computational methods for inverse problems in geophysics
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Computational Methods for Inverse Problems in Geophysics Russell J. - PowerPoint PPT Presentation

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Computational Methods for Inverse Problems in Geophysics Russell J. Hewett Mathematics & CMDA, Virginia Tech Theory and Experience in Solving Inverse Problems in Geophysics Workshop Uppsala University April 9, 2019 RJH (Virginia Tech)

  1. Exploration Seismic Data Acquisition: Receivers Land: Geophone http://web.mit.edu/12.000/www/finalpresentation/experiments/geology.html RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 14 / 55

  2. Exploration Seismic Data Acquisition: Receivers Land: Geophone http://web.mit.edu/12.000/www/finalpresentation/experiments/geology.html RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 14 / 55

  3. Exploration Seismic Data Acquisition: Receivers Marine: Hydrophone https://woodshole.er.usgs.gov/operations/sfmapping/hydrophone.htm RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 14 / 55

  4. Exploration Seismic Data Acquisition: Receivers 4-Component Sensors (4C) https://www.glossary.oilfield.slb.com/Terms/sym/4c seismic data.aspx RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 14 / 55

  5. Exploration Seismic Data Acquisition x=128 t=1024 100 500 200 1000 t t x 300 1500 400 2000 500 100 200 300 400 500 100 200 300 400 500 y y t=1024 RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 15 / 55

  6. Geophysical Inverse Problems I am considering only the seismic inverse problem. Regimes I have neglected: ◮ CSEM (Controlled Source Electromagnetic) ◮ Gravity ◮ LIDAR ◮ etc. RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 16 / 55

  7. Geophysical Inverse Problems I will discuss only the full-waveform inversion problem and I will stay in an “exploration” context. Seismic Data Response RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 17 / 55

  8. Geophysical Inverse Problems I will discuss only the full-waveform inversion problem and I will stay in an “exploration” context. Seismic Data Response RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 17 / 55

  9. Geophysical Inverse Problems I will discuss only the full-waveform inversion problem and I will stay in an “exploration” context. Seismic Data Response RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 17 / 55

  10. (Exploration) Seismic Inverse Problem max $ ( d, k, $) RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 18 / 55

  11. (Exploration) Seismic Inverse Problem max $ ( d, k, $) d : data RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 18 / 55

  12. (Exploration) Seismic Inverse Problem max $ ( d, k, $) d : data k : knowledge RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 18 / 55

  13. (Exploration) Seismic Inverse Problem max $ ( d, k, $) d : data k : knowledge $ : money RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 18 / 55

  14. (Exploration) Seismic Inverse Problem max $ ( d, k, $) d : data k : knowledge $ : money $ : more money RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 18 / 55

  15. (Exploration) Seismic Inverse Problem max $ ( d, k, $) Subproblems: RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 19 / 55

  16. (Exploration) Seismic Inverse Problem max $ ( d, k, $) Subproblems: ◮ arg max $ ( d, k, $) d ◮ Find better data ◮ Engineers and Analysts RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 19 / 55

  17. (Exploration) Seismic Inverse Problem max $ ( d, k, $) Subproblems: ◮ arg max $ ( d, k, $) d ◮ Find better data ◮ Engineers and Analysts ◮ arg max $ ( d, k, $) k ◮ Find better knowledge or use knowledge better ◮ Research Scientists RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 19 / 55

  18. (Exploration) Seismic Inverse Problem max $ ( d, k, $) Subproblems: ◮ arg max $ ( d, k, $) d ◮ Find better data ◮ Engineers and Analysts ◮ arg max $ ( d, k, $) k ◮ Find better knowledge or use knowledge better ◮ Research Scientists ◮ arg min max $ ( d, k, $) $ ◮ Do all this, but cheaper ◮ Managers RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 19 / 55

  19. (Exploration) Seismic Inverse Problem max $ ( d, k, $) Subproblems: ◮ arg max $ ( d, k, $) d ◮ Find better data ◮ Engineers and Analysts ◮ arg max $ ( d, k, $) k ◮ Find better knowledge or use knowledge better ◮ Research Scientists ◮ arg min max $ ( d, k, $) $ ◮ Do all this, but cheaper ◮ Managers RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 19 / 55

