Overview of Mixed-domain Downward Extrapolation

In this section, we will briefly review, from the mathematical point of view, the $\omega$-$\mathbf{x}$ - $\omega$-$\mathbf{k}$ algorithm. This will serve as the starting point for the presentation of the new extrapolation method in the next section.

Let $\mathbf{W}^N$ be the wavefield at depth step N in the $\omega$-$\mathbf{k}$domain and let $\mathbf{W}_l^{N+1}$ be the wavefield extrapolated from depth step N to depth step N+1 using reference velocity V_l. That is:

$$\mathbf{W}_l^{N+1}=\mathbf{W}^Ne^{ik_{z_l}\Delta z},$$

where $\Delta z$ is the depth of the Nth layer and k_zl is given by the dispersion relation:  

$$k_{z_l}=\sqrt{\frac{\omega^2}{V_l^2}-\vert\mathbf{k}\vert^2}$$ (1)

with $\vert\mathbf{k}\vert$ being the magnitude of the horizontal wavenumber vector. PSPI handles the difference between the true velocity and the reference velocity by interpolating the downward-continued wavefields in the $\omega$-$\mathbf{x}$ domain based on the difference between the reference velocities and the model velocity at each $\mathbf{x}$ position. The interpolated wavefield is therefore given by:

$$\mathbf{w}^{N+1}(j)=\sum_{l=1}^{n_v}\sigma_l\mathbf{w}_l^N(j)$$

where $\sigma_l$ is an interpolation factor ($\sum_l^{n_v}\sigma_l=1$), $\mathbf{w}_l^{N+1}(j)$ is the downward-continued wavefield in the $\omega$-$\mathbf{x}$ domain at the spatial location j, and n_v is the number of reference velocities.

Extended split-step adds a correction before the interpolation, the so-called ``thin lens term'':

$$e^{ik_{ss_l}} \mbox{ where, } k_{ss_l}=\frac{\omega}{V}-\frac{\omega}{V_l}$$

where V is the true model velocity. Depending on the choice and number of reference velocities, split-step can make significant improvements in accuracy compared to PSPI.

Other methods, such as pseudo-screen and Fourier finite-difference, increase the accuracy of the result by adding high-order spatial derivatives to the computation of the k_zl term Biondi (2004); Huang et al. (1999); Ristow and Ruhl (1994); Xie and Wu (1999). The more accurate approximation of k_zl relaxes the need for a large number of reference velocities such that with fewer reference velocities similar or even better accuracies can be obtained compared with split-step.

The last step in either of these methods is to Fourier transform the interpolated wavefield to the $\omega$-$\mathbf{k}$ space. This wavefield will then be the input to the propagation at the next depth step.

It should be clear that the accuracy of these methods, especially PSPI and extended split-step, is directly related to the accuracy of the wavefield interpolation and the number and choice of the reference velocities.