Comparison of Nonlinear Two-Dimensional Modeling of Gravimetric Data of Western Anatolia, Turkey, Using Non-Dominated Sorting Genetic Algorithm and Single-Objective Genetic Algorithm

Document Type : Research Article


1 Department of Mining Exploration Engineering, Faculty of Mining, Petroleum and Geophysics Engineering, Shahrood University of Technology, Shahrood, Iran.

2 Department of Mining Engineering, Faculty of Technical Engineering, Malayer University, Malayer, Iran.


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 model’s parameters; 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 minimums. Therefore, in local inverse methods, the objective is to find the minimum of an objective function, and also an objective function might have a few local minimums 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 minimums 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 and single-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 and 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 very different from constraining local optimization that is used for stabilizing inverse simulation. The algorithms used in the present work included a non-dominated sorting genetic algorithm (NSGA-II) and single-objective genetic algorithm, which were used to estimate the depth. 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 addition, the single-objective genetic algorithm (GA) is capable of modeling and solving complex problems. In the present study, both algorithms were verified and validated using the data produced by an imaginary and complex synthetic model. In order for a more precise examination of the performance of both algorithms, the imaginary synthetic data were used both with no noise and with up to 10% Gaussian white noise (GWN). Accordingly, the modeling results indicated a good consistence between the algorithms and the primary model; so that, the root mean square error parameter for the data obtained from the initial data of the synthetic model ranged from 0.05 to 0.35mGal for the NSGA-II and from 0.07 to 0.52mGal for the GA. Also, this parameter didn't exceed 72.4 in the NSGA-II and didn't exceed 93.8 in the GA. Based on the gravimetric modeling of the Western Anatolia, Turkey, the results obtained from both algorithms under similar conditions in terms of parameter settings and number of algorithm executions indicated good performance of the NSGA-II algorithm compared to the single-objective algorithm.


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