Spectral-Element and Adjoint Methods in Seismology

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1 Spectral-Element and Adjoint Methods in Seismology Jeroen Tromp Min Chen, Vala Hjorleifsdottir, Dimitri Komatitsch, Swami Krishnan, Qinya Liu, Alessia Maggi, Anne Sieminski, Carl Tape & Ying Zhou

2 Introduction to the Spectral-Element Method (SEM)

3 Governing Equations Equation of motion: Boundary condition: Initial conditions: Earthquake source:

4 Weak Form Weak form valid for any test vector Boundary conditions automatically included Source term explicitly integrated Finite-fault (kinematic) rupture:

5 Finite-Elements Mapping from reference cube to hexahedral elements: Volume relationship: Jacobian of the mapping: Jacobian matrix:

6 Lagrange Polynomials and Gauss- Lobatto-Legendre (GLL) Points The 5 degree 4 Lagrange polynomials: Degree 4 GLL points: General definition: GLL points are n+1 roots of: Note that at a GLL point:

7 Interpolation Representation of functions on an element in terms of Lagrange polynomials: Gradient on an element:

8 Integration Integration of functions over an element based upon GLL quadrature: Integrations are pulled back to the reference cube In the SEM one uses: interpolation on GLL points GLL quadrature Degree 4 GLL points:

9 The Diagonal Mass Matrix Representation of the displacement: Degree 4 Lagrange polynomials: Representation of the test vector: Weak form: Diagonal mass matrix: Integrations are pulled back to the reference cube In the SEM one uses: interpolation on GLL points GLL quadrature Degree 4 GLL points:

10 Assembly Need to distinguish: local mesh global mesh Efficient routines available from FEM applications Global equations: Global SEM time-marching is based upon an explicit second-order time scheme

11 Parallel Implementation Global mesh partitioning: Regional mesh partitioning: Cubed Sphere: 6 n2 mesh slices n x m mesh slices

12 Basin Code: SPECFEM3D_BASIN Freely available for non-commercial use from geodynamics.org Manual available at geodynamics.org 3D Attenuation 3D Anisotropy Ocean load Topography & bathymetry Kinematic ruptures Movies Models: Harvard 3D Southern California model SOCAL 1D model Homogeneous half-space model Adjoint capabilities

13 Southern California Simulations A A B B Vp(km/s) n x m mesh slices

14 Periods > 2 s Komatitsch et al. 2004

15 June 12, 2005, M=5.1 Big Bear QuickTime and a YUV420 codec decompressor are needed to see this picture.

16 3D Regional Forward Simulations Periods > 6 s 379 km June 12, 2005, M=5.1 Big Bear 100km Qinya Liu

17 San Andreas Rupture Scenario QuickTime and a YUV420 codec decompressor are needed to see this picture.

18 San Andreas Rupture Scenario: Quantitative Seismic Hazard Assessment Swami Krishnan

19 Near Real-Time Applications Automated near real-time simulations of all M>3.5 events ShakeMovies at Soon: CMT solutions Synthetic seismograms

for mesh partitioning & load-balancing Retain the SPECFEM3D_BASIN solver (takes ParMETIS meshes;

20 SPECFEM3D_BASIN: Future Plans Switch to a (parallel) CUBIT hexahedral finite-element mesher (Casarotti, Lee) Topography & bathymetry Major geological interfaces Basins Fault surfaces Use ParMETIS or SCOTCH for mesh partitioning & load-balancing Retain the SPECFEM3D_BASIN solver (takes ParMETIS meshes; Komatitsch) Add dynamic rupture capabilities (Ampuero, Lapusta, Kaneko) Taipei Basin 102km N 88km 100km S.-J. Lee

21 Global Code: SPECFEM3D_GLOBE Freely available for non-commercial use from geodynamics.org Manual available at geodynamics.org 3D Attenuation 3D Anisotropy Ocean load Topography & bathymetry Rotation Self-gravitation (Cowling approximation) Kinematic ruptures Movies Models: 1D models: isotropic PREM, transversely isotropic PREM, AK135, IASP91, 1066A S20RTS Crust2.0 Adjoint capabilities

22 Global Simulations PREM benchmarks Cubed sphere mesh Parallel implementation: 2 6 n mesh slices

23 SEM Implementation of Attenuation Anelastic, anisotropic constitutive relationship: Equivalent Standard Linear Solid (SLS) formulation: Attenuation (3 SLSs) Memory variable equation: Unrelaxed modulus: Modulus defect: Physical dispersion (3 SLSs)

24 Effect of Attenuation

25 Attenuation

26 Effect of Anisotropy

27 SEM Implementation of Anisotropy

28 Antipodal Transverse Record

29 Vertical PKP

30 Rotation & Self-Gravitation Wave equation solved by SPECFEM3D_GLOBE in crust, mantle and inner core: Wave equation solved by SPECFEM3D_GLOBE in fluid outer core: SPECFEM3D_GLOBE uses domain decomposition between the fluid outer core and the solid inner core and mantle matching exactly: continuity of traction continuity of the normal component of displacement

31 Full Gravity Versus Cowling

32 SEM Implementation of Gravity

33 Effect of Ocean

34 SEM Implementation of Ocean Modified boundary condition:

35 3D Mantle Models S20RTS (Ritsema et al. 1999)

36 Crustal and Topographic Models Crust 2.0 (Bassin et al. 2000) ETOPO5

37 Great 2004 Sumatra-Andaman Earthquake Finite slip model (Chen et al., 2005) Main shock & aftershocks (Harvard)

38 Sumatra Surface Waves QuickTime and a YUV420 codec decompressor are needed to see this picture.

39 Surface-Wave Fits Vala Hjorleifsdottir

SPECFEM3D_GLOBE: Future Plans On-demand TeraGrid applications: Automated, near real-time simulations of all M>6 earthquakes Analysis of past events (more than

