Next we get two equations from the plane-wave equation, one differentiating by x, the other by y.
Eliminate p from these two equations by solving for it:
The plane-wave operator L (u) will not vanish unless u is a plane wave going in the direction specified by p. A remarkable property of the function G(u) is that it vanishes for any orientation of plane wave. If we want to test a 2-D field for one-dimensionality, the test L (u) requires us to know p. The test G (u) does not. Generally we do not know p and we need to estimate it by statistical means in a window of some size that we must specify. In principle, G escapes those problems (although it might be worse in practice because it is a nonlinear function of the wavefield).
In differential geometry, a quantity appears that is known as the ``Gaussian curvature''. For small vertical motions u, this Gaussian curvature reduces to our expression G.
The gentle flexure of a sheet of paper follows the principle that the Gaussian curvature vanishes. The deformation must be one dimensional. We were first attracted to Gaussian curvature as a way of interpolating sparse data where the data represents a wavefield or a sedimentary earth model. We would seek the interpolation that was most ``paper like'', which minimized the integral of the square of the Gaussian curvature. Unfortunately, G is already a quadratic function of u even before we square G to minimize a positive value.