Inverse distance weighting models work on the premise that observations further away should have their contributions diminished according to how far away they are. The simplest model involves dividing each of the observations by the distance it is from the target point raised to a power α :
The value kj in this expression is an adjustment to ensure that the weights add up to 1. If the parameter α=1 we have:
Many GIS packages provide this kind of inverse distance model for interpolation, as it is simple to implement and to understand. Often the model is generalized in a number of ways:
|•||a faster rate of distance decay may be provided, by including a power function of distance, α>1, rather than simple linear distance. While any α value convenient for a given application may be used, common practice is to use distance (α=1) or distance squared (α=2)|
|•||since it is possible that the grid intersection could coincide with a data point (especially likely at region corners or edges if MBRs have been used on the original point set), an explicit check or adjustment to the expression is needed to avoid computational errors (overflow). Typically the adjacent point weight is set to 1 (its value is copied) and the remaining weights are set to 0|
|•||if a user-selectable adjustment to the minimum inter-point distance is specified, this can result in smoothed rather than exact interpolation. This may be a simple incremental amount added to the distance, t, or an adjusted distance value such as dij*=√(dij2+t2)|
|•||as with all methods, additional controls may be applied or available: limiting the number of points included; specifying the search directions and search shape; and limiting computations by excluding pre-defined regions, breaklines or faults|
Figure 6‑34 and Figure 6‑35 provide illustrations of the method applied to the test data for Pentland Hills OS NT04. The surface plot shows how simple IDW with no smoothing and power 2 distance decay results in dips and peaks around the data points but is otherwise relatively smooth in appearance.
Figure 6‑34 IDW as surface plot
Figure 6‑35A shows the source data (spot heights and contours) with Figure 6‑35B and Figure 6‑35C illustrating the surface obtained using parameters of α=1 and α=2. Both exhibit the familiar bull’s eye effect of standard IDW. Figure 6‑35D is markedly different and seems much closer to the source contours. In this case we have selected α=3, a smoothing factor of t=2, and an anisotropy (directional bias) of 45 degrees using an elliptical search region with a ratio of 2:1. The selection of these values was made after limited experimentation using simple cross-validation and comparison with additional information.
Some GIS products, such as the ArcGIS Geostatistical Analyst (but not the ArcGIS Spatial Analyst) attempt to select an optimal value for α automatically. The method adopted varies, but typically involves cross-validating the data for incremental values of α, for example α=0(0.25)4, and selecting the value (or locally interpolated value) that yields the minimum RMSE value for the surface as a whole. In the example illustrated ArcGIS estimates α=2.96 and RMSE at approximately 5 meters, although the maximum absolute error is around 100 meters.
Figure 6‑35 Contour plots for alternative IDW methods, OS NT04
A. Source data — contours
B. Inverse distance weighting — 1/d
C. Inverse distance weighting, 1/d2
D. IDW — 1/d3 plus smoothing and anisotropy