Contour Plot

A contour plot (or diagram) is a two-dimensional representation of a three-dimensional surface. It consists of a set of curves called contours formed by projecting the curves of intersection of the surface with planes parallel to one of the coordinate planes. Typically, the surface is the graph of a function, which may be given as a formula or as output of measured data. The contours share the property that on any particular curve, the value of the dependent variable is constant. Normally, the diagram is such that the values between contours vary by a constant amount.

Figure 1 is an example of a contour plot. It was used to fit a logistic curve, that is, a curve of the form

in the least-squares sense to a set of data for the U.S. cumulative underground gas storage capacity from 1932 to 2000 (C is an arbitrary constant of integration). By manipulating the expression for a logistic function, it is possible to write a formula in terms of only two parameters: its growth rate and its inflection point. In order to minimize the corresponding least-square error function, one needs an accurate estimate of those parameters. This usually requires a [Page 186] first estimate, which in this example can be obtained more easily from the contour plot of the error function than from the three-dimensional graph (see Figure 2).

As with all contour plots, the contour plot illustrated above reflects the intersection of planes parallel to the inflection time-growth rate plane (t0, r – plane) and the surface of the error function displayed in Figure 2. Each contour depicts a curve of constant error; that is, the value of the error function remains the same along a curve.

Graphing contour lines by hand is generally impractical. Instead, one uses either graphics software or a mathematical or statistical software package. The plots shown were created in Matlab; instead of labeling values of the error at some curves, values are indicated by a scale given by a color bar.

Contour plots are used in very diverse fields; in particular, they are widely used in the earth sciences. A common application is topographic maps, which display elevation in terms of longitude and latitude. Since the elevation between any two consecutive contours varies by the same amount, proximity of contour lines indicates the rate of change of elevation.