# Robust 3D Joint Inversion of Gravity and Magnetic Data: A High-Performance Computing Approach
## 1 Introduction
If necessary, the new seismic section is further used to simulate resistivity, etc. The approach is advantageous as it is relatively simple and requires much less computer core memory than the simultaneous joint inversion [15]. According to [PERSON] et al. [21], inversion methods can be categorized as follows: Cluster analysis is grouping according to the proximity of samples in the space of properties [22; 23]; the Gaussian classifier is based on the assumption that the vector of properties has an n-dimensional Gaussian distribution of the conditional pdf within each lithological group [23]; K-means clustering implies an iterative search of a set of points (centers) so as to minimize the mean squared distance from each data point to its nearest center in training samples [24]; Discriminant analysis searches the functions of properties that separate groups in an optimal way (in terms of root mean square) in order to choose the combination of linear coefficients to maximize the variance of group centroids while minimizing the variance within the groups [25]. In this regard, the approach of the objective function according to the type of problem to be solved and the way to obtain a solution are fundamental to achieving satisfactory results; thus, a mathematical approach to obtain a solution capable of fitting the needs of each particular case is what we call \"robust\". The evaluation of uncertainty is an important step in inversion [26] since inverse problems always belong to decision-making processes and, by nature, are ill-posed, i.e., there are different solutions (called equivalents) that are compatible with the prior information and fit the observed data within the same error bounds [11; 27; 28; 29]. The main objective of geophysical data inversion, and in this case, gravity and magnetic data, is to solve the inverse problem of geophysics and related geological problems. Such problems include geologic structure, tectonics and geodynamics, volcanism and seismic processes, hydrogeology and mineral resource studies, etc. The advantages of these methods are relatively low cost, high productivity of work, and the possibility to obtain information about large territories in great depth. Many examples of successful applications of potential methods in different areas of the world and for solving both prospecting and theoretical geologic problems are described [30; 31; 32]. In the first stages, the inversion of geophysical data is carried out in two dimensions and for each method separately [33; 34; 35; 36; 37; 38]. The next step may be three-dimensional modeling [39; 40]. The last step may be a joint inversion by two or more methods, performed in two and three dimensions [41; 42; 43]. A serious disadvantage of gravity and geomagnetic methods is the ambiguity of the interpretation of the obtained data. Beyond the reductions as such, data analysis, functional fitting, and regional-local anomaly separation will also introduce errors; however, these remain essentially undefined because the basic assumptions are arbitrary and regional-local separation as such cannot be made in an unambiguous manner [44; 45; 46]. It should be noted that this is a problem for most geophysical methods. One way to overcome this problem, or at least to reduce uncertainty about the results, is to apply different filters and integrate methods [47; 48]. However, this is not always successful either. The best results can currently be achieved by the joint inversion of several methods, significantly improving the reliability of the interpretation [49; 50; 51; 52; 53]. On the other hand, it makes the mathematical procedure of data processing more complex and leads to the need for higher time consumption and more computational power; high-performance computing techniques are an important task to improve the mathematical process of joint data processing in order to obtain solutions in a reasonable time with ordinary personal computers.