Wavefield extrapolation in pseudodepth domain

Appendix B: $\tau$ domain traveltime

A geometrical description of the $\tau$ domain isotropic wavefield can be achieved by looking at its eikonal. A dispersion relation associated with the $\tau$ domain wave equation 15 is obtained by taking Fourier transform of this equation in space and time, specifically, do the substitution $\partial t \rightarrow -i\omega$ and $\partial x_i \rightarrow i k_i$ , the result is

$$\frac{\omega^2}{v^2} = \sum_{i=1}^2 \left(k_i + \sigma_i k_3\right)^2 + \frac{k_3^2}{v_m^2} .$$ (32)

Note here $k_1$ and $k_2$ has units of $\mathrm{rad/m}$ , while $k_3$ has the angular frequency unit of $\mathrm{rad/sec}$ .

We then relate slowness vector $\mathbf{p}$ with wavenumber vector $\mathbf{k}$ by $\mathbf{k} = \omega \mathbf{p}$ , thus the $\tau$ domain isotropic eikonal equation is

$$\frac{1}{v^2} = \sum_{i=1}^2 \left(p_i + \sigma_i p_3\right)^2 + \frac{p_3^2}{v_m^2}$$ (33)

where $p_i = \partial T / \partial x_i$ is the component of slowness vector in the $x_i$ direction, $T$ is traveltime. Similarly, while $p_1$ and $p_2$ has slowness units $\mathrm{sec/m}$ , $p_3$ is dimensionless. Alternatively, Equation G-2 can be derived by applying chain rule 9 to the Cartesian domain eikonal $\sum_{i=1}^3 p_i^2 = 1/v^2$ .

Wavefield extrapolation in pseudodepth domain