## 3 Numerical Examples  
### Inversion Effects in the Ideal Case  
The effects of CNEWI source wavelet estimation and elastic full-waveform inversion under ideal conditions are discussed in this section. The model we used in the numerical tests is a modified Marmousi2 model [43]. The original model was rescaled, and the P-wave velocity and S-wave velocity models we used are shown in Figure 2a and 2b, respectively. The S-wave velocity in the shallow part was set to non-zero values, because we mainly simulate a land seismic exploration case, which is more challenging for source wavelet inversion and FWI. One advantage of the proposed method is that it can overcome the effects of residual surface waves and near-offset bad traces on source wavelet inversion. In some cases, such as sand or soil, the velocity at the near surface can be treated as uniform. The primary focus of this paper revolves around examining the influence of source wavelet estimation on elastic full-waveform inversion. For this purpose, the initial velocity models here adopt the smooth velocity model, illustrated in Figure 3, which represents the achievable accuracy level in tomography. The observed seismic data were generated using a Ricker wavelet characterized by a dominant frequency of 20 Hz. Thirty-two sources were utilized with a source interval of 60 m. The data were sampled at intervals of 1 ms, spanning a recording time of 1.2 s. The observed data corresponding to a single shot are depicted in Figure 4, where Figure 4a,b represent the Z-component and X-component data, respectively. For inversion purposes, data within the 0.2 s timeframe were extracted. Both the L2 EWI and CNEWI methods were applied to invert the source wavelet, and the resulting outcomes are presented in Figure 5. It is observed that, under ideal conditions, the L2 EWI method accurately captures both the amplitude and phase information of the source wavelet, while the CNEWI method performs exceptionally well in extracting phase information. The inverted source wavelets were used to simulate direct waves, illustrated in Figure 6. Specifically, Figure 6a shows the Z-component direct wave record of the observed seismic data. Subsequently, Figure 6b,c displays the simulated direct wave data using the source wavelets inverted by the L2 EWI and CNEWI methods, respectively. Notably, both simulated direct wave datasets align closely with the observed direct wave data, indicating the effectiveness of the source wavelet obtained through both methods. The changes in the objective functions of the two inversion methods are shown in Figure 7. Figure 7a shows the change in the objective function curves of the L2 EWI method for inverting the 32-shot data, and Figure 7b shows the change in the objective function curves of the CNEWI method for inverting the 32-shot data. It can be seen that the CNEWI method reaches convergence in 10 iterations, while the L2 EWI method reaches convergence in about 30 iterations. In  
Figure 1: Cross-talk tests of the inversion algorithm. (**a**) and (**b**) are true P-wave and S-wave velocity models, respectively; (**c**) and (**d**) are initial P-wave and S-wave velocity models, respectively; (**e**) and (**f**) are inverted P-wave and S-wave velocity models, respectively. this case, the computational efficiency of the CNEWI method is three times that of the L2 EWI method. When conducting EFWI, the multiscale inversion strategy is not necessary because the initial velocity model is good enough to avoid the cycle-skipping problem. Therefore, full-band information was used for inversion. During inversion, the P-wave velocity and S-wave velocity were simultaneously updated. The optimization algorithm used is the conjugate gradient method. Two kinds of inverted source wavelets were used to perform elastic full-waveform inversion, and the results are shown in Figure 8. The total iteration time is 100 and the computational cost is 2 h 14 min.