Wavefield extrapolation in pseudodepth domain

Implementation aspects

The Sampling of the $\tau$ axis should be small enough to avoid wavefield aliasing in the $\tau$ domain, for example

$$\Delta \tau \leq \frac{1}{10} \frac{v_{min}}{f_{max}} ,$$ (25)

where $v_{\min}$ is the minimum velocity in the model and $f_{max}$ is the maximum frequency of the wave. Accordingly, the number of samples representing the $\tau$ axis should be chosen to cover the largest expected $\tau$ value,

$$n_\tau \Delta \tau \geq \max_{x,y} \int_0^z \frac{\mathrm{d} z^\prime}{v_m(x,y,z^\prime)} .$$ (26)

The mapping velocity $v_m$ is often chosen as a slightly smoothed version of true velocity $v$ . This is because the $\tau$ domain wave equation involves a differentiation of $v_m$ , for example the first equation in 15.

For second-order wave equations, the operators on the right-hand side can become significantly complicated in the $\tau$ domain, such as Equation 23. Thus, it is more convenient to code up its first-order form 21. For consistency, we will extrapolate the wavefields using the first-order form for all the examples in this paper. Thus, the time derivatives in these equations are approximated by central differences,

$$\frac{\partial p}{\partial t} \approx \frac{p^{n+1} - p^{n-1}}{2\Delta t},$$ (27)

and the spatial derivatives are approximated using the Fourier pseudospectral approuch (Carcione et al., 2002; Gazdag, 1981), as follows

$$\frac{\partial p}{\partial x_1} \approx F_1^{-1}\lbrace\mathrm{i} k_1 F_1\lbrace p \rbrace\rbrace,$$ (28)

where superscript $n$ indicate time steps, $F_i$ is the spatial Fourier transform in the $x_i$ direction. The change of the vertical axis from $z$ to $\tau$ does not affect the stability condition. For both isotropic and VTI extrapolations, the same time-step is used in both the Cartesian and $\tau$ domains.

Wavefield extrapolation in pseudodepth domain