# Two-Dimensional Full-Waveform Joint Inversion of Surface Waves Using Phases and Z/H Ratios
## 4 Inversion Experiments
### Large-Scale Inversion Using Ambient Noise/Earthquake Surface Waves  
Seismic linear arrays with dense station spacing have been deployed around the globe, which makes it possible to apply joint inversion to ambient-noise data or earthquake surface waves recorded by linear seismic arrays. We set up a target model (Figure 6a) with alternating 6% box-shaped high- and low-Vs anomalies in the background model (also used as the initial model in this experiment; see material properties in Table 1), and no Vp or density perturbation was introduced into the target model. The size of the model and the number of uniform mesh nodes were the same as the parameter settings described in Section 2.3. All of the boxes with anomalies had lengths of 40 km, while the widths changed from 8 km in the upper row to 15 km in the middle row to 30 km in the lower row. These different sizes were designed to assess the spatial resolutions of the varying lengths for a Rayleigh wave with different periods. The source-time function was Gaussian, and the dominant frequency of the source was  
Figure 5: Flow chart of the 2D full-waveform joint inversion of the Rayleigh wave phase and the Z/H ratio. 0.2 Hz. Fifty stations were spaced 10 km apart and were located from 150 to 650 km, which could also be considered as a virtual source generating an empirical Green's function of the surface wave. We performed all of the forward and adjoint simulations using SPECFEM2D. Then, we filtered the synthetic and observed waveforms at four central period bands (10, 20, 30, and 40 s) using a Gaussian filter, which could cover the period band of 5 to 50 s quite well. Following the workflow of the joint inversion described in Figure 5, we measured the frequency-dependent travel-time and amplitude misfit between the synthetic and observed waveform and obtained the corresponding adjoint sources of the phase and amplitude, respectively. After that, we calculated the phase-sensitivity kernel and the amplitude kernel of the Z-component and H-component by injecting the adjoint sources at the stations based on the  
Figure 6: (**a**) Target model with alternating 6% box-shaped high- and low-Vs anomalies in the background model. The vs. sensitivity kernels for the first iteration of the (**b**) phase and (**c**) Z/H ratio. adjoint-state method. In addition, the Z/H-ratio kernels were calculated by subtracting the Z-component and H-component amplitude kernels. Finally, all of the event kernels were summed, smoothed, and preconditioned to obtain the final misfit gradient for updating the model. A 2D Gaussian function was used to smooth the summed kernel as a regularization procedure ([PERSON] et al., 2010). The horizontal and vertical widths of the Gaussian function used to smooth the gradient were chosen as 30 km and 10 km, respectively, in the first 10 iterations. Then, they were reduced to smaller values of 10 km and 5 km, reflecting the incorporation of the shorter-period surface-wave measurements. Next, the smoothed kernel was preconditioned using a preconditioner, which was numerically calculated through the interaction between the forward and adjoint acceleration fields; namely, the pseudo-Hessian ([PERSON], 2012). \[P(x)=H_{pp}^{2}(x,\ x)=\sum_{s=1}^{N_{s}}\int\partial_{t}^{2}s(x,t)\cdot \partial_{t}^{2}s^{\dagger}(x,T-t)dt\, \tag{17}\]  
where \(s\) and \(s^{\dagger}\) are the forward and adjoint displacements, respectively, and \(N_{s}\) is the number of sources. Finally, we used the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method ([PERSON], 2012) (see detail shown in Appendix A) to update the model, and a line-search method was used to determine the optimal step length for updating the model.