\) and (2) the maximum amplitude of the auto-correlogram of the synthetic waveform. \(K_{a}^{q},\,K_{\beta}^{q}\), and \(K_{\rho}^{q}\) are the correspondingamplitude-reduction Frechet kernels for the density, P-wave velocity, and S-wave velocity, respectively. The Z/H ratio (\(\eta\)) of a Rayleigh wave is always defined as the Z-component's amplitude (\(A_{Z}\)) over the H-component's amplitude (\(A_{H}\)) at a certain frequency \(\omega\):  
\[\eta(\omega)=\frac{A_{Z}(\omega)}{A_{H}(\omega)}\, \tag{5}\]  
The perturbation of the Z/H ratio can be expressed as the difference between the Z-component and H-component of the amplitude perturbation:  
\[\delta\ln\eta=\delta\left(\ln A^{Z}/A^{H}\right)=\delta\ln A^{Z}-\delta\ln A^ {H}\, \tag{6}\]  
If we consider the fact that the variation in the measurements is only caused by the variation in the S-wave velocity (\(\beta\)), then  
\[\delta\ln\eta=\int K_{\beta}^{\eta}\delta\ln\beta d^{2}x;\ \delta\ln A_{Z}=\int K_{ \beta}^{Z}\delta\ln\beta d^{2}x;\ \delta\ln A_{H}=\int K_{\beta}^{H}\delta\ln\beta d^{2}x, \tag{7}\]  
where \(K_{\beta}^{\eta}\), \(K_{\beta}^{Z}\), and \(K_{\beta}^{H}\) are the sensitivity kernels to the S-wave velocity of the Z/H ratio, the Z-component's amplitude, and the H-component's amplitude, respectively. \[\int K_{\beta}^{\eta}\delta\ln\beta d^{2}x=\int K_{\beta}^{Z}\delta\ln\beta d^ {2}x-\int K_{\beta}^{H}\delta\ln\beta d^{2}x\, \tag{8}\]  
This equation holds true for any volume V, so the Z/H-ratio kernel of the S-wave velocity can be expressed as:  
\[K_{\beta}^{\eta}=K_{\beta}^{Z}-K_{\beta}^{H}, \tag{9}\]  
Similarly, we can obtain the following equations for P-wave velocity (\(\alpha\)) and density (\(\rho\)):  
\[K_{\alpha}^{\eta}=K_{\alpha}^{Z}-K_{\alpha}^{H},K_{\rho}^{\eta}=K_{\rho}^{Z}-K _{\rho}^{H}, \tag{10}\]  
Thus, to generate the kernels above, we only need to calculate the sensitivity kernels of the Z-component's amplitude and the H-component's amplitude, which can also be solved by using the adjoint-state method. The adjoint source of the amplitude is:  
\[f_{q}^{\dagger}(t,\omega)=\frac{w(t,\omega)s(t,\omega)}{\int w(t,\omega)s^{2 }(t,\omega)dt}\, \tag{11}\]  
where \(w(t,\omega)\) is the time window over which the measurement of the amplitude is made, s(\(t,\omega\)) is the waveform recorded by a station, and \(\omega\) is the central frequency of the Gaussian filter. The definition of Z/H ratio in this study is noteworthy. Generally, the previous studies [7; 27] commonly used the maximum amplitude of the envelope to calculate Z/H ratio as follows:  
\[\eta(\omega)=\frac{A_{z}(\omega)}{A_{H}(\omega)}=\frac{\max(E[s_{z}(\omega)] )}{\max(E[s_{H}(\omega)])}\, \tag{12}\]  
where \(s_{z}(\omega)\) and \(s_{H}(\omega)\) are the Z-component and H-component of the seismogram processed using a Gaussian filter at a certain frequency \(\omega\), respectively, and \(E\) is their envelope. In this study, we used the following equation defined in [42]:  
\[\eta(\omega)=\frac{A_{z}(\omega)}{A_{H}(\omega)}=\sqrt{\frac{\int_{t_{1}}^{t_{2 }}s_{Z}^{2}dt}{\int_{t_{1}}^{t_{2}}s_{H}^{2}dt}}, \tag{13}\]  
where \(t_{1}\) and \(t_{2}\) are the start and end of the measurement window, respectively. With respect to physical properties, both definitions in Equations (12) and (13) are robust and acceptable, and could result in very similar Z/H ratios for synthetic tests.