Full Waveform Inversion via Matched Source Extension

Transcription

1 Full Waveform Inversion via Matched Source Extension Guanghui Huang and William W. Symes TRIP, Department of Computational and Applied Mathematics May 1, 215, TRIP Annual Meeting G. Huang and W. W. Symes MSWI 1 / 31

2 Outline 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 2 / 31

3 Outline 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 3 / 31

4 Overview Assume that the received pressure field p(x r, t; x s ) generated by a causal isotropic point radiator at source position x = x s satisfies the constant density acoustic wave equation, 1 2 p v 2 t 2 p = δ(x x s)f(t) in R 2. (1) p t= = p = (2) t t= Let s introduce the forward modeling operator S[v] to relate the velocity v(x, z) and wavelet function f(t) to the scattering field at the receiver x r, S[v, f](x r, t; x s ) = p(x r, t; x s ). (3) G. Huang and W. W. Symes MSWI 4 / 31

5 Full Waveform Inversion Given recorded traces d(x r, t; x s ), find velocity v and wavelet function f such that S[v, f] = d. FWI via data fitting, J FWI [v, f] = 1 2 x r,x s S[v, f](x r, t; x s ) d(x r, t; x s ) 2 dt. (4) FWI objective function is quadratic with respect to f, but highly nonlinear and nonconvex in velocity v (frequency dependent). Cycle skipping problem (eg. Symes, 1994). G. Huang and W. W. Symes MSWI 5 / 31

6 Outline 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 6 / 31

7 Extended Modeling & Null Space Extended Modeling: Let f(x r, x s, t) be the extended model of f(t), define the extended modeling operator S[v, f] = p(x r, t; x s ) where p is the solution of (1)-(2) with source function being f. Null Space (Annihilator): Differential semblance operator A = zs (see Symes (1994)); t-moment operator after source signature deconvolution A = tf 1 (t) (eg. deconvolution-based by Luo and Sava (211), AWI by Warner (214)). G. Huang and W. W. Symes MSWI 7 / 31

8 MSWI Matched Source Waveform Inversion (MSWI) is stated as follows, J MS [v] = 1 A 2 f 2 dt (5) x r,x s s.t. S[v, f](x r, t; x s ) = d(x r, t; x s ). (6) Key feature: Even given wrong velocity, data fitting is perfect, hence no cycle skipping problem! G. Huang and W. W. Symes MSWI 8 / 31

9 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with Outline 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian Analysis of Gradient Local Convexity of Hessian Relation with DSO formulation 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 9 / 31

10 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian Analysis of Gradient Local Convexity of Hessian Relation with DSO formulation 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 1 / 31

11 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with Factorization of Tangent Operator Under single arrival approximation, we have S[v, f](x r, t; x s ) a(x r, x s ) f(x r, x s, t τ(x r, x s )) Then taking the first order variation of S[v, f] formally gives us the desired factorization of operator (D S[v]δv) f(x r, t; x s ) a(x r, x s ) t f ( x r, x s, t τ(x r, x s ) ) ( Dτ[v]δv) S[v, Q[v, δv] f]. where Dτ[v] is the tangent operator of traveltime function, and Q[v, δv] f is bilinear operator with respect to δv and f with Q[v, δv] = (Dτ[v]δv) t. G. Huang and W. W. Symes MSWI 11 / 31

12 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with Backprojection of Traveltime Differences Taking the first order perturbation of J MS, we have DJ MS [v]δv x r,x s A T A f(dτ[v]δv f t )dt Hence the gradient is given by, g ( Dτ[v] T (A T A f) f ) t dt x r,x s Here Dτ[v] T is the adjoint operator of Dτ[v], which backprojects its arguments along rays. G. Huang and W. W. Symes MSWI 12 / 31

13 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with Adjoint Source in the Gradient Assume that data is noise-free and well approximated by geometric optics, d(x r, t; x s ) a (x r, x s )f (t τ[v ](x r, x s )). (7) Denote by τ(x r, x s ) = τ[v ](x r, x s ) τ[v](x r, x s ), then For differential semblance operator A = zs, (A T A f) f ( a ) 2 ( f ) 2dt( ( t dt ) T ) τ[v](z r, z s ). a t z s z s For t-moment operator, we have (A T A f) f t dt (a /a) 2 τ[v](z r, z s ). G. Huang and W. W. Symes MSWI 13 / 31

14 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with Traveltime Tomography Travel tomography attempts to minimize the difference between the computed traveltime and picked traveltime from data, J TT = 1 τ[v](x r, x s ) τ[v ](x r, x s ) 2, 2 x r,x s The gradient of J TT is given by, J TT = x r,x s Dτ[v] T ( τ[v](z r, z s )). G. Huang and W. W. Symes MSWI 14 / 31

