measure is a key capability required in most GIS applications. Perhaps the most commonly used surface type is a digital elevation model of terrain. These datasets are readily available at small scales for various parts of the world. However, as you have read earlier, just about any measure taken at locations across a landscape, subsurface, or atmosphere can be used to generate a continuous surface. A major challenge facing most GIS modelers is to generate the most accurate possible surface from existing sample data as well as to characterize the error and variability of the predicted surface. Newly generated surfaces are used in further GIS modeling and analysis as well as in 3D visualization. Understanding the quality of this data can greatly improve the utility and purpose of GIS modeling. This is the role of the Geostatistical Analyst.
Analyzing the surface properties of nearby locations
Generally speaking, things that are closer together tend to be more alike than things that are farther apart. This is a fundamental geographic principal (Tobler, 1970). Suppose you are a town planner, and you need to build a scenic park in your town. You have several candidate sites, and you may want to model their viewsheds at each location. This will require a more detailed elevation surface dataset for your study area. Suppose you have preexisting elevation data for 1,000 locations throughout the town. You can use this to build a new elevation surface.
When trying to build the elevation surface, you can assume that the sample values closest to the prediction location will be similar.
As you move farther away from the prediction location, the influence of the points will decrease. Considering a point too far away may actually be detrimental because the point may be located in an area that is dramatically different from the prediction location.
One solution is to consider enough points to give a good sample but small enough to be practical. The number will vary with the amount and distribution of the sample points and the character of the surface. If the elevation samples are relatively evenly distrib- uted and the surface characteristics do not change across your landscape, you can predict surface values from nearby points with reasonable accuracy. To account for the distance relation- ship, the values of closer points are weighted more heavily than those farther away.
THE PRINCIPLES OF GEOSTATISTICAL ANALYSIS 51
Visualizing global polynomial interpolation There are other solutions for predicting the values for unmea- sured locations. Another proposed site for the observation area is on the face of a gently sloping hill. The face of the hill is a sloping plane. However, the locations of the samples are in slight depressions or on small mounds (local variation). Using the local neighbors to predict a location may over or underestimate because of the influence of depressions and mounds. Further, you may pick up the local variation and may not capture the overall sloping plane (referred to as the trend). The ability to identify and model local structures and surface trends can increase the accuracy of your predicted surface.
To base your prediction on the overriding trend, you can fit a plane between the sample points. A plane is a special case of a family of mathematical formulas called polynomials. You then determine the unknown height from the value on the plane for the prediction location. The plane may be above certain points and below others. The goal for interpolation is to minimize error. You can measure the error by subtracting each measured point from its predicted value on the plane, squaring it, and adding the results together. This sum is referred to as a least-squares fit. This process is the theoretical basis for the first-order global polynomial interpolation.
But what if you were trying to fit the plane to a landscape that is a valley? You will have a difficult task obtaining a good surface from a plane. However, if you are allowed one bend in the plane (see image below), you may be able to obtain a better fit (get closer to more values). To allow one bend is the basis for second- order global polynomial interpolation. Two bends in the plane would be a third-order polynomial, and so forth. The bends can occur in both directions, possibly resulting in a bowl-shaped surface.
Visualizing local polynomial interpolation
Now what happens if the area slopes, levels off, and then slopes again? Asking you to fit a flat plane through this study site would give poor predictions for the unmeasured values. However, if you are permitted to fit many smaller overlapping planes, and then use the center of each plane as the prediction for each location in the study area, the resulting surface will be more flexible and perhaps more accurate. This is the conceptual basis for local polynomial interpolation.
Visualizing radial basis functions
Radial basis functions enable you to create a surface that
captures global trends and picks up the local variation. This helps in cases where fitting a plane to the sample values will not accurately represent the surface.
force the surface to form nice curves (thin-plate spline), or you can control how tightly you pull on the edges of the surface (spline with tension). This is the conceptual framework for interpolators based on radial basis functions.
THE PRINCIPLES OF GEOSTATISTICAL ANALYSIS 53
Geostatistical solutions
So far, the techniques that we have discussed are referred to as deterministic interpolation methods because they are directly based on the surrounding measured values or on specified mathematical formulas that determine the smoothness of the resulting surface. A second family of interpolation methods consists of geostatistical methods that are based on statistical models that include autocorrelation (statistical relationships among the measured points). Not only do these techniques have the capability of producing a prediction surface, but they can also provide some measure of the certainty or accuracy of the
predictions.
The following example will guide you through the basic steps of geostatistics using ordinary kriging.
Kriging is similar to IDW in that it weights the surrounding measured values to derive a prediction for each location. How- ever, the weights are based not only on the distance between the measured points and the prediction location but also on the overall spatial arrangement among the measured points. To use the spatial arrangement in the weights, the spatial autocorrelation must be quantified.
To solve the geostatistical example, you will walk you through a series of steps.
Calculate the empirical semivariogramkriging, like most interpolation techniques, is built on the assumption that things that are close to one another are more alike than those farther away (quantified here as spatial autocorrelation). The empirical semivariogram is a means to explore this relationship. Pairs that are close in distance should have a smaller measurement differ- ence than those farther away from one another. The extent that this assumption is true can be examined in the empirical semivari- ogram.