GEOG 486
Cartography and Visualization

Inverse Distance Weighting

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Inverse distance weighting is a deterministic, nonlinear interpolation technique that uses a weighted average of the attribute (i.e., phenomenon) values from nearby sample points to estimate the magnitude of that attribute at non-sampled locations. The weight a particular point is assigned in the averaging calculation depends upon the sampled point's distance to the non-sampled location (see Figure 6.cg.25, below). The method is called inverse distance weighting because according to Tobler's first law of geography, (see Interpolation) the similarity of two locations should decrease with increasing distance.

diagram and related equation to explain the idea of Inverse Distance Weighting. More in formation in text.
Figure 6.cg.25 Here, the green points represent known sample points and the red point is an unknown location for which we would like to estimate a value (Z(x)). The distances between sample points and the unknown point are shown in black text, while the attribute values of the sample points are shown in blue. You can see how the distance affects the influence of a point on the estimated value in the equation at the right: nearer points have a significantly greater effect on the estimated value than more distant points.

To use inverse distance weighting interpolation to create a surface, there are several factors that cartographers need to consider. One important question is the type of relationship the phenomenon has with distance (e.g., does it decrease dramatically with distance, or do even relatively distant points have some degree of similarity with the non-sampled location?). Many cartographers choose to specify an inverse distance-squared relationship, where the weight of a point differs with the inverse square of distance (i.e., 1/distance2) rather than a simple inverse distance (i.e., 1/distance) (see Figure 6.cg.26, below).

 

The first (top) map in a series of three to illustrate how inverse distance weighting interpolation works to create a surface. This map uses a distance exponent of one. visual spaceThe second (middle) map in a series of three to illustrate how inverse distance weighting interpolation works to create a surface. This map uses a distance exponent of two.visual spaceThe third (bottom) map in a series of three to illustrate how inverse distance weighting interpolation works to create a surface. This map uses a distance exponent of three.
Figure 6.cg.26 We used a distance exponent of one for the top map, an exponent of two for the middle map, and an exponent of three for the bottom map. You can see that more distant points have a greater effect on the overall pattern of the top map, in that it is a smoother surface than the middle and bottom maps (i.e., it is gives greater weight to points that are farther away (and that have fewer similar attribute values), thereby suppressing the influence of individual peaks). In the bottom map, you can see that individual peaks in rainfall are most clearly visible, as the interpolated values are not influenced as heavily by more distant points that have lower values.

A second important factor is determining how large the neighborhood of influence should be: all points within some fixed distance of the non-sampled location, or whether the neighborhood should consist of some particular number of points, regardless of their distance to the non-sampled location. A variation on this second method might be to specify some combination of distance and number of points (e.g., select the nearest n number of points within 10 km of the non-sampled location). Each of these decisions can have an impact upon the final appearance of the interpolated surface and therefore needs to be carefully considered (see Figure 6.cg.27, below).

The first (top) map in a series of three, this map shows incorporated points within a fixed radius of 25 km.visual spaceThe second (middle) map in a series of three, this map shows incorporated points within a fixed radius of 50 km.visual spaceThe third (bottom) map in a series of three, this map shows incorporated points within a fixed radius of 100 km.
Figure 6.cg.27 In the three maps above, we incorporated all points within a fixed radius of 25, 50 and 100 kilometers in the top, middle and bottom maps, respectively. You can see from the top map that a radius of 25 km is inadequate for this set of samples, as there are several locations that do not have any points within 25 km, and for which the interpolation algorithm cannot make any prediction at all. Generally, as the size of the search radius increases, the number of points included in the calculation increases, which has the effect of smoothing the map pattern.
The first (top) map in a series of three, this map shows use of the nearest 5 points for calculating precipitation at non-sampled locations. visual spaceThe second (middle) map in a series of three, this map shows use of the nearest 12 points for calculating precipitation at non-sampled locations. visual spaceThe third (bottom) map in a series of three, this map shows use of the nearest 25 points for calculating precipitation at non-sampled locations.
Figure 6.cg.28 In the three maps above, we have used the nearest 5, 12 and 25 points for calculating precipitation at non-sampled locations in the top, middle and bottom maps, respectively. Again, you can see that increasing the number of points used in the calculation smooths the map pattern, by decreasing the impact of extreme values on the predicted values (i.e., there are fewer locations with the highest amounts of rainfall predicted).

Because inverse distance weighting is a deterministic technique, it does not take into account the spatial structure (i.e., arrangement) of the sample points. Therefore, the results that you get using this technique can be influenced by the spacing and density of the samples, and it is good to be cautious about the accuracy of the interpolated values.  Also, because inverse distance weighting computes an average value, the value it calculates for a non-sampled point can never be higher than the maximum value for a sample point or lower than the minimum value of the sample point, so if the peaks and valleys of the data are not represented in your sample, this technique may be wildly inaccurate in some locations.