Two-Scale Wave Equation Modeling for Seismic Inversion

Transcription

1 Two-Scale Wave Equation Modeling for Seismic Inversion Susan E. Minkoff Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, MD 21250, USA RICAM Workshop 3: Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment November 21, 2011 Thanks to: Tanya Vdovina, Oksana Korostyshevskaya, Sean Griffith, and Bill Symes

2 Outline I. Motivation: A Seismic Inversion Example II. Operator Upscaling for the Acoustic Wave Equation A. Parallel Cost and Timing Studies B. Numerical Examples (Accuracy) C. Numerical Experiments (Convergence) D. Matrix Analysis To Illustrate Physics of Solution E. Solution of the Inverse Problem III. Conclusions Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 2 / 44

3 Motivation: A Seismic Inversion Example To illuminate a section of the subsurface, geophysicists introduce energy into the ground. Seismic source is always a part of the resulting data. Good estimate of the energy source essential to recovery of mechanical earth parameters. Source s shape (signature) and direction-dependence (radiation pattern) are of no use in themselves. Seismic inverse problems generally do not have unique solutions. The model estimate may have unique average behavior. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 3 / 44

4 Inputs to Numerical Experiments: Gulf of Mexico Data from Exxon Production Research Co. Eleven common-midpoint data gathers. Radon transform to yield 48 plane-wave traces (per gather) with slow ness values from p min =.1158 ms/m to p max = ms/m. Estimate of anisotropic air gun source in a 31-component Legendre expansion in slowness, p. Viscoelastic model (with estimated attenuation coefficient) used as forward simulator for inversion. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 4 / 44

5 Stacked Section of Gulf of Mexico Data Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 5 / 44

6 Seismic data in τ p domain Figure: The τ p transformed seismic data from common midpoint gather 6. Data was filtered by convolving it with a 15 Hz Ricker filter. Plane Trace Trace Plane Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 6 / 44

7 Inversion Algorithm: 1 Estimate P-wave background velocity and elastic parameter reflectivities by Differential Semblance Optimization J DSO [v, r] = 1 2 { S[v, r] S data 2 + λ 2 Wr 2 +σ 2 r/ p 2 } 2 Estimate seismic source and elastic parameter reflectivities (again) by alternation and Output Least Squares Inversion J OLS [r, f ] = 1 2 { S[r, f ] S data 2 + λ 2 W [r, f ] 2 } Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 7 / 44

8 Alternation Algorithm: Repeat until convergence: 1 Given the current source, f c,and current reflectivity, r c, invert for a new estimate of the reflectivity r + using Output Least Squares (i.e., J OLS [r, f ] = 1 2 d pred [r, f ] d obs 2 ) 2 Replace r c by r +. 3 Given the current source and reflectivity guesses, f c, r c, invert for a new estimate of the source f + using OLS. 4 Replace f c by f +. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 8 / 44

9 Two Experiments: 1 Model m = three elastic parameter reflectivities. (Source not updated in inversion) 2 Model m = three elastic parameter reflectivities and seismic source. In both cases the data is the same. The fixed background velocity is the same. The starting guesses for the reflectivities are the same (zero). The algorithmic stopping tolerances are the same. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 9 / 44

10 Sources Figure: Left: air gun model source; Right: anisotropic source from inversion Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 10 / 44

11 Results: Inversion-estimated source appears to allow for a better corresponding reflectivity estimate: Data Fit: 1 Experiment 1 (air gun source) 55% rms error. 2 Experiment 2 (inversion-estimated source) 27% rms error. 3 In fact in Experiment 2 (inversion-estimated source) 10% rms error in region around gas sand target. Model Fit: Experiment 2 reflectivity estimates match well log better than Experiment 1 estimates. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 11 / 44

12 Plane Trace Trace Plane Plane Trace Trace Plane Data Misfit Comparison for Two experiments Figure: Left: 55% data misfit. Right: 29% data misfit Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 12 / 44

13 Well log comparisons for P-wave impedance Figure: Left: air gun model source Right: inversion estimated source 0.25 SHORT SCALE RELATIVE FLUCTUATION IN P WAVE IMPEDANCE 0.2 SHORT SCALE RELATIVE FLUCTUATION IN P WAVE IMPEDANCE reflectivity (dimensionless) reflectivity (dimensionless) way time (ms) way time (ms) The solid line shows the inversion result. The dashed line shows the detrended well log. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 13 / 44