  20. Seismic Inverse Problem Full Waveform Inversion Objective J ( m ) = � d ( t ) − F ( m ( x )) � 2 2 d ( t ) : Data m ( x ) : Unknown physical coefficients F : Modeling operator RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 20 / 55

  21. Seismic Inverse Problem FWI Objective: “Complete” Version � J ( m, f ) = {� g ( d s ) − g ( S s F s ( R s ( m ) , f s )) � + T f ( f s ) } + T m ( m ) s ∈S RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 21 / 55

  22. Seismic Inverse Problem Full Waveform Inversion Objective J ( m ) = � d ( t ) − F ( m ( x )) � 2 2 d ( t ) : Data m ( x ) : Unknown physical coefficients F : Modeling operator RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 22 / 55

  23. Seismic Inversion Full Waveform Inversion Objective J ( m ) = � d ( t ) − F ( m ( x )) � 2 2 d ( t ) : Data m ( x ) : Unknown physical coefficients F : Modeling operator RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 23 / 55

  24. Seismic Inversion Full Waveform Inversion Objective J ( m ) = � d ( t ) − F ( m ( x )) � 2 2 d ( t ) : Data m ( x ) : Unknown physical coefficients F : Modeling operator RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 24 / 55

  25. Seismic Inversion Full Waveform Inversion Problem J ( m ) = � d ( t ) − F ( m ( x )) � 2 arg min 2 m s.t. L [ m ] u = f for F ( m ) = u ◮ L [ m ] u = f is a wave equation ◮ F operator solves wave equations ◮ PDE constrained optimization! ◮ Time-domain, frequency-domain, Laplace-domain, etc. RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 25 / 55

  27. Modeling Operators Example: Second Order Isotropic Acoustics (w/ Constant Density) ◮ m ( x ) = 1 /c 2 ( x ) , where c ( x ) is p-wave velocity ◮ L is self adjoint ◮ For continuous at least. . . ◮ Up to BCs F ( m 0 ) = u 0 ⇔ ( m 0 ∂ tt − △ ) u 0 = f F m 0 δm = δu ⇔ ( m 0 ∂ tt − △ ) δu = − δm∂ tt u 0 � δm, − � q, ∂ tt u 0 � T � Ω F ∗ m 0 r = δm ⇔ s.t. ( m 0 ∂ tt − △ ) q = r � T δm = − � q, ∂ tt u 0 � T = − q ( x, t ) ∂ tt u 0 ( x, t ) dt 0 RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 27 / 55

  28. Optimization Setup J ( m ) = 1 2 || d − F ( m ) || 2 2 ◮ Objective function evaluation ◮ Computation: Solve wave equation ◮ Cost: ∼ 1 RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 28 / 55

  29. Optimization Setup J ( m ) = 1 2 || d − F ( m ) || 2 2 ◮ Objective function evaluation ◮ Computation: Solve wave equation ◮ Cost: ∼ 1 ∇ J ( m 0 ) = − F ∗ m 0 ( d − F ( m 0 )) ◮ Objective gradient evalutation ◮ Computation: Adjoint state method ◮ Cost: ∼ 2+ RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 28 / 55

  30. Optimization Setup J ( m ) = 1 2 || d − F ( m ) || 2 2 ◮ Objective function evaluation ◮ Computation: Solve wave equation ◮ Cost: ∼ 1 ∇ J ( m 0 ) = − F ∗ m 0 ( d − F ( m 0 )) ◮ Objective gradient evalutation ◮ Computation: Adjoint state method ◮ Cost: ∼ 2+ D 2 Jδm = F ∗ m 0 F m 0 δm − < D 2 F δm, d − F ( m 0 ) > ◮ Objective Hessian application ◮ Computation: 2 nd -order adjoint state method ◮ Cost: ∼ 4+ RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 28 / 55

  31. Modeling Operators F ( m 0 ) = u 0 ⇔ L [ m 0 ] u 0 = f ◮ Forward modeling ◮ Cost: 1 wave solve RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 29 / 55

  32. Modeling Operators F ( m 0 ) = u 0 ⇔ L [ m 0 ] u 0 = f ◮ Forward modeling ◮ Cost: 1 wave solve L [ m 0 ] δu = − δL F m 0 δm = δu ⇔ δm [ δm ] u 0 ◮ Linear forward modeling (Born) ◮ Cost: 2 wave solves ◮ Impossible to form as matrix! RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 29 / 55

