GMS:Quadratic Nodal Functions

The nodal functions used in inverse distance weighted interpolation can be higher degree polynomial functions constrained to pass through the scatter point and approximate the nearby points in a least squares manner. Quadratic polynomials have been found to work well in many cases (Franke & Nielson 1980; Franke 1982). The resulting surface reproduces local variations implicit in the data set, is smooth, and approximates the quadratic nodal functions near the scatter points. The equation used for the quadratic nodal function centered at point k is as follows:

$Q_{k}(x,y)=a_{k1}+a_{k2}\left(x-x_{k}\right)+a_{k3}(y-y_{k})+a_{k4}(x-x_{k})^{2}+a_{k5}(x-x_{k})(y-y_{k})+a_{k6}(y-y_{k})^{2}$

To define the function, the six coefficients a_k1..a_k6 must be found. Since the function is centered at the point k and passes through point k, we know beforehand that a_k1=f_k where f_k is the function value at point k. The equation simplifies to:

$Q_{k}\left(x,y\right)=f_{k}+a_{k2}(x-x_{k})+a_{k3}(y-y_{k})+a_{k4}(x-x_{k})^{2}+a_{k5}(x-x_{k})(y-y_{k})+a_{k6}(y-y_{k})^{2}$

Now there are only five unknown coefficients. The coefficients are found by fitting the quadratic to the nearest NQ scatter points using a weighted least squares approach. In order for the matrix equation used to solve for the coefficients to be stable, there should be at least five scatter points in the set.

3D Interpolation

For 3D interpolation, the equation for the quadratic nodal function is:

File:Idw3d eq4.gif $+a_{k5}\left(x-x_{k}\right)(y-y_{k})+a_{k6}(x-x_{k})(z-z_{k})+a_{k7}(y-y_{k})(z-z_{k})$

To define the function, the ten coefficients a_k1..a_k10 must be found. Since the function is centered on point k, we know that a_k1=f_k where f_k is the data value at point k. The equation simplifies to:

$Q_{k}\left(x,y\right)=f_{k}+a_{k2}(x-x_{k})+a_{k3}(y-y_{k})+a_{k4}(z-z_{k})$

Now there are only nine unknown coefficients. The coefficients are found by fitting the quadratic to a subset of the neighboring scatter points in a weighted least squares fashion. In order for the matrix equation used to be solve for the coefficients to be stable, there should be at least ten non-coplanar scatter points in the set.