# Two-Dimensional Full-Waveform Joint Inversion of Surface Waves Using Phases and Z/H Ratios
## 4 Inversion Experiments
### Small-Scale Inversion Using Active Sources  
Another suitable application of this method is retrieving model parameters and characterizing the near-surface region, mainly the shear-wave velocity model using active sources, for engineering applications, geotechnical studies, and physical modeling tests. In this section, we used more realistic synthetic experiments to demonstrate the promising applications of this full-waveform joint-inversion method to small-scale inversions, with an improved resolution. The 600 \(\times\) 100 m target model for elastic S-wave and P-wave velocities used for the numerical experiments in this section is shown in Figure 9. With this model, we used 100 and 20 uniform mesh nodes in the horizontal and vertical directions, respectively, resulting in a total of \((4*100+1)*(4*20+1)=32,481\) unique grid points. The target elastic model included heterogeneous Vp (Figure 9a) and vs. models (Figure 9b) with a homogeneous density of 1000 kg/m\({}^{3}\). Both the Vp and vs. increased with depth from a low-velocity layer, with the exception of one low-velocity zone (LVZ) and two high-velocity zones (HVZs) in the middle. We assumed an initial model with a homogeneous Vp of 1000 m/s, a homogeneous vs. of 500 m/s, and a homogeneous density of 1000 kg/m\({}^{3}\). We set up 50 shots located from 50 to 550 m at a depth 0.5 m below the model's surface, with an equal  
Figure 8: The recovered S-wave velocity models for the three types of inversions. (**a**) The final inverted model using only the Rayleigh wave phase. (**b**) The final inverted model using only the Rayleigh wave Z/H ratio. (**c**) The final inverted model using the joint inversion of the Rayleigh wave phase and Z/H ratio. 10 m spacing. The stations were spaced 5 m apart and were located 5 m to 595 m at a depth of 0.5 m.  
Synthetic data were computed for the target model and the initial model and were simulated using source-time functions with a dominant 10 Hz Ricker wavelet. Then, we conducted the joint inversion following the same inversion procedures described in Figure 5. To avoid the local minima and reduce the non-linearity of the waveform misfit function, we adopted a hierarchical strategy, in which we successively inverted from low to high frequency progressively through 20, 30, 40, 50, 60, and 80 Hz. We also conducted three types of inversions to further demonstrate the advantage of the joint inversion over the separate inversions. For each inversion, we simultaneously inverted both the S-wave and P-wave velocities. First, we used the objective function in Equation (13) to update the model via the phases. The result of using the Rayleigh wave phase is shown in the inverted S-wave (Figure 10a) and P-wave (Figure 11a) velocity models. A better result for the vs. final model was expected, since the surface waves were more sensitive to this elastic parameter. Neither the S- nor the P-wave velocity model converged to a model near the target shown in Figure 9. The most probable reason for this was that the inversion suffered from strong cycle-skipping problems, which easily occur in surface waveform FWI, with incorrect updates leading to a secondary minimum. Next, we replaced the phases with the Z/H ratio in the objective function. That is, we minimized the discrepancy defined by the Z/H ratio in Equation (14). Figures 10b and 11b show the resulting vs. and Vp models. Because the envelope information using the Z/H ratio calculation in this paper mitigated the cycle-skipping effects, working only with the Z/H ratio led to a reasonably successful recovery of the vs. model of the shallow structure, but it did not constrain the deeper structure. Moreover, the capacity of the Z/H ratio to constrain the shallow crust with a complex lateral heterogeneity helped us obtain a significantly better low-velocity layer where the surface wave phase could not resolve the structure.