## 2 Optimization Methods  
### Local Optimization Method  
The individual and joint inversion of DCR and SR data are usually regularized with a smoothing function due to the non-uniqueness and instability associated with the inverse process. In the inversion of both DCR and SR methods, the data misfit can be formulated as follows:  
\[E(m)=\|\mathbf{W}_{d}(\mathbf{d}_{obs}-\mathbf{f}(m))\|^{2}, \tag{1}\]  
\(\mathbf{f}(m)\) is a function that is used to describe forward modeling, \(\mathbf{d}_{obs}\) is the observed field record, and \(\mathbf{W}_{d}\) is the weight matrix used to adjust the data anomaly (e.g., high or low amplitudes). Usually, we use norms to quantify the misfit between the observed and calculated data, and the common one used for this type of inverse problem is the \(L_{2}\) norm because we assume that the error in the data is Gaussian. Therefore, the objective functions for the DCR and SR data are given as follows:  
\[\Phi(\mathbf{m}_{dc})=\|\mathbf{W}_{dc}(\mathbf{d}_{dc}-\mathbf{f}_{dc}(\mathbf{m}_{dc}))\|^{2}+ \alpha_{dc}\|\
abla^{2}\mathbf{m}_{dc}\|^{2} \tag{2}\]  
\[\Phi(\mathbf{m}_{sr})=\|\mathbf{W}_{sr}(\mathbf{d}_{sr}-\mathbf{f}_{sr}(\mathbf{m}_{sr}))\|^{2}+ \alpha_{sr}\|\
abla^{2}\mathbf{m}_{sr}\|^{2} \tag{3}\]  
where \(\Phi\) is the misfit or objective function, \(\mathbf{d}_{dc}\) and \(\mathbf{d}_{sr}\) are the measured data, and \(\mathbf{f}_{dc}\) and \(\mathbf{f}_{sr}\) is the model response for the DCR and SR methods, respectively. Additionally, \(S=\
abla^{2}\mathbf{m}_{sr}^{2}\) is the Laplacian of the model parameter that is transformed into the smoothness matrix by obtaining its Laplacian operator, while \(\alpha_{dc}\) and \(\alpha_{sr}\) are the regularization parameters that determine the level of the smoothness of resistivity and seismic models, respectively. The joint inversion offers conventional ways of integrating data from different geophysical methods in such a manner that the outcome models are consistent and similar. One of the methods used to accomplish this goal is to apply the cross-gradient constraint proposed by [PERSON] et al. [8] in the objective function. The parallel spatial variation in the models (resistivity or seismic velocity) is required to satisfy the cross-gradient constraint [8; 13; 15]. This means that model anomalies or layer boundaries must essentially point in the same or opposite direction. Applying the cross-gradient constraint, Equations (2) and (3) become:  
\[\Phi(\mathbf{m}_{dc},\mathbf{m}_{sr})=\left\| \begin{matrix}\mathbf{W}_{dc}(\mathbf{d}_{dc}-f_{ dc}(\mathbf{m}_{dc}))\\ \mathbf{W}_{sr}(\mathbf{d}_{sr}-\mathbf{f}_{sr}(\mathbf{m}_{sr}))\end{matrix} \right\|^{2}+ \alpha_{dc}\big{\|}\
abla^{2}\mathbf{m}_{dc}\big{\|}^{2}+\overset{\rightarrow}{c} \big{\|}\mathbf{m}_{dc}\big{\|}^{2} \tag{4}\]  
subject to \(\overset{\rightarrow}{c}(\mathbf{m}_{dc},\mathbf{m}_{sr})=\overset{\rightarrow}{0}\). Where \(\overset{\rightarrow}{c}(\mathbf{m}_{dc},\mathbf{m}_{sr})\) is the cross-gradient constraint and can be defined as:  
\[\overset{\rightarrow}{c}(\mathbf{m}_{dc},\mathbf{m}_{sr})=\
abla\mathbf{m}_{dc}(x,z) \times\
abla\mathbf{m}_{sr}(x,z) \tag{5}\]  
Equation (4) can be minimized by applying the appropriate regularized local optimization algorithm similar to the approach used in [15]. Therefore, the model parameter correction vector can be expressed as:  
\[\Delta\mathbf{m}=\mathbf{G}^{-1}\mathbf{n}-\mathbf{G}^{-1}\mathbf{B}^{T}\Big{(}\mathbf{B}\mathbf{G}^{-1} \mathbf{B}^{T}\Big{)}^{-1}\Big{[}\mathbf{B}\mathbf{G}^{-1}\mathbf{n}-\mathbf{B}\Delta\mathbf{m}_{i-1}+ \overset{\rightarrow}{c}(\mathbf{m}_{i-1})\Big{]} \tag{6}\]where \(G\) and \(n\) are defined as:  
\[G=\left(J^{T}W^{T}WJ+\alpha C^{T}C\right) \tag{7}\]  
\[n=\left(J^{T}W^{T}WR-\alpha C^{T}Cm^{i-1}\right) \tag{8}\]  
In Equations (7) and (8), \(J\) is the Jacobean matrix, \(W\) is the weighting matrix, \(R\) is the data residual, \(C\) is the Laplacian operator, and \(B\) is the cross-gradient derivative.