40 SPECFEM3D_GLOBE: Future Plans On-demand TeraGrid applications: Automated, near real-time simulations of all M>6 earthquakes Analysis of past events (more than 20,000 events) Seismology Web Portal (prototype available at this meeting) Petascale simulations: Global simulations at 1-2 Hz New doubling brick (perfect load-balancing)

41 Adjoint Spectral-Element Simulations (ASEM)

42 Adjoint Tomography PDE-constrained waveform tomography: Change in the waveform misfit function:

43 Adjoint Equations Adjoint wavefield: Adjoint equation of motion: Adjoint boundary conditions: Adjoint initial conditions: Adjoint source:

44 Frechet derivative The Frechet derivative may be expressed as: Density and elastic tensor kernels:

45 Isotropic Kernels For isotropic perturbations we have: where and we have defined the strain deviators In terms of wave speeds: where

46 Traveltime Frechet Derivatives Traveltime tomography: Change in the misfit function: Traveltime anomaly in terms of banana-donut kernel (Dahlen): The kernel is a weighted sum of banana-donut kernels :

47 Adjoint Wavefield Kernel in terms of the adjoint wavefield: The adjoint wavefield is generated by the adjoint source Notes: Need simultaneous access to the regular wavefield at time t and the adjoint wavefield at time T t Use of the time-reversed velocity as the source for the adjoint wavefield

48 Numerical Implementation Need simultaneous access to and During calculation of adjoint field, reconstruct by solving the `backward wave equation Need to store from a previous forward simulation: Last snapshot Wavefield absorb on artificial boundaries Challenge: `Undoing attenuation

49 2D Adjoint Tomography

50 Construction of a Banana-Donut Kernel Tape et al. 2006

51 Adjoint Tomography

52 Construction of an Event Kernel Event Kernel: Sum of weighted bananadonut kernels Two simulations per event

53 Construction of the Misfit Kernel Misfit kernel: Sum of all the event kernels Two simulations per event Gradient of misfit function

54 Conjugate Gradient Algorithm

55 Conjugate Gradient Algorithm

56 Joint Structure-Source Inversion

58 Southern California Rayleigh Waves 25 events

59 Toward 3D Tomography: SPECFEM3D Adjoint Capabilities

60 3D Sensitivity Kernels S SS PS PS SPECFEM3D_BASIN Liu & Tromp 2006

61 3D Body-Wave Sensitivity Kernels September 3, 2002, M=4.2 Yorba Linda

62 3D Surface-Wave Sensitivity Kernels Rayleigh (HEC) Love (HEC)

63 Our First 3D Event Kernels!

64 Global 3D Body-Wave Sensitivity Kernels Finite-Frequency effects 20 second P wave 9 second P wave SPECFEM3D_GLOBE Qinya Liu

65 Global 3D Body-Wave Sensitivity Kernels Pdiff PKP ScS SKS Qinya Liu

66 PKP Kernel

67 P P Kernel Liu & Tromp 2006

68 Global 3D Transversely Isotropic Kernels α h P α v α h SH α v β h β v β h β v η ρ η ρ Anne Sieminski

69 Transversely Isotropic Parameters: A, C, L, N, F 1 ζ: J, K, M 2 ζ: G, B, H 3 ζ: D 4 ζ: E Anne Sieminski

70 1D Versus 3D Kernels 1D 3D Ying Zhou

71 Time-Reversal Imaging: Glacial Earthquakes Greenland Carene Larmat

72 Measuring all available Phase & Amplitude Anomalies

74 Measure all suitable phases RAR F2=0.93 dt=1.00 F2=0.83 dt=1.00 F2=0.83 dt=6.00 F2=0.68 dt=2.00 F2=0.78 dt=-1.00 PP PKiKP ppkikp spkikp SKiKP SKSac S SKKSac psksac ps ssksac ss P PcP pp ScS SPn sp PnS SS SKKSdf SKKSac P P df Time (s) Seismograms ANMO F2=0.91 F2=0.93 F2=0.68 F2=0.92 F2=0.91 F2=0.83 F2=0.84 F2=0.83 F2=0.69 F2=0.85 F2=0.91 F2=0.92 F2=0.51 F2=0.61 F2= Time (s) Alessia Maggi

75 Conclusions Adjoint methods: Choose an observable, e.g., waveforms or cross-correlation traveltimes Choose a measure of misfit, e.g., least-squares Determine the appropriate adjoint source for this observable & measurement Use fully 3D reference models Any arrival suitable for measurement No dependence on the number of stations, components, or measurements 3D sensitivity kernels may be calculated based upon two forward simulations for each earthquake Number of simulations: 3 * (# earthquakes) * (# iterations) Full anisotropy for the same cost Attenuation remains a challenge Regional simulations: One 3 minute forward simulation accurate to 1.5 seconds takes 45 minutes on a 75 node cluster 150 events and 3 iterations would require 1800 simulations, i.e., three weeks of dedicated CPU time on 75 nodes Near real-time simulations Global simulations: One 1 hour forward simulation accurate to 20 seconds takes 4 hours on a 75 node cluster 500 events and 3 iterations would require 6,000 simulations, i.e., 100 days on a 750 node cluster Near real-time simulations On-demand global seismology Petascale application

CO2 sequestration crosswell monitoring based upon spectral-element and adjoint methods

CO2 sequestration crosswell monitoring based upon spectral-element and adjoint methods Christina Morency Department of Geosciences, Princeton University Collaborators: Jeroen Tromp & Yang Luo Computational