# CSEM Optimization Using the Correspondence Principle
## 4 Discussion  
The results illustrated in the previous section consolidate the use of the adjoint-state method and the correspondence principle to obtain the gradient (relative to the model conductivities) of the objective function defined in Equation (23). The fictitious domain modeling strategy, already established in the context of solving the forward problem, also proved efficient in obtaining the adjoint fields. For the calculation of the gradient consistent with the perturbation observed in the data due to the presence of a resistor in the model, we  
Figure 16: Curves showing the evolution of the objective function during the inversions for the Marlim model, using the steepest descent method (black curve), momentum (blue curve), and L-BFGS-B (red curve). Figure 15: Resistivity models obtained from the inversions using (**a**) steepest descent, (**b**) momentum, and (**c**) L-BFGS-B. The black outline represents the boundary of the resistor present in the Marlim model. opted for the approach based on obtaining the adjoint sources for each frequency separately, unlike other authors who address the topic, as seen in the work of [36]. We validated the quality of the gradient calculated with the proposed methodology, preconditioning, and regularization using model parameterization through the inversions of simulated data from three different models: the model consisting of two resistors in a conductive background, the MARE model, an adaptation of the example model from the MARE2 DEM package, and the Marlim model, adapted from the MR3D model, constructed by [33]. We used three optimization methods: steepest descent, our proposed modified momentum algorithm, and L-BFGS-B. Regardless of the dataset we used and the inversion methodology employed, we observed interesting aspects in the inversion results, which produced models with well-positioned resistors at depth and without artifacts near the receiver positions, commonly seen in CSEM inversion results. We attribute these achievements to the use of gradient preconditioning and smoothing by reparameterization. For the model with two resistors, all three methods successfully obtained results that included the bodies at different depths, with the positioning of the deeper resistor being slightly better in the case of L-BFGS-B (Figure 9). From the perspective of the number of iterations needed for the objective function to reach its minimum value, L-BFGS-B was slightly slower than the steepest descent, with the momentum inversion requiring the most iterations (Figure 8). The inversions for the MARE model were obtained using synthetic data produced with the MARE2 DEM package, and the analysis of the objective function evolution curves during the inversions indicated slower convergence for the momentum optimization, which reached its lowest level after iteration 30, in contrast to the performance of steepest descent and L-BFGS-B, which did so at around iterations 20 and 15, respectively (Figure 13). The results obtained were similar but presented a subtle difference in the resistivity of the larger resistor, which was slightly overestimated in some cases (Figure 12). This feature was somewhat more pronounced in the model obtained with the steepest descent and less noticeable in the model obtained with momentum. In the inversions for the Marlim model, we observed similar behavior in the convergence curve of the objective function during the first iterations with the steepest descent and momentum; the L-BFGS-B showed a sharper reduction followed by stagnation that lasted for about ten iterations, after which the objective function began to decrease again (Figure 16). The models obtained with the three optimizers differed (Figure 15). While the steepest descent result indicated the presence of a modest and localized resistor, the result obtained with momentum was more consistent with the overall geometry of the reservoir.