Nonlinear two-dimensional modeling of gravimetric data using non-dominated sorting genetic algorithm (NSGA-II) and NSGAII-TOPSIS hybrid technique

Studying the bedrock geometry in mining and oil exploration operations to obtain its 2D pattern requires nonlinear reverse computations. Local optimization methods for solving nonlinear inverse problems are based on linearizing the changes of the model similar to a primary model and finding an objective function of minimum error from the parameters of the model; however, these optimization methods are not able to select a suitable primary function that is close enough to the general optimal value. That is to say, every objective function can have several minimum and maximum solutions. The lowest minimum is called the global minimum while the rest of them are named local minima. Therefore, in local inverse methods, the goal is to find the minimum of an objective function, and also an objective function might have a few local minima with different values. In this case, it is not suitable to use gradient-based methods for exploration purposes, unless the primary model is very close to the actual answer, which is outside the control of geological structures or the geometry of the subsurface. Despite the easy execution and high convergence rate of the local methods, there is the possibility of being trapped in local minima because these methods are dependent on the primary model, and also finding more than one optimized point in 2D or 3D simulations; this is why local optimization methods are considered deterministic algorithms. Multi-objective metaheuristic optimization algorithms are capable of searching the feasible region and they also provide a solution independent of the primary model. Searching the feasible region means finding all the feasible solutions for a problem. Each point in this region is representing a solution that can be ranked based on its value. One of the important differences between local optimization and metaheuristic methods is constraining. Constraining metaheuristic global optimization methods are only used for constraining the feasible region based on previous knowledge or estimation relations, which is different from constraining local optimization that is used for stabilizing inverse simulation. The algorithm used in the present work includes non-dominated sorting genetic algorithm (NSGA-II). The NSGA-II is commonly used to solve problems with multiple, typically conflicting objective functions. This algorithm is capable of being developed and also has a high potential for solving unbounded multi-objective problems. In the present study, NSGA-II algorithm was verified and validated using the data produced by an imaginary and complex synthetic model. In the present research work, a hybrid technique of NSGA-II and TOPSIS algorithms was introduced and utilized as a viable search method for nonlinear modeling of the gravity data, and a substitute for the optimization methods. In order for a more precise examination of the performance of this algorithm, the imaginary synthetic data were used both with no noise and with up to 10% Gaussian white noise (GWN). Based on the gravimetric data of the Moghan basin and Atacama Desert, Chile, the results obtained from algorithm indicated good performance of the NSGA-II and NSGAII-TOPSIS algorithms.