Offset‐dependent geometrical spreading in isotropic laterally homogeneous media using constant velocity gradient models

Abstract

Laterally homogeneous models allow for processing of seismic data in the time domain. Well-known formulas account for the time difference between rays at zero and nonzero offset, and efficiently remove offset-dependent moveout of reflected events. For short spreads, a hyperbolic formula involving rms velocities is an excellent approximation of the kinematics in a stack of laterally homogeneous layers. Time processing also allows for amplitude correction directly in the domain of measured time. The most exact approach to amplitude calculation here would be to invert for interval velocities from rms velocities and to perform ray tracing in the interval model. This approach, however, is relatively costly. The simplest correction using the rms velocity field directly was provided by Newman (1973). The Newman factor V 2 rms t / V S , where V S denotes velocity of the first layer, V rms the rms velocity at vertical time t 0 , and t the traveltime, is exact for t = t 0 only, (i.e., for zero-offset rays). Although often used in practise, using it for rays at nonzero offset lacks theoretical justification. The offset-dependence is more correctly accounted for by the formulas derived by Hubral (1978) and Ursin (1990), which are based on a Taylor expansion of the exact spreading formula for layered media with respect to the offset coordinate. As shown by Ursin (1990), however, the formulas significantly and systematically provide too small amplitudes at nonzero offset, and the more exact ones among them require knowledge of velocity averages other than the rms one.