An Essay on Spatial Interpolation

 Spatial interpolation is the technique of estimating a complete, continuous raster surface based off of a patchwork of known point values captured across a surface in reality.  The surface that is created attempts to represent the gradation, or smooth progression between varying spatial values found in real-world observations.  Interpolation is a powerful visualization method used to make sense of point data that is otherwise very difficult to envision and interpret.  Applications of such raster surfaces include maps of population densities, precipitation levels, temperature differences, etc.  This technique has nearly limitless uses and can create a surface for any spatially-based phenomenon – one of the most notable and easily interpreted of which is an interpolation of elevation values.  It is possible to use interpolation techniques in a vector environment using a triangulated irregular network (TIN), however much of my experience has remained in the world of raster based interpolation.

Spatial interpolation works by estimating values for the unknown, unsampled area between two or more known points.  A basic method of this is to derive isolines between sample points – this can even be done by hand.  I have performed a manual contour interpolation in a meteorology course in the past, where isopleths – lines of equal barometric pressure – were estimated from a map containing only weather station point data.  These lines were sketched, and another layer – frontal boundaries – could then be inferred from the location and orientation of the isolines.  Although this exercise included only a hand full of points, a surprisingly accurate map could be produced by hand, however the addition of more points can quickly force manual interpolation to become time consuming and increase chances of error.  Another method is to create a continuous surface – usually using advanced computer algorithms to automatically calculate the hundreds, if not millions or more raster cells that make up the interpolated surface.  My experience with these types of raster surfaces begin with a project conducted for the Florida State University Environmental Service Program in 2006, which used raster surfaces to model the efficiency of the University’s recycling program on campus, and to suggest new recycling bin locations.  Point locations were collected for centers of student population around campus – classroom buildings, libraries, parking lots, dorms, etc. – as well as for trash can and recycling bin locations.  Each point was assigned an appropriate weight value, and a raster surface was created for each layer.  The interpolations for each layer of centers of student population were combined using a raster calculation, and contour surfaces were created and overlaid with the raster surface of recycling bin distribution on campus.  It was then very easy to observe how the centers of high student traffic compared with the current distribution of recycling facilities on campus.

One final description of spatial interpolation is enforced by Tobler’s First Law of Geography where locations close to each other have more in common than locations which are farther apart.  Spatial autocorrelation, a formulation of Tobler’s Law, continues by measuring how clustered (positive spatial autocorrelation) or dispersed (negative autocorrelation) a set of spatial features exhibit.  These ideas illustrate the significance of neighboring points (or point sets) upon one another and how neighboring points effect the resulting interpolated surface.  More advanced interpolation techniques using these concepts will be discussed.

As with any area of scientific estimation and modeling, there are many issues that affect the accuracy of an interpolated raster or vector surface.  Basic affects stem from any combination of the number of points used for interpolation (the more, the better), and the distribution (positive or negative spatial autocorrelation) or distance between points (the location and proximity of neighbors significantly effect estimations between points).  More advanced effects begin with directional influence, and continue with the presence of barriers, local neighbor under- or overcorrection, and instrument error.  Finally, possibly one of the most important factors affecting the accuracy of an interpolated surface resides with choosing the most appropriate method to perform the surface interpolation.

Due to this great amount of effect that the method of spatial interpolation holds upon the accuracy of the resulting surface, it is important to carefully choose a method that is most appropriate for the nature of the given data.  The class notes discuss in detail the differences between several methods.  Generally, Inverse Distance Weighted (IDW) methods limit the influence of neighbors as their distance increases (implementing Tobler’s Law).  More sample points promote a more smooth surface, however areas of littler or no data will skew the surface toward the overall mean of the dataset, creating holes: an evenly spaced distribution of input points avoids these holes. Thiessen (Voronoi, or natural neighbor) polygons often have odd-shaped boundaries to transition between polygons.  Continuous variables are not well represented.  The trend surface method uses multiple regression (predicting a dependent elevation, Z variable with independent X & Y, location variables) to approximate values; however this method rarely intersects the original point.  Splines are used to interpolate smooth curves, and are best for surfaces that are already smooth.  Kriging is a more in-depth, random, weighted average technique using more advanced algorithms and spatial autocorrelation, best suited when correlated distances are known or if there is directional bias in the data.  Finally, it is important to understand issues associated with exact, and approximate interpolators, as well as deterministic and geostatistical interpolators.

The goal of interpolation is to minimize error.  It is the responsibility of the GIS user to understand the discussed differences between varying techniques used to interpolate surfaces.  Combining a working knowledge of the issues pertaining to spatial interpolation with the resources of the GIS reference information will assist in the creation of accurate representations.