A three-dimensional plane-wave function is a projection of a one-dimensional waveform f(s):
Given an input image , plane-wave estimation finds an approximation of normal vector and a waveform estimation f(s). The naive formulation of the plane-wave estimation problem
The formulation of the cross-product minimization stems from the fact that an image volume is a plane-wave volume with normal (), if and only if at all
The constraint of a nonzero normal vector can be implemented in various ways. The constraint leads to a symmetric formulation in all spatial variables x,y, and z. However, for mathematical convenience, I choose the constraint pz = 1, which excludes vertical plane waves and leads to an asymmetry between the spatial variables. In practice, this particular constrain does not limit the processing of ordinary seismic images since image features are rarely vertical.
The minimization of equation (12) can be solved analytically. Let be the vector of partial x-derivatives of discretized g. Each element represents the derivative at a given location .The vectors and are the corresponding partial y- and z-derivatives. Given the constraint pz = 1, equation (12) can be expressed in matrix form asThe minimization's corresponding normal equation is
If we set the variables in the y dimension - qy and py - to zero, the solution simplifies towhich is identical to Claerbout's 1992 two-dimensional dip picking scheme. My original cross-product formulation (12) expresses the natural symmetry of the plane-wave detection problem that Claerbout's Lomoplan formulation lacks. However, my constraint for the nonzero normal vector (pz = 1) introduces the same asymmetry later.
Stacking of the image volume along planes orthogonal to the estimated normal vector yields a first estimate of the unknown plane-wave waveform f(s). Symes formulates a similar, but more sophisticated, iterative estimation scheme of the normal vector and the waveform f(s).