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Natural neighbor interpolation is one of the most general and robust methods of gridding available. It will produce a conservative, artifice-free (no "bull's eyes") result by finding the areal weighted averages at each grid node of the data values associated with that subset of the data that are natural neighbors of the grid node. The resulting surface is continuous everywhere within the convex hull of the data.
The weights applied to the data points are the natural neighbor local coordinates and are the ratios of the intersection contents of Voronoi polygons. These weights are always positive and will sum to one. Specifically, natural neighbors are determined using all circles that pass through the three or more data points, such that no data point lies within any circle. This will result in a series of overlapping circles with data points at their intersections. Each circle through a datum point is defined by two other points such that no other point is closer to the circumcenter than these three points. These two points are defined as natural neighbors of the datum point. All of the circles that pass through a datum point define all the natural neighbors of the datum point. The circles will have an average radius that is less than it would be for any other selection of triplet points and the triangles defined by the natural neighbors are the most equiangular possible. In general, each datum point will have six natural neighbor polygons. However, at edges of the interpolation, the datum points may have as few as one polygon.
After all of the natural neighbors of the data are calculated, the grid nodes can be interpolated using two methods. The easiest method is to calculate the barycentric coordinates of the grid node within the triplets of natural neighbors that contain the grid node, this is referred to as barycentric interpolation. The other method uses the grid node as an additional data point and calculates new natural neighbors for the grid node and uses these to interpolate the value for the grid node, this is referred to as Voronoi interpolation. With this type of interpolation the minimum inner angle to use for the new natural neighbor triplets calculated using the grid node should also be specified to prevent the triplets from falling along or close to a line.
The resulting surface from natural neighbor interpolation is continuous within the convex hull of the data. To obtain values for grid nodes outside of the convex hull, the natural neighbor triplets must be used to extrapolate the value. Since the grid node does fall within a natural neighbor triplet, the number of triplets to use to extrapolate the value needs to be specified.