14 For More Detail See... 1 Minkoff, S. E. and Symes, W. W., Full Waveform Inversion of Marine Reflection Data in the Plane-Wave Domain, Geophysics, 62 pp , Minkoff, S. E., A Computationally Feasible Approximate Resolution Matrix for Seismic Inverse Problems, Geophysical Journal International, 126, pp , Minkoff, S. E. and Symes, W. W., Estimating the Energy Source and Reflectivity by Seismic Inversion, Inverse Problems 11, pp , Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 14 / 44

15 Problem: Running inversion experiments is expensive... Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 15 / 44

16 Upscaling One would like to simulate wave propagation on a coarser scale than the one on which the parameters are defined. We adapted the subgrid upscaling technique developed for elliptic problems (flow in porous media) to the wave equation. Goal is to be able to solve the problem on the coarser grid while still capturing some of the small scale features internal to coarse grid blocks. Operator upscaling is one option. Original reference: Arbogast, Minkoff & Keenan, An Operator-Based Approach to Upscaling the Pressure Equation, Computational Methods in Water Resources XII (1998). Advantages: No Scale Separation or Periodic Medium Requirements. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 16 / 44

17 Model problem The Acoustic Wave Equation 1 2 ( ) p 1 ρc 2 t 2 ρ p = f p is the pressure c(x, y) is the sound velocity, ρ(x, y) is the density, f is the source of acoustic energy The First Order System v = 1 p in Ω, ρ 1 2 p ρc 2 = v + f in Ω, t2 Boundary conditions v ν = 0, on Ω Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 17 / 44

18 The finite element method Find v V and p W such that ρ v, u = p, u, 1 2 p ρc 2 t 2, w = v, w + f, w for all u V and w W W ={piecewise discontinuous constant functions} V ={piecewise continuous linear functions of the form (a 1 x + b 1, a 2 y + b 2 )} Pressure Acceleration Full fine grid Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 18 / 44

19 Upscaling technique Goal: Capture the fine scale behavior on the coarser grid Pressure Unknowns Acceleration Unknowns Idea: Decompose the solution v = v c + δ v v c V c is the coarse-scale solution δ v δv are the fine-grid unknowns internal to each coarse-grid cell subgrid unknowns Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 19 / 44

20 Upscaling technique Simplifying assumption: δ v ν = 0 on the boundary of each coarse cell. This assumption allows us to decouple the subgrid problems from coarse-grid block to coarse-grid block. We use full fine-grid pressure. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 20 / 44

21 Two steps of upscaling Step 1: On each coarse element solve the subgrid problem: Find δ v δv and p W such that ρ( v c + δ v), δ u = p, δ u, 1 2 p ρc 2 t 2, w = ( v c + δ v), w + f, w for all δ u δv and w W Note: the value of v c is unknown at this stage. The subgrid problem fully determines p. Step 2: Use the solution of subgrid problem to solve the coarse problem: Find v c V c such that ρ( v c + δ v), u c = p, u c, for all u c V c Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 21 / 44

22 Parallel implementation Serial cost (N p + 2N u )N δ N c + O(N c ) Parallel cost (N p + 2N u )N δ N c p + O(N c ) N p and N u are the number of flops required to solve for pressure and acceleration respectively, N δ and N c are the sizes of the subgrid and coarse problems, p is the number of processors. Subgrid problems: Embarrassingly parallel. No communication between processors. No ghost-cell memory allocation required. Coarse problem: Solve in serial. Post-processing: Cheap and fast. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 22 / 44

23 Timing Studies Numerical grid contains fine grid blocks and coarse grid blocks. timings are for 20 time steps. number total subgrid coarse postof processors time problems problem processing References: Vdovina, T., Minkoff, S., and Korostyshevskaya, O., Operator Upscaling for the Acoustic Wave Equation, Multiscale Modeling and Simulation, 4, pp , Vdovina, T. and Minkoff, S., An A Priori Error Analysis of Operator Upscaling for the Acoustic Wave Equation, International Journal of Numerical Analysis and Modeling, 5, pp , Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 23 / 44

24 Timing Studies: number of Finite-difference Upscaling code processors code number of fine blocks per coarse block Note: Performance improves as coarse block size increases. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 24 / 44

25 Numerical Experiment I (Acoustic Wave Equation) Domain is of size km. Fine grid: Coarse grid: Gaussian source, 200 time steps. Sound velocity strips range from 3500 m/s to 7500 m/s. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 25 / 44

26 Vertical Acceleration Full finite-difference soln Reconstructed Upscaled Soln Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 26 / 44

