Extended Diffraction-Slice Theorem for Wavepath Traveltime Tomography

Wei Dai, Ge Zhan, Xin Wang, and G. Schuster

KAUST and University of UtahMulti-source Least Squares Migration and Waveform Inversion This is an overview of some salient results from our 2003 UTAM research.OutlineFast Multisource+Precond. TheoryMultisource Waveform InversionMultisource Least Squares MigrationConclusionRTM Problem & Possible Soln.Problem: RTM computationally costly

Partial Solution: Multisource LSM RTM 3

Preconditioning speeds convergence by factor 2-3LSM reduces crosstalkMy talk is organized in the following way:

1. The first part is motivation. I will talk about a least squares migration (LSM ) advantages and challenges.2. The second part is theory for a deblurring filter, which is an alternative method to LSM.3. In the third part, I will show a numerical result of a deblurring filter.

4. The fourth is the main part of my talk.Deblurred LSM (DLSM) is a fast LSM with a deblurring filter.I will explain how to use the filter in LSM algorithm.

5. Then I will show numerical results of the DLSM.6. Then I will conclude my presentation.

Each figure has a slide number is shown at the footer.

3Multisource Migration:mmig=LTdForward Model:m =[LTL]-1LTdMultisrc-Least Sq. Migration :Multisource Least Squares Migration m = m - LT[Lm - d]f ~ [LTL]-1fSteepest DescentPreconditioned

d +d =[L +L ]m1221multisource preconditionermultisource modeler+adjointL{d{OutlineFast Multisource+Precond. TheoryMultisource Waveform InversionMultisource Least Squares MigrationConclusion9SEG/EAGE Salt Model

01640

Depth (km)X (km)15004500Velocity (m/s)

Time (s)X (km)X (km)CSGMultisource CSGRandom Time Shifted CSG and Add :m = m - LT[Lm - d] + reg.Multisource Least Squares Migration Workflowfd =d + d12Compute Preconditioner : f =[LTL]-1Iterate Preconditioned Regularized CG:

*f = Generate ~200 CSGs, Born approx:

d and d123Z (km)8

01.5Z (km)ModelLS M (30 its)Kirchhoff Migration

Model, KM, and LSM Images0 3kmLSM 10 srcs (5 its)KM 10 Srcs

LSM 10 srcs (30 its)90x1x1.5x9x0.1xThose are velocity and density models.

Velocity and density values are assigned to the layers based on the rock and fluid types.

I created a zero-offset section from these models.I will explain how I created the data in the next a couple of figures.

83Z (km)9

01.5Z (km)ModelKirchhoff Migration

LSM 10 srcs (5 its)KM 40 Srcs

LSM 40 srcs (30 its)

90x1x1.5x2.5x0.02xModel, KM, and LSM ImagesLS M (30 its)0 3kmThose are velocity and density models.

Velocity and density values are assigned to the layers based on the rock and fluid types.

I created a zero-offset section from these models.I will explain how I created the data in the next a couple of figures.

9Did Deblurring Help?CG deblurringStandard precond. CG01.4030Iteration #||Data Residual||Those are velocity and density models.

Velocity and density values are assigned to the layers based on the rock and fluid types.

I created a zero-offset section from these models.I will explain how I created the data in the next a couple of figures.

10Conclusions1. Empirical Results: Multisrc. LSM effective in suppressing crosstalk for up to 40 source supergather, but at loss of subtle detail. Did not achieve breakeven 2.5x > 1x. 3. Blending Limitation: Overdetermined>Undetermined2. Deblurring precond. >> Standard 1/r precond.2T4. Future: Better deblurring [L L] and regularizer-1Second I compute reflectivity model within this offset range from the velocity and density models.

I also created a source wavelet that mimics an air gun source signature.

Fdom = 25 Hz.

11OutlineFast Multisource+Precond. TheoryMultisource Waveform InversionMultisource Least Squares MigrationConclusionMultiscale Waveform Tomography1. Collect data d(x,t)2. Generate synthetic data d(x,t) by FD methodsyn.3. Adjust v(x,z) until ||d(x,t)-d(x,t) || minimized by CG.syn.24. To prevent getting stuck in local minima: a). Invert early arrivals initially

mute

7 b). Use multiscale: low freq. high freq.Multi-Source Waveform Inversion Strategy(Ge Zhan) Generate multisource field data with known time shift

Generate synthetic multisource data with known time shift from estimated velocity modelMultisource deblurring filter Using multiscale, multisource CG to update the velocity model with regularization

Initial velocity model144 shot gathers 12-Source Waveform Tomogram

0X(m)19100Z (m)59550002000m/s

Single-Source Waveform Tomogram0X(m)19100Z (m)59550002000m/s

Marmousi ModelAcoustic Marmousi Model and Multiscale Waveform Inversion12xSmooth Starting Model

50 iterations2000200012-Source Misfit Gradient vs Deblurred Gradient

Standard 12-Src Gradient19.5% Error7.1% ErrorDeblurred 12-Src Gradient

Residual Gradient vs # of Shots

SummaryMultisource+Precond. +CG Reduces CrosstalkMultisource Waveform Inversion: reduces computation by 12x for Marmousi

Multisource LSM: Reduces LSM computation $$ but still costs > standard mig. Problem: Need Formulas for S/N vs dxPotential O(10) speedup with 3DOutlineFast Multisource+Precond. TheoryMultisource Waveform InversionMultisource Least Squares MigrationMultisource MVAConclusion