## 3 Synthetic Test
### Synthetic Results  
To obtain good local optimization (LOM) results, an appropriate regularization parameter was used in addition to adding the smoothing term in the objective function [46; 47]. The regularization parameter was determined by obtaining the maximum value of the diagonal matrix in the singular value decomposition of the Jacobian matrix in both the individual and joint inversion of the DCR and SR data. The value of the regularization parameter was modified using a cooling approximation at each iteration. The inversion process started with the initial guess of the model parameters that improved with each iteration. This procedure was used to estimate the model parameter correction vector as applied by [15; 16]. The LOM was terminated when there was no decrease in error and the RMS dropped below a certain threshold value (convergence criteria). In the combined global optimization (CGO) techniques, a forward modeling algorithm was used to estimate the theoretical data in both DCR and SR methods; thereafter, the calculated data \begin{table}
\begin{tabular}{c c c c} \hline \hline
**S/N** & **Inversion Type** & **Abbreviation** & **Description** \\ \hline
3 & Individual inversion method & IIM & Inversion of a dataset from one geophysical method using the local optimization, e.g., SR data inversion. \\ \hline
4 & Joint inversion method & JIM & Inversion of datasets from more than one geophysical method using the local optimization, e.g., SR and DCR data inversion. \\ \hline
5 & Single-objective optimization & SOO & Processing of a dataset from one geophysical method using global optimization, e.g., SR data inversion. \\ \hline
6 & Multi-objective optimization & MOO & Processing of dataset from more than one geophysical method using global optimization, e.g., SR and DCR data inversion. \\ \hline
7 & Combined (local plus global) & CGO & Inversion of a dataset from either one or more geophysical methods using the combination of local (to define a search space) with global optimization, e.g., DCR, or SR and DCR data inversion. \\ \hline
8 & (Global conventional) & GOM & Processing a dataset from either one or more geophysical methods using only the global optimization, e.g., DCR, or SR and DCR data inversion. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. were compared to the observed field data to compute their misfits. The data misfit (from Equations (10) and (11)) for the DCR method is presented in a simplified form as:  
\[M_{dc}=\frac{100\times\mathbf{W}_{dc}\times(\left\|d_{dc}-td_{dc}\right\|_{2})}{ \left\|d_{dc}\right\|_{2}} \tag{12}\]  
where \(d_{dc}\) is the observed real data, \(\mathbf{W}_{dc}\) is the weighting matrix of the real data, and \(td_{dc}\) is the theoretical data. The misfit function for the SR method is defined with a similar annotation as:  
\[M_{sr}=\frac{100\times\mathbf{W}_{sr}\times(\left\|d_{sr}-td_{sr}\right\|_{2})}{ \left\|d_{sr}\right\|_{2}} \tag{13}\]  
The combined misfits form the objective function for the multi-objective optimization algorithm that is represented as:  
\[M_{dc\ sr}=[M_{dc}\ M_{sr}] \tag{14}\]  
Notice that for the multi-objective (MOO) algorithm in Equation (14), the individual misfits are not added together but are simply concatenated to make a two-column matrix of the misfits. In addition, selecting the number of populations and generations is important to improve the model resolution in the CGO algorithm for both methods. Conventionally, the population size of 50 is recommended for the GA, with the number of decision variables less than or equal to five. Otherwise, 200 populations should be used when the decision variable is greater than five. The CGO algorithm is terminated when the average change in the spread of Pareto solutions is less than the function tolerance (i.e., 1 \(\times\) 10\({}^{-4}\) for multi-objective optimization, MOO) and the specified number of generations (i.e., for single objective optimization, SOO).