15 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian Analysis of Gradient Local Convexity of Hessian Relation with DSO formulation 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 15 / 31

16 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with Local Convexity of Hessian The Hessian of J MS at consistent data A f = is given by D 2 J[v ](δv, δv) [A, Q](v, δv) f 2 dt x r,x s where [A, Q] = AQ QA. for A = zs, ( f D 2 J[v ) 2dt ( ](δv, δv) Dτ[v ]δv 2 ) t z s x r,x s for A = tf 1 (t), D 2 J[v ](δv, δv) Dτ[v ]δv 2 x r,x s Hessian of J MS is proportional to the Hessian of a traveltime objective function, and is as convex as tomographic objective. G. Huang and W. W. Symes MSWI 16 / 31

17 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian Analysis of Gradient Local Convexity of Hessian Relation with DSO formulation 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 17 / 31

18 Overview Matched Source Waveform Inversion Analysis of Gradient Analysis and of Hessian Gradient Numerical Local Convexity ExamplesofConclusion Hessian Relation and Discuss with Relation with DSO formulation DSO formulation seeks the balance between data fitting and annihilator term, i.e. J α [v, f] = 1 2 S[v, f] d 2 dt + α A 2 f 2 dt. x r,x s x r,x s As α, the gradient and Hessian of 1 α J α is the same as J MS. As α +, A f =, then it s equivalent to minimize J FWI G. Huang and W. W. Symes MSWI 18 / 31

19 Outline 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 19 / 31

20 Example 1: Velocity Scan of J MS 1 Objective function value Slowness (s/km) Figure: J MS [v] for homogeneous velocity v,.25 s/km v 1.75 s/km. Correct velocity is 2 km/s. G. Huang and W. W. Symes MSWI 2 / 31

21 Example 2: Gaussian Lens In this example, the target velocity consists of two Gaussian velocity anomalies embedded in a v = 2km/s background: v(x, z) = 2.6e (x.25) 2 +(z.3) 2 (.2) 2.6e (x.25)2 +(z.7) 2 (.1) 2, where x [,.5]km, z [, 1]km. The initial model is given by the constant velocity v = 2 km/s. Data is consisted of 5 shots and 99 receivers for each shot, which are uniformly distributed. G. Huang and W. W. Symes MSWI 21 / 31

22 Example 2: Gaussian Lens Time (s) Time (s) Distance (m) Distance (m) G. Huang and W. W. Symes MSWI 22 / 31

23 Example 2: Gaussian Lens Depth (m) Depth (m) Distance (m) Distance (m) G. Huang and W. W. Symes MSWI 23 / 31

24 Example 2: Gaussian Lens Time (s) Distance (m) Figure: Extended source functions G. Huang and W. W. Symes MSWI 24 / 31

25 Example 3: Big Gaussian Figure: Low velocity Gaussian anomaly model with radius 25m 15m embedded in the constant background velocity v = 2m/s G. Huang and W. W. Symes MSWI 25 / 31

26 Example 3: Big Gaussian Figure: Receivers are uniformly distributed long x r = 199m from z r = 1m to x r = 99m for each shots, shots interval is 2m. The zero-phase ricker wavelet with main frequency f = 1Hz for generating the data. G. Huang and W. W. Symes MSWI 26 / 31

27 Example 3: Big Gaussian Figure: Inverted velocity after 5 iterations by MSWI (top) and FWI (bottom) G. Huang and W. W. Symes MSWI 27 / 31

28 Outline 1 Overview 2 Matched Source Waveform Inversion 3 Analysis of Gradient and Hessian 4 Numerical Examples 5 Conclusion and Discussion G. Huang and W. W. Symes MSWI 28 / 31

29 Conclusion and Discussion Conclusion Establish the relation between waveform inversion and traveltime tomography; Convexity of objective function is independent of frequency content. MSWI still get stuck in local minima due to the strong multipaths. Discussion Does perfect data fitting (no cycle skipping) mean avoiding local minima; How do we choose the extended model space and the related annihilator. G. Huang and W. W. Symes MSWI 29 / 31

30 To be cont d Figure: Inverted velocity after 1 iterations and 2 iterations G. Huang and W. W. Symes MSWI 3 / 31

31 Thanks Total E&P USA, for funding me The Rice Inversion Project, for supporting me TRIP members, for helping me All of you, for listening to me G. Huang and W. W. Symes MSWI 31 / 31