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Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Dynamical Inverse Scattering Roland Potthast Deutscher Wetterdienst University of Reading (Uni G ottingen) Linz 24.11.2011 1/81 Introduction Orthogonality

  1. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Time-Domain to Frequency Domain ◮ We assume that we take a windowed Fast Fourier Transform of our available time-domain data. ◮ This leads to a large set of wave numbers for which data in the frequency domain is available. ◮ Usually we will have a ◮ low number of directions of incidence (sources) ◮ larger number of measurement points for the scattered or total wave. ◮ many wavenumbers u ∞ (ˆ x j , d ℓ , κ ξ ) , j = 1 , ..., N , ℓ = 1 , ..., M , ξ = 1 , ..., Q . (1) 8/81

  2. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Time-Domain to Frequency Domain ◮ We assume that we take a windowed Fast Fourier Transform of our available time-domain data. ◮ This leads to a large set of wave numbers for which data in the frequency domain is available. ◮ Usually we will have a ◮ low number of directions of incidence (sources) ◮ larger number of measurement points for the scattered or total wave. ◮ many wavenumbers u ∞ (ˆ x j , d ℓ , κ ξ ) , j = 1 , ..., N , ℓ = 1 , ..., M , ξ = 1 , ..., Q . (1) 8/81

  3. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Time-Domain to Frequency Domain ◮ We assume that we take a windowed Fast Fourier Transform of our available time-domain data. ◮ This leads to a large set of wave numbers for which data in the frequency domain is available. ◮ Usually we will have a ◮ low number of directions of incidence (sources) ◮ larger number of measurement points for the scattered or total wave. ◮ many wavenumbers u ∞ (ˆ x j , d ℓ , κ ξ ) , j = 1 , ..., N , ℓ = 1 , ..., M , ξ = 1 , ..., Q . (1) 8/81

  4. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Time-Domain to Frequency Domain ◮ We assume that we take a windowed Fast Fourier Transform of our available time-domain data. ◮ This leads to a large set of wave numbers for which data in the frequency domain is available. ◮ Usually we will have a ◮ low number of directions of incidence (sources) ◮ larger number of measurement points for the scattered or total wave. ◮ many wavenumbers u ∞ (ˆ x j , d ℓ , κ ξ ) , j = 1 , ..., N , ℓ = 1 , ..., M , ξ = 1 , ..., Q . (1) 8/81

  5. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Orthogonality Sampling Method A LGORITHM (O NE - WAVE OS, M ULTI - WAVE OS) For fixed wave number κ one-wave orthogonality sampling calculates � � � e i κ ˆ � x · y u ∞ (ˆ � µ ( y , κ ) = x ) ds (ˆ x ) (2) � � S y ∈ R m from the knowledge of the far field pattern u ∞ on on a grid G of points ˜ the unit sphere S . For fixed wave number κ multi-direction orthogonality sampling calculates � � � � e i κ ˆ � x · y u ∞ (ˆ � µ ( y , κ ) = x , θ ) ds (ˆ x ) � ds ( θ ) (3) � S S y ∈ R m from the knowledge of the far field pattern on a grid G of points ˜ u ∞ (ˆ x , θ ) for ˆ x , θ ∈ S . 9/81

  6. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Orthogonality Sampling Method A LGORITHM (O NE - WAVE OS, M ULTI - WAVE OS) For fixed wave number κ one-wave orthogonality sampling calculates � � � e i κ ˆ � x · y u ∞ (ˆ � µ ( y , κ ) = x ) ds (ˆ x ) (2) � � S y ∈ R m from the knowledge of the far field pattern u ∞ on on a grid G of points ˜ the unit sphere S . For fixed wave number κ multi-direction orthogonality sampling calculates � � � � e i κ ˆ � x · y u ∞ (ˆ � µ ( y , κ ) = x , θ ) ds (ˆ x ) � ds ( θ ) (3) � S S y ∈ R m from the knowledge of the far field pattern on a grid G of points ˜ u ∞ (ˆ x , θ ) for ˆ x , θ ∈ S . 9/81

