1. Surface wave tomography： part3: waveform inversion, adjoint tomography
Huajian Yao
USTC
May 24, 2013

2. Previous lectures: inversion of Vs structure from travel times of surface wave propagation, i.e., from phase or group velocity dispersion data.
This lecture: inversion of Vs structure from surface wave waveform methods: partitioned waveform inversion and adjoint tomography

3. Partitioned waveform inversion (PWI) (Nolet, 1990, JGR)
Seismic waves (spectrum) at station j as a sum of surface wave modes (n)
Lateral
homogeneous
Lateral
heterogeneous

4. Average wavenumber perturbation along the surface wave path Pj

5. Rewrite the seismic signal (spectrum) as:
Rewrite the velocity perturbation as a basis function:
We have the linear relationship between wavenumber perturbation along the ray path and the model basis functions:

6. PWI ---- Step 1 : Waveform inversion for path averaged structure
Determine γj (or path average model) along the Pj path
dk(t): observed data
Rk: windowing and filtering operator
wk: weighting of the various data
Inversion Method: conjugate gradient (Nolet, 1987, GRL)
Use finite differences to compute Hessian Matrix H

7. Example of waveform inversion (Simons et al., 1999, Lithos)

8. PWI ---- Step 2: Tomographic inversion for 3-D structure from path averaged models
Introduce new parameters:
Orthogonality condition:

9. Example of tomographic inversion for 3-D structure from path averaged models (Simons et al., 1999, Lithos)

11. Adjoint tomography
Calculate 3-D sensitivity kernels of data (waveforms, traveltimes, etc) to model parameters (e.g., density, elastic parameters) from 3D models using the adjoint method
This step requires computation of wavefields twice (forward wavefield and adjoint wavefield) using methods of FD, FEM, SEM, etc
Perform tomographic inversion based on 3D adjoint kernels (conjugate gradient, Newton’s method, Gauss- Newton’s method, …)
Update the model, re-compute the adjoint kernels, then iterate a number of times to obtain the final model
FD: finite difference
FEM: finite element
SEM: spectral element

12. Some reference papers
Tarantola, A., A strategy for nonlinear elastic inversion of seismic reflection data. Geophysics 51, 1893–1903.
Tromp, J., Tape, C.H., Liu, Q., Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophysical Journal International 160, 195–216.
Liu, Q., Tromp, J., Finite-frequency kernels based on adjoint methods. Bulletin of the Seismological Society of America 96, 2283–2397.
Tape, C.H., Liu, Q., Tromp, J., Finite-frequency tomography using adjoint methods — methodology and examples using membrane surface waves. Geophysical Journal International 168, 1105–1129
Fichtner, A., Bunge, H.P., Igel, H., 2006a. The adjoint method in seismology — I. Theory. Physics of the Earth and Planetary Interiors 157, 86–104.
Fichtner, A., Bunge, H.P., Igel, H., 2006b. The adjoint method in seismology — II. Applications:traveltimes and sensitivity functionals. Physics of the Earth and Planetary Interiors 157, 105–123.
Liu, Q, Y.J. Gu, Seismic imaging: From classical to adjoint tomography. Tectonophysics,

13. Adjoint kernels (Tromp et al. 2005)
Waveform misfit:
Fréchet derivatives:
Waveform adjoint field:
Waveform
adjoint source
Time reversed data residual

14. Isotropic Medium
kernels

15. Traveltime misfit:
The Frechet derivative of traveltime is defined in terms
of cross-correlation of an observed and synthetic waveform

16. kernels
Traveltime adjoint field
adjoint source

17. Traveltime misfit kernels
Combined traveltime
adjoint field
Combined traveltime
adjoint source

18. Example of 2-D adjoint tomography using surface waves based on traveltime misfits (Tape et al. 2007, GJI)
Sequence of interactions between the regular and adjoint wavefields during the construction of a traveltime cross-correlation event kernel K(x) for one event-receiver case

20. Construction of an event kernel for this target model for multiple receivers, thereby incorporating multiple measurements

21. Construction of a misfit kernel
Construction of a misfit kernel.(a)–(g)Individual event kernels, (h) The misfit kernel is simply the sum of the 25 event kernels. (i) The source–receiver geometry and target phase-speed model.

22. Iterative improvement of the reference phase-speed model using the conjugate gradient algorithm

23. Example of 3-D adjoint tomography using based on traveltime misfits (Tape et al. 2009, Science; Tape et al. 2010, GJI)

24. Starting model (m00): 3-D reference model
Earthquakes: point sources (origin time, hypocenter, moment tensor from previous studies)
Traveltime measurements: 3 components data, 3 bands, cross-correlation traveltime differences
Inversion method: conjugate gradient (Tape et al. 2007)

26. Frequency-dependent data fitting
6-30 s
m16
3-30 s
m16
2-30 s
m16
2-30 s
1-D model

29. extra: extra earthquakes
tomo: earthquakes (143) and stations (203) used in the tomographic inversion (tomo)
extra: extra earthquakes
(91) used in validating the final tomographic model, but not used in the tomographic inversion
top: Travel time differences
Bottom: amp. differences

30. Multipathing of Rayleigh waves and complex kernels