Velocity Analysis Using Shaping Regularization Sergey Fomel UTAM Sponsor Meeting 2009

“If I have seen a little further it is by standing on the shoulders of Giants.” Isaac NewtonRobert Hooke February 5, 1676

Linearization Hooke’s law Hooke’s law elastic theory elastic theory seismic waves seismic waves Newton’s method Newton’s method gradient optimization gradient optimization seismic tomography seismic tomography  

The Marmousi experience (Versteeg, 1994)

Outline Shaping framework for nonlinear inversion Shaping framework for nonlinear inversion R-algorithm (Goldin, 1986) R-algorithm (Goldin, 1986) Iterative thresholding (Daubechies et al., 1994) Iterative thresholding (Daubechies et al., 1994) Toy problem Toy problem 1-D velocity estimation 1-D velocity estimation local slopes local slopes Collaborative research environments Collaborative research environments

Forward and Backward m model (what we want to find) m model (what we want to find) d data (what we measure) d data (what we measure) d = F[m] forward operator (accurate) d = F[m] forward operator (accurate) m 0 = B[d] backward operator (affordable) m 0 = B[d] backward operator (affordable) m = m + B[d] - BF[m] m = m + B[d] - BF[m] m k+1 = m k + B[d] - BF[m k ] m k+1 = m k + B[d] - BF[m k ] Landweber iteration (R-algorithm) Landweber iteration (R-algorithm)

Sergey V. Goldin ( )

Landweber Iteration m k+1 = m k + B[d] - BF[m k ] m k+1 = m k + B[d] - BF[m k ] converges to the solution of B[d]=BF[m] converges to the solution of B[d]=BF[m] if spectral radius (I-BF) < 1 if spectral radius (I-BF) < 1 alternatively iteration in the data space alternatively iteration in the data space d k+1 = d k + d - FB[d k ] d k+1 = d k + d - FB[d k ] and m k = B[d k ] and m k = B[d k ] “adding the noise back” (Osher et al., 2005)

Regularized Iteration m k+1 = S[m k + B[d] - BF[m k ]] m k+1 = S[m k + B[d] - BF[m k ]] S shaping operator S shaping operator Example: sparsity-constrained inversion Example: sparsity-constrained inversion (Daubecheis et al., 2004) (Daubecheis et al., 2004) Thresholding Thresholding ║ d - F m ║ 2 + ║ m ║ 1 ║ d - F m ║ 2 + ║ m ║ 1 m = S[m + B[d] - BF[m]] m = S[m + B[d] - BF[m]] m = [I + S (BF-I)] -1 SB[d] m = [I + S (BF-I)] -1 SB[d]

Shaping Regularization m = [I + S (BF-I)] -1 SB[d] m = [I + S (BF-I)] -1 SB[d] (Fomel, 2007; 2008) (Fomel, 2007; 2008) Shaping

Shaping versus Tikhonov/Bayes m = [I + S (BF-I)] -1 SB d m = [I + S (BF-I)] -1 SB d m = M F * [F M F * +N] -1 d m = M F * [F M F * +N] -1 d S = [I+ M -1 ] -1 S = [I+ M -1 ] -1 B = F * N -1 B = F * N -1

Outline Shaping framework for nonlinear inversion Shaping framework for nonlinear inversion R-algorithm (Goldin, 1986) R-algorithm (Goldin, 1986) Iterative thresholding (Daubechies et al., 1994) Iterative thresholding (Daubechies et al., 1994) Toy problem Toy problem 1-D velocity estimation 1-D velocity estimation local slopes local slopes Collaborative research environments Collaborative research environments

Toy problem Model m Model m Data and forward operator d = F[m] Data and forward operator d = F[m] Backward operator m 0 = B[d] Backward operator m 0 = B[d] local slopes local slopes Shaping operator S[m] Shaping operator S[m] smooth models smooth models blocky models blocky models

Forward Operator  zz (Stoffa et al, 1982)

Predict each trace from its neighbor Predict each trace from its neighbor Minimize prediction error (least squares) Minimize prediction error (least squares) Apply shaping regularization Apply shaping regularization

Backward Operator (Fomel, 2007)

Backward Operator Interval velocity as a data attribute! Interval velocity as a data attribute! T( ,p), v( ,p) T( ,p), v( ,p)

Toy problem Assumptions Assumptions 1-D 1-D no multiples no multiples no AVO no AVO Forward operator Forward operator ray tracing ray tracing Backward operator Backward operator local slopes local slopes hyperbolic moveout hyperbolic moveout Shaping operator smooth blocky Extensions 3-D waveform modeling stereoslopes anisotropy

Outline Shaping framework for nonlinear inversion Shaping framework for nonlinear inversion R-algorithm (Goldin, 1986) R-algorithm (Goldin, 1986) Iterative thresholding (Daubechies et al., 1994) Iterative thresholding (Daubechies et al., 1994) Toy problem Toy problem 1-D velocity estimation 1-D velocity estimation local slopes local slopes Collaborative research environments Collaborative research environments

Collaborative Research Environment “Reproducible research” “Reproducible research” Open source Open source Open community Open community Open science Open science

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Conclusions Shaping regularization is a general framework for linear and nonlinear inverse problems. Shaping regularization is a general framework for linear and nonlinear inverse problems. Velocity analysis and velocity model building Velocity analysis and velocity model building Local slope estimation Local slope estimation Nonstationary matching filtering Nonstationary matching filtering Shared research environments enable world-wide collaboration and computational reproducibility. Shared research environments enable world-wide collaboration and computational reproducibility. “Standing on the shoulders of Giants” “Standing on the shoulders of Giants”