Parameter estimation for a linear velocity function

For a horizontally layered medium with isotropic layers with constant velocity gradient, it is possible to estimate the velocity function (gradient and velocity at the top of the layer) and thickness of each layer. From large-offset PP seismic reflections one can estimate three traveltime parameters: the zero-offset two-way traveltime, the NMO velocity and a heterogeneity coefficient, using the shifted hyperbola approximation or a fractional approximation. From the estimated traveltime parameters at the top and bottom of a layer, it is possible to compute the thickness and velocity function of the layer. It is necessary to solve a simple nonlinear equation which also can be solved approximately. There are two solutions, corresponding to a positive and negative velocity gradient. Therefore, the sign of the velocity gradient must be chosen. When there is one solution, the velocity gradient in the layer is zero, and the result is the standard Dix equations.