27 Numerical Experiment II (Elastic Wave Equation) Fine grid: Coarse grid: Source is Gaussian in space, Ricker in time. Wave propagates about 7 wavelengths. Mixture of two materials with Vp = 2.5 km/s or 3 km/s; Vs = 1.5 and 1.75 km/s respectively, ρ = and kg/km3. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 27 / 44

28 yz slices of first component of velocity solution z z x y y x Full finite-element soln Augmented Upscaled Soln Reference: Vdovina, T., Minkoff, S., and Griffith, S., A Two Scale Solution Algorithm for the Elastic Wave Equation, SIAM Journal on Scientific Computing, 31, pp , Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 28 / 44

29 xy slices of first component of velocity solution x x y 16 0 y x x Full finite-element soln Augmented Upscaled Soln Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 29 / 44

30 Comparison of time traces for the first component of velocity Red curve = full finite element solution. Green curve = coarse solution. Blue curve = full finite element solution for the homogeneous medium with a single (average) value for each of the three input parameter. x10-10 x fisrt component of velocity (km/s) fisrt component of velocity (km/s) time (s) Receiver at (16 km, 8 km, 12 km) time (s) Receiver at (16 km, 8 km, 18 km) Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 30 / 44

31 Order of Accuracy Experiment 1 Number of coarse grid blocks is fixed at 100 in x and y. Number of fine blocks per coarse block changes in each experiment. P p 0 U u 0 Number of Number of Time Number of fine blocks coarse blocks step (s) time steps p 1 u Linear convergence in pressure (h). Reference: Vdovina, T. and Minkoff, S., An A Priori Error Analysis of Operator Upscaling for the Acoustic Wave Equation, International Journal of Numerical Analysis and Modeling, 5, pp , Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 31 / 44

32 Order of Accuracy Experiment 2 Number of fine grid blocks is fixed at 6400 in x and y. Number of coarse blocks varies in each experiment. Number of fine blocks Number of coarse blocks P p 0 U u 0 p 1 u Linear convergence in acceleration (H). Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 32 / 44

33 An Alternate Analysis: Reference: Korostyshevskaya, O. and Minkoff, S., A Matrix Analysis of Operator-Based Upscaling for the Wave Equation, SIAM J. Numer. Anal., 44, pp , Recall two steps of upscaling algorithm: Subgrid problems: On each coarse element solve the subgrid problem: Find δ v δv and p W such that ρ( v c + δ v), δ u = p, δ u, 1 2 p ρc 2 t 2, w = ( v c + δ v), w + f, w for all δ u δv and w W Coarse problem: Use the solution of subgrid problem to solve the coarse problem: Find v c V c such that ρ( v c + δ v), u c = p, u c, for all u c V c Use finite element expansions to obtain the matrix-vector form Subgrid problems: A cf v c x + A ff δv x = B f p W 2 p t 2 = (Bc ) T v c x (B f ) T δv x (v y terms) + F Coarse problem: A cc v c x + (A cf ) T δv x = B c p Matrix entries al,n ff = ρ(δux)n, (δux)l acf 1 w l,m = wm, wl ρc2 f l = f, w l l,i = ρ(ux c ) i, (δu x) l bl,m f = w m, (δux)l x al,i cc = ρ(ux c ) i, (ux c ) l bl,m c = w m, (uc x ) l x Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 33 / 44

34 Matrix equation Idea: Eliminate the subgrid acceleration from the coarse equation Subgrid problem Coarse problem δv x = (A ff ) 1 A cf v c x + (A ff ) 1 B f p (A cc (A cf ) T (A ff ) 1 A cf )v c x = (B c (A cf ) T (A ff ) 1 B f )p Obtain a matrix equation for the coarse acceleration and pressure only Uv c x = Dp Find explicit formulas for the entries of U and D to derive a difference equation Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 34 / 44

35 Differential equation Differential equation (from Taylor series expansion) ρ ups v c x = p x ρ ups is the upscaled density given by the average of the density values on the boundary of the coarse cell. Pl Pr Recall one of two original continuous pde s in our system: v = 1 ρ p Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 35 / 44

36 Pressure Equation Analysis Assuming appropriate smoothness on pressure p and density ρ, the upscaling algorithm solves the following differential equation for pressure inside a coarse block: 1 2 p ρc 2 t 2 = p (ρ 1 x x ) + p (ρ 1 y y ) + f, [order of approximation O(h 2 x + h 2 y )]. Along coarse block edges upscaling satisfies the following equation for pressure: 1 2 p ρc 2 t 2 = ( ) ( ) (ρ ups 1 p ) + (ρ ups ) 1 2 p K+ ( ) (ρ ups 1 p ) +f, x x x y y y [order of approximation O(h x + h y )] and K is a constant which depends on the location of the pressure node within a single coarse cell. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 36 / 44