  7. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Orthogonality Sampling Method A LGORITHM (O NE - WAVE OS, M ULTI - WAVE OS) For fixed wave number κ one-wave orthogonality sampling calculates � � � e i κ ˆ � x · y u ∞ (ˆ � µ ( y , κ ) = x ) ds (ˆ x ) (2) � � S y ∈ R m from the knowledge of the far field pattern u ∞ on on a grid G of points ˜ the unit sphere S . For fixed wave number κ multi-direction orthogonality sampling calculates � � � � e i κ ˆ � x · y u ∞ (ˆ � µ ( y , κ ) = x , θ ) ds (ˆ x ) � ds ( θ ) (3) � S S y ∈ R m from the knowledge of the far field pattern on a grid G of points ˜ u ∞ (ˆ x , θ ) for ˆ x , θ ∈ S . 9/81

  8. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Multi-frequency Orthogonality Sampling A LGORITHM (M ULTI -F REQUENCY ) The multi-frequency orthogonality sampling calculates � κ 1 � � � e i κ ˆ � x · y u ∞ (ˆ � µ ( y , θ ) = x , θ ) ds (ˆ x ) � d κ (4) � κ 0 S y ∈ R m from the knowledge of the far field pattern u ∞ on a grid G of points ˜ κ (ˆ x ) for ˆ x ∈ S and κ ∈ [ κ 0 , κ 1 ] . Here also multi-direction multi-frequency sampling is possible by adding the indicator functions for several directions of incidence. 10/81

  9. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation One Wave, one frequency: the simplest setting Graphics: Orthogonality sampling with κ = 1 or κ = 3 for fixed frequency, one direction of incidence 11/81

  10. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Multi-direction Ortho Sampling Graphics: Orthogonality sampling, many directions of incidence, fixed frequency 12/81

  11. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Multi-frequency Ortho Sampling Graphics: Orthogonality sampling, many directions of incidence, fixed frequency 13/81

  12. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Resolution Study: Large Scale Graphics: Multi-frequency Orthogonality sampling with κ between 0 . 1 and 1, i.e. with a frequency between λ = 6 and λ = 60, one direction of incidence 14/81

  13. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Resolution Study: Medium Scale Graphics: MDMF Orthogonality sampling with κ between 3 and 4, i.e. with a frequency between λ = 1 . 5 and λ = 2 15/81

  14. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Resolution Study: Medium Scale Graphics: MDMF Orthogonality sampling with κ between 6 and 15, i.e. with a frequency between λ = 0 . 4 and λ = 1 16/81

  15. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Resolution Study: Fine Scale Graphics: MDMF Orthogonality sampling with κ between 10 and 20, i.e. with a frequency between λ = 0 . 3 and λ = 0 . 6 17/81

  16. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Resolution Study: Very Fine Scale Graphics: MDMF Orthogonality sampling with κ between 20 and 40, i.e. with a frequency between λ = 0 . 15 and λ = 0 . 3 18/81

  17. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Resolution Study: Very Fine Scale Graphics: MDMF Orthogonality sampling with κ between 20 and 40, i.e. with a frequency between λ = 0 . 15 and λ = 0 . 3 19/81

  18. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Medium Reconstructions I Graphics: Orthogonality sampling for medium reconstruction, MD, fixed frequency κ = 9. 20/81

  19. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Medium Reconstructions II Graphics: Orthogonality sampling for medium reconstruction, MDMF . 21/81

  20. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Medium Reconstructions III Graphics: Orthogonality sampling for medium reconstruction, MDMF . 22/81

  21. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Medium Reconstructions IV Graphics: Orthogonality sampling for medium reconstruction, MDMF . 23/81

  22. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Neumann BC I Graphics: Orthogonality sampling for the Neumann BC, MF . 24/81

  23. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Neumann BC II Graphics: Orthogonality sampling for the Neumann BC, MDMF . 25/81

  24. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Neumann BC II Graphics: Orthogonality sampling for the Neumann BC, MDMF . 26/81

