Interpolation

Interpolation is a procedure for computing the values of a function at unsampled locations using the known values at sampled points. When using GIS, spatial distributions of physical and socioeconomic phenomena can often be approximated by continuous, singlevalued functions that depend on location in space. Typical examples are heights, temperature, precipitation, soil properties, or population densities. Data that characterize these phenomena are usually measured at points or along lines (profiles, contours), often irregularly distributed in space and time. On the other hand, visualization, analysis, and modeling of this type of fields within GIS are often based on a raster representation. Interpolation is therefore needed to support transformations between different discrete representations of spatial and spatiotemporal fields, typically to transform irregular point or line data to raster representation (see Figure 1) or to resample between different raster resolutions. In general, interpolation from points to raster is applied to data representing continuous fields. Different approaches are used for transformation of data that represent geometric objects (points, lines, polygons) using discrete categories.

The interpolation problem can be defined formally as follows. Given the n values of a studied attribute measured at discrete points within a region of a d-dimensional space, find a d-variate function that passes through the given points but that extends into the “empty” unsampled locations between the points and so yields an estimate of the attribute value in these locations. An infinite number of functions fulfills this requirement, so additional conditions have to be imposed to construct an interpolation function. These conditions can be based on geostatistical concepts (as in kriging), locality (nearest-neighbor and finiteelement methods), smoothness and tension (splines), or ad hoc functional forms (polynomials, multiquadrics). The choice of additional conditions depends on the type of modeled phenomenon and the application. If the point sample data are noisy as a result of measurement error, the interpolation condition is relaxed and the function is required only to pass close [Page 238]to the data points, leading to approximation rather than interpolation. Approximation by lower-order polynomials is known in the literature as a trend surface.

Figure 1 Interpolation is used to compute the unknown values at the centers of the grid cells using the values measured at the scattered points (shown as circles)

Interpolation in GIS applications poses several challenges. First, the modeled fields are usually complex; the data are spatially heterogeneous; and significant noise or discontinuities can be present. Second, data sets can be very large (thousands to millions of points), originating from various sources, and have different accuracies. Interpolation methods suitable for GIS applications should therefore satisfy several important requirements relating to accuracy and predictive power, robustness and flexibility in describing various types of phenomena, smoothness for noisy data, applicability to large data sets, computational efficiency, and ease of use.