## 4 Conclusions
Utilising this equation, we computed the partial derivatives of the reflectivity with respect to the model parameters, and \(\frac{\partial\mathbf{r}}{\partial m_{iz}}\) is discretised as  
\[\frac{\partial\mathbf{r}}{\partial m_{iz}}=\frac{1}{2}\int_{-\infty}^{\infty} \frac{\partial R}{\partial m_{iz}}e^{i\omega t}d\omega. \tag{11}\]  
The partial derivative \(\frac{\partial R}{\partial m}\) in the frequency-angle domain can be expressed as  
\[\frac{\partial\mathbf{R}}{\partial m}=\Big{[}\frac{\partial R(\theta_{1})}{ \partial m_{z}}\quad\cdots\quad\frac{\partial R(\theta_{k})}{\partial m_{iz}} \Big{]}^{T}. \tag{12}\]  
In this context, \(\theta_{1}\) to \(\theta_{k}\) represent \(k\) different angles of incidence used during the inversion process, assuming \(\theta_{ik}\) represents the \(ik\)-th angle of incidence, where \(\frac{\partial R(\theta_{ik})}{\partial m_{iz}}\) is an \(N_{f}\times 1\) dimensional matrix, with \(N_{f}\) representing the number of frequencies. \[\frac{\partial R(\theta_{ik})}{\partial m_{iz}}=\Big{[}\frac{\partial R( \theta_{ik},\omega_{1})}{\partial m_{iz}}\quad\cdots\quad\frac{\partial R \big{(}\theta_{ik},\omega_{N_{f}}\big{)}}{\partial m_{iz}}\Big{]}^{T}. \tag{13}\]  
The first-order partial derivatives of the reflection transmission coefficient matrix with respect to the model can be expressed as  
\[ \begin{array}{l}\frac{\partial\mathbf{R}}{\partial m_{iz}}=\Big{[}\mathbf{A}_{1}- \mathbf{F}_{1,1}\frac{nz-1}{i_{2}=1}\mathbf{B}_{iz}^{new}/\mathbf{F}_{nz,1}\mathbf{A}_{2,n} \Big{]}^{-1}\\ \begin{pmatrix}\mathbf{F}_{1,1}\mathbf{B}_{1}^{new}\mathbf{B}_{z}^{new}\quad\cdots\quad \left(\begin{array}{c}\frac{\partial\mathbf{B}_{iz-1}^{new}}{\partial m_{iz}} \mathbf{B}_{iz}^{new}\\ +\mathbf{B}_{iz-1}^{new}\frac{\partial\mathbf{B}_{iz}^{new}}{\partial m_{iz}}\end{array} \right)\quad\cdots\mathbf{B}_{nz}^{new}\quad/\mathbf{F}_{nz,1}\end{pmatrix}\mathbf{A}_{2,n }\mathbf{r}.\end{array}  \tag{14}\]  
where \(\frac{\partial\mathbf{B}_{iz-1}^{new}}{\partial m_{iz}}\) and \(\frac{\partial\mathbf{B}_{iz}^{new}}{\partial m_{iz}}\) are defined as follows:  
\[ \begin{array}{l}\frac{\partial\mathbf{B}_{iz-1}^{new}}{\partial m_{iz}}=F_{iz-1, 1}\mathbf{E}_{iz-1,1}\backslash\big{(}\frac{\partial\mathbf{F}_{iz,2}}{\partial m_{iz} }/\mathbf{E}_{iz,2}-F_{iz,2}/\mathbf{E}_{iz,2}\frac{\partial\mathbf{E}_{iz,2}}{\partial m _{iz}}/\mathbf{E}_{iz,2}\big{)},\end{array}  \tag{15}\]  
\[ \begin{array}{l}\frac{\partial\mathbf{B}_{iz-1}^{new}}{\partial m_{iz}}=-(\mathbf{F} _{iz,1}\mathbf{E}_{iz,1})\backslash\big{(}\frac{\partial\mathbf{F}_{iz,1}}{\partial m _{iz}}\mathbf{E}_{iz,1}+\mathbf{F}_{iz,1}\frac{\partial\mathbf{E}_{iz,1}}{\partial m_{iz} }\big{)}\big{/}\\ \begin{pmatrix}\mathbf{F}_{iz,1}\mathbf{E}_{iz,1}\mathbf{F}_{iz+1,2}/\mathbf{E}_{iz+1,2}+(\mathbf{ F}_{iz,1}\mathbf{E}_{iz,1})\backslash\\ \begin{pmatrix}\mathbf{\partial\mathbf{F}}_{iz+1,2}/\mathbf{E}_{iz+1,2}-\mathbf{F}_{iz+1,2}/\bm {E}_{iz+1,2}\frac{\partial\mathbf{E}_{iz+1,2}}{\partial m_{iz}}/\mathbf{E}_{iz+1,2} \end{pmatrix}.\end{array} \end{array} \tag{16}\]  
where \(\frac{\partial\mathbf{F}_{iz,2}}{\partial m_{iz}}\), \(\frac{\partial\mathbf{F}_{iz,1}}{\partial m_{iz}}\), \(\frac{\partial\mathbf{F}_{iz+1,2}}{\partial m_{iz}}\), \(\frac{\partial\mathbf{F}_{iz,2}}{\partial m_{iz}}\), \(\frac{\partial\mathbf{F}_{iz,1}}{\partial m_{iz}}\), and \(\frac{\partial\mathbf{E}_{iz+1,2}}{\partial m_{iz}}\) are defined as follows:  
\[\begin{array}{l}\frac{\partial\mathbf{F}_{iz,2}}{\partial m_{iz}}=\mathrm{i} \omega