Statistical Travel-time Tomography in Terms of Stacking Velocity

— Velocity evaluation is a key step in seismic analysis. The covariance of the true velocity field must be known when interpolating or simulating velocities from well measurements using geostatistical methods. In addition, inversion procedures often require information pertaining to this covariance. Traditionally it has been taken to be the covariance of stacking velocities. We present a simple example to show that this approximation can lead to significant errors. Better methods, such as those of Touati (1996) and Iooss (1998), use the variance of prestack picked travel times as a function of offset to infer that of the velocities. In this paper we extend their results on the estimation of the covariance of the reflected traveltimes, and obtain an explicit expression for the covariance of the square of the stacking slowness as a function of the covariance of the velocities. Although we are not able to invert the formula analytically to yield an explicit estimator for these parameters, the results obtained using it furnish a good and quick estimation of the velocity's covariance. This is illustrated with synthetic examples.