Interpolation:
Inverse Distance Weighting |
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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/d^{p}

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.