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Mtm Quantitative Analysis of Watershed Geomorphology Mtm _■_______---! Vo|. 38, No. 6 Transactions. American Geophysical Union December 1957 • going into suspension , but part of the same Quantitative Analysis of Watershed Geomorphology mt in the low velocities kwater areas. Examples Arthur N. Strahler een noted close to river ere sand splays or dunes, Abstract—Quantitative geomorphic methods developed within the past few years provide t greatly different from means of measuring size and form properties of drainage basins. Two general classes of de ve been deposited on the 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 jam bed. While it is not ratios of length measures, whereby the shapes of analogous units can be compared irrespec is paper to evaluate the tive of scale. of the other bed-load Linear scale measurements include length of stream channels of given order, drainage manner, it is suggested density, constant of channel maintenance, basin perimeter, and relief. Surface and cross- as to the applicability of sectional areas of basins are length products. If two drainage basins are geometrically similar, bed-load and total-load all corresponding length dimensions will be in a fixed ratio. Dimensionless properties include stream order numbers, stream length and bifurcation sand-bed streams a great ratios, junction angles, maximum valley-side slopes, mean slopes of watershed surfaces, terial is carried into sus- channel gradients, relief ratios, and hypsometric curve properties and integrals. If geomet rical similarity exists in two drainage basins, all corresponding dimensionless numbers will -, it may be found practi- be identical, even though a vast size difference may exist. Dimensionless properties can be •ediment load of a water- correlated with hydrologic and sediment-yield data stated as mass or volume rates of flow erosion. Slope is a factor per unit area, independent of total area of watershed. rally recognized bed-load tree, 1955, p. 114]. Exten- Introduction—Until about ten years ago the principles of scale-model similarity [Strahler, 1954a, ccssary to determine the ..omorphologist operated almost entirely on a p. 343; 1957]. Figure 1 illustrates the concept of features as vegetative iescriptive basis and was primarily concerned with geometrical similarity, with which we are pri dumped rock dams, and :he history of evolution of landforms as geological marily concerned in topographical description. ould act as major dctcn- eatures. With the impetus given by Horton [1945], Basins A and B are assumed to be geometrically ice the slope, Bid under the growing realization that the classical similar, differing only in size. The-larger may be cd to gully erosion and iescriptive analysis had very limited value in designated as the prototype, the smaller as the ere direct methods are .radical engineering and military applications, a model. All measurements of length between cor heir quantities, estimates few geomorphologists began to attempt quantifica- responding points in the two basins bear a fixed ust be based upon past :;ra of landform description. scale ratio, X. Thus, if oriented with respect to a upon the best of the bed- This paper reviews progress that has been made common center of similitude, the basin mouths Q' therefrom. n quantitative landform analysis as it applies to and Q are located at distances r' and r, respectively, normally developed watersheds in which running from C; the ratio of r' to r is X. In short, all cor :_xces vater and associated mass gravity movements are responding length measurements, whether they ; 'he chief agents of form development. The treat ction of basic data on sedfc be of basin perimeter, basin length or width, stream on Water Resource;;, ///. ment cannot be comprehensive; several lines of length, or relief (h' and h in lower profile), are in a /, pp. 53-62. October 1951. •tudy must be omitted. Nevertheless, this paper fixed ratio, if similarity exists. ee, C. EL, Computations of . may suggest what can be done by systematic ap- All corresponding angles are equal in prototype Niobrara River near Cody, .!•• v. Paper 1357. 187 pp.. 1955. i proach to the problem of objective geometrical and model (Fig. 1). This applies to stream junction s-u. W. M.. Estimating bed analysis of a highly complex surface. angles a' and a, and to ground slope angles $' and phys. Union, 32, 121-12.5. Most of the work cited has been carried out at /3. Angles are dimensionless properties; hence the ussion of future of reser- I Columbia University over the past five years under generalization ■ that in two geometrically similar Civ. Eng., Ill, 1258-1262. j i contract with the Office of Naval Research, systems all corresponding dimensionless num 1 Geography Branch, Project NR 389-042 for the bers or products describing the geometry must be 1 -tudy of basic principles of erosional topography. equal. District, Tuisa, Oklalwnui References cited below give detailed explanations Studies of actual drainage basins in differing "' techniques and provide numerous examples \ 1. 