Quantitative Analysis of Watershed Geomorphology

Quantitative geomorphic methods developed within the past few years provide means of measuring size and form properties of drainage basins. Two general classes of de scriptive numbers are (1) linear scale measurements, whereby geometrically analogous units of topography can be compared as to size; and (2) dimensionless numbers, usually angles or ratios of length measures, whereby the shapes of analogous units can be compared irrespec tive of scale. Linear scale measurements include length of stream channels of given order, drainage density, constant of channel maintenance, basin perimeter, and relief. Surface and crosssectional areas of basins are length products. If two drainage basins are geometrically similar, all corresponding length dimensions will be in a fixed ratio. Dimensionless properties include stream order numbers, stream length and bifurcation ratios, junction angles, maximum valley-side slopes, mean slopes of watershed surfaces, channel gradients, relief ratios, and hypsometric curve properties and integrals. If geomet rical similarity exists in two drainage basins, all corresponding dimensionless numbers will be identical, even though a vast size difference may exist. Dimensionless properties can be correlated with hydrologic and sediment-yield data stated as mass or volume rates of flow per unit area, independent of total area of watershed. Introduction—Until about ten years ago the ..omorphologist operated almost entirely on a iescriptive basis and was primarily concerned with :he history of evolution of landforms as geological eatures. With the impetus given by Horton [1945], Bid under the growing realization that the classical iescriptive analysis had very limited value in .radical engineering and military applications, a few geomorphologists began to attempt quantifica:;ra of landform description. This paper reviews progress that has been made n quantitative landform analysis as it applies to normally developed watersheds in which running vater and associated mass gravity movements are ; 'he chief agents of form development. The treat ment cannot be comprehensive; several lines of •tudy must be omitted. Nevertheless, this paper . may suggest what can be done by systematic api proach to the problem of objective geometrical analysis of a highly complex surface. Most of the work cited has been carried out at I Columbia University over the past five years under j i contract with the Office of Naval Research, 1 Geography Branch, Project NR 389-042 for the 1 -tudy of basic principles of erosional topography. References cited below give detailed explanations "' techniques and provide numerous examples *^ken from field and map study. j Dimensional analysis and geometrical similarity— i "e have attempted to base a system of quantita:ve geomorphology on dimensional analysis and principles of scale-model similarity [Strahler, 1954a, p. 343; 1957]. Figure 1 illustrates the concept of geometrical similarity, with which we are pri marily concerned in topographical description. Basins A and B are assumed to be geometrically similar, differing only in size. The-larger may be designated as the prototype, the smaller as the model. All measurements of length between cor responding points in the two basins bear a fixed scale ratio, X. Thus, if oriented with respect to a common center of similitude, the basin mouths Q' and Q are located at distances r' and r, respectively, from C; the ratio of r' to r is X. In short, all cor responding length measurements, whether they be of basin perimeter, basin length or width, stream length, or relief (h' and h in lower profile), are in a fixed ratio, if similarity exists. All corresponding angles are equal in prototype and model (Fig. 1). This applies to stream junction angles a' and a, and to ground slope angles $' and /3. Angles are dimensionless properties; hence the generalization ■ that in two geometrically similar systems all corresponding dimensionless num bers or products describing the geometry must be equal. Studies of actual drainage basins in differing environments show that in many comparisons in homogeneous rock masses, geometrical similarity is closely approximated when mean values are considered, whereas in other comparisons, where geologic inhomogeneity exists, similarity is def.!••