Slanted deconvolution and inversion

Because of the wide offsets used in practice, it has become clear that seismologists must pay attention to differences in the sea floor from bounce to bounce. A straightforward and appealing method of doing so was introduced by Taner [1980]--that was his radial-trace method. A radial trace is a line cutting through a common-shot profile along some line of constant $r\,=\,h/t$.Instead of deconvolving a seismogram at constant offset, we deconvolve on a radial trace. The deconvolution can be generalized to a downward-continuation process. Downward continuation of a radial trace may be approximated by time shifting. Unfortunately, there is a problem when the data on the line consists of both sea-floor multiples and peglegs, because these require different trajectories. The problem is resolved, at least in principle, by means of Snell waves. Estevez, in his dissertation [1977], showed theoretically how Snell waves could also be used to resolve other difficulties, such as diffraction and lateral velocity variation (if known). An example illustrating the relevance of the differing depths of the sea floor on different bounces, is shown in Figure 15.

 

estevez
Figure 15 Time of multiple depends on sum of all times. (Estevez [1977])

Incompleteness of the data causes us to have problems with most inversion methods. Data can be incomplete in time, space, or in its spectrum. Any recursive method must be analyzed to ensure that an error made at shallow depths will not compound uncontrollably during descent. All data is spectrally incomplete. Also, with all data there is uncertainty about the shot waveform. At the p-values for which pegleg multiples are a problem, the first sea-floor bounce usually occurs too close to the ship to be properly recorded. To solve this problem, Taner built a special auxiliary recording system.

It is an advantage for Snell wave methods that slant stacking creates some signal-to-noise enhancement from the raw field data, but it is a disadvantage that the downward continuation must continue to all depths. The methods to be discussed next are before-stack methods, but they do not require downward continuation much below the sea floor.