The nonuniqueness of the determination of interval velocities from moveout velocities

In earlier papers (Krey, 1976; Hubral and Krey, 1980) I described how to obtain an equation for v_n, the nth interval velocity in an isovelocity layered earth having interfaces with arbitrary dips and curvatures, provided the velocities v₁, v₂, … to v_n−1 for the first n − 1 layers and the depths of the first n − 1 interfaces z_K(x,y), K = 1, 2, …, n − 1, are known and have continuous derivatives. Moreover, we assume that the zero-offset time for the reflection from the base of the nth layer and gradient of the traveltime with respect to the horizontal coordinates are known. Finally, the normal moveout (NMO) velocity v_nmoϕ for the nth interface is observed in one arbitrary azimuth (one only), defined by ϕ, the angle between the x-axis and the seismic profile.