Interpolation: Inverse Distance Weighting

Inverse distance weighting is the simplest interpolation method. A neighborhood about the interpolated point is identified and a weighted average is taken of the observation values within this neighborhood. The weights are a decreasing function of distance. The user has control over the mathematical form of the weighting function, the size of the neighborhood (expressed as a radius or a number of points), in addition to other options.

Weighting function

The simplest weighting function is inverse power:

w(d)= 1/dp


with p>0. The value of p is specified by the user. The most common choice is p= 2. For p= 1, the interpolated function is "cone-like" in the vicinity of the data points, where it is not differentiable .

Shepard's method (Shepard, 1968) is a variation on inverse power, with two different weighting functions using two separate neighborhoods. The default weighting function for Shepard's method is an exponent of 2 in the inner neighborhood and an exponent of 4 in the outer neighborhood. The form of the outer function is modified to preserve continuity at the boundary of the two neighborhoods.


Neighborhood size

The neighborhood size determines how many points are included in the inverse distance weighting. The neighborhood size can be specified in terms of its radius (in km), the number of points, or a combination of the two. If a radius is specified, the user also can specify an override in terms of a minimum and/or maximum number of points. Invoking the override option will expand or contract the circle as needed. If the user specifies the number of points, an override of a minimum and/or maximum radius can be included. It also is possible to specify an average radius based upon a specified number of points. Again, there is an override to expand or contract the neighborhood to include a minimum and/or maximum number of points. In Shepard's method, there are two neighborhoods; the inner neighborhood is taken to be one-third the radius of the outer radius.

Anisotropy correction

In many instances, the observation points are not uniformly spaced about the interpolation points, with several in a particular direction and fewer in others. This situation produces a spatial bias of the estimate, as the clustered points carry an artificially large weight. The anisotropy corrector permits the weighted average to downweight clustered points that are providing redundant information. The user selects this option by setting the anisotropy factor to a positive value. A value of 1 produces its full effects, while a value of 0 produces no correction.
This correction factor is defined by computing the angle between every pair of observation points in the neighborhood, relative to the observation point.

Gradient correction

A disadvantage of the inverse weighted distance functions is that the function is forced to have a maximum or minimum at the data points (or on a boundary of the study region). The gradient corrector permits a non-zero gradient at the observation points. The implementation is such that the interpolated value is the sum of two values.


References

Cressman (1968)

Fisher, N. I., T. Lewis, and B. J. J. Embleton (1987) Statistical Analysis of Spherical Data, Cambridge University Press, 329 pp.

Shepard, D. (1968) A two-dimensional interpolation function for irregularly-spaced data, Proc. 23rd National Conference ACM, ACM, 517-524.