## 2 Methodology of the Forward Modelling
### General Equations of Inversion Problem  
A fundamental problem in inversion theory is the minimisation of various functionals. When the forward operator \(\mathbf{G}\) is nonlinear, it can only be solved using iterative methods. The second norm is the following misfit function, which is commonly used in geophysical inversions:  
\[\phi(\mathbf{m})=\|\mathbf{G}(\mathbf{m})-\mathbf{d}\|^{2}. \tag{29}\]  
where \(\phi(\mathbf{m})\) and \(\mathbf{m}\) represent the fitted data error or error functional and model parameters, respectively, \(\mathbf{d}\) denotes the geophysical observational data, and \(\mathbf{G}\) denotes the nonlinear operator of the forward modelling. To obtain a solution to the inversion problem \(\mathbf{G}(\mathbf{m})=\mathbf{d}\) for minimising the functional \(\phi(\mathbf{m})\), it is necessary to perform a Taylor expansion with respect to the model parameters and to neglect the higher-order terms, as follows:  
\[\phi(\mathbf{m})\approx\phi(\mathbf{m}_{0})+\frac{\partial\phi(\mathbf{m}_{0})}{\partial \mathbf{m}}\Delta\mathbf{m}+\frac{1}{2}\frac{\partial^{2}\phi(\mathbf{m}_{0})}{\partial \mathbf{m}^{2}}\Delta\mathbf{m}^{2}. \tag{30}\]where \(\frac{\partial\phi(\mathbf{m})}{\partial\mathbf{m}}\) can be represented as  
\[\frac{\partial\phi}{\partial\mathbf{m}}=\left(\frac{\partial G}{\partial\mathbf{m}} \right)^{T}(G(\mathbf{m}_{0})-\mathbf{d}). \tag{31}\]  
where \(\mathbf{m}_{0}\) represents the initial model. \(\frac{\partial G}{\partial\mathbf{m}}\) is also referred to as the Frechet derivative of \(\mathbf{G}\). The Frechet derivative is an important concept of differentiation in function spaces, which describes the rate of change in the function \(\mathbf{G}\) at a specific point \(\mathbf{m}\). For the calculation of the Frechet matrix in the new propagation matrix approach, please refer to Appendix A. The second-order partial derivative of the objective function with respect to parameter m can be expressed as  
\[\frac{\partial^{2}\phi}{\partial\mathbf{m}^{2}}=\left(\frac{\partial^{2}\mathbf{G}}{ \partial\mathbf{m}^{2}}\right)^{T}(\mathbf{G}(\mathbf{m}_{0})-\mathbf{d})+\left(\frac{ \partial\mathbf{G}}{\partial\mathbf{m}}\right)^{T}\left(\frac{\partial\mathbf{G}}{ \partial\mathbf{m}}\right). \tag{32}\]  
Because the second-order partial derivatives of the forward operator with respect to the model parameters are extremely complex to compute, they are typically omitted from practical calculations. \[\frac{\partial^{2}\phi}{\partial\mathbf{m}^{2}}\approx\left(\frac{\partial\mathbf{G}} {\partial\mathbf{m}}\right)^{T}\left(\frac{\partial\mathbf{G}}{\partial\mathbf{m}}\right) =\mathbf{H}(\mathbf{m}). \tag{33}\]  
The updated quantity for the quasi-Newton method is computed as follows:  
\[\Delta\mathbf{m}=-\mathbf{H}(\mathbf{m})^{-1}\frac{\partial\phi}{\partial\mathbf{m}}. \tag{34}\]  
The iterative formula for updating the model can be expressed as  
\[\mathbf{m}_{k+1}=\mathbf{m}_{k}-\alpha\Delta\mathbf{m}, \tag{35}\]  
where \(\alpha\) represents the updating step size. Seismic AVA inversion is often a pathological problem due to the influence of noise and other factors. Regularisation techniques are commonly used to mitigate the issue of multiple solutions in the inversion and to improve the stability of the inversion results. Tikhonov regularisation is a relatively common regularisation method [31]. The objective function of a regularised least-squares problem can be expressed as  
\[\phi(\mathbf{m})=\|\mathbf{G}(\mathbf{m})-\mathbf{d}\|^{2}+\alpha\|\mathbf{L}\mathbf{m}\|, \tag{36}\]  
where \(\mathbf{L}\mathbf{m}\) is a finite difference approximation. \[\mathbf{L}= \begin{bmatrix}1&-2&1&&&&\\ &1&-2&1&&\\ &&\ddots&\ddots&\ddots&&\\ &&&1&-2&1&\\ &&&&1&-2&1\end{bmatrix} . \tag{37}\]  
The Frechet matrix of Equation (36) can be expressed as  
\[\mathbf{K}(\mathbf{m})= \begin{bmatrix}\mathbf{J}(\mathbf{m})\\ \alpha\mathbf{L}\end{bmatrix} , \tag{38}\]where \(\mathbf{J(m)}\) denotes the Frechet matrix for the \(\mathbf{G(m)}\).