  25. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Orthogonality Sampling Convergence Dirichlet Case Theorem (Convergence or Ortho-Sampling, P 2007/08) The orthogonality sampling algorithm with the Dirichlet boundary condition for one-wave fixed frequency reconstructs the reduced scattered field, i.e. � j 0 ( κ | x − y | ) ∂ u ( y ) u s x ∈ R m . red ( x ) = ds ( y ) , (5) ∂ν ( y ) ∂ D Convergence analysis of the method can be based on the Funk-Hecke formula. 27/81

  26. Introduction Orthogonality Sampling Time-Domain Probe Method Data Assimilation Literature Potthast, R.: Acoustic Tomography by Orthogonality Sampling, Institute of Acoustics Spring Conference, Reading, UK 2008. Potthast, R: Orthogonality Sampling for Object Visualization, Inverse Problems 2010. Griesmaier, R: Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems (2011) 28/81

  27. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Outline Introduction Orthogonality Sampling Time-Domain Probe Method Field & Shape Reconstruction Time-Domain Probe Data Assimilation 3dVar 4dVar Dynamic Inverse Scattering 29/81

  28. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Outline Introduction Orthogonality Sampling Time-Domain Probe Method Field & Shape Reconstruction Time-Domain Probe Data Assimilation 3dVar 4dVar Dynamic Inverse Scattering 30/81

  29. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Setup of Inverse Rough Surface Scattering ◮ Measurements on some surface Γ h , A ◮ Unknown surface Γ below measurement surface and above zero-surface. Dirichlet boundary condition. ◮ Measure total scattered field v from one time-harmonic incident field G ( · , z ) with source point z above or on Γ h , A . 31/81

  30. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Tasks of Inverse Rough Surface Scattering Tasks: 1. Reconstruct the total field u or scattered field u s . Since the incident field u i = G ( · , z ) is known, these tasks are equivalent. 2. Reconstruct the scattering surface Γ or any surface which generates the data for the given incident field u i = G ( · , z ) . Remark. If we do this for sufficiently many incident waves simultaneously, we have uniqueness of the reconstruction. 32/81

  31. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Tasks of Inverse Rough Surface Scattering Tasks: 1. Reconstruct the total field u or scattered field u s . Since the incident field u i = G ( · , z ) is known, these tasks are equivalent. 2. Reconstruct the scattering surface Γ or any surface which generates the data for the given incident field u i = G ( · , z ) . Remark. If we do this for sufficiently many incident waves simultaneously, we have uniqueness of the reconstruction. 32/81

  32. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Method of Kirsch-Kress or Potential Method I The idea of the Kirsch-Kress method is to calculate an approximation of the scattered field by a single-layer approach S ϕ defined on a subset of an auxiliary surface Γ t . It is carried out by minimization of the Tikhonov functional J α, B = � SP B ϕ − v � 2 + α � P B ϕ � 2 . (6) L 2 (Γ h , A ) � �� � � �� � = regularization = measured data The unknown scattering surface Γ by the minimization of the approximated total field � u � L 2 (Γ) = � G ( · , z ) + SP B ϕ � L 2 (Γ) (7) � �� � = field on surface over some suitable set U of admissible surfaces Γ . 33/81

  33. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Method of Kirsch-Kress or Potential Method I The idea of the Kirsch-Kress method is to calculate an approximation of the scattered field by a single-layer approach S ϕ defined on a subset of an auxiliary surface Γ t . It is carried out by minimization of the Tikhonov functional J α, B = � SP B ϕ − v � 2 + α � P B ϕ � 2 . (6) L 2 (Γ h , A ) � �� � � �� � = regularization = measured data The unknown scattering surface Γ by the minimization of the approximated total field � u � L 2 (Γ) = � G ( · , z ) + SP B ϕ � L 2 (Γ) (7) � �� � = field on surface over some suitable set U of admissible surfaces Γ . 33/81

  34. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Method of Kirsch-Kress or Potential Method I The idea of the Kirsch-Kress method is to calculate an approximation of the scattered field by a single-layer approach S ϕ defined on a subset of an auxiliary surface Γ t . It is carried out by minimization of the Tikhonov functional J α, B = � SP B ϕ − v � 2 + α � P B ϕ � 2 . (6) L 2 (Γ h , A ) � �� � � �� � = regularization = measured data The unknown scattering surface Γ by the minimization of the approximated total field � u � L 2 (Γ) = � G ( · , z ) + SP B ϕ � L 2 (Γ) (7) � �� � = field on surface over some suitable set U of admissible surfaces Γ . 33/81

