Combining asymptotic linearized inversion and full waveform inversion

S U M M A R Y A method for combining the asymptotic operator designed by Beylkin (Born migration operator) for the solution of linearized inverse problems with full waveform inversion is presented. This operator is used to modify the standard L2 norm that measures the distance between synthetic and observed data. The modified misfit function measures the discrepancy of the synthetic and observed data after they have been migrated using the Beylkin operator. The gradient of this new misfit function is equal to the cross-correlation of the single scattering data with migrated/demigrated residuals. The modified misfit function possesses a Hessian operator that tends asymptotically towards the identity operator. The trade-offs between discrete parameters are thus reduced in this inversion scheme. Results on 2-D synthetic case studies demonstrate the fast convergence of this inversion method in a migration regime. From an accurate estimation of the initial velocity, three and five iterations only are required to generate high-resolution P-wave velocity estimation models on the Marmousi 2 and synthetic Valhall case studies.

[1]  A. Tarantola.  Inversion of seismic reflection data in the acoustic approximation , 1984 .

[2]  Gregory Beylkin,et al.  Linearized inverse scattering problems in acoustics and elasticity , 1990 .

[3]  Gregory Beylkin,et al.  Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform , 1985 .

[4]  N. Bleistein.  On the imaging of reflectors in the earth , 1987 .

[5]  Andreas Fichtner,et al.  Theoretical background for continental‐ and global‐scale full‐waveform inversion in the time–frequency domain , 2008 .

[6]  R. Plessix,et al.  Full waveform inversion and distance separated simultaneous sweeping: a study with a land seismic data set , 2012 .

[7]  Mark Noble,et al.  Wavelengths of Earth structures that can be resolved from seismic reflection data , 1989 .

[8]  M. Warner,et al.  Anisotropic 3D full-waveform inversion , 2013 .

[9]  Romain Brossier,et al.  Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation , 2013 .

[10]  William W. Symes,et al.  Velocity inversion by differential semblance optimization , 1991 .

[11]  P. Podvin,et al.  Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools , 1991 .

[12]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[13]  Jean Virieux,et al.  An overview of full-waveform inversion in exploration geophysics , 2009 .

[14]  A. Devaney.  A filtered backpropagation algorithm for diffraction tomography. , 1982, Ultrasonic imaging.

[15]  Gary Martin,et al.  Marmousi2 An elastic upgrade for Marmousi , 2006 .

[16]  S. Operto,et al.  Fast full waveform inversion with source encoding and second-order optimization methods , 2015 .

[17]  Ludovic Métivier,et al.  Full waveform inversion and the truncated Newton method: quantitative imaging of complex subsurface structures , 2014 .

[18]  A. Tarantola,et al.  Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results , 1986 .

[19]  S. Operto,et al.  Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 2: imaging compressive-wave and shear-wave velocities , 2013 .

[20]  Mark Noble,et al.  3-D preserved amplitude prestack depth migration on a workstation , 1999 .

[21]  Stéphane Operto,et al.  3 D ray + Born migration / inversion — Part 1 : Theory , 2003 .

[22]  Tariq Alkhalifah,et al.  A recipe for practical full-waveform inversion in anisotropic media: An analytical parameter resolution study , 2014 .

[23]  Hongkai Zhao,et al.  A fast sweeping method for Eikonal equations , 2004, Math. Comput..

[24]  Jean Virieux,et al.  Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt , 2004 .

[25]  Mario Bebendorf,et al.  Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems , 2008 .

[26]  S. Operto,et al.  Combining Diving and Reflected Waves for Velocity Model Building by Waveform Inversion , 2014 .

[27]  Ludovic Métivier.  Interlocked optimization and fast gradient algorithm for a seismic inverse problem , 2011, J. Comput. Phys..

[28]  Tariq Alkhalifah,et al.  Efficient traveltime compression for 3D prestack Kirchhoff migration , 2011 .

[29]  Denes Vigh,et al.  3D Full waveform inversion on a Gulf of Mexico WAZ data set , 2010 .

[30]  G. Herman,et al.  Three-dimensional, nonlinear, asymptotic seismic inversion , 1996 .

[31]  Stéphane Operto,et al.  3D ray+Born migration/inversion—Part 1: Theory , 2003 .

[32]  R. Plessix.  A review of the adjoint-state method for computing the gradient of a functional with geophysical applications , 2006 .

[33]  Stéphane Operto,et al.  Fast 2-D ray+Born migration/inversion in complex media , 1999 .

[34]  T. Moser.  Review of ray-Born forward modeling for migration and diffraction analysis , 2012, Studia Geophysica et Geodaetica.

[35]  C. Shin,et al.  Improved amplitude preservation for prestack depth migration by inverse scattering theory , 2001 .

[36]  L. Sirgue,et al.  3D Frequency Domain Waveform Inversion Using Time Domain Finite Difference Methods , 2008 .

[37]  R. Plessix,et al.  Thematic Set: Full waveform inversion of a deep water ocean bottom seismometer dataset , 2010 .

[38]  Jack K. Cohen,et al.  Computational and asymptotic aspects of velocity inversion , 1985 .

[39]  Ludovic Métivier,et al.  Full Waveform Inversion and the Truncated Newton Method , 2013, SIAM J. Sci. Comput..

[40]  Andrew Ratcliffe,et al.  Anisotropic 3 D full-waveform inversion , 2013 .

[41]  Patrick R. Amestoy,et al.  Multifrontal parallel distributed symmetric and unsymmetric solvers , 2000 .

[42]  Jean Charles Gilbert,et al.  Numerical Optimization: Theoretical and Practical Aspects , 2003 .

[43]  Hicks,et al.  Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion , 1998 .

[44]  Jean Virieux,et al.  Quantitative imaging of complex structures from dense wide‐aperture seismic data by multiscale traveltime and waveform inversions: a case study , 2004 .

[45]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[46]  Acoustic multiparameter Full-waveform Inversion of diving and reflected waves through a hierachical scheme , 2014 .

[47]  Romain Brossier,et al.  On the footprint of anisotropy on isotropic full waveform inversion: the Valhall case study , 2011 .

[48]  Spyros Lazaratos,et al.  Improving the Convergence Rate of Full Wavefield Inversion Using Spectral Shaping , 2011 .

[49]  Jean Virieux,et al.  Crustal seismic imaging from multifold ocean bottom seismometer data by frequency domain full waveform tomography: Application to the eastern Nankai trough , 2006 .

[50]  Toward Gauss-Newton and Exact Newton Optimization for Full Waveform Inversion , 2012 .

[51]  Fuchun Gao,et al.  Waveform tomography at a groundwater contamination site: Surface reflection data , 2007 .

[52]  S. Operto,et al.  Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion , 2009 .

[53]  Ludovic Métivier,et al.  A 2D nonlinear inversion of well-seismic data , 2011 .

[54]  R. Plessix,et al.  Waveform acoustic impedance inversion with spectral shaping , 2013 .

[55]  Ludovic Métivier,et al.  The SEISCOPE optimization toolbox: A large-scale nonlinear optimization library based on reverse communication , 2016 .

[56]  Ludovic Métivier,et al.  A guided tour of multiparameter full-waveform inversion with multicomponent data: From theory to practice , 2013 .

[57]  F. Kroode.  A wave-equation-based Kirchhoff operator , 2012 .

[58]  J. Vidale.  Finite-difference calculation of travel times , 1988 .

[59]  S. Operto,et al.  Mixed‐grid and staggered‐grid finite‐difference methods for frequency‐domain acoustic wave modelling , 2004 .

[60]  William W. Symes,et al.  Migration velocity analysis and waveform inversion , 2008 .