# CSEM Optimization Using the Correspondence Principle
## 3 Numerical Experiments
### Inversion Results
#### 3.2.2 MARE Model  
The model we will call MARE consists of an adaptation of the isotropic conductivity model available in one of the versions of the MARE2 DEM package for modeling and inversion (the MARE2 DEM package [32] is a CSEM 2.5D data modeling and inversion software suite based on finite elements. The code was initially developed as part of the SEMC consortium (Seafloor Electromagnetic Methods Consortium) at Scripps Institution of Oceanography, University of California, San Diego, USA). It simulates the presence of two bodies: a smaller one with a resistivity of 50 ohm.m and a larger one, to the left of the first, of 4 ohm.m. These resistors are embedded in a medium formed by conductive layers and an extended structure with 100 ohm.m, located in the lower portion of the model. The adaptation involved replacing the bathymetry present in the original model with a horizontal seabed at a depth of 1.2 km, as we can see in Figure 10.  
The acquisition used in the simulation of observed data consisted of 25 receivers positioned on the seabed, regularly spaced between 1 and 25 km, and sources located 50 m above the receivers, from coordinate 0 to coordinate 26 km, with sampling every 200 m. We computed the synthetic data using the MARE2 DEM modeling code [34]. The field used corresponds to the inline electric field. The receiver data were limited to offsets greater than 1 km and those with field amplitude greater than 10\({}^{-15}\) V/m. The inversions used 0.25 Hz data.  
To perform the inversions, we limited the vertical extension of the model to a depth of 6 km on a grid with a spacing of 100 m (vertical) by 200 m (horizontal), as seen in the initial model (shown in Figure 11a), constructed by smoothing the background formed by the conductive layers present in the reference model. Figure 11b illustrates the gradient relative to vertical conductivity at a frequency of 0.25 Hz, calculated with the model presented in  
Figure 9: Results of the inversions with (**a**) the steepest descent, (**b**) the momentum method, and (**c**) the L-BFGS-B method uses the observed data simulated from the model with two resistors. Black boxes represent the edge of the resistors.  
Figure 11a. There is a strong correlation between the regions where the gradient stands out and the locations where the resistors are present in the reference model that produced the observed data.  
Next, we present the best results obtained with each methodology presented. In the case of the steepest descent with line search (Figure 12a) and the inversion with momentum (Figure 12b), we used the gradient with precondition 2. In contrast, in the inversion with L-BFGS-B (Figure 12c), the best result was obtained using precondition 1. Figure 13 shows the convergence curves for the objective function. As we can see, the steepest descent and L-BFGS-B present the best convergence behavior compared to the inversion with the momentum algorithm. The estimated models are very similar, especially concerning the smaller and more resistive body on the right, which is well-defined by the three inversions (with resistivity values slightly closer to the reference model in the results obtained with momentum and L-BFGS-B). Regarding the larger resistor on the left, it is possible to notice that there is an overestimation of resistivity in the three results, with a slightly pronounced effect in the result of the inversion with the steepest descent and L-BFGS-B.  
Figure 11: (**a**) Initial model used in the inversions of data modeled from the MARE model. (**b**) Gradient relative to vertical conductivity at a frequency of 0.25 Hz, obtained from the initial model used in the inversions.  
Figure 10: The MARE model used to simulate observed data in the inversions. Triangles represent the receivers, and stars represent the sources used in the simulation. Not all sources are represented, and their height relative to the seafloor is illustrative.