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Problem Statement

The first FWT case study will be a synthetic test with relatively low compute cost to verify the employed methodology. In this case study, we use a 1-D velocity model as the true model and attempt to recover it from an initial homogeneous model. The computational domain will be 60km X 40km X 20km with a test configuration including 12 receivers (3 X 4, equally spaced spacing at every 10km) and 6 sources (2 X 3, equally spaced spacing at every 10km ); all the sources and receivers are uniformly embedded at 10km depth). The wavefields are generated by numerical solutions of the 3-D elastodynamic/ visco-elastodynamic equations according to a specific velocity model. The synthetic "observed" data will be generated using the 1-D velocity model (treated as the true model) and synthetic data will be generated using the homogeneous model initially, and the current model in subsequent iterations. The misfit between the "observed" and synthetic data will then be calculated and used in the inversion. After a number of inversion iterations, the final inverted model using FWT is expected to match the 1-D velocity model well.

Tasks

Forward modeling of the elastic wave propagation for the 1-D model to create the synthetic "observed" data.

• Choosing a 1-D velocity model as the true model.

• Choosing the cell size for the staggered grid using in forward simulation (0.2 km) and a stable time step according to the CFL criteria.

• Specifying station and source locations; recorded components of the wavefields at the station locations.

• Describing the source signal: using a Ricker wavelet with the central frequency of .5Hz, specifying the recorded time and the time delay for the source.
• Forward modeling of the elastic wave propagation for the 1-D model to create the synthetic "observed" data at receivers.

Inversion: specify the number of iterations, optimization method or optimal step length (if applicable) and follow the steps for doing inversion at one iteration:

• Forward modeling of the elastic wave propagation for the homogeneous model to create the synthetic data and storing the forward velocity wavefields at every grid cell and for every subsampled time step.

• Calculate the displacement residual between the "observed" and synthetic data.

• Backward propagation of the displacement residual at all stations for each and every source and store the backward velocity wavefields at every grid cell and for every subsampled time step.

• Calculation of the sensitive kernels (using 3 components of ground motion velocity wavefields or 6 components of stress wavefields).

• Preconditioning of the kernels to suppress the near source and receiver artifacts.

• Update of the model with the optimal search direction (using fixed or optimized step lengths; applying Conjugate Gradient or BFGS methods; implementing regularization).

Schedule

Verification

Can interrogation of the final inverted velocity model and a comparison to the true velocity model be the first method of verification?

Verifying the inverted result using FWT - Adjoint Wavefield method with FWT-Scattering Integral or SASW methods.