# Lp-Norm Inversion of Gravity Data Using Adaptive Differential Evolution
###### 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  
Generally, according to the kind of model parameters selected, there are two basic approaches to gravity inversion--parametric inversion (e.g., [7; 8; 9; 10]) and physical property inversion (e.g., [2; 3; 11; 12; 13]). The methods of parametric inversion determine the geometrical properties of unknown density sources with a fixed density contrast, and offer a homogeneous body of the postulated density. Compared with parametric inversion, 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. Therefore, the inversion for physical properties is currently important and prevalent. As we know, 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. Moreover, considering the inherent ambiguity of inverse problem for the potential field, it is necessary to choose an appropriate starting model and add regularization constraints (see [2; 14]). To avoid the weak points of the linearly iterative optimization procedures, global optimization techniques, such as genetic algorithm (GA), simulated annealing (SA), particle swarm optimization (PSO), differential evolution (DE), etc., are considered appropriate to solve the inverse problems [6; 7; 15; 16; 17]. Among them, genetic algorithms have been successfully used for physical property inversion of gravity data (e.g., [6; 18]). Differential evolution, which has been illustrated as a simple but efficient evolution algorithm, was first proposed by [PERSON] and [PERSON] to solve Chebyshev polynomial problems [19]. In recent years, it has been verified that DE provides better accuracy, more robustness, and fast convergence speed than other evolution algorithms, such as GA and PSO [20; 21; 22]. Hence, it has attracted great attention in many scientific and engineering fields, such as neural network training [23; 24; 25; 26], data mining [27; 28], and image processing [29; 30; 31; 32], etc. With regards to geophysical inverse problems, the applications of DE are mainly non-gravity methods. [PERSON] [33] implemented an inversion program for self-potential (SP) and vertical electrical sounding (VES) by DE; the synthetic data and field data demonstrate the effectiveness of DE in geoelectrical inversion. [PERSON] et al.