1957; presented as pari environments show that in many comparisons in itershed Erosion and Sed»" *^ken from field and map study. homogeneous rock masses, geometrical similarity /-Seventh Annual Meeting. j Dimensional analysis and geometrical similarity— is closely approximated when mean values are 1. 1956; open for formal i "e have attempted to base a system of quantita- 58.) considered, whereas in other comparisons, where :ve geomorphology on dimensional analysis and geologic inhomogeneity exists, similarity is def- 05 MILE order bo./. -o.o- o 2 ORDER, F,G. 3 - Regression of number of stream segments on stream order; data from Smith (1953, Plate found bifurcation ratios of first-order to second- designating stream orders order streams to range from 4.0 to 5.1; ratios of , 1954a, p. 344) second-order to third-order streams to range from measure of the changing length of channel seg 2.8 to 4.9. These values differ little from Strahler's ments as order changes. Because this is a non ;i952, p. 1134]. etry through use of order linear variation, the assumption is implicit that 3 in drainage-network anal- Frequency distribution of stream lengths—Length )f stream segments of each of stream channel is a dimensional property which geometrical similarity is not preserved with in 1 by analysis of the way in can be used to reveal the scale of units comprising creasing order of magnitude of drainage basin. earn segments change with die drainage network. One method of—length Drainage basin areas—Area of a given watershed or drainage basin, a property of the square of analysis is the measurement of length of_____h length, is a prime determinant of total runoff or 'orlon's [1945, p. 291] law of segment of channel of a given stream order. For a sediment yield and is normally eliminated as a that the numbers of stream ■_iven watershed these lengths can be studied by variable by reduction to unit area, as in annual r form an inverse geometric frequency distribution analysis [Schumm, 1956, sediment loss in acre-feet per square mile. In order number. This is generally p. 607]. Stream lengths are strongly skewed right, to compare drainage basin areas in a meaningful ted data [Strahler, 1952, p. but this may be largely corrected by use of loga rithm of length. Arithmetic mean, estimated way, it is necessary to compare basins of the same p. 603] and is conveniently order of magnitude. Thus, if we measure the areas Figure 3. A regression of population variance, and standard deviation serve of drainage basins of the second order, we are of streams of each order as standards of description whereby different order (abscissa) generally measuring corresponding elements of the systems. drainage nets can be compared and their differ If approximate geometrical similarity exists, the ences tested statistically [Strahler, 1954b]. plot with very little scatter area measurements will then be indicators of the Relation of stream length to stream order—Still though the function relating size of the landform units, because areas of similar .ed only for integer values of another means of evaluating length relationships forms are related as the square of the scale ratio. in a drainage network is to relate stream length to .le, a regression line is fitted; Basin area increases exponentially with stream or regression coefficient b is stream order. A regression of logarithm of total order, as stated in a law of areas [Schumm, 1956, p. thm of b is equivalent to stream length for each order on logarithm of ratio rb and in this case has order may be plotted (Fig. 4). Again, the function 606], paraphrasing Horton's law of stream lengths. Schumm [1956, p. 607] has shown histograms of s means that on the average is defined' only for integer values of order. Several the areas of basins of the first and second orders e-half times as many streams such plots of length data made to date seem to and of patches of ground surface too small to have next higher order. yield consistently good fits to a straight line, but channels of their own. Basin area distributions are tt the bifurcation ratio would the general applicability of the function is not yet mensionless number for ex- established, as in the case of the law of stream strongly skewed, but this is largely corrected by a drainage system. Actually numbers. use of log of area. Area is measured by planimeter • stable and shows a small The slope of the regression line b (Fig. 4) is the from a topographic map, hence represents pro i region to region or environ- exponent in a power function relating the two jected, rather than true surface area.
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