presentation

1. presentation Seismic wave Propagation and Imaging in Complex media: a European network • DANIEL PETER • Early Stage Researcher • Host Institution: ETH Zurich • Place of Origin: Sargans, Switzerland • Appointment Time: August 2004 • Project: Membrane waves, sensitivity kernels, global tomography. • Task Groups: TG Planetary Scale • Cooperation: Oxford University, IPG Paris

2. title Surface wave tomography: membrane waves and adjoint methods Daniel Peter In collaboration with Lapo Boschi, Carl Tape*, John H. Woodhouse** ETH Zürich, Switzerland *Caltech, California, USA **University of Oxford, England

3. Motivation Motivation Born sensitivity kernels consider single scattering in the Earth and should improve tomographic models, especially for surface waves. Computational costs of both numerical and analytical kernel construction techniques in 3D are greatly reduced if membrane waves as an analogue for surface waves or adjoint methods are used.

4. Overview Overview Forward problem: Membrane wave model Sensitivity kernels Global surface wave tomography Inverse problem:

5. Membrane theory Membrane waves Theory • Equations of motion with a Love wave ansatz become the 2D wave equation for scalar wave potentials on a spherical membrane [Tanimoto, 1990] [Tromp & Dahlen, 1993] [Tape, 2003] [Tape, 2003]

6. Membrane wave Membrane waves Wave propagation • Analytical source on the membrane • Simulation error by comparison with analytical solution in a homogeneous background Earth [Tape, 2003] Time(s) Scalar potential s Model time (s)

7. Membrane wave Membrane waves Wave propagation • Analytical source on the membrane • Simulation error by comparison with analytical solution in a homogeneous background Earth [Tape, 2003] Without filtering Love waves 150 s Phase shift (%) Grid refinement level

8. Membrane wave movie Membrane waves Wave propagation • Simulation on the whole sphere (no boundary conditions needed) in the presence of a single low velocity spot

9. Sensitivity theory Sensitivity kernels Theory • Born scattering: Phase anomalies are given as linear relation between phase velocity perturbations and sensitivity kernel values (aka. “banana-doughnut” kernel/function)

10. Sensitivity theory Sensitivity kernels Theory • “Brute-force” approach: (1) we define phase velocity perturbations (2) measure the phase anomaly by cross-correlation after the simulation and (3) obtain the sensitivity kernel value at the location of the perturbation

11. Sensitivity direct vs analytical Sensitivity kernels Homogeneous background Earth “Brute-force” approach vs. analytically derived kernel Love waves at 150 s period, source/station placed on equator [Spetzler et al., 2002]

12. Sensitivity theory Sensitivity kernels Theory • “Adjoint-method” approach: (1) after a first simulation, we define the “adjoint source” (2) a second, back-propagation is needed to compute the “adjoint seismogram” and (3) we obtain the sensitivity kernel value at every location on the sphere after only these two simulations by

13. Sensitivity direct vs adjoint Sensitivity kernels Homogeneous background Earth “Brute-force” approach and “adjoint-method” approach cross-section 10º cross-section 45º Love waves at 150 s period Sensitivity kernel latitude latitude

14. Sensitivity heterogeneous Sensitivity kernels Heterogeneous background Earth phase velocity map by CRUST 2.0 for Love waves at 150 s projected on the membrane

15. Sensitivity heterogeneous Sensitivity kernels Heterogeneous background Earth homogeneous vs. heterogeneous phase velocity map Love waves at 150 s kernel difference

16. Tomography theory Surface wave tomography Theory • Linearized inverse problem x: phase velocity map d: phase anomaly measurements where matrix A uses sensitivity kernels

17. Tomography data Surface wave tomography Data • Harvard database with relative phase anomalies measured by [Ekström et al, 1997] 150 s Love wave observations ray-theoretical hit count map (3º x 3º pixel map)

18. Tomography homogeneous Surface wave tomography Trade-off analysis • Homogeneous starting model L-curves for different roughness damping (150 s Love waves) curvatures from L-curves misfit Ray theory Analytic kernels Adjoint kernels Normalized image roughness curvature Normalized image roughness

19. Tomography homogeneous Surface wave tomography Phase velocity maps • Homogeneous starting model 150 s Love waves

20. Tomography heterogeneous Surface wave tomography Trade-off analysis • Heterogeneous starting model L-curves for different roughness damping (150 s Love waves) curvatures from L-curves misfit Adjoint heterogeneous Analytic homogeneous Adjoint homogeneous Normalized image roughness curvature Normalized image roughness

21. Tomography heterogeneous Surface wave tomography HOMOGENEOUS Phase velocity maps • Heterogeneous & homogeneous starting model 150 s Love waves HETEROGENEOUS ADJOINT

22. Conclusions Conclusions • The membrane wave model can be used to derive numerically 2D sensitivity kernels relating surface-wave phase anomaly data to phase velocity heterogeneities • The tomographic results and trade-off analysis suggest that this method is at least compatible with existing approaches (ray theory, analytical Born theory) • Improvement may become more significant in 3D when inverting for shear velocity in the Earth’s upper mantle [Zhou, 2005]

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24. Membrane theory Membrane waves Theory • 2D wave equation for scalar wave potentials on a spherical membrane can be solved by a finite-difference approximation [Heikes & Randall, 1994] [Tape, 2003]

25. Membrane grid Membrane waves Spherical grid • Initial points: dodecahedron & icosahedron • “Subfolding” method divides a single triangle into smaller ones [Tape, 2003]

26. Membrane grid Membrane waves Spherical grid • Distortion produces numerical artifacts • Error in Laplacian by comparison with spherical harmonic functions

27. Membrane grid Membrane waves Spherical grid • Distortion produces numerical artifacts • Error in Laplacian by comparison with spherical harmonic functions

28. Membrane wave Membrane waves Wave propagation • Phase velocity maps with a single scatterer can be used to derive “sensitivity functions”: depending on the location of the scatterer, the seismogram at the receiver station will vary in phase and amplitude

29. Membrane wave Membrane waves Wave propagation • Parallel computation on the SEG cluster (up to 16 AMD Opteron 64-bit processors, 2.0 GHz clock speed) Grid spacing ~70 km

30. Sensitivity theory Sensitivity kernels Theory - drawback • “Brute-force” approach needs around 64’442 simulations to compute a complete kernel (over the whole sphere) reduction to 181 simulations can only be done for a homogeneous background phase velocity map (reduces computation time from around 3 days to 12 mins)

31. Scope Project ScopeGlobal surface wave tomography: membrane waves and adjoint methods Born sensitivity kernels take account of single scattering in the Earth and should improve tomographic models especially for surface waves. Computational costs of both numerical and analytical kernel construction techniques in 3D are greatly reduced if membrane waves as an analogue for surface waves or adjoint methods are used.

32. Results Main Results Membrane waves together with adjoint methods can be used to derive numerical sensitivity kernels including back-scattering. Grid analysis Sensitivity kernels Benchmarking adjoint analytical direct

33. Outlook Outlook Membrane kernels might slightly improve the resolution due to back-scattering and avoidance of far-field approximations. Love 150s direct analytical direct Ray theory analytical Full 3D sensitivity kernels should allow for even better tomographic images. Adjoint methods may be a way to reach this. A next step will be the construction of 3D kernels by normal modes and spectral element codes… Ray theory

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