A local level-set method for 3D inversion of gravity-gradient data

ABSTRACTWe have developed a local level-set method for inverting 3D gravity-gradient data. To alleviate the inherent nonuniqueness of the inverse gradiometry problem, we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain D so that we may convert the original inverse problem into a domain inverse problem. Because the unknown domain D may take a variety of shapes, we parametrized the domain D by a level-set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level-set function. Because the convergence of the level-set algorithm relied heavily on initializing the level-set function to enclose the gravity center of a source body, we applied a weighted L1-regularization method to locate such a gravity center so that the level-set function can be properly initialized. To rapidly compute the gradient of the nonlinear functional arising in the level-...

[1]  Michael Herty,et al.  Identification of uncertainties in the shape of geophysical objects with level sets and the adjoint method , 2011 .

[2]  Michael S. Zhdanov,et al.  Potential field migration for rapid imaging of gravity gradiometry data , 2011 .

[3]  Hongkai Zhao,et al.  Imaging of location and geometry for extended targets using the response matrix , 2004 .

[4]  M. Burger.  A level set method for inverse problems , 2001 .

[5]  J. Qian,et al.  A level-set method for imaging salt structures using gravity data , 2016 .

[6]  F. Santosa,et al.  Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set , 1998 .

[7]  S. Osher,et al.  A level set-based Eulerian approach for anisotropic wave propagation , 2003 .

[8]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[9]  Eric L. Miller,et al.  A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography , 2007 .

[10]  Michael S. Zhdanov,et al.  Focusing geophysical inversion images , 1999 .

[11]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[12]  Stanley Osher,et al.  Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension , 2006, J. Comput. Phys..

[13]  M. Fedi,et al.  SCALFUN:3D Analysis of Potential Field Scaling Function to Determine Independently Or Simultaneously Structural Index And Depth to Source. , 2006 .

[14]  Jianliang Qian,et al.  Adaptive Finite Difference Method For Traveltime And Amplitude , 1999 .

[15]  S. Osher,et al.  A LEVEL SET METHOD FOR THREE-DIMENSIONAL PARAXIAL GEOMETRICAL OPTICS WITH MULTIPLE POINT SOURCES ⁄ , 2004 .

[16]  Shingyu Leung,et al.  A Fast Local Level Set Method for Inverse Gravimetry , 2011 .

[17]  J. Kisabeth,et al.  Joint 3-D Inversion of Gravity, Magnetic and Tensor Gravity Fields For Imaging Salt Formations in the Deepwater Gulf of Mexico , 2000 .

[18]  Shingyu Leung,et al.  A fast local level set adjoint state method for first arrival transmission traveltime tomography with discontinuous slowness , 2013 .

[19]  H. Brezis.  SOLUTIONS WITH COMPACT SUPPORT OF VARIATIONAL INEQUALITIES , 1974 .

[20]  Michael S. Zhdanov,et al.  Three-dimensional regularized focusing inversion of gravity gradient tensor component data , 2004 .

[21]  Dominique Lesselier,et al.  Reconstruction of a 2-D binary obstacle by controlled evolution of a level-set , 1998 .

[22]  O. Dorn,et al.  Level set methods for inverse scattering , 2006 .

[23]  D. Oldenburg,et al.  Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method , 2003 .

[24]  M. Fedi,et al.  Inversion of potential field data using the structural index as weighting function rate decay , 2008 .

[25]  Shingyu Leung,et al.  Eulerian Gaussian Beams for High Frequency Wave Propagation , 2007 .

[26]  X. Li.  Efficient 3 D Gravity and Magnetic Modeling , 2010 .

[27]  Yaoguo Li.  3-D Inversion of Gravity Gradiometer Data , 2001 .

[28]  Yaoguo Li.  Processing Gravity Gradiometer Data Using an Equivalent Source Technique , 2001 .

[29]  M. Pilkington,et al.  Understanding imaging methods for potential field data , 2012 .

[30]  F. Condi,et al.  Resolution And Efficient Inversion of Gravity Gradiometry , 1999 .

[31]  C. Farquharson,et al.  2-D reconstruction of boundaries with level set inversion of traveltimes , 2013 .

[32]  D. Oldenburg,et al.  3-D inversion of gravity data , 1998 .

[33]  F. Santosa.  A Level-set Approach Inverse Problems Involving Obstacles , 1995 .

[34]  Jianliang Qian,et al.  An adaptive finite-difference method for traveltimes and amplitudes , 2002 .

[35]  Jianliang Qian,et al.  A level-set adjoint-state method for crosswell transmission-reflection traveltime tomography , 2014 .

[36]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[37]  Colin Farquharson,et al.  Three-dimensional modelling of gravity data using finite differences , 2009 .

[38]  S. Osher,et al.  Compressed modes for variational problems in mathematics and physics , 2013, Proceedings of the National Academy of Sciences.

[39]  Yaoguo Li,et al.  3D inversion of airborne gravity gradiometry data in mineral exploration: A case study in the Quadrilátero Ferrífero, Brazil , 2013 .

[40]  K. Kubik,et al.  Compact gravity inversion , 1983 .

[41]  J. B. Lee.  Falcon Gravity Gradiometer Technology , 2001 .

[42]  Jianliang Qian,et al.  A Local Level Set Method for Paraxial Geometrical Optics , 2006, SIAM J. Sci. Comput..

[43]  Stanley Osher,et al.  A survey on level set methods for inverse problems and optimal design , 2005, European Journal of Applied Mathematics.

[44]  M. Pilkington.  Analysis of gravity gradiometer inverse problems using optimal design measures , 2012 .

[45]  Uri M. Ascher,et al.  Multiple Level Sets for Piecewise Constant Surface Reconstruction in Highly Ill-Posed Problems , 2010, J. Sci. Comput..

[46]  Shingyu Leung,et al.  A THREE-DIMENSIONAL INVERSE GRAVIMETRY PROBLEM FOR ICE WITH SNOW CAPS , 2013 .

[47]  3D Inversion of Airborne Gravity Gradiomentry For Iron Ore Exploration In Brazil , 2010 .

[48]  W. Symes,et al.  Finite‐difference quasi‐P traveltimes for anisotropic media , 2001 .

[49]  Shingyu Leung,et al.  A level set based Eulerian method for paraxial multivalued traveltimes , 2004 .

[50]  G. Barnes,et al.  Imaging geologic surfaces by inverting gravity gradient data with depth horizons , 2012 .

[51]  Maurizio Fedi,et al.  DEXP: A fast method to determine the depth and the structural index of potential fields sources , 2007 .

[52]  Yaoguo Li,et al.  Inversion of gravity data using a binary formulation , 2006 .