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

3-D case

Equation 11 can be rewritten in the following form

$$v=\sqrt[4]{\det\mathbf{F}(\det\mathbf{Q})^2},$$ (17)

where $\mathbf{F}$ is the left-hand side of equation 11. As in 2-D, we rewrite system 9 in the time-domain coordinates $(\mathbf{x}_0,t_0)$ . Then we get

$$\mathbf{Q}_{t_0} = v^2\mathbf{P},$$ (18)

$$\mathbf{P}_{t_0} = -\frac{1}{v}\mathbf{Q}^{-1}\left[ \nabla\left(\mathbf{Q}^{-1}\nabla v\right)^T\right]\mathbf{Q},$$ (19)

where $v$ is given by equation 17, and the gradients are taken with respect to $\mathbf{x}_0$ . Then the PDE for $\mathbf{Q}$ is

$$\left(\frac{1}{v^2}\mathbf{Q}_{t_0}\right)_{t_0}= -\frac{1}{v}\mathbf{Q}^{-1}\left[ \nabla\left(\mathbf{Q}^{-1}\nabla v\right)^T\right]\mathbf{Q}.$$ (20)

The initial conditions are $\mathbf{Q}(\mathbf{x}_0,0)=\mathbf{I}_2$ , $\mathbf{Q}_{t_0}(\mathbf{x}_0,0)=\mathbf{0}$ . The required input $\sqrt{\det\mathbf{F}}$ is well-approximated by the squares of the Dix velocity obtained from the 3-D prestack time migration. We emphasize that despite the fact that $\mathbf{Q}$ is a matrix in 3-D, scalar data are enough for its computation.

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