  33. Modeling Operators F ( m 0 ) = u 0 ⇔ L [ m 0 ] u 0 = f F ∈ R m × n ◮ Forward modeling ◮ Cost: 1 wave solve L [ m 0 ] δu = − δL F m 0 δm = δu ⇔ δm [ δm ] u 0 ◮ Linear forward modeling (Born) ◮ Cost: 2 wave solves ◮ Impossible to form as matrix! RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 29 / 55

  34. Modeling Operators F ( m 0 ) = u 0 ⇔ L [ m 0 ] u 0 = f F ∈ R m × n ◮ Forward modeling 3D Survey ◮ Cost: 1 wave solve ◮ 10 × 1000 rcv (small) ◮ 8s recording (short) L [ m 0 ] δu = − δL ◮ 8ms sampling (long) F m 0 δm = δu ⇔ δm [ δm ] u 0 ◮ m = 10,000,000 samples ◮ Linear forward modeling (Born) ◮ Cost: 2 wave solves ◮ Impossible to form as matrix! RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 29 / 55

  35. Modeling Operators F ( m 0 ) = u 0 ⇔ L [ m 0 ] u 0 = f F ∈ R m × n ◮ Forward modeling 3D Survey ◮ Cost: 1 wave solve ◮ 10 × 1000 rcv (small) ◮ 8s recording (short) L [ m 0 ] δu = − δL ◮ 8ms sampling (long) F m 0 δm = δu ⇔ δm [ δm ] u 0 ◮ m = 10,000,000 samples ◮ Linear forward modeling (Born) 3D Modeling ◮ Cost: 2 wave solves ◮ 876 × 1001 × 750 dof ◮ Impossible to form as matrix! ◮ n = 657,657,000 dof RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 29 / 55

  36. Modeling Operators F ( m 0 ) = u 0 ⇔ L [ m 0 ] u 0 = f F ∈ R m × n ◮ Forward modeling 3D Survey ◮ Cost: 1 wave solve ◮ 10 × 1000 rcv (small) ◮ 8s recording (short) L [ m 0 ] δu = − δL ◮ 8ms sampling (long) F m 0 δm = δu ⇔ δm [ δm ] u 0 ◮ m = 10,000,000 samples ◮ Linear forward modeling (Born) 3D Modeling ◮ Cost: 2 wave solves ◮ 876 × 1001 × 750 dof ◮ Impossible to form as matrix! ◮ n = 657,657,000 dof Matrix Size ◮ IEEE single precision. . . ◮ ∼ 23.4PB Storage ◮ n wave solves ◮ ∼ 18 years @ 100k/day RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 29 / 55

  37. Modeling Operators F ( m 0 ) = u 0 ⇔ L [ m 0 ] u 0 = f F ∈ R m × n ◮ Forward modeling 3D Survey ◮ Cost: 1 wave solve ◮ 10 × 1000 rcv (small) ◮ 8s recording (short) L [ m 0 ] δu = − δL ◮ 8ms sampling (long) F m 0 δm = δu ⇔ δm [ δm ] u 0 ◮ m = 10,000,000 samples ◮ Linear forward modeling (Born) 3D Modeling ◮ Cost: 2 wave solves ◮ 876 × 1001 × 750 dof ◮ Impossible to form as matrix! ◮ n = 657,657,000 dof Matrix Size q, − δL � � δm [ δm ] u 0 F ∗ Ω × T ◮ IEEE single precision. . . m 0 r = δm ⇔ s.t. L ∗ [ m 0 ] q = r ◮ ∼ 23.4PB Storage ◮ n wave solves ◮ Adjoint modeling ◮ ∼ 18 years ◮ “Migration” or imaging operator @ 100k/day ◮ Cost: 1+ wave solves RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 29 / 55

  38. Solving FWI ◮ Solve with gradient-based optimization Generalized Gradient Scheme 1. Given m 0 . 2. While i < MaxIter 2.1 g i = ∇ J ( m i ) 2.2 s i = h ( m i , g i ) 2.3 α = LineSearch ( m i , g i , s i ) 2.4 m i +1 = m i + αs i ◮ Gradient descent? ◮ L-BFGS? ◮ Hessian/Quasi-Newton schemes? RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 30 / 55