  35. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Method of Kirsch-Kress or Potential Method II Burkard, C. and Potthast, R.: A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Problems Vol. 26, No. 4, 2010. Heinemeyer, E., Linder, M. and Potthast, R.: Convergence and numerics of a multi-section method for scattering by three-dimensional rough surfaces, SIAM J. Numer. Anal. 46, 1780 (2008), 1780-1798. Chandler-Wilde, S., Heinemeyer, E. and Potthast, R.: Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions. SIAM J. Appl. Math Vol. 66, Issue 3 (2006), 1001-1026. Chandler-Wilde, S.N., Heinemeyer, E. and Potthast, R. A well-posed integral equation formulation for three-dimensional rough surface scattering Proceedings of the Royal Society a-Mathematical Physical and Engineering Sciences, 462 (2006), 3683-3705 34/81

  36. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Numerical Examples I 35/81

  37. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Numerical Examples I 36/81

  38. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Numerical Examples II 37/81

  39. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Numerical Examples II 38/81

  40. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Numerical Examples III 39/81

  41. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Numerical Examples IV 40/81

  42. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Outline Introduction Orthogonality Sampling Time-Domain Probe Method Field & Shape Reconstruction Time-Domain Probe Data Assimilation 3dVar 4dVar Dynamic Inverse Scattering 41/81

  43. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Time-Domain Probe Method: the Idea ◮ Incident time-dependent pulse coming from some point z ∈ D . ◮ When the pulse reaches some point of the scattering surface, a scattered field starts to evolve. ◮ By reconstructing the time-dependent field we can probe the region and determine those points where a scattered field evolves right at the moment when the incident pulse first reaches a particular point. ◮ Use the potential method of Kirsch-Kress or the point-source method of the author to reconstruct U s ( x , t ) for x ∈ Ω , t ∈ R . 42/81

  44. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Idea 43/81

  45. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Idea 44/81

  46. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Idea 45/81

  47. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, References Chandler-Wilde, S. and Lines, C.: Inverse Scattering by Rough Surfaces in the Time Domain, Waves 2003 Chandler-Wilde, S. and Lines, C.: A Time Domain Point Source Method for Inverse Scattering by Rough Surfaces, Computing, Volume 75, Numbers 2-3, (2005), 157-180 Luke, D.R. and Potthast, R.: The point source method for inverse scattering in the time domain. Math. Meth. Appl. Sci. Volume 29, Issue 13 (2006) 1501-1521 Burkard, C. and Potthast, R.: A Time-Domain Probe Method for Three-dimensional Rough Surface Reconstructions, Inverse Problems and Imaging, Volume 3, No. 2 (2009) 46/81

  48. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Characteristics We need to study the range of influence of a time-dependent acoustic field ... 47/81

  49. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Characteristics We need to study the range of influence of a time-dependent acoustic field ... 47/81

  50. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Characteristics We need to study the range of influence of a time-dependent acoustic field ... 47/81

  51. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence I The basic idea behind a convergence proof: 48/81

  52. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence I The basic idea behind a convergence proof: 48/81

  53. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence II For a point x ∈ Ω we define the first hitting time with respect to the incident field U i by t ≥ 0 | U i ( x , t ) | > ρ, T ( x ) := inf (8) where we usually employ ρ = 0 or small ρ > 0 in dependence of the particular choice of the incident field. Lemma Let U i be an incident spherical pulse. For every point x ∈ Ω we have that U s ( x , t ) = 0 for all t < T ( x ) . (9) 49/81

  54. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence II For a point x ∈ Ω we define the first hitting time with respect to the incident field U i by t ≥ 0 | U i ( x , t ) | > ρ, T ( x ) := inf (8) where we usually employ ρ = 0 or small ρ > 0 in dependence of the particular choice of the incident field. Lemma Let U i be an incident spherical pulse. For every point x ∈ Ω we have that U s ( x , t ) = 0 for all t < T ( x ) . (9) 49/81

