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