Time-to-depth conversion and seismic velocity estimation using time-migration velocity

Spectral Chebyshev method

Alternatively, we solve our PDE in the form given by equation 15 by a spectral Chebyshev method Boyd (2001). Using cubic splines, we define the input data at $N_{coef}$ Chebyshev points. We compute the Chebyshev coefficients and the coefficients of the derivatives in the right-hand side of equation 15. Then we use a smaller number $N_{eval}$ of the coefficients for function evaluation. We need to do such Chebyshev differentiation twice. Finally we perform the time step using the stable third-order Adams-Bashforth method Boyd (2001), which is

$$u^{n+1}=u^n+\Delta t\left(\frac{23}{12}F^n- \frac{4}{3}F^{n-1}+\frac{5}{12}F^{n-2}\right),$$ (23)

where $F^n\equiv F(u^{n},x,t^{n})$ is the right-hand side. In numerical examples, we tried $N_{coef}\ge 100$ and $N_{eval}\le 25$ . This method allows larger time steps than the finite difference, and has the adjustable parameter $N_{eval}$ .

For step 2, we use a Dijkstra-like solver introduced in Cameron et al. (2007). It is an efficient time-to-depth conversion algorithm motivated by the Fast Marching Method (Sethian, 1996). The input for this algorithm is $v(x_0,t_0)$ and the outputs are the seismic velocity $v(x,z)$ and the transition matrices from time-domain to depth-domain coordinates $x_0(x,z)$ and $t_0(x,z)$ . We solve the eikonal equation with an unknown right-hand side coupled with the orthogonality relation

$$\vert\nabla t_0\vert=\frac{1}{v(x_0(x,z),t_0(x,z))},\quad \nabla t_0\cdot\nabla x_0=0.$$ (24)

The orthogonality relation means that the image rays are orthogonal to the wavefronts. Such time-to-depth conversion is very fast and produces the outputs directly on the depth-domain grid.

Time-to-depth conversion and seismic velocity estimation using time-migration velocity