(13) |

Next we get two equations from the plane-wave equation,
one differentiating by *x*, the other by *y*.

(14) | ||

(15) |

Eliminate *p* from these two equations by solving for it:

(16) |

(17) |

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.

4/27/2000