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Surface wave tomography : part3: waveform inversion, adjoint tomography. Huajian Yao USTC May 24, 2013. Previous lectures: inversion of Vs structure from travel times of surface wave propagation, i.e., from phase or group velocity dispersion data. - PowerPoint PPT Presentation
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Surface wave tomography:part3: waveform inversion, adjoint
tomography
Huajian Yao
USTC May 24, 2013
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
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
Average wavenumber perturbation along the surface wave path Pj
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:
PWI ---- Step 1 : Waveform inversion for path averaged structure
Determine γj (or path average model) along the Pj path
dk(t): observed dataRk: windowing and filtering operatorwk: weighting of the various data
Inversion Method: conjugate gradient (Nolet, 1987, GRL)Use finite differences to compute Hessian Matrix H
Example of waveform inversion (Simons et al., 1999, Lithos)
PWI ---- Step 2: Tomographic inversion for 3-D structure from path averaged modelsIntroduce new parameters:
Orthogonality condition:
Example of tomographic inversion for 3-D structure from path averaged models (Simons et al., 1999, Lithos)
Adjoint tomographyCalculate 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
Some reference papers Tarantola, A., 1986. A strategy for nonlinear elastic inversion of
seismic reflection data. Geophysics 51, 1893–1903.
Tromp, J., Tape, C.H., Liu, Q., 2005. Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophysical Journal International 160, 195–216.
Liu, Q., Tromp, J., 2006. Finite-frequency kernels based on adjoint methods. Bulletin of the Seismological Society of America 96, 2283–2397.
Tape, C.H., Liu, Q., Tromp, J., 2007. 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, 2012. Seismic imaging: From classical to adjoint tomography. Tectonophysics, http://dx.doi.org/10.1016/j.tecto.2012.07.006
Adjoint kernels (Tromp et al. 2005)
Waveform misfit:
Fréchet derivatives:
Waveform adjoint field:
Waveform adjoint source
Time reversed data residual
Isotropic Medium
kernels
Traveltime misfit:
The Frechet derivative of traveltime is defined in terms of cross-correlation of an observed and synthetic waveform
kernels
Traveltime adjoint field
adjoint source
Traveltime misfit kernels
Combined traveltimeadjoint field
Combined traveltimeadjoint source
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
Construction of
an event kernel
for this target
model for
multiple
receivers,
thereby
incorporating
multiple
measurements
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.
Iterative improvement of the reference phase-speed model using
the conjugate gradient algorithm
Example of 3-D adjoint tomography using based on traveltime misfits (Tape et al. 2009, Science; Tape et al.
2010, GJI)
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)
Frequency-dependent data fitting
6-30 sm16 3-30 s
m16
2-30 sm16 2-30 s
1-D model
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 differencesBottom: amp. differences
Multipathing of Rayleigh waves and complex kernels
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