**4. Results**

We use two numerical experiments on (1) the Overthrust model and (2) the Marmousi model to validate the proposed method. An O(2, 8) time-space-domain finite-difference staggered-grid solution [36] of the modified AEC equation is used to generate both forward- and back-propagated 4C wavefields. A convolution perfectly matched layer absorbing boundary [37,38] is used around the calculation area without consideration of the sea surface. In these experiments, a Ricker wavelet is adopted to generate pure P-wave sources with a peak frequency of 8 Hz (the bandwidth in [2 Hz, 20 Hz]).

#### *4.1. Overthrust Model Test*

We first use the Overthrust model to demonstrate the effectiveness of the proposed PTGN algorithm for OBC 4C data. The true *V p* and *Vs* models are shown in Figure 4. The density model is set to be 1000 kg/m<sup>3</sup> in the water layer and 2000 kg/m<sup>3</sup> below the seabed. The model is sampled as 801 × 166 grids with intervals of 12.5 m in both horizontal and vertical directions. The initial parameters are generated from the true ones with a smoothing window of 250 m. The acquisition geometry includes 101 shots with an interval of 100 m below the sea surface and 801 OBS receivers at the seabed for each shot. The total recording time is 4.0 s, and the temporal sampling rate is 1.0 ms. Observed seismic data are shown in Figure 5, including horizontal and vertical displacements and pressure components. As a comparison, the inversion is also performed using a preconditioned conjugate gradient (PCG). The maximum number of the loop for parameter update is 21, and that of the inner loop in the PTGN algorithm is 10.

**Figure 4.** True and initial parameters of the Overthrust model: (**<sup>a</sup>**,**<sup>c</sup>**) *V p* and (**b**,**d**) *Vs*.

**Figure 5.** Observed multicomponent seismic data: (**a**) horizontal and (**b**) vertical displacement and (**c**) pressure components.

Figure 6 displays the multiparameter gradients of the 1*th* iteration. We observe that the PCG gradients have insufficient illuminations for the deep part of the model, while the PTGN gradients are much improved and behave as amplitude-preserving subsurface images. The final *V p* and *Vs* models are displayed in Figure 7. The inverted *V p* and *Vs* using PTGN give better descriptions of structural boundaries with a higher interface continuity and fewer vertical artifacts. The vertical profiles (Figure 8) and the root mean square (RMS) errors (Table 1) can further demonstrate that PTGN can provide more accurate multiparameter inversions. Multicomponent data residuals between observed and simulated are shown in Figure 9. The residuals of the PTGN method are much weaken than those of PCG, which demonstrates that the PTGN can better interpret the observed multicomponent data. The convergence curves (Figure 10) shows that PTGN has a higher decreasing rate and eventually converges to a lower misfit value.

**Figure 6.** The multiparameter gradients of the 1*th* iteration: preconditioned conjugate gradients (PCGs) of (**a**) *V p* and (**b**) *Vs*, and preconditioned truncated Gauss–Newton (PTGN) gradients of (**c**) *V p* and (**d**) *Vs*.

**Figure 7.** Multiparameter inversion results: (**a**) *V p* and (**b**) *Vs* using PCG, and (**c**) *V p* and (**d**) *Vs* using PTGN.

**Table 1.** Root mean square (RMS) errors of inversion results using PCG and PTGN.


**Figure 8.** Vertical profiles of the inverted *V p* (solid) and *Vs* (dashed) using the PTGN and PCG methods at horizontal distances of 2 km, 5 km, and 8 km.

**Figure 9.** Multicomponent data residuals obtained by inverted models in Figure 7. The scale is consistent with the shot gathers in Figure 5 seismic data. (**<sup>a</sup>**–**<sup>c</sup>**) The residuals of PCG and (**d**–**f**) the residuals of PTGN.

**Figure 10.** Convergence profiles of the misfit function using PTGN (solid) and PCG (dashed).

#### *4.2. Marmousi Model Test*

Next, we use the elastic Marmousi model to demonstrate the effectiveness of the AEC-EFWI method. Smoothed versions of true *V p*, *Vs*, and *ρ* (see Figure 1) with a window of 300 m are taken as the initial models (see Figure 11). The dimension of the model is 601 × 201, and the intervals are 15 m in the horizontal and vertical directions. We have 61 shots with an interval of 150 m and 601 OBS receivers for each shot. The total recording time is 4.8 s, and the temporal sampling rate is 1.2 ms. A Ricker wavelet with a peak frequency of 8 Hz is adopted to generate a pure P-wave source. Observed multicomponent seismic data are simulated using the modified AEC equation, as shown in Figure 12. As a comparison, an EFWI method for horizontal and vertical displacement components are performed. A maximum of 15 iterations is used for the PTGN loop for both AEC-EFWI and EFWI.

**Figure 11.** Initial models of *V p* (**a**) and *Vs* (**b**).

**Figure 12.** Observed multicomponent seismic data: (**a**) horizontal and (**b**) vertical displacement and (**c**) pressure components.

Figure 13 displays the inverted *V p* and *Vs* models using the two methods. In the *V p* results (Figure 13a,c), AEC-EFWI provides better-resolved structures, i.e., anticlines, faults, lithologic interfaces, and high-speed bodies. For *Vs* (Figure 13b,d), however, the results using the two methods are comparable. With incorporation of the RMS errors in Table 2, we can find that considering the pressure data may not take as grea<sup>t</sup> effects on *Vs* as *V p*. It may be because the pressure data are more sensitive to the *V p* perturbations.

**Figure 13.** Multiparameter inversion results: (**a**) *V p* and (**b**) *Vs* using modified acoustic-elastic coupled-elastic full-waveform inversion (AEC-EFWI), and (**c**) *V p* and (**d**) *Vs* using elastic full-waveform inversion (EFWI).

The vertical profiles extracted at the horizontal distances of 3.0, 4.5, and 6.0 km are displayed in Figure 14. The AEC-EFWI results (marked by red lines) can precisely illustrate the deep reflectors with narrower sidelobes, and they are very close to the true models (marked by black lines). In contrast, as displayed in the corresponding wavenumber spectrum (Figure 15), EFWI underestimates the perturbations, especially for the interfaces with sharp parameter contrasts (see green lines in Figure 14). The data residuals simulated by the inverted results (Figure 16) and the convergence profiles of the misfit function (Figure 17) prove that the AEC-EFWI can better match the observed data and have a higher convergence rate.

**Table 2.** RMS errors of the inversion results using AEC-EFWI and elastic full-waveform inversion (EFWI) methods for the Marmousi model test.


**Figure 14.** Vertical profiles of *V p* (solid) and *Vs* (dashed) at the horizontal distances of 3.0, 4.5, and 6 km.

**Figure 15.** The 1D wavenumber spectrum of vertical profiles, corresponding to Figure 14. Black lines denote the true model, red lines indicate the AEC-EFWI results, and green lines are the EFWI results.

**Figure 16.** Multicomponent data residuals obtained by inverted models in Figure 13. The scale is consistent with the shot gathers in Figure 12. Panels (**<sup>a</sup>**–**<sup>c</sup>**) denote the residuals of EFWI, and panels (**d**–**f**) indicate those of AEC-EFWI.

**Figure 17.** Convergence profiles of the AEC-EFWI (solid) and EFWI (dashed) methods.
