Jie Hou. Education. Research Interest True Amplitude Seismic Imaging and Inversion Acceleration of Least Squares Migration Inversion Velocity Analysis

Transcription

1 Jie Hou Education Rice University Ph.D. Candidate in Geophysics, Earth Science 09/2012 Present China University of Petroleum(East China) 09/ /2012 B.S. in Exploration Geophysics Thesis: High Order Finite-difference Modeling of Acoustic and Elastic Wave Research Interest True Amplitude Seismic Imaging and Inversion Acceleration of Least Squares Migration Inversion Velocity Analysis

2 .. An Approximate Inverse to the Extended Born Modeling Operator Jie Hou 2014 Review Meeting May 1st, 2015 Slides based on same title paper submitted to Geophysics

3 Linearized Inverse Problem Model m F[m] = d F 1 [d] = m Data d Born Approximation = Linearized Seismic Inverse Problem Model Separation First Order Approximation m = m 0 + δm F[m] F[m 0 ] + F[m 0 ]δm Linearized Map F[m 0 ]δm δd 1

4 From Imaging to Inversion Given m 0 (x), δd(x r, t; x s ), find δm(x) to fit the data: F[m 0 ]δm δd Imaging Locate the reflector Kinematically Adjoint Operator F RTM Inversion Recover the reflector Kinematically & Dynamically Inverse Operator F 1 True Amplitude RTM 2

5 Born Modeling and its Adjoint. Born Modeling and Migration Operator. F[v]δv(x r, t; x s ) = 2 t 2 dx F [v]d(x) = 2 v 3 (x). dτ 2δv(x) v 3 (x) G(x, t τ; x r)g(x, τ; x s ) dx s dx r dtdτg(x, τ; x s ) 2 d(x r, t; x s ) t 2 G(x, t τ; x r ) S R S R (a) Born Modeling (b) Born Migration 3

6 Extended Model Extended Model M E[m] = m X[ m] = m F m = d F 1 d = m Model M Fm = d F 1 d = m Data D M = physical model space M = bigger extended model space F : M D extended modeling operator Extension Property: E[M] M; m M Fm = Fm 4

7 Subsurface offset Extension S h h R Subsurface Extension : 2h = Difference between subsurface scattering points (subsurface offset) Physical meaning : action at a positive distance Extend the operator by permitting δv to also depend on (half) offset h.. Extended Born Modeling and Migration Operator. F[v]δv(x s, x r, t) = 2 t 2 F d(x, h) = 2. v 3 (x) dxdhdτg(x + h, t τ; x r ) 2δv(x, h) v 3 G(x h, τ; x s ) (x) dx s dx r dtdτg(x h, τ; x s )G(x + h, t τ; x r ) 2 d(x r, t; x s ) t 2 5

8 Extended Kirchhoff Operator Fons ten Kroode (2012) constructed the inverse of the extended Kirchhoff Operator (in asymptotic sense) :. Fons ten Kroode,2012. Ki = 1 dxdhdωe iωt G(x r, x + h, ω) 2π Ĩd = 32. πv 2 (x) i(x, h) G(x h, x s, ω) z dx r dx s dω( iω) G (x + h, x r, ω) z r d(x r, x s, ω) G (x s, x h, ω) z s ( Can we construct a similar operator to extended Born Modeling Operator? 6

9 Construction of the Inverse Operator Asymptotic Analysis of the Normal Operator F[v] F[v]δv(x, h). Extended Born Modeling Operator and its Adjoint. F[v]δv = 2 2δv(x, h) t 2 dxdhdτg(x + h, t τ; x r ) v 3 G(x h, τ; x s ) (x) F [v]d = 2. v 3 dx s dx r dtdτg(x h, τ; x s )G(x + h, t τ; x r ) 2 d(x r, t; x s ) (x) t 2 Step 1 High Frequency Approximation Step 2 Principle of Stationary Phase Step 3 Modify adjoint operator by some Scaling and Filters 7

10 Where miracle happens Key element left a 2 s a2 r det Hess Relation between amplitudes and Beylkin determinant (Bleistein, N.; Zhang, Y.; Xu, S.; Zhang, G.; Gray, S, 2005) x s θ s Surface θr x r a 2 sa 2 r cosθ s cosθ r det Hess v s v r x h x + h 8

11 An Approximate Inverse F[v 0 ] = W 1 model [v 0] F[v 0 ] W data [v 0 ]. W 1 model = 4v5 0 LP, W data = I 4 t D zs D zr L = 2 (x,z) 2 (h,z) P is integral operator with computable kernel (P 1 near h = 0 or if horizontal velocity variation is small) I t is the time integral D zs, D zr are the source and receiver depth derivative. 9

