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**Multi-source Least Squares Migration and Waveform Inversion**

Wei Dai, Ge Zhan, Xin Wang, and G. Schuster KAUST and University of Utah This is an overview of some salient results from our 2003 UTAM research.

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**Outline Fast Multisource+Precond. Theory**

Multisource Least Squares Migration Multisource Waveform Inversion Conclusion

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**RTM Problem & Possible Soln.**

Problem: RTM computationally costly Partial Solution: Multisource LSM RTM Preconditioning speeds convergence by factor 2-3 LSM reduces crosstalk My 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. 3 3

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**Multisource Migration: Multisrc-Least Sq. Migration :**

Multisource Least Squares Migration d { L { d +d =[L +L ]m 1 2 Forward Model: Multisource Migration: mmig=LTd m =[LTL]-1LTd Multisrc-Least Sq. Migration : multisource preconditioner multisource modeler+adjoint m’ = m LT[Lm - d] f ~ [LTL]-1 f Steepest Descent Preconditioned

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**Outline Fast Multisource+Precond. Theory**

Multisource Least Squares Migration Multisource Waveform Inversion Conclusion

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**SEG/EAGE Salt Model 4500 Velocity (m/s) Depth (km) 4 X (km) 16 1500 9**

Velocity (m/s) Depth (km) 4 X (km) 16 1500 CSG Multisource CSG Time (s) 9 X (km) X (km)

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**Generate ~200 CSGs, Born approx: Random Time Shifted CSG and Add :**

Multisource Least Squares Migration Workflow d and d 1 2 Generate ~200 CSGs, Born approx: d =d + d 1 2 Random Time Shifted CSG and Add : *f = Compute Preconditioner : f = [LTL] -1 Iterate Preconditioned Regularized CG: m’ = m LT[Lm - d] + reg. f

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**Model, KM, and LSM Images Z (km) 1.5 0 3km Model LS M (30 its)**

Kirchhoff Migration 90x 1x 1.5x 9x 0.1x Z (km) Z (km) 3 1.5 km LSM 10 srcs (5 its) LSM 10 srcs (30 its) KM 10 Srcs 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. 8 8

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**Model, KM, and LSM Images Z (km) 1.5 0 3km Model LS M (30 its)**

Kirchhoff Migration 90x 1x 1.5x 2.5x 0.02x Z (km) Z (km) 3 1.5 km LSM 10 srcs (5 its) LSM 40 srcs (30 its) KM 40 Srcs 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. 9 9

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**Did Deblurring Help? Iteration # Standard precond. CG CG deblurring**

1.4 Standard precond. CG ||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. CG deblurring Iteration # 30 10

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Conclusions 1. 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. 2. Deblurring precond. >> Standard 1/r precond. 2 3. Blending Limitation: Overdetermined>Undetermined Second 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. T 4. Future: Better deblurring [L L] and regularizer -1 11

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**Outline Fast Multisource+Precond. Theory**

Multisource Least Squares Migration Multisource Waveform Inversion Conclusion

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**Multiscale Waveform Tomography**

1. Collect data d(x,t) 2. Generate synthetic data d(x,t) by FD method syn. 3. Adjust v(x,z) until ||d(x,t)-d(x,t) || minimized by CG. syn. 2 4. To prevent getting stuck in local minima: a). Invert early arrivals initially mute b). Use multiscale: low freq high freq. 7

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**Multi-Source Waveform Inversion Strategy**

(Ge Zhan) Generate multisource field data with known time shift 144 shot gathers Initial velocity model Generate synthetic multisource data with known time shift from estimated velocity model Using multiscale, multisource CG to update the velocity model with regularization Multisource deblurring filter

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**Acoustic Marmousi Model and Multiscale Waveform Inversion**

Single-Source Waveform Tomogram Marmousi Model m/s 5000 Z (m) 2000 595 X(m) 1910 Smooth Starting Model 12-Source Waveform Tomogram m/s 12x 5000 50 iterations Z (m) 2000 595 X(m) 1910

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**12-Source Misfit Gradient vs Deblurred Gradient**

Standard 12-Src Gradient 19.5% Error 2000 Deblurred 12-Src Gradient 7.1% Error 2000

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**Residual Gradient vs # of Shots**

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**Summary Multisource+Precond. +CG Reduces Crosstalk**

Multisource 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 dx Potential O(10) speedup with 3D

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**Outline Fast Multisource+Precond. Theory**

Multisource Least Squares Migration Multisource Waveform Inversion Multisource MVA Conclusion

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