Lp-Norm Inversion of Gravity Data Using Adaptive Differential Evolution

Tao Song,Xing Hu,Qian Li,Lianzheng Cheng,Wei Du,Tiaojie Xiao

doi:10.3390/app11146485

Abstract

As a popular population based heuristic evolutionary algorithm, differential evolution (DE) has been widely applied in various science and engineering problems. Similar to other global nonlinear algorithms, such as genetic algorithm, simulated annealing, particle swarm optimization, etc., the DE algorithm is mostly applied to resolve the parametric inverse problem, but has few applications in physical property inversion. According to our knowledge, this is the first time DE has been applied in obtaining the physical property distribution of gravity data due to causative sources embedded in the subsurface. In this work, the search direction of DE is guided by better vectors, enhancing the exploration efficiency of the mutation strategy. Besides, to reduce the over-stochastic of the DE algorithm, the perturbation directions in mutation operations are smoothed by using a weighted moving average smoothing technique, and the Lp-norm regularization term is implemented to sharpen the boundary of density distribution. Meanwhile, in the search process of DE, the effect of Lp-norm regularization term is controlled in an adaptive manner, which can always have an impact on the data misfit function. In the synthetic anomaly case, both noise-free and noisy data sets are considered. For the field case, gravity anomalies originating from the Shihe iron ore deposit in China were inverted and interpreted. The reconstructed density distribution is in good agreement with the one obtained by drill-hole information. Based on the tests in the present study, one can conclude that the Lp-norm inversion using DE is a useful tool for physical property distribution using gravity anomalies.

Highlights

The physical property inversion methods allow the densities of the elements of a regular subsurface partition to vary
Inversion methods search for the possible solutions employing optimization techniques with linearly iterative approaches such as steepest descent method, conjugate gradients, etc
Li et al [34] proposed a modified Boltzmann Annealing Differential Evolution (BADE) algorithm, which uses an annealing strategy to avoid the local minima and solve the inversion problem in the directional resistivity logging-while-drilling (DRLWD) measurements; the results show robustness and immunity to the non-uniqueness inversion problem of their method

Summary

Introduction

The physical property inversion methods allow the densities of the elements of a regular subsurface partition to vary. In this case, the solution can give great flexibility to recover the depths and complex shape of sources. Inversion methods search for the possible solutions employing optimization techniques with linearly iterative approaches such as steepest descent method, conjugate gradients, etc. These optimization techniques have traditionally proven very difficult to solve the highly non-linear mathematical formulation since the iteration process can be prone to fall into the local minima. Genetic algorithms have been successfully used for physical property inversion of gravity data (e.g., [6,18])

Methods

Results

Conclusion