37 Solving the Inverse Problem Want to find c given some observed p. Easier to find m = 1 c 2, squared slowness. The least squares functional is given by J (m) = 1 2 [ T s,r 0 ] (S s,r p d s,r ) 2 + (T s,r ( p) b s,r ) 2 dt. s and r are indices over source and receiver position. b s,r and d s,r are observed values for p and p. S s,r and T s,r are sampling operators, restricting the values of p and p to locations with observed values. (Reference: Plessix, A Review of the Adjoint State Method For Computing the Gradient of a Functional with Geophysical Applications) Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 37 / 44

38 The Formal Optimization Problem Find min m J (φ, m) s.t. F (φ, m) = 0. φ is the state variable (pressure). m is the control variable (squared slowness). F (φ, m) = 0 is the forward problem (the wave equation). Method: δ m J = δ φ J δ m φ + δmj e. δ e mj is the explicit variation of J w.r.t. m. Problem: How do we find δ m φ? (Reference: Gunzburger s Perspectives in Flow Control and Optimization.) Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 38 / 44

39 The Adjoint Problem The adjoint problem is: 1 2 λ c 2 t 2 λ = Ss,r (S s,r p d s,r ) r r λ t=t = 0 λ t t=t = 0 ( λ + T s,r (T s,r ( p) b s,r ) ) ν = 0 on Γ. T s,r (T s,r ( p) b s,r ) The auxiliary conditions are: µ 0 = m λ t t=0 µ 1 = mλ t=0 ψ = λ on Γ. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 39 / 44

40 Obtaining an Expression for the Gradient Question: How do we get D m J (n) (x, y)? To get D m J (n) (x, y) from δ m J, we let m be δ (x, y). This gives: (D m J ) (x, y) = s [ T 0 2 ] p (x, y) t 2 λ (x, y) dt. To calculate the gradient, we must have access to both the forward and adjoint solutions at all time steps. Use a checkpointing scheme to optimally determine what to store and what to recalculate. (Reference: Gunzburger s Perspectives in Flow Control and Optimization.) Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 40 / 44

41 An Optimization Algorithm Beginning with m (0) (x, y), we repeat the following: 1 Solve the forward problem to get the state variables p (n) (x, y, t) and then solve the adjoint problem to get the adjoint variable λ (n) (x, y, t). 2 Calculate the gradient D m J (n) (x, y) = [ ] T 2 p(x,y) s 0 t λ (x, y) dt. 2 3 Update squared slowness for the next step: [ T m (n+1) (x, y) = m (n) (x, y) + β n using some step size β n. s 0 ] 2 p (x, y) t 2 λ (x, y) dt, Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 41 / 44

42 The Upscaled Adjoint Problem Introduce χ = λ and decompose χ into δχ + χ c. Then, for each subgrid problem, (χ c + δχ, δu) = (λ, δu) ( 1 2 ) λ c 2 t 2, w = ( (χ c + δχ), w) ( + Ss,r (S s,r p d s,r ) r r T s,r (T s,r ( p) b s,r ), w ) for w W and δu δv. And, for each coarse problem, (χ c + δχ, u c ) = (λ, u c ) for u c V c. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 42 / 44

43 Conclusions To speed up an expensive iterative process like seismic inversion, operator upscaling can be applied to the wave equation. The upscaled solution captures some of the sub-wavelength heterogeneities of the full solution. For the acoustic implementation, the method exhibits good parallel performance due to the independence of the subgrid problems (most expensive part of the algorithm). A matrix analysis of the algorithm indicates that, in fact, the upscaling technique is solving the continuous differential equation with density given by averaging along coarse block edges. The pressure equation corresponds to the standard acoustic wave equation at nodes internal to coarse blocks. Along coarse cell boundaries, the upscaled solution solves a modified wave equation which includes a mixed second-derivative term. Using the adjoint method we can reuse much of the upscaling code to solve the inverse problem. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 43 / 44

44 Acknowledgments This research was performed with funding from the Collaborative Math-Geoscience Program at NSF (grant #EAR ), the Computational Mathematics Program at NSF (grant #DMS ), a GAANN grant from the U.S. Department of Education (Award #P200A030097), a generous fellowship provided by NASA Goddard s Earth Sciences and Technology Center (GEST), and UMBC s ADVANCE grant (NSF #SBE ). Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 44 / 44