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

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

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

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 = . 36468 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

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

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. 10 20 30 40 Plane Trace 0 0 200 200 400 400 600 600 800 800 1000 1000 1200 1200 1400 1400 1600 1600 1800 1800 2000 2000 2200 2200 2400 2400 2600 2600 2800 2800 3000 3000 Trace 10 20 30 40 Plane Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 6 / 44

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

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 O utput L east S quares (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

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

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

Results: Inversion-estimated source appears to allow for a better corresponding reflectivity estimate: Data Fit: Experiment 1 (air gun source) 1 55% rms error. Experiment 2 (inversion-estimated source) 2 27% rms error. In fact in Experiment 2 (inversion-estimated source) 3 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

Data Misfit Comparison for Two experiments Figure: Left: 55% data misfit. Right: 29% data misfit Plane 10 20 30 40 Plane 10 20 30 40 Trace Trace 0 0 0 0 200 200 200 200 400 400 400 400 600 600 600 600 800 800 800 800 1000 1000 1000 1000 1200 1200 1200 1200 1400 1400 1400 1400 1600 1600 1600 1600 1800 1800 1800 1800 2000 2000 2000 2000 2200 2200 2200 2200 2400 2400 2400 2400 2600 2600 2600 2600 2800 2800 2800 2800 3000 3000 3000 3000 Trace Trace 10 20 30 40 10 20 30 40 Plane Plane Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 12 / 44

Well log comparisons for P-wave impedance Figure: Left: air gun model source Right: inversion estimated source SHORT−SCALE RELATIVE FLUCTUATION IN P−WAVE IMPEDANCE SHORT−SCALE RELATIVE FLUCTUATION IN P−WAVE IMPEDANCE 0.25 0.2 0.2 0.15 0.15 0.1 0.1 reflectivity (dimensionless) reflectivity (dimensionless) 0.05 0.05 0 0 −0.05 −0.05 −0.1 −0.1 −0.15 −0.15 −0.2 −0.2 −0.25 −0.25 1800 1900 2000 2100 2200 2300 2400 2500 2600 1800 1900 2000 2100 2200 2300 2400 2500 2600 2−way time (ms) 2−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

For More Detail See... Minkoff, S. E. and Symes, W. W., “Full Waveform Inversion of Marine Reflection 1 Data in the Plane-Wave Domain”, Geophysics , 62 pp. 540–553, 1997. Minkoff, S. E., “A Computationally Feasible Approximate Resolution Matrix for 2 Seismic Inverse Problems”, Geophysical Journal International , 126 , pp. 345–359, 1996. Minkoff, S. E. and Symes, W. W., “Estimating the Energy Source and Reflectivity 3 by Seismic Inversion”, Inverse Problems 11 , pp. 383–395, 1995. Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 14 / 44

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

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

Model problem The Acoustic Wave Equation ∂ 2 p � 1 � 1 ∂ t 2 − ∇ · ρ ∇ p = f ρ c 2 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 � ρ ∇ p in Ω , ∂ 2 p 1 ∂ t 2 = −∇ · � v + f in Ω , ρ c 2 Boundary conditions � v · ν = 0 , on ∂ Ω Susan E. Minkoff (UMBC) Wave Equation Upscaling 11/21/11 17 / 44

The finite element method Find � v ∈ V and p ∈ W such that � ρ� v ,� u � = � p , ∇ · � u � , � 1 ∂ 2 p � ∂ t 2 , w = −�∇ · � v , w � + � f , w � ρ c 2 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

Upscaling technique Goal : Capture the fine scale behavior on the coarser grid Pressure Unknowns Acceleration Unknowns Idea : Decompose the solution v c + δ� � v = � v v c ∈ V c is the coarse-scale solution � δ� v ∈ δ V 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

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

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