Topographic Map
Interpretation

TECHNIQUE NUMBER 10

TREND SURFACES

X, Y, and Z axes

A trend surface is a three-dimensional surface fit to a set of data points. The simplest trend surface to visualize is that which is determined in a three-point problem: a planar surface floating in space. Perhaps the most difficult trend surface to visualize is that displayed on any topographic map: a convoluted surface, fit to numberless data points, but still more simplified than reality. In between lie the type of trend surfaces with which you might work: those determined on the basis of a finite number of data points which lie on the surface depicted on a topographic map. Examples of surface defined from the present ground surface but not part of it might be water tables (defined by lakes and swamps presumed to be interconnected by groundwater) and the regional Glaciation Threshold (defined by the Equilibrium Line Altitudes of a number of mountain glaciers).

A trend surface may be fitted to a data set by a rigorous statistical procedure in which the equation which defines a first, second, third, etc. order surface (planar, curved, or complexly folded) is generated. Alternatively, the empirical approach is just to contour the points.

Contouring points on a map is simple (see any introductory lab manual for instructions). You need only to follow the rules of contours and use the same contour interval as the base map, or a larger one. The more points you have, the easier it is to establish the position of a given contour, and the more accurate and complex your trend surface.

I will not ask you to generate a trend surface in this example, but try to visualize what one would be like if you used only the elevations at the section corners. Can you see that the southern portion of the Bozeman quadrangle would be greatly simplified, while the northern portion would be little changed?

There are sophisticated micro-computer programs for fitting trend surfaces (either equations or smoothed contours) to three-dimensional data, such as "Surfer" by Golden Software.