MORE ON GAUSSIAN CURVATURE

Given a function u(x,y), its x-derivative u_x, its y-derivative u_y, and a slope parameter p, we have the planewave operator L

$$0 \quad =\quad L (u) = u_x + p u_y$$ (13)

which vanishes when u(x,y) is not really a two-dimensional function but is a one-dimensional function u=f(x-py).

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

$$\begin{eqnarray} 0 \quad =\quad L (u_x) &=& u_{xx} +p u_{xy} \\ 0 \quad =\quad L (u_y) &=& u_{yx} +p u_{yy}\end{eqnarray}$$ (14) (15)

Eliminate p from these two equations by solving for it:

$$-p \quad =\quad {u_{xx} \over u_{xy}} \quad =\quad {u_{yx} \over u_{yy}}$$ (16)

or

$$G(u) \quad =\quad u_{xx}u_{yy} - u_{xy}u_{yx} \quad =\quad0$$ (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.