  55. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence II For a point x ∈ Ω we define the first hitting time with respect to the incident field U i by t ≥ 0 | U i ( x , t ) | > ρ, T ( x ) := inf (8) where we usually employ ρ = 0 or small ρ > 0 in dependence of the particular choice of the incident field. Lemma Let U i be an incident spherical pulse. For every point x ∈ Ω we have that U s ( x , t ) = 0 for all t < T ( x ) . (9) 49/81

  56. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence II For a point x ∈ Ω we define the first hitting time with respect to the incident field U i by t ≥ 0 | U i ( x , t ) | > ρ, T ( x ) := inf (8) where we usually employ ρ = 0 or small ρ > 0 in dependence of the particular choice of the incident field. Lemma Let U i be an incident spherical pulse. For every point x ∈ Ω we have that U s ( x , t ) = 0 for all t < T ( x ) . (9) 49/81

  57. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence III Since U s ( x , t ) = − U i ( x , t ) according to the Dirichlet boundary condition, we know that | U s ( x , t ) | > ρ ≥ 0 , T ( x ) < t < T ( x ) + ǫ, for x ∈ ∂ Ω , | U s ( x , t ) | = 0 , T ( x ) < t < T ( x ) + ǫ, for x �∈ ∂ Ω for ǫ > 0 sufficiently small. This can be used to detect the boundary ∂ Ω . Theorem (Convergence of Time-Domain Probe Method) The continous version of the Time-Domain Probe Method provides a complete reconstruction of the surface Γ above the rectangle Q. 50/81

  58. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence III Since U s ( x , t ) = − U i ( x , t ) according to the Dirichlet boundary condition, we know that | U s ( x , t ) | > ρ ≥ 0 , T ( x ) < t < T ( x ) + ǫ, for x ∈ ∂ Ω , | U s ( x , t ) | = 0 , T ( x ) < t < T ( x ) + ǫ, for x �∈ ∂ Ω for ǫ > 0 sufficiently small. This can be used to detect the boundary ∂ Ω . Theorem (Convergence of Time-Domain Probe Method) The continous version of the Time-Domain Probe Method provides a complete reconstruction of the surface Γ above the rectangle Q. 50/81

  59. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Convergence III Since U s ( x , t ) = − U i ( x , t ) according to the Dirichlet boundary condition, we know that | U s ( x , t ) | > ρ ≥ 0 , T ( x ) < t < T ( x ) + ǫ, for x ∈ ∂ Ω , | U s ( x , t ) | = 0 , T ( x ) < t < T ( x ) + ǫ, for x �∈ ∂ Ω for ǫ > 0 sufficiently small. This can be used to detect the boundary ∂ Ω . Theorem (Convergence of Time-Domain Probe Method) The continous version of the Time-Domain Probe Method provides a complete reconstruction of the surface Γ above the rectangle Q. 50/81

  60. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Reconstruction 1 51/81

  61. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Reconstruction 2 52/81

  62. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Reconstruction 3 53/81

  63. Introduction Orthogonality Sampling Field & Shape Reconstruction Time-Domain Probe Method Time-Domain Probe Data Assimilation Inverse Rough Surface Scattering Time-domain probe method, Reconstruction 4 54/81

  64. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Outline Introduction Orthogonality Sampling Time-Domain Probe Method Field & Shape Reconstruction Time-Domain Probe Data Assimilation 3dVar 4dVar Dynamic Inverse Scattering 55/81

  65. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Data Assimilation Task ◮ Dynamical system M : ϕ k �→ ϕ k + 1 , states at time t k , k = 1 , 2 , 3 , ... ◮ Measurement Operator H : ϕ k �→ f k with measurements f k at time t k ◮ Reconstruct ϕ k using the knowledge of M and of f k at t k ! Basic Notation: ◮ We call the reconstruction at time t k the analysis ϕ ( a ) . k ◮ The propagated state ϕ ( b ) k + 1 := M ( ϕ ( a ) is called background. k 56/81

