Numeric implementation of wave-equation migration velocity analysis operators

Wave-equation migration velocity analysis MVA is a technique similar to wave-equation tomography because it is designedtoupdatevelocitymodelsusinginformationderivedfrom full seismic wavefields. On the other hand, wave-equation MVA is similar to conventional, traveltime-based MVAbecause it derives the information used for model updates from properties of migrated images, e.g., focusing and moveout. The main motivation for using wave-equation MVAis derived from its consistency with the corresponding wave-equation migration, which makes this technique robust and capable of handling multipathing characterizing media with large and sharp velocity contrasts. The wave-equation MVAoperators are constructed using linearizations of conventional wavefield extrapolation operators, assuming small perturbations relative to the background velocity model. Similar to typical wavefield extrapolation operators, the wave-equation MVAoperators can be implemented in the mixed space-wavenumber domain using approximations of different orders of accuracy.As for wave-equation migration, wave-equation MVA can be formulated in different imaging frameworks, dependingonthetypeofdatausedandimageoptimizationcriteria. Examples of imaging frameworks correspond to zero-offset migrationdesigned for imaging based on focusing properties of the image, survey-sinking migration designed for imaging based on moveout analysis using narrow-azimuth data, and shot-record migration also designed for imaging based on moveout analysis, but using wide-azimuth data. The wave-equation MVA operators formulated for the various imaging frameworks are similar because they share elements derived from linearizations of the single square-root equation. Such operators represent the core of iterative velocity estimation based on diffractionfocusingorsemblanceanalysis,andtheirapplicabilityin practice requires efficient and accurate implementation. This tutorial concentrates strictly on the numeric implementation of thoseoperatorsandnotontheiruseforiterativemigrationvelocityanalysis.

[1]  Sergey Fomel,et al.  Theory of 3-D angle gathers in wave-equation imaging , 2004 .

[2]  W. Symes,et al.  Angle‐domain common‐image gathers for migration velocity analysis by wavefield‐continuation imaging , 2004 .

[3]  Paul Sava,et al.  Image Differencing and Focusing in Wave-Equation Velocity Analysis , 2006 .

[4]  Guy Chavent,et al.  Migration‐based traveltime waveform inversion of 2-D simple structures: A synthetic example , 2001 .

[5]  Peng Shen,et al.  Differential Semblance Velocity Analysis Via Shot Profile Migration , 2005 .

[6]  Gilles Lambaré,et al.  Stereotomography: a semi‐automatic approach for velocity macromodel estimation , 2004 .

[7]  Paul Sava,et al.  Offset and angle-domain common image-point gathers for shot-profile migration , 2002 .

[8]  D. Ristow,et al.  Fourier finite-difference migration , 1994 .

[9]  Paul Sava,et al.  Wave-Equation Migration Velocity Analysis , 2004 .

[10]  Oz Yilmaz,et al.  Migration velocity analysis by wave-field extrapolation , 1984 .

[11]  A. J. Berkhout,et al.  Seismic Migration: Imaging of Acoustic Energy by Wave Field Extrapolation , 1980 .

[12]  R. Pratt.  Seismic waveform inversion in the frequency domain; Part 1, Theory and verification in a physical scale model , 1999 .

[13]  Peter Mora,et al.  Inversion = Migration + Tomography , 1988, Shell Conference.

[14]  R. G. Pratt.  Velocity Models From Frequency Domain Waveform Tomography – Past, Present and Future , 2004 .

[15]  Biondo Biondi,et al.  Angle-domain common image gathers by wave-equation migration , 1999 .

[16]  Josef Paffenholz,et al.  Subsalt Multiple Attenuation And Imaging: Observations From the Sigsbee2B Synthetic Dataset , 2002 .

[17]  Ru-Shan Wu,et al.  Extracting Angle Domain Information From Migrated Wavefield , 2002 .

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

[19]  Uwe Albertin,et al.  Localized image‐difference wave‐equation tomography , 2007 .

[20]  A. Tarantola.  Inverse problem theory : methods for data fitting and model parameter estimation , 1987 .

[21]  A. J. Berkhout,et al.  Angle-dependent reflectivity by means of prestack migration , 1990 .

