Institute of Geophysics, University of TehranJournal of the Earth and Space Physics2538-371X42220160822Gravity data inversion using L1-norm stabilizerGravity data inversion using L1-norm stabilizer3373485481610.22059/jesphys.2016.54816FASaeedVatankhahJournal Article20150218In this paper the inversion of gravity data using L1–norm stabilizer is considered. The inversion is an important step in the interpretation of data. In gravity data inversion, the goal is to estimate density and geometry of the unknown subsurface model from a set of known observation measured on the surface. Commonly, rectangular prisms are used to model the subsurface under the survey area. The unknown density contrasts within each prism are the parameters which should be estimated. The inversion of gravity data is an example of underdetermined and ill-posed problem, i.e. the solution can be non-unique and unstable. Thus, in order to find an acceptable solution regularization should be imposed. Solution is usually obtained by minimizing a global objective function consisting of two terms, data misfit and the regularization term. Data misfit measures how well an obtained model can reproduce the observed data. Usually, it is assumed noise in gravity data is Gaussian, therefore a L2–norm measure of the error between observed and predicted data is well suited for data misfit. There are several choices for a stabilizer, depends on type of features one wants to see from inverted model. A typical choice is a L2 –norm of a low-order differential operator applied to the model, which also a priori information and depth weighting can be incorporated (Li and Oldenburg, 1996). In this case the objective function is quadratic, then minimization of the function results a linear system to be solved. However, the models recovered in this way are characterized by smooth feature which are not always consistent with the real geological structures. There are situations in which the sources are localized and separated by sharp, distinct interfaces. To deal with this problem, during last decades, researchers have proposed a few types of stabilizer. Last and Kubik (1983) presented a compactness criterion for gravity inversion that seeks to minimize the area (or volume in 3D) of the causative body. Portniaguine and Zhdanov (1999) based on this stabilizer, who named the minimum support (MS), developed the minimum gradient support (MGS) stabilizer. For both constraint, the regularization term can be written as the weighted L2–type norm of the model. Therefore, the problem of the minimization of the objective function can be treated same as conventional Tikhonov functional. The only difference is that a priori variable weighting matrix for model parameters incorporated in the regularization term. Thus the Iteratively Reweighted Least Square (IRLS) algorithm is required to solve the problem. Other possibility for stabilizer is the minimization of the L1-norm of model or gradient of model, the latter indicates total variation regularization. The L1–norm stabilizer allows occurrence of large elements in the inverted model among mostly small values. Therefore, it can be used to obtain sharp boundaries and blocky features. Although the L1–norm stabilizer has favorable properties, in reconstruction of sparse models, its numerical implementation in a minimization problem can be difficult because its derivatives with respect to an element is not defined at zero. To overcome this difficulty, in this paper, the L1–norm stabilizer is approximated by a reweighted L2 –norm term. The algorithm is extended to gravity inverse problem, which needs depth weighting and other priori information to be included in the objective function. For estimating the regularization parameter, which balances between two terms of objective function, the Unbiased Predictive Risk Estimator (UPRE) method is used. The solution of the resulting objective functional is found using Generalized Singular Value Decomposition (GSVD), also provides for efficient determination of the regularization parameter at each iteration. Simulation using synthetic data of a dipping dike demonstrates that the method is capable to reconstruct focused image, boundaries and slop of the reconstructed model are close to those of the original model. The method is applied on gravity data acquired over the Gotvand dam site, in the south-west of Iran. The results show rather good agreement with those obtained from the boreholes.In this paper the inversion of gravity data using L1–norm stabilizer is considered. The inversion is an important step in the interpretation of data. In gravity data inversion, the goal is to estimate density and geometry of the unknown subsurface model from a set of known observation measured on the surface. Commonly, rectangular prisms are used to model the subsurface under the survey area. The unknown density contrasts within each prism are the parameters which should be estimated. The inversion of gravity data is an example of underdetermined and ill-posed problem, i.e. the solution can be non-unique and unstable. Thus, in order to find an acceptable solution regularization should be imposed. Solution is usually obtained by minimizing a global objective function consisting of two terms, data misfit and the regularization term. Data misfit measures how well an obtained model can reproduce the observed data. Usually, it is assumed noise in gravity data is Gaussian, therefore a L2–norm measure of the error between observed and predicted data is well suited for data misfit. There are several choices for a stabilizer, depends on type of features one wants to see from inverted model. A typical choice is a L2 –norm of a low-order differential operator applied to the model, which also a priori information and depth weighting can be incorporated (Li and Oldenburg, 1996). In this case the objective function is quadratic, then minimization of the function results a linear system to be solved. However, the models recovered in this way are characterized by smooth feature which are not always consistent with the real geological structures. There are situations in which the sources are localized and separated by sharp, distinct interfaces. To deal with this problem, during last decades, researchers have proposed a few types of stabilizer. Last and Kubik (1983) presented a compactness criterion for gravity inversion that seeks to minimize the area (or volume in 3D) of the causative body. Portniaguine and Zhdanov (1999) based on this stabilizer, who named the minimum support (MS), developed the minimum gradient support (MGS) stabilizer. For both constraint, the regularization term can be written as the weighted L2–type norm of the model. Therefore, the problem of the minimization of the objective function can be treated same as conventional Tikhonov functional. The only difference is that a priori variable weighting matrix for model parameters incorporated in the regularization term. Thus the Iteratively Reweighted Least Square (IRLS) algorithm is required to solve the problem. Other possibility for stabilizer is the minimization of the L1-norm of model or gradient of model, the latter indicates total variation regularization. The L1–norm stabilizer allows occurrence of large elements in the inverted model among mostly small values. Therefore, it can be used to obtain sharp boundaries and blocky features. Although the L1–norm stabilizer has favorable properties, in reconstruction of sparse models, its numerical implementation in a minimization problem can be difficult because its derivatives with respect to an element is not defined at zero. To overcome this difficulty, in this paper, the L1–norm stabilizer is approximated by a reweighted L2 –norm term. The algorithm is extended to gravity inverse problem, which needs depth weighting and other priori information to be included in the objective function. For estimating the regularization parameter, which balances between two terms of objective function, the Unbiased Predictive Risk Estimator (UPRE) method is used. The solution of the resulting objective functional is found using Generalized Singular Value Decomposition (GSVD), also provides for efficient determination of the regularization parameter at each iteration. Simulation using synthetic data of a dipping dike demonstrates that the method is capable to reconstruct focused image, boundaries and slop of the reconstructed model are close to those of the original model. The method is applied on gravity data acquired over the Gotvand dam site, in the south-west of Iran. The results show rather good agreement with those obtained from the boreholes.https://jesphys.ut.ac.ir/article_54816_7cef661ac7a645cd13f9ab36c325e804.pdf