Plane-wave estimation

A three-dimensional plane-wave function is a projection of a one-dimensional waveform f(s):   where ${\bf p}$ is normal to the constant planes. Additionally, ${\bf p}$ is non-zero (). Such a projection spreads the value f(s) along the plane perpendicular to the normal direction ${\bf p}$. For any given point , the projection yields the argument of the function f based on the geometric interpretation of the vector product.

Given an input image , plane-wave estimation finds an approximation of normal vector ${\bf p}$ and a waveform estimation f(s). The naive formulation of the plane-wave estimation problem   is nonlinear and highly non-convex (). Alternatively, the problem can be reformulated into two well-behaved estimations. The estimation   finds the best-fitting vector ${\bf p}$ that is normal to g's gradient. The subsequent estimation

(11)

uses the normal ${\bf p}$ estimated by the first minimization to find the waveform f(s).

The formulation of the cross-product minimization stems from the fact that an image volume is a plane-wave volume with normal ${\bf p}$ (), if and only if at all   If g is indeed a plane-wave function with a normal proportional to ${\bf p}$, then

and, consequently,

Similarly, any image that satisfies equation (12) can be shown to be a plane-wave volume as defined in equation (8).

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 p_z = 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 p_z = 1, equation (12) can be expressed in matrix form as

The minimization's corresponding normal equation is   The variables   estimate the gradient slope in the x and y direction of g. Substitution of the gradient expressions (14) into the normal equation (13) yields

Finally, the equation system can be solved for the x- and y-component of the plane wave's normal vector:

Claerbout 1992 shows how the gradient slopes q_x and q_y can be computed from the input image using finite-difference approximations of the partial derivative expressions , and .The solution fully specifies the normal vector ${\bf p}$ since the vector's z component p_z is constrained to 1.

If we set the variables in the y dimension - q_y and p_y - to zero, the solution simplifies to

which 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 (p_z = 1) introduces the same asymmetry later.

Stacking of the image volume along planes orthogonal to the estimated normal vector ${\bf p}$ 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 ${\bf p}$ and the waveform f(s).