An SVD in Spherical Surface Wave Tomography Chemnitz University of Technology, Faculty of Mathematics An SVD in Spherical Surface Wave Tomography Michael Quellmalz (joint work with Ralf Hielscher and Daniel Potts) Chemnitz University of Technology Faculty of Mathematics New Trends in Parameter Identification for Mathematical Models Chemnitz Symposium on Inverse Problems on Tour Rio de Janeiro, 2 November 2017 2 November 2017 · Michael Quellmalz 1 / 24 tu-chemnitz.de/ ∼ qmi
An SVD in Spherical Surface Wave Tomography Content 1. Introduction Motivation 2. Arc transform Definition Singular value decomposition 3. Special cases Arcs starting in a fixed point Recovery of local functions Arcs with fixed length 2 November 2017 · Michael Quellmalz 2 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Content 1. Introduction Motivation 2. Arc transform Definition Singular value decomposition 3. Special cases Arcs starting in a fixed point Recovery of local functions Arcs with fixed length 2 November 2017 · Michael Quellmalz 3 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Funk–Radon transform ◮ Sphere S 2 = { ξ ∈ R 3 : � ξ � = 1 } ◮ Function f : S 2 → C ◮ Funk–Radon transform (a.k.a. Funk transform or spherical Radon transform) F : C ( S 2 ) → C ( S 2 ) , � F f ( ξ ) = f ( η ) d λ ( η ) � ξ , η � =0 Theorem [Funk 1911] Any even function f can be reconstructed from F f . 2 November 2017 · Michael Quellmalz 4 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Funk–Radon transform ◮ Sphere S 2 = { ξ ∈ R 3 : � ξ � = 1 } ◮ Function f : S 2 → C ◮ Funk–Radon transform (a.k.a. Funk transform or spherical Radon transform) F : C ( S 2 ) → C ( S 2 ) , � F f ( ξ ) = f ( η ) d λ ( η ) � ξ , η � =0 Theorem [Funk 1911] Any even function f can be reconstructed from F f . 2 November 2017 · Michael Quellmalz 4 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Funk–Radon transform ◮ Sphere S 2 = { ξ ∈ R 3 : � ξ � = 1 } ◮ Function f : S 2 → C ◮ Funk–Radon transform (a.k.a. Funk transform or spherical Radon transform) F : C ( S 2 ) → C ( S 2 ) , � F f ( ξ ) = f ( η ) d λ ( η ) � ξ , η � =0 Theorem [Funk 1911] Any even function f can be reconstructed from F f . 2 November 2017 · Michael Quellmalz 4 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Motivation Spherical surface wave tomography ◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S 2 Method ◮ Measure the traveltimes of surface waves between many pairs of epicenter and detector ◮ Reconstruct the local speed of propagation Assumption A wave propagates along the arc of a great circle. 2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Motivation Spherical surface wave tomography ◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S 2 Method ◮ Measure the traveltimes of surface waves between many pairs of epicenter and detector ◮ Reconstruct the local speed of propagation Assumption A wave propagates along the arc of a great circle. 2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Motivation Spherical surface wave tomography ◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S 2 Method ◮ Measure the traveltimes of surface waves between many pairs of epicenter and detector ◮ Reconstruct the local speed of propagation Assumption A wave propagates along the arc of a great circle. 2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Motivation Spherical surface wave tomography ◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S 2 Method ◮ Measure the traveltimes of surface waves between many pairs of epicenter and detector ◮ Reconstruct the local speed of propagation Assumption A wave propagates along the arc of a great circle. 2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Motivation Spherical surface wave tomography ◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S 2 Method ◮ Measure the traveltimes of surface waves between many pairs of epicenter and detector ◮ Reconstruct the local speed of propagation Assumption A wave propagates along the arc of a great circle. 2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Motivation Spherical surface wave tomography ◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S 2 Method ◮ Measure the traveltimes of surface waves between many pairs of epicenter and detector ◮ Reconstruct the local speed of propagation Assumption A wave propagates along the arc of a great circle. 2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Selected references P. Funk. ¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien. Math. Ann. , 74(2): 278 – 300, 1913. J. H. Woodhouse and A. M. Dziewonski. Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms. J. Geophys. Res. Solid Earth , 89(B7):5953–5986, 1984. A. Amirbekyan, V. Michel, and F. J. Simons. Parametrizing surface wave tomographic models with harmonic spherical splines. Geophys. J. Int. , 174(2):617–628, 2008. R. Hielscher, D. Potts and M. Quellmalz. An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models . Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284 2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Selected references P. Funk. ¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien. Math. Ann. , 74(2): 278 – 300, 1913. J. H. Woodhouse and A. M. Dziewonski. Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms. J. Geophys. Res. Solid Earth , 89(B7):5953–5986, 1984. A. Amirbekyan, V. Michel, and F. J. Simons. Parametrizing surface wave tomographic models with harmonic spherical splines. Geophys. J. Int. , 174(2):617–628, 2008. R. Hielscher, D. Potts and M. Quellmalz. An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models . Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284 2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Selected references P. Funk. ¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien. Math. Ann. , 74(2): 278 – 300, 1913. J. H. Woodhouse and A. M. Dziewonski. Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms. J. Geophys. Res. Solid Earth , 89(B7):5953–5986, 1984. A. Amirbekyan, V. Michel, and F. J. Simons. Parametrizing surface wave tomographic models with harmonic spherical splines. Geophys. J. Int. , 174(2):617–628, 2008. R. Hielscher, D. Potts and M. Quellmalz. An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models . Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284 2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/ ∼ qmi
Introduction Selected references P. Funk. ¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien. Math. Ann. , 74(2): 278 – 300, 1913. J. H. Woodhouse and A. M. Dziewonski. Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms. J. Geophys. Res. Solid Earth , 89(B7):5953–5986, 1984. A. Amirbekyan, V. Michel, and F. J. Simons. Parametrizing surface wave tomographic models with harmonic spherical splines. Geophys. J. Int. , 174(2):617–628, 2008. R. Hielscher, D. Potts and M. Quellmalz. An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models . Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284 2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/ ∼ qmi
Arc transform Content 1. Introduction Motivation 2. Arc transform Definition Singular value decomposition 3. Special cases Arcs starting in a fixed point Recovery of local functions Arcs with fixed length 2 November 2017 · Michael Quellmalz 7 / 24 tu-chemnitz.de/ ∼ qmi
Arc transform Definition The arc transform ◮ Function f : S 2 → R ◮ Surface waves: f = 1 c ( c ... speed of sound) ◮ ξ , ζ ∈ S 2 not antipodal ◮ γ ( ξ , ζ ) great circle arc Definition � t ( ξ , ζ ) = f d γ γ ( ξ , ζ ) 2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/ ∼ qmi
Arc transform Definition The arc transform ◮ Function f : S 2 → R ◮ Surface waves: f = 1 c ( c ... speed of sound) ◮ ξ , ζ ∈ S 2 not antipodal ◮ γ ( ξ , ζ ) great circle arc Definition � t ( ξ , ζ ) = f d γ γ ( ξ , ζ ) 2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/ ∼ qmi
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