Discussion and Future Work

The issues below are important for improving the quality of wave-equation migration directly from topography images. In some cases, we discuss ideas not yet implemented, while in others we speculate on directions of future research.

Muting direct arrivals:

We did not mute out the direct arrivals from the shot gathers, which probably introduced artifacts. In principle, a first-arrival mute is fairly easy to implement; however, their non-linear moveout requires introducing more complicated muting functions. In the future, we will eliminate this source of image contamination.

Using improved mapping weighting functions:

We speculate that amplitudes could become more uniform along the reflectors through the use of a better weighting function. The image is currently interpolated to the Cartesian domain using sinc-function operators, where the image points are weighted by the mapping fold. A better weighting function should include the Jacobian of the transformation between the two coordinate systems.

Including dynamic propagation terms:

We have incorporated only the second-order partial differential terms in the phase-screen approximation for extrapolation direction wavenumber, $k_\tau$. Including the remaining two dynamic terms should lead to reflectors of more uniform intensity, since these terms contribute to wavefield amplitudes.

Incorporating multiple reference media:

The above image was generated using one reference medium (i.e., we performed Taylor expansions about a₀ and b₀). However, in practice many reference media (e.g., velocities) are often used to generate images through the PSPI approach. Noting that the variability of coordinate spacing is significant (and functions a and b thereby), we surmise that the incorporation of multiple reference media is likely necessary to eliminate existing kinematic errors and to improve diffraction focusing.

Implementing a seperate wavefield datuming step:

By extension, we have shown that this procedure works as a datuming procedure. For example, a coordinate system generated by conformal mapping could be used in an upward continuation scheme to establish the wavefield at a uniform level above topography. Standard Cartesian migration technology could then be applied directly to migrate the datumed wavefield.