  39. Solving FWI (a) True (b) Initial (c) Final ◮ 50 L-BFGS iterations w/ Locally 1D Time Solver (L. Zepeda, RJH, M. Rao, L. Demanet (SEG 2013)) RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 31 / 55

  40. Real-world Utility ◮ Interpreters don’t actually use inverted material parameters ◮ They still want to look at seismic “images” ◮ Found by “migration” or imaging algorithms ◮ Kirchoff migration ◮ (One-way) wave equation migration ◮ Reverse-time migration ◮ Linearized inversion RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 32 / 55

  41. Real-world Utility ◮ Interpreters don’t actually use inverted material parameters ◮ They still want to look at seismic “images” ◮ Found by “migration” or imaging algorithms ◮ Kirchoff migration ◮ (One-way) wave equation migration ◮ Reverse-time migration ◮ Linearized inversion ◮ Migration is essentially back-propagation ◮ Or, we can consider it as a gradient calculation in FWI RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 32 / 55

  42. Real-world Utility ◮ Interpreters don’t actually use inverted material parameters ◮ They still want to look at seismic “images” ◮ Found by “migration” or imaging algorithms ◮ Kirchoff migration ◮ (One-way) wave equation migration ◮ Reverse-time migration ◮ Linearized inversion ◮ Migration is essentially back-propagation ◮ Or, we can consider it as a gradient calculation in FWI ◮ In any case, it is computed at much higher-frequency ◮ Looking at reflectivity, not material parameters RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 32 / 55

  43. Seismic Image Netherlands Block F3 - Crossline 900 https://ghassanalregibdotcom.files.wordpress.com/2018/05/amir aapg2018 slides.pdf RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 33 / 55

  44. Real World Results “An offshore Gabon full-waveform inversion case study,” Xiao, et al., Interpretation , November 2016. Data from CGG. RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 34 / 55

  45. Real World Results “An offshore Gabon full-waveform inversion case study,” Xiao, et al., Interpretation , November 2016. Data from CGG. RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 34 / 55

  46. Real World Results “An offshore Gabon full-waveform inversion case study,” Xiao, et al., Interpretation , November 2016. Data from CGG. RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 34 / 55

  47. Real World Results “An offshore Gabon full-waveform inversion case study,” Xiao, et al., Interpretation , November 2016. Data from CGG. RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 34 / 55

  48. Challenges for FWI ◮ Due to the physical formulation ◮ Due to mathematical formulation ◮ Due to computational requirements ◮ Due to business decisions RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 35 / 55

  49. Cycle Skipping Jean Virieux ◮ Due to the physical and mathematical formulations ◮ Manifests as global nonconvexity ◮ Partially resolved by working in frequency domain (or in Laplace domain) ◮ Time-domain wave equation becomes Helmholtz equation RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 36 / 55

  50. FWI In Frequency Domain 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ◮ max. 512 wavelengths in domain ◮ PML width: 2.5 wavelengths ◮ 64 × 64 domain decomposition RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 37 / 55

  51. FWI In Frequency Domain 0 20 40 60 80 100 120 140 0 100 200 300 400 500 0 20 40 60 80 100 120 140 0 100 200 300 400 500 0 20 40 60 80 100 120 140 0 100 200 300 400 500 ◮ Frequency continuation RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 38 / 55

  52. Realistic and Practical Physics ◮ I have shown the idea behind FWI using constant density acoustic physics ◮ Of course, the earth is not constant density, nor acoustic ◮ Massive increase in computational – and software – costs ◮ Does better physics drive the need for more compute? ◮ Or is more compute driving the availability of better physics? RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 39 / 55

  53. Realistic and Practical Physics Physical Models ∂ tt u = a △ u + f Physics Solutions Parameters Computation Iso-Aco (const. ρ ) (2nd) 1 1 – RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 40 / 55

  54. Realistic and Practical Physics Physical Models � p � � p � g � � � � − c ∇· ∂ t = + a ∇ v v h Physics Solutions Parameters Computation Iso-Aco (const. ρ ) (2nd) 1 1 – Iso-Aco (1st) 4 2 1x-2x RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 40 / 55

