Natural Neighbor

Natural neighbor is as simple to use as Nearest Neighbor and provides more precise results; however, is only available for two-dimensional interpolations. Natural neighbor requires that a grid be defined. (See Setting Up the Grid.) Once the grid has been defined, select **Natural Neighbor** from the combo box in the **Geo** tab of the Control Panel. SADA is now ready for you to press the **Estimates **or **Risk** mapping buttons on the main toolbar.

Natural Neighbor interpolation is a weighted moving average technique that uses geometric relationships in order to choose and weight nearby points.

The equation for the Natural Neighbor (NN) interpolation is:

where: G(x,y) is the NN estimation at (x,y);

n is the number of nearest neighbors used for interpolation;

f(xi,yi) is the observed value at (xi,yi); and

wi is the weight associated with f(xi,yi).

The number of natural neighbors is determined by constructing natural neighbor circles, called circumcircles. Two points are natural neighbors if they lie on the same natural neighbor circle. Delaunay triangulation is then used to determine the weights in order to interpolate. The weights (wi) depend on the area about each of the data points (Voronoi polygons) instead of the distance between data points, as with Inverse Distance Weighting (IDW).

The natural neighbor circles are constructed under the following constraints:

• No data are within a natural neighbor circle;

• No other datum is closer to the centroid of the circle;

• Smallest radius criterion for any group of three data; and

• Each natural neighbor circle passes through three data points.

For more detailed information about Natural Neighbor, refer to the following sources:

Owen, S.J., An Implementation of Natural Neighbor Interpolation in Three Dimensions, Thesis, Brigham Young University, 1992.

Sibson, R., “A Brief Description of Natural Neighbor Interpolation,” Chapter 2 in Interpolating multivariate data, John Wiley & Sons, New York, 1981, pp. 2136.

Watson, D.F., “Natural Neighbor Sorting,” The Australian Computer Journal, vol. 17, no. 4, 1995.

Watson, D.F., nngridr: An Implementation of Natural Neighbor Interpolation, published by David Watson, Australia, 1994.