Effective filtering of artifacts for implicit finite‐difference paraxial wave equation migration¹

Abstract

Implicit finite-difference implementations of the paraxial wave equation are widely used in industrial prestack and post-stack migration programs for imaging and velocity analysis. This type of implementation gives rise to numerical artifacts which, in general, do not degrade image quality but which do impede effective velocity analysis. This paper reviews the artifacts generated by the paraxial approximation and a post-extrapolation, spatially varying filtering scheme is described which completely eliminates these artifacts. The method is illustrated with numerous examples.

ADI plus interpolation: accurate finite‐difference solution to 3D paraxial wave equation

“…We have seen that , applied in the interpolation step, acts as a 2D low‐pass spatial filter. A spatial filter needs to be applied to the conventional ADI result to remove the undesirable artefact (Bunks 1995). Since the interpolation of is implemented together with the post‐ADI filtering, the ADIPI scheme incurs no significant extra computational cost, compared with the conventional ADI method plus filtering.…”

“…It will be shown here that the interpolation step, after the ADI implementation, also acts as a 2D spatial filter applied to the ADI result to suppress the non‐physical evanescent waves, which are artefacts in the form of cardioid arches caused by the paraxial wave equation diverging from the exact one‐way wave equation. The 2D spatial filter should be applied in any case to the conventional in‐line/cross‐line splitting method (Bunks 1995). Since the ADIPI scheme realizes the interpolation within the 2D spatial filtering, it requires no extra computational cost when compared with the standard ADI method plus filtering.…”

The stabilization of high-order multistep schemes for the Laguerre one-way wave equation solver

“…This results in an improper propagation of the evanescent mode which should exponentially decay. For stabilizing the real Padé approximation there are a few approaches [17,18,19] that allow suppressing unstable components of a wave field. On the other hand, setting the coefficients γ s , β s to be complex [20,21,22,23], a better consistency of the right-hand and the left-hand sides of equation (2) can be attained.…”

Salvo: Seismic imaging software for complex geologies

“…The impulse problem should produce a semi-circle (see Figure 10 on p. 35), but because of the square-root approximation, the 45 degree equation will produce a cardioid (similar to the two-dimensional poststack image without phase correction shown in Table 4 on p. 97) where the evanescent waves have looped back to the center of the image. A good discussion of this problem is provided by Bunks [7]. In Section 3.3, methods to correct this difficulty are presented.…”

“…encoding parm Chirp encoding requires a stretching parameter which may vary between the approximate range [2][3][4][5][6][7][8][9][10]. Linear phase encoding requires the desired time shift between the shots, -to,in seconds.…”