  55. Realistic and Practical Physics Physical Models � ˜ � � ˜ � g − A T D ∗ R � � � � σ σ ∂ t = + a R T DA v v h Physics Solutions Parameters Computation Iso-Aco (const. ρ ) (2nd) 1 1 – Iso-Aco (1st) 4 2 1x-2x TTI-Aco (1st) 5 (+?) 4 3x RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 40 / 55

  56. Realistic and Practical Physics Physical Models � σ � � σ � g − C D ∗ � � � � ∂ t = + a D v v h Physics Solutions Parameters Computation Iso-Aco (const. ρ ) (2nd) 1 1 – Iso-Aco (1st) 4 2 1x-2x TTI-Aco (1st) 5 (+?) 4 3x Iso-Ela (1st) 9 3 ∼ 40x RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 40 / 55

  57. Realistic and Practical Physics Physical Models � σ � � σ � g − C D ∗ � � � � ∂ t = + a D v v h Physics Solutions Parameters Computation Iso-Aco (const. ρ ) (2nd) 1 1 – Iso-Aco (1st) 4 2 1x-2x TTI-Aco (1st) 5 (+?) 4 3x Iso-Ela (1st) 9 3 ∼ 40x VTI-Ela (1st) 9 8 ∼ 40x RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 40 / 55

  58. Realistic and Practical Physics Physical Models � σ � � σ � g − C D ∗ � � � � ∂ t = + a D v v h Physics Solutions Parameters Computation Iso-Aco (const. ρ ) (2nd) 1 1 – Iso-Aco (1st) 4 2 1x-2x TTI-Aco (1st) 5 (+?) 4 3x Iso-Ela (1st) 9 3 ∼ 40x VTI-Ela (1st) 9 8 ∼ 40x TTI-Ela (1st) 9 36 > 100x RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 40 / 55

  59. Realistic and Practical Physics Physical Models � σ � � σ � g − C D ∗ � � � � ∂ t = + a D v v h Physics Solutions Parameters Computation Iso-Aco (const. ρ ) (2nd) 1 1 – Iso-Aco (1st) 4 2 1x-2x TTI-Aco (1st) 5 (+?) 4 3x Iso-Ela (1st) 9 3 ∼ 40x VTI-Ela (1st) 9 8 ∼ 40x TTI-Ela (1st) 9 36 > 100x Elastic (1st) 9 81 > 100x RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 40 / 55

  60. Software Considerations ◮ I have greatly simplified the problem ◮ Only considering one simple PDE and an “academic” objective function ◮ My personal results in this talk are only embarassingly parallel ◮ Essentially data parallel in shot record ◮ There is still model parallelism to exploit, which requires high-end HPC software ◮ In addition to. . . RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 41 / 55

  61. Software Considerations FWI Objective J ( m ) = � d − F ( m ) � 2 2 d ( t ) : Data m ( x ) : Unknown physical coefficients F : Modeling operator f s ( x, t ) : RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 42 / 55

  62. Software Considerations FWI Objective � � d s − F s ( m ) � 2 J ( m ) = 2 s ∈S d s ( t ) : Data for shot s m ( x ) : Unknown physical coefficients F s : Modeling operator for shot s f s ( x, t ) : RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 42 / 55

  63. Software Considerations FWI Objective � � d s − S s F s ( m ) � 2 J ( m ) = 2 s ∈S d s ( t ) : Data for shot s m ( x ) : Unknown physical coefficients F s : Modeling operator for shot s f s ( x, t ) : S s : Data sampling operator for shot s RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 42 / 55

  64. Software Considerations FWI Objective � J ( m ) = � d s − S s F s ( m ) � s ∈S d s ( t ) : Data for shot s m ( x ) : Unknown physical coefficients F s : Modeling operator for shot s f s ( x, t ) : S s : Data sampling operator for shot s �·� : (Arbitrary) residual norm RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 42 / 55

  65. Software Considerations FWI Objective � J ( m, f ) = � d s − S s F s ( m, f s ) � s ∈S d s ( t ) : Data for shot s m ( x ) : Unknown physical coefficients F s : Modeling operator for shot s f s ( x, t ) : S s : Data sampling operator for shot s �·� : (Arbitrary) residual norm f s ( x, t ) : Unknown seismic source function for shot s RJH (Virginia Tech) Computation & Geophysical Inversion Uppsala / April 9, 2019 42 / 55

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