Method

A simple way to represent a wave traveling at constant velocity with slowness s is as an expanding circle:  

$$t^2 = \tau^2 + {x^2}{s^2},$$ (1)

where t is traveltime, $\tau$ is traveltime depth and x is offset Claerbout (1995). Differentiating with respect to x at constant traveltime depth $\tau$ we obtain:  

$$s^2 = \frac{t}{x} \frac{dt}{dx},$$ (2)

where dt/dx is Snell's parameter p. Snell's parameter is related to the apparent horizontal velocity. It can also be regarded as a measure of the local stepout (dip) at any given time and offset along a hyperbola.Therefore, from equation (3) it can be seen that multiplying the local dip of hyperbolas in the (x,t) plane by the ratio of their time and space cordinates yields an estimate of slowness squared. This estimate of slowness is independent of where it is measured along the event in the (x,t) plane; consequently, a NMO correction of the slowness squared section will result in horizontal lines of constant slowness.

In order to obtain a dip estimate for the events in the plane we employ the method of Fomel (2002). This technique estimates local stepouts with plane wave destructor filters. Only one dip is estimated at every time and offset position, which makes this method sensitive to crossing events or coherent noise. A solution to this problem is to estimate multiple dips at every location and to select those of interest Fomel (2002). Once the dips have been estimated, the slowness can be computed in a straightforward manner by mulitplying each dip estimate by t/x. A map of local slowness (squared) is then obtained that need only be converted to a velocity profile for a given CMP gather.

To achieve this goal, a NMO correction with an approximate velocity trend can be applied to roughly flatten the hyperbolas. Finally, a median stack over the x coordinate provides a reasonable estimate of s² as a function of $\tau$. To ensure local bad dip estimates do not skew the results of the method, data points corresponding to atypical slowness values are disregarded, and the final result is smoothed in time. Once the estimate of slowness squared is obtained, we convert it to V_RMS.