Kantorovich-Rubinstein misfit for inverting gravity-gradient data by the level-set method

Abstract

We have developed a novel Kantorovich-Rubinstein (KR) norm-based misfit function to measure the mismatch between gravity-gradient data for the inverse gradiometry problem. Under the assumption that an anomalous mass body has an unknown compact support with a prescribed constant value of density contrast, we implicitly parameterize the unknown mass body by a level-set function. Because the geometry of an underlying anomalous mass body may experience various changes during inversion in terms of level-set evolution, the classic least-squares (L2-normbased) and the L1-norm-based misfit functions for governing the level-set evolution may potentially induce local minima if an initial guess of the level-set function is far from that of the target model. The KR norm from the optimal transport theory computes the data misfit by comparing the modeled data and the measured data in a global manner, leading to better resolution of the differences between the inverted model and the target model. Combining the KR norm with the level-set method yields a new effective methodology that is not only able to mitigate local minima but is also robust against random noise for the inverse gradiometry problem. Numerical experiments further demonstrate that the new KR norm-based misfit function is able to recover deep dipping flanks of SEG/EAGE salt models even at extremely low signal-to-noise ratios. The new methodology can be readily applied to gravity and magnetic data as well. INTRODUCTION The gravity gradiometer measures the gravity gradient tensor. Correct interpretation of gravity-gradient data provides essential guidelines in analyzing the composition of the earth and in delineating subsurface source bodies, such as mineral deposits. However, because gravity-gradient data are higher order derivatives of the potential, manual interpretation of such data is extremely challenging, whereas automatic interpretation calls for developing efficient inversion methods. In the past several decades, a lot of effort has been made to develop various inversion methods for gravity or gravitygradient data (Last and Kubik, 1983; Li and Oldenburg, 1998; Condi and Talwani, 1999; Portniaguine and Zhdanov, 1999; Jorgensen and Kisabeth, 2000; Li, 2001a, 2001b, 2010; Routh et al., 2001; Zhdanov et al., 2004; Krahenbuhl and Li, 2006; Barnes et al., 2008; Martinez et al., 2010, 2013; Barnes and Barraud, 2012). In these methods, prior geologic constraints on density models are usually enforced so that the resulting solutions conform to realistic earth models (Last and Kubik, 1983; Li and Oldenburg, 1998; Condi and Talwani, 1999; Portniaguine and Zhdanov, 1999). Because most inversion techniques provide quantitative descriptions for subsurface structures, one has to extract the position of a source body from the resulting solutions, which is not an easy task. The level-set method (Osher and Sethian, 1988) has been widely used as a suitable and powerful tool for shape-optimization problems mainly due to its abilities in automatic interface merging and splitting. Assuming that a homogeneous density-contrast distribution with the value of density contrast specified a priori was supported on an unknown bounded domain, we can convert the inverse gravity gradient problem into a domain inverse problem by implicitly parameterizing the unknown anomalous mass body with a level-set function, and we apply the steepest-descent method to minimize the least-squares objective so as to find the corresponding level-set function. Along this line, a series of works by Isakov et al. (2011, 2013), Manuscript received by the Editor 9 November 2018; revised manuscript received 17 April 2019; published ahead of production 30 May 2019; published online 14 August 2019. Michigan State University, Department of Mathematics, East Lansing, Michigan 48824, USA. E-mail: [email protected]. Harbin Institute of Technology (Shenzhen), College of Science, Shenzhen, China. E-mail: [email protected]. Michigan State University, Department of Mathematics and Department of CMSE, East Lansing, Michigan 48824, USA. E-mail: [email protected] (corresponding author). © 2019 Society of Exploration Geophysicists. All rights reserved. G55 GEOPHYSICS, VOL. 84, NO. 5 (SEPTEMBER-OCTOBER 2019); P. G55–G73, 19 FIGS., 2 TABLES. 10.1190/GEO2018-0771.1 D ow nl oa de d 09 /0 6/ 19 to 1 72 .5 .4 3. 19 2. R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg / Lu et al. (2015), Lu and Qian (2015), Li et al. (2016, 2017), and Li and Qian (2016) have developed various level-set methods to delineate subsurface source bodies; see also Li and Qian (2016) and Zheglova et al. (2017) for applying the level-set method to carry out the joint inversion of traveltime and gravity data. However, because the geometry of the underlying domains may evolve in a very complicated way, these level-set methods based on the classical leastsquares objective are vulnerable to local minima induced by the misfit function and sensitive to random noise as shown by Lu et al. (2015), Lu and Qian (2015), and Li et al. (2016, 2017). The optimal transport theory (Villani, 2003) has been applied in seismic inversion to build novel objective functions for mitigating the cycle-skipping-induced local minimum issues (Virieux and Operto, 2009), where the cycle skipping is caused by the mismatch of the traveltime shift in full-waveform inversion problems (Benamou and Brenier, 2001; Engquist and Froese, 2014; Metivier et al., 2016). One of the important features of the optimal transport distance is to measure the difference between two probability distribution functions by rearranging one to the other in a global way (Engquist et al., 2016), and this feature leads to effective capturing of shift and dilation changes between these two functions. Consequently, we are motivated to apply the optimal transport theory to develop new misfit functions to measure the data mismatch in the inverse gradiometry problems so that possible local minimum issues can be mitigated. Because it is hard to satisfy all of the requirements needed for applying the original optimal transport theory to inverse problems (Engquist and Froese, 2014; Metivier et al., 2016), such as the nonnegativity of data, we appeal to a variant of optimal transport distances, the so-called Kantorovich-Rubinstein (KR) norm (Villani, 2003; Lellmann et al., 2014), to measure the data misfit for inverse gravity-gradient problems in that the KR norm does not require nonnegativity of data. Although we choose to work on gravity-gradient problems here, the methodology will be naturally applicable to other potential data as well. The rest of the paper is organized as follows. We present the methodology by reviewing the least-squares-based level-set method, developing the KR norm-based level-set method for gravity-gradient data, and further illustrating some properties of the new KR normbased misfit function for gravity-gradient inversion. Numerical experiments demonstrate the performance and effectiveness of the new method for reconstructing complex salt bodies. METHODOLOGY The level-set method for inverse gradiometry problem Gravity gradiometry measures the gradient of each component of the gravity field on a prescribed acquisition surface. To simplify the presentation, we only consider inverting the zz-component of the gravity gradient, which can be modeled by GzzðrÞ 1⁄4 γ Z Ωρ Kzzðr; r 0Þρðr 0Þdr 0; r ∈ Γr; (1) where γ 1⁄4 6.67408 × 10−8 cm g−1 s−2 is the universal gravitational constant, ρðrÞ is the density-contrast function with compact support below the acquisition surface Γr, r 0 denotes the coordinates ðx; zÞ in 2D or ðx; y; zÞ in 3D, and the kernel Kzzðr; r 0Þ is given by Kzzðr; r 0Þ 1⁄4 1 jr−r 0j4 ð2jz − z 0j2 − jr − r 0j2Þ in R; 1 jr−r 0j5 ð3jz − z 0j2 − jr − r 0j2Þ in R: (2)