Three‐dimensional potential field data inversion with L0 quasinorm sparse constraints

Abstract

The quantitative explanation of the potential field data of three‐dimensional geological structures remains one of the most challenging issues in modern geophysical inversion. Obtaining a stable solution that can simultaneously resolve complicated geological structures is a critical inverse problem in the geophysics field. I have developed a new method for determining a three‐dimensional petrophysical property distribution, which produces a corresponding potential field anomaly. In contrast with the tradition inverse algorithm, my inversion method proposes a new model norm, which incorporates two important weighting functions. One is the L0 quasi norm (enforcing sparse constraints), and the other is depth‐weighting that counteracts the influence of source depth on the resulting potential field data of the solution. Sparseness constraints are imposed by using the L0 quasinorm on model parameters. To solve the representation problem, an L0 quasinorm minimisation model with different smooth approximations is proposed. Hence, the data space (N) method, which is much smaller than model space (M), combined with the gradient‐projected method, and the model space, combined with the modified Newton method for L0 quasinorm sparse constraints, leads to a computationally efficient method by using an N × N system versus an M × M one because N ≪ M. Tests on synthetic data and real datasets demonstrate the stability and validity of the L0 quasinorm spare norms inversion method. With the aim of obtaining the blocky results, the inversion method with the L0 quasinorm sparse constraints method performs better than the traditional L2 norm (standard Tikhonov regularisation). It can obtain the focus and sparse results easily. Then, the Bouguer anomaly survey data of the salt dome, offshore Louisiana, is considered as a real case study. The real inversion result shows that the inclusion the L0 quasinorm sparse constraints leads to a simpler and better resolved solution, and the density distribution is obtained in this area to reveal its geological structure. These results confirm the validity of the L0 quasinorm sparse constraints method and indicate its application for other potential field data inversions and the exploration of geological structures.

A sigmoid stabilizing function for fast sparse 3D inversion of magnetic data

“…Last and Kubik ; Barbosa and Silva ; Portniaguine and Zhdanov ; Camacho, Montesinos and Vieira ; Bertete‐Aguirre, Cherkaev and Oristaglio ; Pilkington ; Zhao, Yu and Zhang ; Xiang et al . ; Meng ; Meng et al . ).…”

“…The second category of the stabilizing functions belongs to a class of functions that produce solutions with focused images (e.g. Last and Kubik 1983;Barbosa and Silva 1994;Portniaguine and Zhdanov 1999;Camacho, Montesinos and Vieira 2000;Bertete-Aguirre, Cherkaev and Oristaglio 2002;Pilkington 2009;Zhao, Yu and Zhang 2016;Xiang et al 2017;Meng 2018;Meng et al 2018). However, one of the most popular stabilizing functions for non-smooth inversion of geophysical data is minimum support (MS) stabilizing function which was used by different researchers (e.g.…”

Comment on “Three‐dimensional potential field data inversion with L0 quasinorm sparse constraints” by Z. Meng, Geophysical Prospecting 66, 626–646

“…The combination of the L0‐norm stabilizer with depth weighting has subsequently been used by a number of authors, as referenced, for example, in Pilkington () and Vatankhah, Ardestani and Renaut (). The motivation for the L0‐norm constraint presented in Meng () is very close to that of the compactness or minimum support constraints. For the benefit of other readers in the following brief note, we expand on the relationship between these constraint conditions.…”

“…Matrix W is a parameter‐dependent diagonal matrix which is updated at each iteration using model parameters obtained at the previous iteration. Specifically, adopting the notation of Meng (), Last and Kubik () used …”

Reply to comment by S. Vatankhah and R.A. Renaut: “Three‐dimensional potential field data inversion with L0 quasinorm sparse constraints”

“…In Vatankhah's comment, there may be some inaccurate understanding. Function (equation (3) in Vatankhah's comment, corresponding to the function used in Meng ()) is better suited for potential field data inversion comparing with (equation (1) in Vatankhah's comment, introduced by Last and Kubik, ). In Vatankhah's Figure 1(a), for large values of σ and for big absolute values of m , has smaller values than , and therefore is smoother than .…”