## 4 Results
### Synthetic Case: Bishop Model  
In order to test the developed algorithm, we performed the inversion by depth for the Bishop model, a synthetic basement model created by upscaling and shifting the topographic data of an area of Bishop, CA, USA [36]. This model contains a variety of structures that resemble the complexity of real basement relief: two long faults, en echelon arrays of smaller faults, transfer zones between faults, and an unfaulted deep basin area. This features allows the model to be exploited by various authors to test new methods for depth to the basement estimation using magnetic data [36; 37] and gravity data [38; 39]. In the north-east region of the model, the shallower parts of the basement (\(\approx\)100 m) are presented and, in the south-west region of the model, the deepest parts of the basement are located up to \(\approx\)9300 m, as seen in the 3D view of Figure 3 and the 2D floor view of Figure 4a, which acts as an unknown true depth model in the inversion. A total of \(n_{x}\times n_{y}\times 2=61\times 61\times 2=3162\) prisms are considered in a region of interest \(0\leq x\leq 380\) km for the east and \(0\leq y\leq 402\) km for the north, given a spacing of \(\Delta x=6.33\) km and \(\Delta y=6.7\) km(width and length) for each prism. The density for the upper prism is \(\rho_{1}=2100\) g/cm\({}^{3}\), corresponding to sedimentary rocks above the basement of density \(\Delta\rho_{2}=2750\) g/cm\({}^{3}\) for the lower prisms. 3D gravity forward modeling was performed, measuring along all of the grid points on the surface, giving a total of \(n_{x}\times n_{y}\times 2=61\times 61=1581\) gravimetric stations (length of the observed data vector \(d^{\text{obs}}\)). The gravity data show anomalies corresponding to the high and lower depths of the Bishop depth to the basement model (Figure 4b). For geophysical data inversion, _a priori_ information was not considered; therefore, \(\mathbf{C}_{m}=0\) and the parameter of regularization were chosen using the \(L\)-curve method for Tikhonov regularization. We performed 10 iterations of the CGLS method (Equation (6)) starting from a constant depth model \(\mathbf{m}_{0}^{z}\) of 1000 m. It is worth mentioning that the starting model does not affect the final solution, but it can help for a fast convergence of the problem. The depth model after 10 iterations of the CGLS is shown in Figure 4c, and the final synthetic response due to this model is shown in Figure 4d. The result obtained clearly exhibits a successful recovery of the structure of the basement with the correct depths and, henceforth, the gravimetric anomalies were fitted properly, with few discrepancies. These differences can be observed in Figure 4e,f for the model error and the gravity residual, respectively. For the depth model, errors up to \(\approx\)\(\pm\)0.7 km appear, mainly located at regions where there is an abrupt change in lateral variation in density (see Figure 3); however, in regions where the basement behaves predominantly horizontal, the errors are almost negligible (\(\approx\)\(\pm\)0.01 km or \(\approx\)\(\pm\)10 m) as seen in the scale of Figure 4a,c. Although this error happens where lateral variation is presented, we infer that this is a product of the Tikhonov regularization constraint; the smoothness interposed does not recover parts where the total variation in the model is presented. We address [PERSON] & Zhdanov [40] for an optional regularization where a focused model is needed. Consequently, this error in the model also produces a gravity residual in such lateral-variation regions of up to 2.15 mGals, which is not harmful for the final results. A single 1D depth to the basement profile is presented, located at \(y=201\) km for an \(x\)-variation as seen in Figure 5a. Again, the structure of the Bishop model is well-recovered in regions where there is not an abrupt change in density (\(x=280\) km).