## 2 Methods
### Source Wavelet Inversion by Cross-Correlation Norm Elastic Waveform Inversion (CNEWI)  
In order to overcome the influences of source wavelet amplitude errors, which may be caused by surface wave interference, noise interference, data processing, etc., the following objective function can be constructed:  
\[\sigma_{4}(\mathbf{s})=\sum_{i}^{ns}\sum_{j}^{nr}\Bigl{[}-\hat{\mathbf{U}}_{i, j}^{x}(\mathbf{s})\cdot\hat{\mathbf{D}}_{i,j}^{x}-\hat{\mathbf{U}}_{i,j}^{z}( \mathbf{s})\cdot\hat{\mathbf{D}}_{i,j}^{z}\Bigr{]}, \tag{17}\]  
where \(\hat{\mathbf{U}}_{i,j}^{x}\) and \(\hat{\mathbf{U}}_{i,j}^{z}\) denote the normalized x- and z-component of the synthetic direct waves, respectively, i.e., \(\hat{\mathbf{U}}_{i,j}^{x}=\mathbf{U}_{i,j}^{x}/\left\|\mathbf{U}_{i,j}^{x}\right\|\), \(\hat{\mathbf{U}}_{i,j}^{z}=\mathbf{U}_{i,j}^{z}/\left\|\mathbf{U}_{i,j}^{z} \right\|\), and \(\hat{\mathbf{D}}_{i,j}^{x}\) and \(\hat{\mathbf{D}}_{i,j}^{z}\) denote the normalized x- and z-component of the observed direct waves, respectively, i.e., \(\hat{\mathbf{D}}_{i,j}^{x}=\mathbf{D}_{i,j}^{x}/\left\|\mathbf{D}_{i,j}^{x} \right\|\), \(\hat{\mathbf{D}}_{i,j}^{z}=\mathbf{D}_{i,j}^{z}/\left\|\mathbf{D}_{i,j}^{z} \right\|\). The derivative of the objective function \(\sigma_{4}\) with respect to the source wavelet function can be written as  
\[\frac{\partial\sigma_{4}}{\partial\mathbf{s}_{i}}=\sum_{j}^{nr}\left[\left( \frac{\partial\mathbf{U}_{i,j}^{x}}{\partial\mathbf{s}_{i}}\right)^{T}\frac{1 }{\left\|\mathbf{U}_{i,j}^{x}\right\|}\left(\hat{\mathbf{U}}_{i,j}^{x}\left( \hat{\mathbf{U}}_{i,j}^{x}\cdot\hat{\mathbf{D}}_{i,j}^{x}\right)-\hat{\mathbf{ D}}_{i,j}^{x}\right)+\left(\frac{\partial\mathbf{U}_{i,j}^{z}}{\partial \mathbf{s}_{i}}\right)^{T}\left(L^{-1}\right)^{T}\frac{1}{\left\|\mathbf{U}_{i,j}^{z}\right\|}\left(\hat{\mathbf{U}}_{i,j}^{z}\left(\hat{\mathbf{U}}_{i,j}^{ z}\cdot\hat{\mathbf{D}}_{i,j}^{z}\right)-\hat{\mathbf{D}}_{i,j}^{z}\right) \right]. \tag{18}\]  
Utilizing the adjoint-state method, Equation (18) can be computed as follows:  
\[\frac{\partial\sigma_{4}}{\partial\mathbf{s}_{i}}=\sum_{j}^{nr}\left[\left( \frac{\partial\mathbf{F}}{\partial\mathbf{s}_{i}}\right)^{T}\left(L^{-1} \right)^{T}\frac{1}{\left\|\mathbf{U}_{i,j}^{x}\right\|}\left(\hat{\mathbf{U} }_{i,j}^{x}\left(\hat{\mathbf{U}}_{i,j}^{x}\cdot\hat{\mathbf{D}}_{i,j}^{x} \right)-\hat{\mathbf{D}}_{i,j}^{x}\right)+\left(\frac{\partial\mathbf{F}}{ \partial\mathbf{s}_{i}}\right)^{T}\left(L^{-1}\right)^{T}\frac{1}{\left\| \mathbf{U}_{EFWI can be conducted. If the observed data are multi-component seismic data, all data components should be used when performing the inversion to reduce the multi-solution nature of the inversion and improve the inversion accuracy and resolution. However, if some data components are of poor quality, they should be removed to avoid introducing interference in the inversion. Theoretically, inversion based on the cross-correlation objective function mainly uses the phase information of seismic data to overcome the influence of amplitude inaccuracy. Therefore, even if the CNEWI method cannot obtain accurate source wavelet amplitude information, it will not have a significant impact on subsequent EFWI, because the cross-correlation objective function can also be used in the EFWI process; this has been verified in the subsequent numerical examples. There are some source-independent inversion strategies to overcome the effects of source wavelet errors on EFWI. In time-domain EFWI, the most common strategy is based on the convolved wavefields [26; 35]. However, the convolution process may amplify noise in some cases because it is a multiplication process in the frequency domain. Also, for real seismic data from a common source, the convolution signal model is only an approximation. For far-offset data, the approximation effect is relatively poor. The reference trace selection will also affect source-independent inversion.