  66. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Data Assimilation Task ◮ Dynamical system M : ϕ k �→ ϕ k + 1 , states at time t k , k = 1 , 2 , 3 , ... ◮ ◮ Measurement Operator H : ϕ k �→ f k with measurements f k at time t k ◮ Reconstruct ϕ k using the knowledge of M and of f k at t k ! Basic Notation: ◮ We call the reconstruction at time t k the analysis ϕ ( a ) . k ◮ The propagated state ϕ ( b ) k + 1 := M ( ϕ ( a ) is called background. k 56/81

  67. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Data Assimilation Task ◮ Dynamical system M : ϕ k �→ ϕ k + 1 , states at time t k , k = 1 , 2 , 3 , ... ◮ ◮ Measurement Operator H : ϕ k �→ f k with measurements f k at time t k ◮ Reconstruct ϕ k using the knowledge of M and of f k at t k ! Basic Notation: ◮ We call the reconstruction at time t k the analysis ϕ ( a ) . k ◮ The propagated state ϕ ( b ) k + 1 := M ( ϕ ( a ) is called background. k 56/81

  68. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Data Assimilation Task ◮ Dynamical system M : ϕ k �→ ϕ k + 1 , states at time t k , k = 1 , 2 , 3 , ... ◮ ◮ Measurement Operator H : ϕ k �→ f k with measurements f k at time t k ◮ Reconstruct ϕ k using the knowledge of M and of f k at t k ! Basic Notation: ◮ We call the reconstruction at time t k the analysis ϕ ( a ) . k ◮ The propagated state ϕ ( b ) k + 1 := M ( ϕ ( a ) is called background. k 56/81

  69. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Data Assimilation Task ◮ Dynamical system M : ϕ k �→ ϕ k + 1 , states at time t k , k = 1 , 2 , 3 , ... ◮ ◮ Measurement Operator H : ϕ k �→ f k with measurements f k at time t k ◮ Reconstruct ϕ k using the knowledge of M and of f k at t k ! Basic Notation: ◮ We call the reconstruction at time t k the analysis ϕ ( a ) . k ◮ The propagated state ϕ ( b ) k + 1 := M ( ϕ ( a ) is called background. k 56/81

  70. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Data Assimilation Task ◮ Dynamical system M : ϕ k �→ ϕ k + 1 , states at time t k , k = 1 , 2 , 3 , ... ◮ ◮ Measurement Operator H : ϕ k �→ f k with measurements f k at time t k ◮ Reconstruct ϕ k using the knowledge of M and of f k at t k ! Basic Notation: ◮ We call the reconstruction at time t k the analysis ϕ ( a ) . k ◮ The propagated state ϕ ( b ) k + 1 := M ( ϕ ( a ) is called background. k 56/81

  71. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Data Assimilation Task ◮ Dynamical system M : ϕ k �→ ϕ k + 1 , states at time t k , k = 1 , 2 , 3 , ... ◮ ◮ Measurement Operator H : ϕ k �→ f k with measurements f k at time t k ◮ Reconstruct ϕ k using the knowledge of M and of f k at t k ! Basic Notation: ◮ We call the reconstruction at time t k the analysis ϕ ( a ) . k ◮ The propagated state ϕ ( b ) k + 1 := M ( ϕ ( a ) is called background. k 56/81

  72. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Outline Introduction Orthogonality Sampling Time-Domain Probe Method Field & Shape Reconstruction Time-Domain Probe Data Assimilation 3dVar 4dVar Dynamic Inverse Scattering 57/81

  73. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Basic Approach Let H be the data operator mapping the state ϕ onto the measurements f . Then we need to find ϕ by solving the equation H ϕ = f (10) When we have some initial guess ϕ ( b ) , we transform the equation into H ( ϕ − ϕ ( b ) ) = f − H ϕ ( b ) (11) with the incremental form ϕ = ϕ ( b ) + H − 1 ( f − H ϕ ( b ) ) . (12) 58/81

