# 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
Once the optimal step length was determined from the minimum of the total function for all of the bands, the model parameter m was updated as follows:  
\[ln\ (m_{i+1}/m_{i})=\
u_{i}\ d_{i}\, \tag{18}\]  
where \(m_{i+1}\) and \(m_{i}\) are the models of the (i + 1)th iteration and the \(i\)th iteration, respectively; and \(\
u_{i}\) and di are the optimal step length and the search direction (i.e., summed event kernels) of the \(i\)th iteration, respectively. To demonstrate the advantage of our joint-inversion framework, we also conducted two additional separate inversions using either only the Rayleigh-wave phase data or only the Z/H-ratio data. The separate inversions also began with the background model and used the same inversion parameters as the joint inversion, including the smoothing width and step lengths. Both data sets were mostly sensitive to the shear-wave velocity. We only inverted for the vs. structures. In total, we conducted three inversions: (1) adjoint tomography of the Rayleigh-wave phase data (i.e., phase inversion); (2) adjoint tomography of the Rayleigh-wave Z/H-ratio data (i.e., Z/H-ratio inversion); and (3) the joint inversion using the two data sets (i.e., joint inversion). Figure 6b,c show the gradients of the vs. for the first iteration of the phase and Z/H-ratio inversions. The gradients show that the two Rayleigh wave measurements had different and complimentary sensitivities to the crustal structure. The phase inversion was more sensitive to the deeper structure than the Z/H-ratio inversion, while the Z/H-ratio inversion had a strong sensitivity to the shallow structure. The total misfit evolution of the phase and Z/H-ratio inversions and the joint inversion are presented in Figure 7a,b. We can see that the joint inversion had a slower convergence rate and slightly larger misfits than those of the separate inversions, which was reasonable, since the joint-inversion scheme tried to fit both data sets simultaneously. The joint inversion converged after 32 iterations when the misfit changed over the last iteration because both the phase and Z/H-ratio data were less than 3%. Figure 7c,d show the waveform fitting of the observed Z-component and X-component data and the two synthetic computed data sets based on the initial and final models for the source-receiver pairs located at (750 km, 0 km) and (150 km, 0 km), and significant improvements in the travel-time and amplitude fits after joint inversion can be observed. To compare the results of the three types of inversions, we show their final vs. models in Figure 8. The Z/H-ratio information for the surface was mostly sensitive to the vs. structures at shallow depths (\(<\)30 km), which was limited by the period range of the Rayleigh wave in this study. Thus, the recovered vs. model (Figure 8b) was better resolved in the shallow regions, and the upper and middle rows show the correct pattern with the anomaly close to target model. Compared with the Z/H-ratio inversion, the phase inversion was sensitive to much deeper vs. structures and could constrain the broader range of the vs. structures at depths of \(\sim\)20 km to 60 km (Figure 8a). The pattern of the anomaly in the middle and lower rows is almost recovered; however, the recovered pattern is rather smoothed and lacks constraints in the shallow structure, which was caused by the sensitivity characteristic of the phases shown in Figure 3a. Benefiting from the sensitivity to the vs. structures concentrated near the station, the Z/H-ratio inversion helped to better resolve the structures in the shallow areas that were not well resolved by the phase inversion. Thus, the addition of the Z/H ratio in the joint inversion helped to constrain the crustal structure at shallow depths. At greater depths, the joint inversion had a better resolution than the phase inversion with the correct amplitude recovery.