12 Setup for Numerical Examples 2-8 Finite Difference Hz Bandpass Wavelet dx = dz = dz = 10m, dt = 1ms Dense sampled sources (every 40m) Fixed Spread Receivers Absorbing Boundary, except free surface on the top 10

13 Numerical Test I Reflectivity Model One-shot Born Data 11

14 Extended Migration Result 12

15 Extended Inversion Result 13

16 Resimulated Data Resimulated Data Data Residual (= 13.6% observed data ) 14

17 One trace Comparison Figure: One trace (middle) comparison between the original data(blue) and resimulated data(green). The difference is shown as the red line. 15

18 From extended to nonextended model Model M E[m] = m X[ m] = m Extended Model M x E[δv] : δv(x, h) = δv(x)δ(h) X [ δv] : δv(x) = δv(x, h)φ(h)dh z where Φ(0) = 1 16

19 From extended to nonextended model Model M E[m] = m X[ m] = m Extended Model M h x E[δv] : δv(x, h) = δv(x)δ(h) X [ δv] : δv(x) = δv(x, h)φ(h)dh z where Φ(0) = 1 16

20 Non-extended Inversion Result Non-extended Inversion Result Model Residual (= 19.1% model ) δv(x) = h δv(x, h) 17

21 One trace Comparison Figure: One trace (middle) comparison between the reflectivity model (blue) and non-extended inversion result (green). The difference is shown as the red line. 18

22 Extended Migration Result-Wrong Background 19

23 Extended Inversion Result-Wrong Background 20

24 Resimulated Data-Wrong Background Resimulated Data Data Residual 21

25 One trace Comparison-Wrong Background Figure: One trace (middle) comparison between the original data(blue) and resimulated data(green). The difference is shown as the red line. 22

26 Stacked Image-Wrong Background Stacked Image Reflectivity Model 23

27 Apply D zs D zr -Naive Implementation S R z = z 2 S + R + z = + z 2 d 11 = data(s, R ) d 12 = data(s, R + ) d 21 = data(s +, R ) d 22 = data(s +, R + ) D zs D zr data = d11 d12 d21+d22 ( z) 2 Reflector 24

28 Apply D zs D zr -Free Surface Simulation S R z = z 2 Surface z = 0 S + R + z = + z 2 +data(s, R ) data(s, R + ) data(s +, R ) +data(s +, R + ) D zs D zr data = Reflector freesurface data ( z) 2 25

29 Numerical Test II Velocity Model Wavefronts and Rays 26

30 Extended Inversion 27

31 Resimulated Data Resimulated Data Data Difference =10.4% observed data 28

32 One trace Comparison Figure: One trace (middle) comparison between the original data(blue) and resimulated data(green). The difference is shown as the red line. 29

33 Non-extended Inversion Non-extended Inversion Result Model Difference =21.3% model 30

34 One trace Comparison Figure: One trace (middle) comparison between the reflectivity model (blue) and non-extended inversion result (green). The difference is shown as the red line. 31

35 Marmousi Model Marmousi Model Background Velocity Model 32

36 Extended Inversion Result 33

37 Resimulation Comparison Original Data 34

38 Resimulation Comparison Resimulated Data 34

39 Resimulation Comparison Data Difference 34

40 One Trace comparison Figure: One trace (middle) comparison between the original data(blue) and resimulated data(green) 35

41 Marmousi Model Nonextended Inversion Result 36

42 Marmousi Model Reflectivity Model 36

43 One Trace comparison Figure: One trace (middle) comparison between the original data(blue) and resimulated data(green)) 37

44 SEG/EAGE Salt Model Reflectivity Model 38

45 SEG/EAGE Salt Model Background Model 39

46 SEG/EAGE Salt Model Background Model (Salt Removed) 40

47 SEG/EAGE Salt Model Extended Inversion Result 41

48 SEG/EAGE Salt Model-Salt Removed Extended Inversion Result 42

49 SEG/EAGE Salt Model Nonextended Inversion Result 43

50 SEG/EAGE Salt Model-Salt Removed Nonextended Inversion Result 44

51 Conclusion Takeaway Messages Subsurface offset extended RTM can be modified into an asymptotic inverse to the extended Born Modeling Operator Although the derivation is based on asymptotic theory, the implementation doesn t involve any ray computation The new inverse operator can approximate the ELSM result The new inverse operator can also produce non-extended inversion, which can approximate LSM 45

52 Acknowledgement Fons ten Kroode, Jon Sheiman, Henning Kuehl, Peng Shen, Yujin Liu Members and Sponsors Shell International Exploration and Production Madagascar, SU,TACC, RCSG Thank you for listening

53 SEG/EAGE Salt Model-Difference Nonextended Inversion Result Difference