# Robust 3D Joint Inversion of Gravity and Magnetic Data: A High-Performance Computing Approach
## 4 Discussion  
Here we list the main discussion issues related to the solution to the problem posed in the paper. In brief, they can be summarized as follows:  
It is possible to further optimize the computational implementation, e.g., by CPU parallelization using MPI; however, this only decreases the processing time (objective already met) without enhancing memory management. On the other hand, by incorporating variable resolution both horizontally and vertically through cells of varying dimensions, we would be able to differentiate the domain more efficiently, thereby reducing memory usage.  
At this point in development, the program is only capable of working with an equal amount of data for both types of input data; likewise, the resolution of the joint inversion models is the same for both geophysical techniques, limiting its application to the spatial coverage of both data types. Additionally, the way we construct our 3D domain does not consider topography yet.  
The shown processing times are representative, as they depend on the quantity of data to be processed, how the calculations were implemented, the compiler and the packages used, as well as the capabilities of the hardware employed. Thus, processing times may vary as any of these factors change.  
The main advantages of this implementation are (1) the approach to obtain inversion solutions (parametric functional) is versatile, applicable to a wide variety of study objects because the linear regularizers that it integrates seek to promote general characteristics in the solutions, (2) the structural similarity regularizer allows reducing the uncertainty of the results, integrating two geophysical methods within the process of obtaining the solution, overcoming one of the great disadvantages of any geophysical method (ambiguity), finally (3) saves computational costs by reducing processing times which improves the efficiency of the process to obtain results and makes better use of available memory allowing to solve joint inversion problems for larger domains with less powerful hardware.  
The use of other linear regularizers, such as the case of L, M, and N, can be incorporated without altering the performance too much since they are part of the expression \(I_{A}\) (Equation (8)), which is calculated only once at the beginning of the inversion process. Therefore, changing the behavior of the solutions with other linear regularizers has no repercussions on the overall performance.  
On the other hand, as can be seen in the convergence graphs (Figures 6, 10 and 13), the iterative process is unstable; that is, the models change with respect to the previous iteration by a high percentage. We associate this characteristic to the method we employ to minimize the parametric functional but overall to the use of many regularizers within the minimization process, as we noticed solutions with fewer regularizers active tend to be more stable, however, as also shown in the same figures, there is a tendency to stabilize the solution as the number of iterations increases, while in the RMS graph, it can be seen that the fit of the solutions is very good since early iterations. Therefore, this instability is not a problem for the quality of the results.