Large‐offset approximation to seismic reflection traveltimes

Abstract

Conventional approximations of reflection traveltimes assume a small offset‐to‐depth ratio, and their accuracy decreases with increasing offset‐to‐depth ratio. Hence, they are not suitable for velocity analysis and stacking of long‐offset reflection seismic data. Assuming that the offset is large, rather than small, we present a new traveltime approximation which is exact at infinite offset and has a decreasing accuracy with decreasing offset‐to‐depth ratio. This approximation has the form of a series containing powers of the offset from 1 to −∞. It is particularly accurate in the presence of a thin high‐velocity layer above the reflector, i.e. in a situation where the accuracy of the Taner and Koehler series is poor. This new series can be used to gain insight into the velocity information contained in reflection traveltimes at large offsets, and possibly to improve velocity analysis and stacking of long‐offset reflection seismic data.

The application of two-parameter velocity and slowness functions in approximating seismic reflection travel times

“…The approximation can be improved at small offsets by using more terms of the series, but the high powers of offset in the series make it diverge rapidly with increasing offset (AI-Chalabi 1973). Causse et at (2000) note that in the presence of a high-velocity layer, the hyperbolic approximation gives relatively large travel time errors even for offsets approximately equal to the reflector depth. Considering the same figures 5 and 7 the Causse series (equation (2» fits the exact travel time curve very well at large offsets, especially for models containing a high-velocity layer.…”

“…Solutions for all four types of velocity or slowness have been derived. The new method was compared with the approximations of Taner and Koehler (1969) and Causse et al (2000) for planelayered isotropic media. It provides a much better match to exact travel times at all offsets, even though the functions only consist of two terms.…”

“…We now compare the accuracy of the functions derived above with those series used by Causse et al (2000) and Taner and Koehler (1969), which we shall henceforth refer to as the Causse and T and K series, respectively. We use the same models as Causse et al (2000), and calculate the approximate travel times for P-waves reflected at the bottom of three different plane-layer models (A-C) with a reflector depth of 2500 m (table 1).…”

“…We use the same models as Causse et al (2000), and calculate the approximate travel times for P-waves reflected at the bottom of three different plane-layer models (A-C) with a reflector depth of 2500 m (table 1). Given the ratio of offset to depth, the parameters (initial velocity and/or slowness and their gradients) in velocity or slowness functions can be obtained for known models.…”

“…To obtain a travel time approximation with good accuracy at large offsets, Causse et al (2000) derived a series based on a large-offset approximation:…”