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  A three-dimensional model of a spatial variable, defined as a single-valued function of either latitude and longitude or grid coordinates. Although geographical surfaces can be represented and analysed as three-dimensional solid objects, cost and convenience usually dictate display in two dimensions, as on a paper map (see map image and map) or computer monitor.

The most common geographical phenomenon treated as a surface is elevation, typically portrayed on topographic maps by contour lines (see isolines). In most instances elevation contours are captured by a photogrammetrist who traces the intersections of a three-dimensional, stereoscopic model of terrain with horizontal planes of constant elevation (Moffitt and Mikhail, 1980, pp. 335-417). The elevations of these planes and their corresponding contour lines are integer multiples of a fixed contour interval, chosen to strike an appropriate balance between information and clutter. Although elevation is a continuous variable, which in principle can be measured anywhere within the horizontal plane, the contour map is a generalization that reflects the relief and complexity of the terrain, the scale of the photogrammetrist\'s original map, and the reliability of the aerial photography and plotting instruments.

Many surfaces are estimated from discrete measurements at representative or conveniently accessible sample points. In meteorology and climatology, for instance, configurations of surfaces representing temperature, pressure and other atmospheric variables are interpolated from sparsely scattered observations, either manually or by computer. Although the informed intuition of a trained meteorologist can reflect predictable effects of land cover as well as avoid conceptual inconsistencies in the patterns of isotherms and isobars, interpolation by computer allows precise inverse-distance weighting and the consistent treatment of local trends (Lam, 1983). Interpolation can also generate isoline maps for population density, per capita income and other area data assigned to presumably representative points within census tracts or counties.

Computational statistics offers several specialized estimation methods. trend surface analysis yields generalized maps of salient trends. Kriging, a geostatistical technique developed to estimate the grade and tonnage of ore deposits, affords potentially insightful interactive solutions (Oliver and Webster, 1990). fractal surface simulation can generate hypothetical surfaces useful in testing interpolation algorithms and describing theoretical landscapes (Lam and De Cola, 1993).

Surfaces coded for electronic display and analysis are usually represented as either (1) matrices of elevation points aligned along the rows and columns of a uniform grid, or (2) lists of point coordinates describing the location and shape of individual contour lines. Array storage as a digital elevation model allows the ready calculation of slope, aspect and other measures of surface geometry and promotes the ready integration of elevation and slope with other data in a geographical information system.

Oblique views of terrain and other surfaces that mimic solid three-dimensional models can be revealingly dramatic (see visualization). Although taller, closer parts of surfaces often hide shorter, more distant features, high-interaction graphics workstations can provide the rotation and tilt needed for a more complete picture as well as cross-sectional and dynamic fly-by views. (MM)

References Lam, N. 1983: Spatial interpolation methods: a review. The American Cartographer 10: 129-49. Lam, N.S-N. and De Cola, L. 1993: Fractal simulation and interpolation. In N.S-N. Lam and L. De Cola, eds, Fractals in geography. Englewood Cliffs, NJ: Prentice-Hall, 56-83. Moffitt, F.H. and Mikhail, E.M. 1980: Photogrammetry, 3rd edn. New York: Harper and Row. Oliver, M.A. and Webster, R. 1990: Kriging: a method of interpolation for geographical information systems. International Journal of Geographical Information Systems 4: 313-32.



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