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| Problem Statement |
Full Waveform Tomography (FWT) based on the Adjoint-Wavefield (AW-FWT) and Generalized Seismological Data Functionals (GSDF) methods to iteratively invert an initial 3-D crustal model for the Canterbury region. In FWT, the wavefields are generated by numerical solutions of the 3-D elastodynamic/ visco-elastodynamic equations according to a specific velocity model and then compared with the observed data to extract the misfit between the current model with the true model. For inversion, a number of misfit measurements based on 146 earthquake seismograms for 43 seismic stations in the Canterbury region are used as the observed data. After a number of inversion iterations, the final inverted model using FWT has good potential for more accurate simulation of ground motion for the Canterbury region, and provides a platform for extension of this method to the wider New Zealand region. |
| Project Members |
Andrei Nguyen, Brendon Bradley, and Robin Lee |
Description (Objectives / Outcomes) |
Synthetic study of FWT for a 1-D velocity model. Inversion of the crustal velocity model for Canterbury region.
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Tasks |
Forward modeling of the elastic wave propagation and storing the forward velocity wavefields at every grid cell and for every subsampled time step: Source description. Velocity model. Station list.
Development of the adjoint simulation based on backward propagation of the displacement residual at all stations or use of the time-reversed velocity field at one particular station as the adjoint source: Calculate the displacement residual at all stations Do backward simulation for NS (number of sources).
Pick-up part of the observed data for a specific channel (eg. SH wavefield or P-SV wavefield) to be used as the adjoint source. Determine the station-specific GSDF adjoint fields , which involves NR (number of stations) simulations.
Calculation of the sensitive kernels and update of the models: Do gradient calculation separately from the backward simulation OR do adjoint simulation together with reconstruction of the forward wavefield and calculation of the sensitive kernels. Do gradient preconditioning and regularization. Choose step lengths (fixed percentage of perturbing the current model or optimal step length). Choose stop criteria for the iteration process (after 20 iterations or no more optimal step length found).
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Comparison of existing FWT methods
Existing FWT methods | Adjoint Wavefield Method | Scattering Integrals Method |
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Forward modelling | Generate the forward wavefields as time series at every cell in the spatial domail (3 ground velocity componets or 6 stress components) for every source as well as the synthetic data according to the station locations | | Storage of the wavefield | Store the forward and backward wavefields seperately and canculate the kernels after that. The displacement residual are calculated from syhtnetic and observed data. Store the forward wavefield and do kernel calculation on the fly together with backward simulations. Store the last state of the forward wavefield and do kernel calculation on the fly together with backward simulations and reconstruction of the forward wavefield.
| Store the equivalent body forces for every source and calculate/ store the Jacobian matrix according to each receiver. The velocity residual are calculated from synthetic and observed data for convolving with the source to remove the influence of the source signature. Store the backward wavefield according to each receiver as an adjoint source for one specific source. Using Green’s tensor estimation to generate the backward wavefield according to the remaining sources.
| Kernal calculation | | Calculate the gradient matrix on the fly by summing up the product of Jacobian matrix and the convolved residual. Estimate the Hessian matrix from the stored Jacobian matrice. Get the search direction by multiply the inverted Hessian matrix to the gradient. Calculate the sensitive kernels from the wavefields like in AW method
| Adjoint source | Displacement residual, multi-channel source (applied at all receiver locations) | Use a reference source (true source, Ricker wavelet, etc..) Use a velocity time serie from one channel of the observed data picked-up by GSDF method.
| Number of simulations (as minimum required) | 2 X Ns (or 6 X Ns?) | Ns+3*Nr | Number of time integrations | | | Optimization algorithm | Conjugate - Gradient | Gauss-Newton | Optimal step length
(Number of iterations needed to match one Gauss–Newton step) | Yes | No |
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Schedule |