Fast Least Squares Migration with a Deblurring Filter 30 October 2008 Naoshi Aoki 1

Outlines Motivation Deblurring filter theory A numerical result of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions 2

Outlines Motivation Deblurring filter theory A numerical result of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions 3

Forward and Inverse Problems for Acoustic Wavefield Forward problem: where d is data, L is forward modeling operator, and m is reflectivity model. Inverse problem: where L T is an adjoint of forward modeling operator, and [L T L] -1 is the inverse of Hessian. 4

Alternatives to Direct Inversion Migration LSM (e.g., Nemeth, Wu and Schuster,1999) where 5

The U Model Test 3D U Model Model Description Model size: –1.8 x 1.8 x 1.8 km U shape reflectivity anomaly Cross-spread geometry –Source : 16 shots, 100 m int. –Receiver : 16 receivers, 100 m int. Depth (m)Reflectivity ● Source ● Receiver U model is designed for testing Prestack 3D LSM with arbitrary 3D survey geometry. Data 0 5 TWT (s) X (m) 6

Depth Slices from Migration and LSM (c) Z = 250 m(e) Z = 750 m(g) Z=1250m (a) Actual Reflectivity Kirchhoff Migration Images (b) Test geometry (d) Z=250m LSM Images after 30 Iterations (f) Z=750m(h) Z=1250m ● Source ● Receiver 7

Challenges in LSM Processing Estimation of modeling operators –Velocity Model –Source wavelet Computational Cost –LSM typically requires 10 or more iterations. –Each LSM iteration requires about 3 times higher computational cost than that of the migration. 8

Outlines Motivation Deblurring filter theory A numerical result of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions 9

An Alternative to LSM Deblur the migration image with a local non-stationary filtering –Migration deconvolution (Hu and Schuster, 2001), –Deconvolution of migration operator by a local non-stationary filter (Etgen, 2002, Guitton 2004), –FFT based approach(e.g., Lecomte(2008); Toxopeus et al, (2008)). 10

Deblurring Filter Theory Actual Migration Image: Compute a reference migration image from a reference model m’: Find a deblurring operator with a matching filter (He, 2003) : Apply the operator to the actual migration image The computational cost is about one iteration of LSM 11

Outlines Motivation Deblurring filter theory A numerical result of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions 12

0 2.5 Z (km) X (km) Actual Reflectivity Model Point Scatterer Model Test TWT (sec) X (km) CSG Example F dominant = 5 Hz; λ=200 m Scatterer: 50 m x 50 m V=1000 m/s ▼▼▼▼▼▼▼▼▼▼▼▼▼ 13

Migration Image Z (km) X (km) Actual Reflectivity Image Z (km) X (km) Migration Image The Rayleigh resolution limit = 200 m 14

Deblurred Migration Image Z (km) X (km) Actual Reflectivity Image Z (km) X (km) Deblurred Migration Image

LSM Image Z (km) X (km) Actual Reflectivity Image Z (km) X (km) LSM Image after 30 Iterations 16

Horizontal Image of the Scatterer Reflectivity X(km) 17

Migration Deblurring Test Summary Deblurring filter improves spatial resolution of migration image about double. The computational cost is about one iteration of LSM. The deblurred migration image is slightly noisier than that in the LSM image. 18

Outlines Motivation Deblurring filter theory A numerical results of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions 19

Deblurred LSM Theory DLSM is a fast LSM with a deblurring filter. 2 types of DLSM algorithms are proposed: 1. Regularized DLSM (or RDLSM) where m apri is a skeletonized version of, and γ is a regularization parameter. 2. Preconditioned DLSM (or PDLSM) 20

Outlines Motivation Deblurring filter theory A numerical results of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions 21

Numerical Results A synthetic data set from the Marmousi2 model. A 2D marine data set from the Gulf of Mexico. 22

Marmousi2 Model Geological Cross Section (Martin et. al., 2006) 23

Velocity and Density Models Z (km) X (km) P wave Velocity Model Velocity (km/s) Z (km) X (km) Density Model Density (g/cc) 24

Traveltime Field Computation Z (km) X (km) P wave Velocity Model Velocity (km/s) Z (km) X (km) Traveltime Field Example 4 1 Velocity (km/s) (UTAM ray- tracing code written by He, 2002) 25

Reflectivity Model and Data Time (msec) Amplitude Source Wavelet Reflectivity Model 0 3 Z (km) 612 X (km) Fdom = 25 Hz 26

Reflectivity Model and Data Zero-offset Data 0 3 TWT (s) 612 X (km) Reflectivity Model 0 3 Z (km) 612 X (km)

Migration Image Poststack Migration 0 3 Z (km) 612 X (km) Actual Reflectivity Model 0 3 Z (km) 612 X (km) CPU time = 10 minutes on a dual processor 2.2 GHz Velocity: m/s Wavelength : m 28

Deblurring Filter with the Exact Model Step1: Compute Matching Operator Actual Migration Image 0 3 Z (km) 612 X (km) Exact Model 0 3 Z (km) 612 X (km) f 29

Deblurring Filter with the Exact Model Step2: Apply the Operator Deblurred Migration Image 0 3 Z (km) 612 X (km) Actual Migration Image 0 3 Z (km) 612 X (km) f 30

DLSM Convergence Curves PDLSM Iteration Number Residual Iteration Number Residual 8 19 Damping parameter: Γ= x0.5n-1, n=1,2,…,30 RDLSM 31

DLSM Images with the Exact Model 0 3 Z (km) 612 X (km) PDLSM after 8 Iterations 0 3 Z (km) 612 X (km) RDLSM after 19 Iterations 32