[22]  R. T. Cutler,et al.  Tomographic determination of velocity and depth in laterally varying media , 1985 .

[23]  Marta Woodward,et al.  Wave-equation tomography , 1992 .

[24]  Paul Sava,et al.  Wave-equation migration velocity analysis. I. Theory , 2004 .

[25]  Paul Sava,et al.  Angle-domain common-image gathers by wavefield continuation methods , 2003 .

[26]  Kamal M. Al-Yahya,et al.  Velocity analysis by iterative profile migration , 1987 .

[27]  Mark Noble,et al.  Migration velocity analysis from locally coherent events in 2‐D laterally heterogeneous media, Part I: Theoretical aspects , 2002 .

[28]  Paul Sava,et al.  Wave‐equation migration velocity analysis. II. Subsalt imaging examples , 2004 .

[29]  Robert Soubaras.  Angle Gathers for Shot-Record Migration by Local Harmonic Decomposition , 2003 .

[30]  Ru-Shan Wu,et al.  Extended local Born Fourier migration method , 1999 .

[31]  Paul Sava,et al.  Wave‐equation migration velocity analysis , 2004 .

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

[33]  J. Sherwood,et al.  The Wave Equation Applied to Migration , 1976 .

[34]  Paul L. Stoffa,et al.  Split-Step Fourier Migration , 1990 .

[35]  Jon F. Claerbout,et al.  Imaging the Earth's Interior , 1985 .

[36]  Biondo Biondi,et al.  Equivalence of source-receiver migration and shot-prole migration , 2003 .

[37]  Guy Chavent,et al.  Determination of background velocities by multiple migration fitting , 1995 .

[38]  Zhenyue Liu,et al.  Migration Velocity Analysis. , 1995 .

[39]  Paul Sava,et al.  Wave-equation migration velocity analysis by focusing diffractions and reflections , 2005 .

[40]  Robert Soubaras,et al.  Velocity model building by semblance maximization of modulated‐shot gathers , 2006 .

[41]  Robert Soubaras.  Angle gathers for shot-record migration by local harmonic decomposition in 73rd Ann , 2003 .

[42]  William W. Symes,et al.  Reverse time migration with optimal checkpointing , 2007 .

[43]  Gilles Lambaré,et al.  Practical aspects and applications of 2D stereotomography , 2003 .

[44]  R. Shipp,et al.  Seismic waveform inversion in the frequency domain, Part 2: Fault delineation in sediments using crosshole data , 1999 .

[45]  Alexander M. Popovici.  Prestack migration by split‐step DSR , 1996 .

[46]  Christof Stork,et al.  REFLECTION TOMOGRAPHY IN THE POSTMIGRATED DOMAIN , 1992 .

[47]  Paul Sava,et al.  Wave-equation migration velocity analysis — I : Theory Geophysical Prospecting , accepted for publication , .

[48]  Scott A. Morton,et al.  Differential Semblance Velocity Analysis by Wave Equation Migration, Applications to High Resolution HAFB Dataset , 2003 .

[49]  R. Gerhard Pratt,et al.  Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies , 2004 .

[50]  Mark Noble,et al.  Migration velocity analysis from locally coherent events in 2-D laterally heterogeneous media, Part II: Applications on synthetic and real data , 2002 .

[51]  Paul Sava,et al.  Adjoint wave-equation velocity analysis , 2006 .

[52]  Robert Soubaras,et al.  Velocity model building by semblance maximization of modulated-shot gathers , 2007 .

[53]  Biondo Biondi,et al.  3D angle‐domain common‐image gathers for migration velocity analysis , 2004 .

[54]  Paul Sava,et al.  Time-shift imaging condition in seismic migration , 2006 .

[55]  J. Claerbout,et al.  Incorporating geologic information into reflection tomography , 2004 .

[56]  Charles C. Mosher,et al.  Common angle imaging conditions for pre-stack depth migration , 2000 .

[57]  Christiaan C. Stolk,et al.  Kinematic artifacts in prestack depth migration , 2004 .

[58]  Oz Yilmaz,et al.  Migration velocity analysis by wave-field extrapolation , 1984 .