A single dipping event in the t − x domain ideally maps onto a single trace in the τ − p domain that represents the dip of that event (Figure 6.3-13). However, because of the discrete sampling along the p-axis and because only a finite number of p traces are spanned from a finite number of offset traces, an imperfect mapping results. When plotted with a higher gain, the slant-stack gather in Figure 6.3-13 seems surprisingly different (Figure 6.3-17b). The linear streaks are contributions from end points E and F of the dipping event in the t − x domain. More specifically, point E maps onto A and B when p is set to its minimum and maximum values, respectively. For any intermediate value of p, point E maps along AB. Similarly, the other end point F maps along CD.
Figure 6.3-17 Panels (a), (d), and (g) are the input CMP gathers, which contain a single dipping event EF. Panels (b), (e), and (h) are the corresponding slant-stack gathers. Panels (c), (f), and (i) are the reconstructed offset gathers. The slant-stack and reconstructed gathers are displayed at a higher gain than the input gathers.
Linear streaks that result from end effects associated with cable truncation are only one type of artifact encountered when constructing slant stacks. Another type of artifact is the high-frequency wavetrain that is especially apparent on traces with large p values as in Figure 6.3-17a. It occurs because the dipping event is sampled along steep slanted paths.
Several practical considerations affect the artifact level in slant stacks. A short cable length in the t − x domain enhances the end effects and, thus causing poor reconstruction as demonstrated in Figure 6.3-17. Start with an offset gather that contains a single dipping event EF in panel (a). Panel (b) is the τ − p gather and panel (c) is the reconstructed t − x gather from it. To emphasize the artifacts, the last two panels of the sets of three are displayed at a higher gain compared to the original. Using two-thirds of the offset gather, panel (d), the τ − p gather and the reconstruction from it were obtained as shown in panels (e) and (f). Finally, using only one-third of the original gather, panel (g), panels (h) and (i) are obtained. Note that short cables produce artifacts G and H on the slant-stack and reconstructed gathers. Accurate construction of slant-stack gathers usually requires sufficiently long cable length and adequately small offset interval.
To study the sampling along the p-axis and the range of p values used in constructing a slant-stack gather, consider the synthetic gather shown in Figure 6.3-18a, which consists of hyperbolic events. These events are mapped along the ellipses in the slant-stack gather (Figure 6.3-18b). The following values were chosen: the number of p traces np equal to the number of x traces nx; the minimum p value, pmin = 0; and the maximum p value, pmax set to the largest dip present in the data. Reconstruction using these parameters produced an accurate result (Figure 6.3-18c). There is some difference in the 2-D amplitude spectra of the original, panel (d), and the reconstructed gathers panel (e), because pmin was set to zero.
What happens when the p-axis is undersampled? Figure 6.3-18f shows the slant-stack gather and Figure 6.3-18g shows the reconstructed gather that is obtained by setting np = nx/2 and keeping (pmin, pmax) the same as in panel (b); thus, the p-increment is twice as large as in panel (b). The input gather is the same as it was in panel (a). Note that undersampling along the p-axis introduces some noise, labeled as A in Figure 6.3-18g, into the reconstructed gather.
Figure 6.3-18 (a) Input gather, (b) slant-stack gather, (c) reconstructed offset gather, (d) f − k spectrum of panel (a), (e) f − k spectrum of panel (c). Panels (f), (h), and (j) are the slant-stack gathers derived from the input gather (a) using different numbers of p-values and ranges, while panels (g), (i), and (k) are reconstructions from them. Input (a) is the same for all cases. See text for details.
Figure 6.3-19 The same sequence of panels as in Figure 6.3-18, except that the input gather contains spatially aliased frequency components. Note wraparound in the f − k spectrum (d).
Figure 6.3-20 (a) A field data set with strong ground-roll energy A, its backscattered component B, guided waves C, and a strong reflection D; (b) τ − p gather obtained from this field data set; (c) reconstruction of the field record using the portion to the left of the solid vertical line in (b) (zone E); (d) dip-filtered data obtained by subtracting the gather in (c) from the original data in (a); (e) the original data set (a) after f − k dip filtering. (Data courtesy Turkish Petroleum Corporation.)
Consider the opposite situation of oversampling along the p-axis as in Figure 6.3-18h. Here, np = 2nx and the (pmin, pmax) range is the same as in Figure 6.3-18b. Note that oversampling in the p-axis does no harm, but gains nothing either (Figure 6.3-18i). Although not shown here, further experiments show that regardless of spread length, oversampling in the p-domain does not improve the quality of the reconstructed gather.
In practice, we may encounter an inappropriate choice of the (pmin, pmax) range, meaning that pmax corresponds to a larger dip than is present in the input gather (Figure 6.3-18j). Here, np = nx, Pmin = 0, pmax is twice as large as the value chosen in Figure 6.3-18b, and the p-increment is the same as in Figure 6.3-18f. Thus, the right half of the p-gather (Figure 6.3-18j) does not contain dip components that are present in the input data (Figure 6.3-18a). Instead, the right half contains noise resulting from cable truncation and sampling along steep slanted paths with p-values associated with dips not contained in the offset data. This results in some noise in the reconstructed gather, labeled as B in Figure 6.3-18k. In practice, suitable muting in the p-domain can eliminate the artifacts caused by spurious p traces — as in the right half of Figure 6.3-18j.
Now consider a synthetic gather that contains spatially aliased events. Figure 6.3-19 shows panels that are equivalent to those in Figure 6.3-18, except that the input gather (Figure 6.3-19a) has spatially aliased frequency components. It is clear that the artifacts observed in Figure 6.3-19 are more pronounced. However, note that if the (pmin, pmax) range, np and the p-increment are chosen properly (Figure 6.3-19b), then reconstruction is quite accurate, even with spatially aliased data. The amplitude spectrum of the original gather (Figure 6.3-19d) and that of the reconstructed gather (Figure 6.3-19e) are almost identical, except that the latter does not contain unaliased energy for p < 0, which was not included in constructing Figure 6.3-19b.
Again, we see the case of an undersampled p-axis causing some noise in the reconstructed gather, the case of oversampled p-axis causing no harm, and finally, inappropriate inclusion of the p-values corresponding to dips absent from the input gather causing noise in the reconstructed gather.
This experimental study and other similar studies of the parameters involved in slant-stack processing lead to the following empirical statements:
- np = nx is a good, general rule.
- The (pmin, pmax) range should only span the dip components of interest in the data. For example, for marine CMP data, pmin = 0, pmax = (1/1500) s/m.
- The p-increment then is (pmax − pmin)/nx. Sampling along the p-axis also can be done with equal increment in horizontal phase velocity 1/p.
- Physical aspects of slant stacking
- Slant-stack transformation
- Practical aspects of slant stacking
- Time-variant dip filtering
- Slant-stack multiple attenuation