  74. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Basic Approach Let H be the data operator mapping the state ϕ onto the measurements f . Then we need to find ϕ by solving the equation H ϕ = f (10) When we have some initial guess ϕ ( b ) , we transform the equation into H ( ϕ − ϕ ( b ) ) = f − H ϕ ( b ) (11) with the incremental form ϕ = ϕ ( b ) + H − 1 ( f − H ϕ ( b ) ) . (12) 58/81

  75. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Basic Approach Let H be the data operator mapping the state ϕ onto the measurements f . Then we need to find ϕ by solving the equation H ϕ = f (10) When we have some initial guess ϕ ( b ) , we transform the equation into H ( ϕ − ϕ ( b ) ) = f − H ϕ ( b ) (11) with the incremental form ϕ = ϕ ( b ) + H − 1 ( f − H ϕ ( b ) ) . (12) 58/81

  76. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Regularization 1 Consider an equation H ϕ = f (13) where H − 1 is unstable or unbounded. H ϕ = f H ∗ H ϕ = H ∗ f ⇒ ( α I + H ∗ H ) ϕ = H ∗ f . ⇒ (14) where ( α I + H ∗ H ) has a stable inverse! Tikhonov Regularization : Replace H − 1 by the stable version R α := ( α I + H ∗ H ) − 1 H ∗ (15) with regularization parameter α > 0. 59/81

  77. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Regularization 1 Consider an equation H ϕ = f (13) where H − 1 is unstable or unbounded. H ϕ = f H ∗ H ϕ = H ∗ f ⇒ ( α I + H ∗ H ) ϕ = H ∗ f . ⇒ (14) where ( α I + H ∗ H ) has a stable inverse! Tikhonov Regularization : Replace H − 1 by the stable version R α := ( α I + H ∗ H ) − 1 H ∗ (15) with regularization parameter α > 0. 59/81

  78. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Regularization 1 Consider an equation H ϕ = f (13) where H − 1 is unstable or unbounded. H ϕ = f H ∗ H ϕ = H ∗ f ⇒ ( α I + H ∗ H ) ϕ = H ∗ f . ⇒ (14) where ( α I + H ∗ H ) has a stable inverse! Tikhonov Regularization : Replace H − 1 by the stable version R α := ( α I + H ∗ H ) − 1 H ∗ (15) with regularization parameter α > 0. 59/81

  79. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Regularization 2: Least Squares Tikhonov regularization is equivalent to the minimization of � α � ϕ � 2 + � H ϕ − f � 2 � J ( ϕ ) := (16) The normal equations are obtained from first order optimality conditions ! ∇ ϕ J = 0 . (17) Differentiation leads to 0 = 2 αϕ + 2 H ∗ ( H ϕ − f ) 0 = ( α I + H ∗ H ) ϕ − H ∗ f , ⇒ (18) which is our well-known Tikhonov equation ( α I + H ∗ H ) ϕ = H ∗ f . 60/81

  80. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Regularization 2: Least Squares Tikhonov regularization is equivalent to the minimization of � α � ϕ � 2 + � H ϕ − f � 2 � J ( ϕ ) := (16) The normal equations are obtained from first order optimality conditions ! ∇ ϕ J = 0 . (17) Differentiation leads to 0 = 2 αϕ + 2 H ∗ ( H ϕ − f ) 0 = ( α I + H ∗ H ) ϕ − H ∗ f , ⇒ (18) which is our well-known Tikhonov equation ( α I + H ∗ H ) ϕ = H ∗ f . 60/81

  81. Introduction 3dVar Orthogonality Sampling 4dVar Time-Domain Probe Method Dynamic Inverse Scattering Data Assimilation Regularization 2: Least Squares Tikhonov regularization is equivalent to the minimization of � α � ϕ � 2 + � H ϕ − f � 2 � J ( ϕ ) := (16) The normal equations are obtained from first order optimality conditions ! ∇ ϕ J = 0 . (17) Differentiation leads to 0 = 2 αϕ + 2 H ∗ ( H ϕ − f ) 0 = ( α I + H ∗ H ) ϕ − H ∗ f , ⇒ (18) which is our well-known Tikhonov equation ( α I + H ∗ H ) ϕ = H ∗ f . 60/81

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