Mapping prestack depth‐migrated coherent signal and noise events back to the original time gathers using Fermat’s principle

Abstract

Because of its computational efficiency, prestack Kirchhoff depth migration is currently the method of choice in both 2-D and 3-D imaging of seismic data. The most algorithmically complex component of the Kirchhoff family of algorithms is the calculation and manipulation of accurate traveltime tables for each source and receiver point. Once calculated, we sum the seismic energy over all possible ray paths, allowing us to accurately image both specular and nonspecular scattered energy. Any seismic events that fall within the velocity passband, including reflected and diffracted signal, mode conversions, multiples, head waves, and aliases of surface waves, are imaged in depth. The transformation of time gathers to depth gathers can be quite complicated and nonintuitive to all but the seasoned imaging expert. In particular, easily recognized head‐wave events on common‐shot gathers are often difficult to differentiate from undermigrated coherent reflections on common‐reflection‐point depth gathers. In contrast, subsalt multiples that have propagated along complex ray paths are often easily recognized on common‐offset depth gathers but are indistinguishable from the distorted primaries on the input common‐shot or common‐midpoint time gathers. In a related area, seismic reflection traveltime tomography is currently the workhorse for 2-D and an active area of research and development for 3-D migration‐driven velocity analysis. The objective function for this “velocity inversion” problem is to either minimize the temporal difference between picked and modeled time picks, or to maximize the similarity between, or flatness of, common‐reflection‐point depth picks. Once picked and associated with the correct reflector, time picks never need to be modified during the velocity‐model updating steps that ultimately lead to a feasible solution. In practice, such time picks are nearly impossible to make in those structurally complex areas that justify the use of prestack depth migration. Instead, we almost always use the second objective function and pick reflector events in depth, where we can use our geologic insight to differentiate between signal and noise and where the difficulty of associating a picked event with the velocity/depth model horizon completely disappears. The major drawback of picking in depth is that these events need to be repicked each time any part of the overlying velocity/depth model has been updated. We show that by applying Fermat’s principle, and by reusing the same traveltime tables used in seismic prestack Kirchhoff depth imaging, we can map interpreted events on the depth gathers to corresponding interpreted events on the original time gathers. This technique, first introduced by J. van Trier in 1990, is considerably more stable and, because we reuse the already computed migration traveltime tables, more economic than two‐point ray‐trace methods. In our first application of coherent noise suppression, we show how we can relate imaging artifacts seen on the depth image to their causative coherent noise on the original time gathers. Once identified, these noise events can be safely suppressed using conventional filtering techniques. In our second application of reflection tomography, we show how we can pick partially focused reflectors in depth, and map them back to time, undoing the effect of the incorrect velocity/depth model used in prestack Kirchhoff depth migration such that the events never need to be repicked during subsequent velocity model updates.