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Next: Finite difference method Up: Cameron, Fomel, Sethian: Velocity Previous: Cauchy problem for elliptic

Inversion Methods

Our numerical reconstruction of true seismic velocity $ v(\mathbf{x})$ in depth-domain coordinates from the Dix velocity given in the time-domain coordinates $ (\mathbf{x}_0,t_0)$ consists of two steps:
Step 1.
Compute the geometrical spreading of the image rays in the time-domain coordinates from the Dix velocity by solving equation 14 in 2-D and 20 in 3-D. Then find $ v(\mathbf{x}_0,t_0)$ from equation 10 in 2-D and equation 17 in 3-D.
Step 2.
Convert the seismic velocity $ v(\mathbf{x}_0,t_0)$ in the time-domain coordinates to the depth-domain coordinates $ \mathbf{x}$ using the time-to-depth conversion algorithm, which was presented by Cameron et al. (2007). It is a fast and robust Dijkstra-like solver motivated by the Fast Marching method (Sethian, 1999,1996).

We performed step 1 in two ways: a finite difference method and a spectral Chebyshev method.