# 8.5 Spatial Interpolation for Spatial Analysis

A surface is a vector or raster dataset that contains an attribute value for every locale throughout its extent. In a sense, all raster datasets are surfaces, but only some vector datasets are surfaces. Surfaces are commonly used in a geographic information system (GIS) to visualize phenomena such as elevation, temperature, slope, aspect, rainfall, and more. In a GIS, surface analyses are usually carried out on either raster datasets or TINs (Triangular Irregular Network; Chapter 5 “Geospatial Data Management,” Section 5.3.1 “Vector File Formats”), but isolines or point arrays can also be used. Interpolation is used to estimate the value of a variable at an unsampled location from measurements made at nearby or neighboring locales. Spatial interpolation methods draw on the theoretical creed of Tobler’s first law of geography, which states that “everything is related to everything else, but near things are more related than distant things.” Indeed, this basic tenet of positive spatial autocorrelation forms the backbone of many spatial analyses (Figure 8.9 “Positive and Negative Spatial Autocorrelation”).

# Creating Surfaces

The ability to create a surface is a valuable tool in GIS. However, the creation of raster surfaces often starts with the creation of a vector surface. One standard method to create such a vector surface from point data is the generation of Thiessen (or Voronoi) polygons. Thiessen polygons are mathematically generated areas that define the sphere of influence around each point in the dataset relative to all other points (Figure 8.10 “A Vector Surface Created Using Thiessen Polygons”). Polygon boundaries are calculated as the perpendicular bisectors of the lines between each pair of neighboring points. The derived Thiessen polygons can then be used as crude vector surfaces that provide attribute information across the entire area of interest. A typical example of Thiessen polygons is the creation of a rainfall surface from an array of rain gauge point locations. These Thiessen polygons can be easily converted to equivalent raster representations by some basic reclassification techniques.

While creating Thiessen polygons results in a polygon layer whereby each polygon, or raster zone, maintains a single value, interpolation is a potentially complex statistical technique that estimates the value of all unknown points between the known points. The primary methods used to create interpolated surfaces are spline, inverse distance weighting (IDW), and trend surface. The spline interpolation method forces a smoothed curve through the set of known input points to estimate the unknown intervening values. IDW interpolation estimates the values of unknown locations using the distance to proximal, known values. The weight placed on the value of each proximal value is in inverse proportion to its spatial distance from the target locale. Therefore, the farther the proximal point, the less weight it carries in defining the target point’s value. Finally, trend surface interpolation is the most complex method. It fits a multivariate statistical regression model to the known points, assigning a value to each unknown location based on that model.

Other highly complex interpolation methods exist, such as kriging. Kriging is a complex geostatistical technique, like IDW, that employs semi-variograms to interpolate the values of an input point layer and is more akin to regression analysis (Krige, 1951). The specifics of the kriging methodology will not be covered here as this is beyond the scope of this text. For more information on kriging, consult review texts such as Stein (1999).

Inversely, raster data can also be used to create vector surfaces. For instance, isoline maps consist of continuous, nonoverlapping lines connecting points of equal value. Isolines have specific monikers depending on their model information (e.g., elevation = contour lines, temperature = isotherms, barometric pressure = isobars, wind speed = isotachs). Figure 8.11 “Contour Lines Derived from a DEM” shows an isoline elevation map. As the elevation values of this digital elevation model (DEM) range from 450 to 950 feet, the contour lines are placed at 500, 600, 700, 800, and 900 feet elevations throughout the extent of the image. In this example, the contour interval, defined as the vertical distance between each contour line, is 100 feet. The user determines the contour interval during the creation of the surface.