Abstract
As the spread length increases, rms velocities determined from seismic data differ from those given by the Dix relation. We have examined the nature of time-distance curves and the velocities found from them, using as our model of the subsurface beds separated by plane horizontal interfaces or a subsurface with an interval velocity that is a continuous function of depth.
Our conclusions are as follows: (1) For a given reflection, the slope of the t2-x2 time-distance curve at any spread distance x is equal to (Vrms)-2 where Vrms(x) is the rms velocity computed along an appropriate raypath. Vrms(0) is the velocity given by the Dix relation. As x increases, Vrms(x) approaches a limiting value equal to the maximum value of the interval velocity in the section of the earth transversed by the raypath. (2) The t2-x2 curve is such that Vrms(x) is a monotonically increasing function of x. Conclusions (1) and (2) permit us to place bounds on the moveout velocity. (3) If the interval velocity is assumed to increase exponentially with depth, the Taylor series expansion for t2-x2 about x=0 converges for all valid values of x.
To illustrate the effect of the x4 term on moveout velocity, subsurfaces were generated from several continuous velocity logs and moveout velocities computed with the usual Dix relation and with a three-term t2(x) expansion. In most cases, the errors made by dropping the x4 term were less than 2 percent.