Model Sensitivity Test Exact model: –the actual model Geological model: –Skeletonized Migrated Image Grid model: –The region is divided into sections; each section has a point scatterer in the center. Exact Model 0 3 Z (km) 612 X (km) Geological Model 0 3 Z (km) 612 X (km) Zoom View of Grid Model 1 2 Z (km) X (km) 250 x 250 m 33

Deblurring Filter with the Geological Model Step1: Compute Matching Operator Reference Migration Image 0 3 Z (km) 612 X (km) Geological Model 0 3 Z (km) 612 X (km) f 34

Deblurring Filter with the Geological Model Step2: Apply the Operator Deblurred Migration Image 0 3 Z (km) 612 X (km) Actual Migration Image 0 3 Z (km) 612 X (km) f 35

DLSM Convergence Curves Preconditioned DLSM Iteration Number Residual Regularized DLSM Iteration Number Residual Damping parameter: Γ= x0.5n-1, n=1,2,…,30 36

DLSM Images with the Geological Model 0 3 Z (km) 612 X (km) PDLSM after 12 Iterations 0 3 Z (km) 612 X (km) RDLSM after 20 Iterations 37

Zoom View of Grid Model 1 2 Z (km) X (km) Deblurring Filter with the Grid Model Step1: Compute Matching Operator Reference Migration Image 0 3 Z (km) 612 X (km) f 38

Deblurring Filter with the Grid Model Step2: Apply the Operator Deblurred Migration Image 0 3 Z (km) 612 X (km) Actual Migration Image 0 3 Z (km) 612 X (km) f 39

Regularized DLSM Iteration Number Residual Damping parameter: Γ= x0.5n-1, n=1,2,…,30 DLSM Convergence Curves Preconditioned DLSM Iteration Number Residual

DLSM Images with the Grid Model 0 3 Z (km) 612 X (km) RDLSM after 20 Iterations 0 3 Z (km) 612 X (km) PDLSM after 10 Iterations 41

Marmousi2 Test Summary (1) The deblurring filter can expedite the computation of an LSM image. –RDLSM and PDLSM provide acceptable LSM images with about 2/3 and 1/3 the cost of standard LSM, respectively. Controlling the model dependency is required. –RDLSM can control the model dependency with a regularization parameter. –In the PDLSM algorithm, not using a deblurring filter after several iteration is recommended. 42

Marmousi2 Test Summary (2) DLSM with the geological model –Computation of an LSM image can be expedited by a human interpretation. –A risk is an erroneous interpretation. The model dependency should be carefully controlled. DLSM with the grid model –The result is not good as that from a better geological model. –An advantage is that no expense of a human interpretation is required for the model building. 43

The Gulf of Mexico Data Test 8 4 TWT(s) X (km) D Poststack Marine Data 44

The Gulf of Mexico Data Test Both the regularization and preconditioning schemes are employed in the DLSM. A geological model is created by the following way: 1.A deblurred migration image is obtained with a grid model. 2.A geological model is created by cosmetic filtering and skeletonizing the deblurred migration image. 45

Zero-offset Data from for a Grid Model 8 4 TWT( ｓ ) X (km) 18 0 Scatterer Interval: 500 m x 500 m 46

Zoom View of Reference Migration Image for a Grid Model Z (km) X (km)

Kirchhoff Migration Z (km) X (km)

Deblurred Migration Image Z (km) X (km)

Geological Model Z (km) X (km) Reflectivity 50

Comparison of Imaging Results Z (km) 813 X (km) Kirchhoff Migration 51

Box A: Comparison of Images Z (km) X (km) Migration Z (km) X (km) LSM after 3 Iterations Z (km) X (km) DLSM after 3 Iterations Z (km) X (km) LSM after 10 Iterations 52

Box B: Comparison of Images Migration 1 1.2Z (km) 1112 X (km) LSM after 3 Iterations 1.2 Z (km) 1112 X (km) 1 DLSM after 3 Iterations 1.2 Z (km) 1112 X (km) 1 LSM after 10 Iterations 1.2 Z (km) 1112 X (km) 1 53

Total Computational Cost Migration 1 1.2Z (km) 1112 X (km) LSM after 3 Iterations 1.2 Z (km) 1112 X (km) 1 DLSM after 3 Iterations 1.2 Z (km) 1112 X (km) 1 LSM after 10 Iterations 1.2 Z (km) 1112 X (km)

Total Computational Cost Migration 1 LSM 3 Iterations9 LSM 10 Iterations30 DLSM 3 Iterations19+ –Deblurring with the grid model3 –Deblurring with the geological model 4+ –DLSM 3 Iterations12 55

The GOM Data Test Summary DLSM can successfully provide an improved LSM image with an affordable computer expense. 56

Outlines Motivation Deblurring filter theory A numerical results of the deblurring filter Deblurred LSM theory Numerical results of the deblurred LSM Conclusions 57

Conclusions A deblurring filter provides a fine apriori model for a regularized LSM, and it can also be used as an effective preconditioning filter. The DLSM algorithms provids acceptable LSM images with 1/3 – 2/3 the cost of standard LSM. 58

Future Works The deblurring filter requires some improvement in quality and efficiency. A computer-aided skeletonization method is required for reducing an expense of a human interpretation. 59

Acknowledgements I would like to thank Prof. Gerard T. Schuster for his encouragement throughout my stay at the University of Utah. I also want to thank my group colleagues for their academic discussions and personal help. I also thank JOGMEC and JAPEX for supporting my study at the University of Utah. 60

Thanks 61