Implicit 3-D depth migration with helical boundary conditions James Rickett, Jon Claerbout & Sergey Fomel Stanford University

Implicit 3-D depth migration with helical boundary conditions James Rickett, Jon Claerbout & Sergey Fomel Stanford University

Implicit 3-D depth migration with helical boundary conditions Implicit extrapolation 45  equation 2D vs 3D Helical boundary conditions Lateral velocity variations

Isotropic impulse response

Wavefield extrapolation Ideally: Explicit: Implicit:

Advantages of implicit extrapolators –Unitary –More accurate for a given filter size BUT: –Need to inverse filter Wavefield extrapolation

Implicit extrapolation with the 45  equation Differential equation: Matrix equation:

Implicit extrapolation with the 45  equation where

2-D implicit depth migration Matrix D is tridiagonal –easily invertible (cost N) 2-D implicit depth migration widely used

3-D implicit depth migration Matrix D is blocked tridiagonal –NOT easily invertible –Splitting methods 3-D implicit not widely used –Explicit methods

2D filter1D filter Helical boundary conditions

Rapid multi-D recursive inverse filtering: 1.Remap filter to 1-D 2.Factor 1-D filter into CCF of 2 minimum- phase filters 3.Divide by 2 minimum-phase filters Helical boundary conditions

3-D implicit depth migration PROBLEM: 2-D inverse filtering Non-causal 1-D filter Causal/anti-causal filter pair LU decomposition Helix 2-D filter1-D filter Spectral factorization

3-D implicit depth migration

Spectral factorization Estimate a minimum-phase function with a given spectrum Algorithm requirements: –Cross-spectra –Filter-size specified a priori

Extension to cross-spectra BUT: Frequency domain –Non-zero coefficients cannot be specified a priori Kolmogoroff factorization

Newton's iteration for square roots: Wilson-Burg factorization Generalized to polynomials (time series):

Iterative –Quadratic convergence Cross-spectra Non-zero coefficients specified a priori Wilson-Burg factorization

3-D impulse response Broad-band Dip-limited Cross- sections:

3-D impulse response Time-slices:

Lateral velocity variations Advantage of f-x vs f-k –Factor spatially variable filters –Non-stationary inverse filtering Rapid –Factors can be precomputed/tabulated Approximation –Similar to explicit methods

Lateral velocity variations Alternative method –Wilson-Burg factorization of non-stationary filters –More accurate –More expensive

3-D depth migration model

3-D depth migration results

Conclusions Shown how helical boundary conditions enable implicit 3-D wavefield extrapolation Lateral variations in velocity are handled by non-stationary inverse filtering

Conclusions Demonstrated 3-D depth migration with 45  wave equation Helical boundary conditions applicable for full range of implicit migration methods