# Gradient Plane Nodal Functions

A limitation of Shepard's method is
that the interpolating surface is a simple weighted average of the data
values of the scatter points and is constrained to lie between the extreme
values in the data set. In other words, the surface does not infer local
maxima or minima implicit in the data set. This problem can be overcome
by generalizing the basic form of the equation for Shepard's method in
the following manner:

where Qi are nodal functions or individual functions defined at each
scatter point (Franke 1982; Watson & Philip 1985). The value of an
interpolation point is calculated as the weighted average of the values
of the nodal functions at that point. The standard form of Shepard's method
can be thought of as a special case where horizontal planes (constants)
are used for the nodal functions. The nodal functions can be sloping planes
that pass through the scatter point. The equation for the plane is as
follows:

where fx and fy are partial derivatives at the scatter point that have
been previously estimated based on the geometry of the surrounding scatter
points. Gradients are estimated in GMS by first triangulating the scatter
points and computing the gradient at each scatter point as the average
of the gradients of each of the triangles attached to the scatter point.

The planes represented by the above equation are sometimes called "gradient
planes". By averaging planes rather than constant values at each
scatter point, the resulting surface infers extremities and is asymptotic
to the gradient plane at the scatter point rather than forming a flat
plateau at the scatter point.

## 3D Interpolation

The 3D equivalent of a gradient plane is a "gradient hyperplane."
The equation of a gradient hyperplane is as follows:

where fx, fy, and fz are partial derivatives at the scatter point that
are estimated based on the geometry of the surrounding scatter points.
The gradients are found using a regression analysis which constrains the
hyperplane to the scatter point and approximates the nearby scatter points
in a least squares sense. At least five non-coplanar scatter points must
be used.

**Related Links:**

Inverse Distance Weighted Interpolation

Computation of Interpolation Weights

Subset Definition