# CSEM Optimization Using the Correspondence Principle
## 3 Numerical Experiments  
### Gradient Calculation  
We validate our implementation to calculate the gradient of the objective function using the data from a 3D isotropic model measuring 18 km in the \(x\) direction, 10 km in the \(y\) direction, and 3 km deep (in the \(z\) direction), inspired by [31]. This model simulates a 2D distribution of conductivities consisting of a resistive body with a resistivity of 25 ohm.m, in a homogeneous background medium with 1 ohm.m, and a water layer of 800 m, with its central section shown in Figure 1. We also use a uniform grid with cell dimensions of 100 m (vertical) \(\times\) 200 m \(\times\) 200 m (in this work, we use a 3D modeling code, and to ensure compatibility with the 2D inversion codes, we adopt the strategy of replicating the models obtained during the optimization process along the third dimension). With the model illustrated in Figure 1, we simulate, using reciprocity, the survey along its central section to use the horizontal component of the electric field (at frequencies of 0.25 Hz, 0.50 Hz, and 0.75 Hz) measured by the receivers on the seafloor with a regular spacing of 1 km, from coordinates 6 km to 12 km. The sources, located 100 m above the receivers, are distributed at intervals of 200 m, between 1.1 km and 16.9 km. Figure 2 illustrates the data associated with the central receiver at the three frequencies (since we use reciprocity, which allows the interchange between sources and receivers, the data illustrated in Figure 2 correspond to the field measured at the source positions for a source attributed to the position of the central receiver). The estimation of the adjoint source involves solving the system of Equation (46) for the residual (right-hand side of the equation) at each offset, such that at the end of the process, the conjugate gradient routine provides a pulse in the (fictitious) time domain (vector **m**) as the solution. An important aspect of the approach we adopted is related to the need to fit the residuals at each frequency separately, in contrast to the more intuitive strategy of simultaneously fitting the residuals at the frequencies where the gradient is to be calculated, as we can see in [17]. The adoption of this approach is essential to obtain the results that will be presented below. Next, we analyze the accuracy of our adjoint source estimation, considering the residuals between the observed data and the data calculated with a model consisting only of the 1 ohm.m background present in the model shown in Figure 1. Figure 3 illustrates the estimated pulses (at each frequency) for source 60 and the data from receiver number 4 and  
Figure 1: Resistivity model used to obtain the synthetic data to validate the gradient calculation. 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. Figure 2: (a) Amplitude and (b) phase curves of the in-line electric field at frequencies 0.25 Hz, 0.50 Hz, and 0.75 Hz associated with the central receiver in the data from the model shown in Figure 1.  
their amplitude spectra, along which we indicate the residuals (in the frequency domain) adjusted during obtaining adjoint sources. We also illustrate the gradient calculation relative to the vertical component of the conductivity tensor based on data for receiver 1 at a frequency of 0.50 Hz. Figure 4 shows the absolute value (in log scale) of the \(E_{z}\) component of the electric field normalized by the source and the field \(\lambda_{E_{z}}\), computed using forward and adjoint modeling, respectively, as well as their product, calculated according to Equation (27). To improve the visualization, we apply muting in a depth range corresponding to five samples below the seafloor. The \"complete\" gradient is obtained by summing the gradients calculated from the data of each receiver. Figure 5 shows the results for frequencies of 0.25, 0.50, and 0.75 Hz, which have similar characteristics in shape but different depth ranges.