Numerical Results

Synthetic tests associated with a fan model (Figure a) are presented to demonstrate the effectiveness of the proposed RTM antialiasing filter. The velocity model shown in Figure a has a homogeneous background velocity of $2000~m/s$ perturbed by a series of dipping events and deep scatterers with a higher velocity of $2500~m/s$ . Figure b displays a typical CSG for this model calculated by a finite-difference (FD) solution to the two-way wave equation. Fifty-one CSGs are computed with the shot interval of $160~m$ , and the first shot is fired at $X=2~km$ . A fixed spread is used and 301 traces are recorded along the surface with a trace interval of $40~m$ . The peak frequency of the Ricker wavelet is $15~Hz$ and the record length is $6~s$ . Both reflections and diffractions can be clearly seen in the CSG.

Figure: Synthetic velocity model and associated shot gather. a) is the fan model, with the background velocity (gray color) of $2000~m/s$ and the perturbed velocity (white color) of $2500~m/s$ . b) shows a typical shot gather at $X=7.6~km$ .

Figure: Kirchhoff migration (KM) examples.

Before doing RTM, I first apply the Kirchhoff migration (KM) to this synthetic data set to illustrate the migration aliasing problem for this model. Figure a shows a CSG with the surface shot at $sx=7.6~km$ , where the direct waves are removed. Figure b depicts the KM operator, which is a hyperbola for a fixed trial image point at ${\bf {x}}$ (located at $X=6~km$ and $Z=3~km$ ). The KM operation for a trial image point ${\bf {x}}$ can be viewed as summing energy along this hyperbola. Figures c and d are the corresponding stacked KM images for 51 CSGs with and without applying an antialiasing filter. Aliasing artifacts are visible in Figure c and are then suppressed with a KM antialiasing filter proposed by Lumley et al. (1994).

The RTM antialiasing filter described in the theory part is now applied to the same synthetic data. Figure b presents the RTM operator for the same trial image point and CSG shown in Figure b. Unlike the hyperbola traveltime curve for the KM operator, the RTM operator contains both primary and multiples along with amplitude and phase information. Figure c shows the stacked RTM image of this model. Aliasing artifacts seen in the standard RTM image are similar to those in Figure c. The RTM antialiasing filter is then constructed using the proposed approach and applied to the RTM operator to eliminate the offending high-frequency components. Figure d displays the RTM image with the antialiasing filtering applied. Compared to Figure c, the aliasing artifacts are largely eliminated with this filter.