Fundamentals of Interpolation

Interpolation includes the set of mathematical and statistical procedures applied to derive estimates at unsampled locations within the spatial coverage of the sample values available. Interpolation is considered to be an effective method of converting data from point observations to a continuous surface, whether it is represented as isopleths, irregular tiles or a regular grid. Burrough and McDonnell (1998) state that interpolation is necessary in three broad situations. The case that reflects the current situation is that “the data we have do not cover the domain of interest completely”.

Fundamentally, interpolation is rooted in Tobler’s First Law of Geography that attribute values are more likely to be closely related to those features in close proximity rather than those that are distant. Interpolation methods can be subdivided in two separate manners. Firstly, techniques can be categorized as being either exact or inexact interpolators. Exact interpolators consider the original sample data to be an ‘exact’ measurement of the true surface and thus retain them in the predicted surface for their respective locations. Inexact interpolators consider that localized variation may be such that an original sample value is equivalent to a measurement of the true surface plus noise. New values are estimated at sample points, as well as unsampled points, corresponding to the best fitting surface. The second classification of interpolation procedures is between global and local methods. Global interpolation techniques utilize all available data points for the study area, whereas local interpolation techniques only operate with the nearest data points within a defined neighbourhood.

Regression methods are a common type of global interpolation. Regression techniques attempt to establish a function that describes the relationship among attributes. Trend surface analysis does this based solely on the geographical coordinates of sample locations. Transfer functions are based on the relationship between sample data and a second spatially variable attribute. In both cases, the study area is modeled by a smooth mathematical function. For example, trend surface analysis fits a polynomial function to the attribute versus geographical location surface using the least squares method (Burrough and McDonnell, 1998). This approach can be very useful in minimizing the cost of a study and maximizing the predictive performance of the surface model by attempting to construct an empirical relationship between the variable of interest and a second variable, which is either cheaper to map or for which data is already more readily available (Burrough and McDonnell, 1998).

Burrough and McDonnel (1998) define four steps in any local interpolation procedure. Firstly, the neighbourhood around a prediction point must be defined. Secondly, the existing sample data points within this search area must be determined. Thirdly, a mathematical function must be applied to represent the variation among these limited data points. Fourthly, this must be evaluated for a point on the grid being used. All the grid points can be calculated in this manner.

Local interpolation procedures can be categorized into three major classes: nearest neighbours, inverse distance weighting, and splines. However, there are also a few less commonly used methods. The most common form of nearest neighbour interpolation is the Theissen polygon method (Theissen, 1911) in which each predicted grid cell is assigned the same value as its closest neighbour. The resulting “surface” of discrete polygons intuitively seems unsuited to the development of precipitation fields, yet for almost 50 years, it was the predominantly used interpolation procedure even for precipitation (Pardo-Iguzquiza, 1998). This was superseded by the development of geostatistics in the 1960s.

Inverse distance weighting (IDW) improves upon the Theissen method by including the effects of more than one neighbour and weighting their influence based on proximity. Weighting is based on inverse distance; nearer stations hold greater influence than more distant ones, in accordance to Tobler’s First Law. The technique can be adjusted by altering the power to which distance is raised and the number of neighbours considered. As the power increases, there is a more rapid decrease in influence with distance from a particular sample value. Most commonly, a power of two is utilized (Inverse Squared Distance) (Goovaerts, 2000). The number of neighbours to be considered is regulated by explicitly choosing the number or setting a search radius for each interpolated point, either of which must be defined a priori. Splines are a piece-wise function that fits a polynomial curve to a small number of local points. For further discussion on splines and subtypes, refer to Burrough and McDonnell (1998).

It should be noted that a digital elevation model (DEM) itself is an interpolated surface, though they are generally based upon a much greater density of sample points, whether they be from ground sampling or aerial photo analysis. Further discussion of DEM issues will not be included in this report.