GMS:Shepard's Method

The simplest form of inverse distance weighted interpolation is sometimes called "Shepard's method" (Shepard 1968). The equation used is as follows:

where n is the number of scatter points in the set, f_i are the prescribed function values at the scatter points (e.g., the data set values), and w_i are the weight functions assigned to each scatter point. The classical form of the weight function is:

$w_{i}={\frac {h_{i}^{-p}}{\displaystyle \sum _{j=1}^{n}h_{j}^{-p}}}$

where p is an arbitrary positive real number called the weighting exponent and is defaulted to 2. The weighting exponent can be modified by turning on the Use classic weight function option. h_i is the distance from the scatter point to the interpolation point or

$h_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}}}$

where (x,y) are the coordinates of the interpolation point and (x_i,y_i) are the coordinates of each scatter point. The weight function varies from a value of unity at the scatter point to a value approaching zero as the distance from the scatter point increases. The weight functions are normalized so that the weights sum to unity.

Although the weight function shown above is the classical form of the weight function in inverse distance weighted interpolation, the following equation is used in GMS:

$w_{i}={\frac {\left[{\frac {R-h_{i}}{Rh_{i}}}\right]^{2}}{\displaystyle \sum _{j=1}^{n}\left[{\frac {R-h_{i}}{Rh_{i}}}\right]^{2}}}$

where h_i is the distance from the interpolation point to scatter point i, R is the distance from the interpolation point to the most distant scatter point, and n is the total number of scatter points. This equation has been found to give superior results to the classical equation (Franke & Nielson, 1980).

The weight function is a function of Euclidean distance and is radially symmetric about each scatter point. As a result, the interpolating surface is somewhat symmetric about each point and tends toward the mean value of the scatter points between the scatter points. Shepard's method has been used extensively because of its simplicity.

3D Interpolation

The 3D equations for Shepard's method are identical to the 2D equations except that the distances are computed using:

$h_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}}}$

where (x,y,z) are the coordinates of the interpolation point and (x_i,y_i,z_i) are the coordinates of each scatter point.