1. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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 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. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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 title

3. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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. Motivation

4. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Overview 1.Membrane wave model 2.Sensitivity kernels 3.Global surface wave tomography Overview Forward problem: Inverse problem:

5. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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] Membrane theory [Tape, 2003]

6. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Membrane waves Wave propagation  Analytical source on the membrane  Simulation error by comparison with analytical solution in a homogeneous background Earth Membrane wave [Tape, 2003] Time(s) Model time (s) Scalar potential s

7. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Membrane waves Wave propagation  Analytical source on the membrane  Simulation error by comparison with analytical solution in a homogeneous background Earth Membrane wave [Tape, 2003] Phase shift (%) Grid refinement level Without filtering Love waves 150 s

8. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Membrane waves Wave propagation  Simulation on the whole sphere (no boundary conditions needed) in the presence of a single low velocity spot Membrane wave movie

9. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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) Sensitivity theory

10. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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 Sensitivity theory

11. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Sensitivity kernels Homogeneous background Earth “Brute-force” approach vs. analytically derived kernel Love waves at 150 s period, source/station placed on equator Sensitivity direct vs analytical [Spetzler et al., 2002]

12. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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 Sensitivity theory

13. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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 direct vs adjoint latitude Sensitivity kernel

14. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Sensitivity kernels Heterogeneous background Earth phase velocity map by CRUST 2.0 for Love waves at 150 s projected on the membrane Sensitivity heterogeneous

15. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Sensitivity kernels Heterogeneous background Earth homogeneous vs. heterogeneous phase velocity map Love waves at 150 s kernel difference Sensitivity heterogeneous

16. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Theory  Linearized inverse problem x: phase velocity map d: phase anomaly measurements where matrix A uses sensitivity kernels Tomography theory Surface wave tomography

17. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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) Tomography data Surface wave tomography

18. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Trade-off analysis  Homogeneous starting model L-curves for different roughness damping (150 s Love waves) curvatures from L-curves Tomography homogeneous Surface wave tomography Normalized image roughness curvature misfit Ray theory Analytic kernels Adjoint kernels

19. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Phase velocity maps  Homogeneous starting model 150 s Love waves Tomography homogeneous Surface wave tomography

20. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Trade-off analysis  Heterogeneous starting model L-curves for different roughness damping (150 s Love waves) curvatures from L-curves Tomography heterogeneous Surface wave tomography Normalized image roughness curvature misfit Adjoint heterogeneous Analytic homogeneous Adjoint homogeneous

21. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Surface wave tomography Phase velocity maps  Heterogeneous & homogeneous starting model 150 s Love waves Tomography heterogeneous HETEROGENEOUS HOMOGENEOUS ADJOINT

22. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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] Conclusions

23. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland For many more details, please just ask… Thanks Thank you for your attention!

24. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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] Membrane theory [Tape, 2003]

25. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Membrane waves Spherical grid  Initial points: dodecahedron & icosahedron  “Subfolding” method divides a single triangle into smaller ones [Tape, 2003] Membrane grid

26. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Membrane waves Spherical grid  Distortion produces numerical artifacts  Error in Laplacian by comparison with spherical harmonic functions Membrane grid

27. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Membrane waves Spherical grid  Distortion produces numerical artifacts  Error in Laplacian by comparison with spherical harmonic functions Membrane grid

28. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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 Membrane wave

29. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Membrane waves Wave propagation  Parallel computation on the SEG cluster (up to 16 AMD Opteron 64-bit processors, 2.0 GHz clock speed) Membrane wave Grid spacing ~70 km

30. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland 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) Sensitivity theory

31. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Project Scope Global 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. Scope

32. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Main Results Membrane waves together with adjoint methods can be used to derive numerical sensitivity kernels including back-scattering. adjoint analytical direct Grid analysis Benchmarking Sensitivity kernels Results

33. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Outlook 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… Membrane kernels might slightly improve the resolution due to back-scattering and avoidance of far-field approximations. Love 150s analytical direct Ray theory direct analytical Ray theory Outlook

34. www.spice-rtn.org SPICE Research and Training Workshop III, July 22-28, Kinsale, Ireland Thanks Thank you for your attention!