# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## Appendix A  
When the objective function is quadratic, the quasi-Newton method can be adopted to calculate the search direction. Using the changes in model parameters and gradients between adjacent iteration steps, the search direction can be obtained to achieve convergence. Assume that the objective function \(Q(\mathbf{m})\) has second-order continuous partial derivatives. Expand \(Q(\mathbf{m})\) in a Taylor series at \(\mathbf{m}_{k}\), and we obtain:  
\[Q(\mathbf{m}_{k}+\Delta\mathbf{m})=Q(\mathbf{m}_{k})+\Delta\mathbf{m}^{T}g_{k }+\frac{1}{2}\Delta\mathbf{m}^{T}H_{k}\Delta\mathbf{m}+o\Big{(}\|\Delta \mathbf{m}\|^{2}\Big{)} \tag{1}\]  
In this context, \(\Delta\mathbf{m}\) is the model update amount, and \(H_{k}\) is the second-derivative Hessian matrix (\(\
abla^{2}Q(\mathbf{m}_{k})\)). By neglecting the high-order term \(o(\mid\mid\Delta\mathbf{m}\mid\mid^{2})\), we can obtain:  
\[Q(\mathbf{m}_{k}+\Delta\mathbf{m})=Q(\mathbf{m}_{k})+\Delta\mathbf{m}^{T}g_{k }+\frac{1}{2}\Delta\mathbf{m}^{T}H_{k}\Delta\mathbf{m} \tag{2}\]  
The quasi-Newton method attempts to approximate the inverse Hessian matrix in the equation. Currently, the L-BFGS algorithm is regarded as one of the most effective algorithms for constructing an approximate inverse Hessian matrix. This algorithm only needs to store the gradient and model parameter information obtained in several adjacent iterations to construct a matrix that approximates the inverse of the Hessian matrix. The gradient and model information obtained in the early iterations will no longer be used in the new iteration process and can be deleted to efficiently utilize the memory space. Thus, both the computational cost and memory consumption are relatively low. Let \(\mathbf{B}_{k}^{0}\) denote the initial inverse Hessian matrix, and \(\mathbf{B}_{k}^{0}=\left(\mathbf{y}_{k-1}\mathbf{s}_{k-1}^{T}\right)/\left( \mathbf{y}_{k-1}^{T}\mathbf{y}_{k-1}\right)\), where \(\mathbf{s}_{k}=\mathbf{m}_{k+1}-\mathbf{m}_{k}\), \(\mathbf{y}_{k}=\mathbf{g}_{k+1}-\mathbf{g}_{k}\), and \(M\) is the number of terms to be retained, where in this paper \(M=5\). Then, the general representation form of the approximate Hessian matrix is [29; 30; 31]  
\[ \begin{array}{l}B_{k}=\left(V_{k-1}^{T}\ldots V_{k-M}^{T}\right)B_{k}^{0}(V_ {k-M}\ldots V_{k-1})\\ \qquad+\rho_{k-M}\left(V_{k-1}^{T}\ldots V_{k-M+1}^{T}\right)s_{k-M}s_{k-M}^{T} (V_{k-M+1}\ldots V_{k-1})\\ \qquad+\rho_{k-M+1}\left(V_{k-1}^{T}\ldots V_{k-M+2}^{T}\right)s_{k-M+1}s_{k-M+ 1}^{T}(V_{k-M+2}\ldots V_{k-1})\\ \qquad+\ldots+\rho_{k-1}s_{k-1}s_{k-1}^{T}\end{array}  \tag{11}\]  
where \(\rho_{k}=1/\mathbf{y}_{k}^{T}\mathbf{s}_{k}\), and \(\mathbf{V}_{k}=\mathbf{I}-\rho_{k}\mathbf{y}_{k}\mathbf{s}_{k}^{T}\). The iteration of the L-BFGS algorithm to solve FWI is shown below (Algorithm A1). ``` Given: \(\text{m}_{0}\)// initial model Initialization:  Set \(k\gets 0\) \(M\)// history step of \(Q(\text{m}_{k})\), round number When the convergence condition is not reached: \(Q_{k}\gets Q(\text{m}_{k})\)// objective function \(\text{g}_{k}\leftarrow\
abla_{m}Q(\text{m}_{k})\)// gradient \(Q_{\text{ref}}\leftarrow\{Q



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