AN ANALYSIS OF THE LAWS OF DRAINAGE NET COMPOSITION

Transcription

1 AN ANALYSIS OF THE LAWS OF DRAINAGE NET COMPOSITION L.E. MILTON Australia M.S. submitted February 1965 ABSTRACT Two of the geomorphic Laws of Drainage Net Composition are discussed and it is shown that they are essentially abstract mathematical relationships that hold for any bifurcating system. Some of the geomorphic factors that determine the value of the characteristic coefficients that are obtained by application of the Laws to single nets are also discussed. RÉSUMÉ Deux des lois géomorphiques de la composition des réseaux de drainage sont discutées et il est montré qu'elles sont essentiellement des relations mathématiques abstractes qui valent pour tout système à bifurcation. Quelques-uns des facteurs géomorphiques qui déterminent la valeur des coefficients caractéristiques qui sont obtenus par l'application des lois à des réseaux sont aussi discutées. GLOSSARY OF MATHEMATICAL SYMBOLS U Order number. N u Number of streams of order U. Lu Length of any one stream of order U. lib Bifurcation ratio = N u +i/n u. Lu Mean length of streams of order U_ in a_given net. RL Ratio of Mean Stream Lengths = L B+ i/l«. Zxj u+n Mean angle of junction between streams of order U and streams of order U+n. INTRODUCTION In recent years there has been a vigorous trend towards quantification in geomorphology and mathematical study of the relationships between landforms. A particularly fruitful field has been that of river patterns, generally called "drainage nets", and it is here that geomorphology becomes closely associated with hydrology (see Morisawa, 1962, and Farvolden, 1963). Horton (1945) gave the lead by defining the usable notion of stream order (later modified by Strahler, 1953), and formulating laws of "drainage composition" that relate stream order to net characteristics such as length of streams. This work has been followed by many others, and there is now a host of such laws. In this article, some of the laws of drainage net composition are analysed, and an attempt made to sift the geomorphic meaning from the mathematical equations that are used to express them. Two methods are used to investigate the laws. Firstly, a real drainage basin, the Miller Basin near Alexandra, Victoria, was analysed in detail (fig. 1). Having such data, it is possible to see how the laws can predict or at least approximate to reality. The second method is to analyse many random walk nets (Leopold and Langbein, 1962). Analysis of these completely artificial and random nets shows what properties of the laws are merely corollaries of statistical methods employed or expressions of probability, and comparison of random nets ana real drainage nets is used to illuminate the properties that are due to geomorphic factors. 51

2 Fig. 1 Drainage Net of the Miller Basin, Central Victoria (Milton, 1965). rq uz FTpj n F.""V-J-F LT! r i H i i r h h r ' r^ rii,tj t H-Hj q rtc [t[]f per; Fig. 2 A Random Walk Net. 52

3 THE LAW OF STREAM NUMBERS Horton (1945, p. 291) stated the following law: "The numbers of streams of different orders in a given drainage basin tend closely to approximate a geometric series in which the first term in unity and the ratio is the bifurcation ratio Rt>. " This law may be expressed mathematically by an equation of the following form (after Maxwell, 1955): Note that logio N u i + niu (1) R b =log^(-nx) The Law of Stream Numbers has been tested on drainage nets by many workers and found satisfactory. In the sixth order Miller Basin, for example, equation O has a correlation coefficient of 0.98 and a standard error of estimate (S v. x ) of only 0.10 log units. Three questions may be raised about equation ( x ): 1. Why is the form of the equation semi-logarithmic? 2. Why should the bifurcation ratio be a constant? 3. What determines the value of «i (i.e. Ri>)7 Leopold and Laogbein (1962) describe the preparation and analysis of random walk nets. They found that the Law of Stream Numbers holds true for such nets, and this has been verified by the writer in five different random walk nets. Leopold and Langbein stated, in effect (p.a16), that because the semilogarithmic relationship between order and number of streams holds even for random walk nets, it is one of optimum probability; i.e., in any net, the most probable distribution of first, second, third order streams, etc., is such that there will be a constant bifurcation ratio. This answers question 1 above; the law is a probability function, and such functions are typically semi-logarithmic. However, question 2 is not answered by the random walk net studies, but it is obvious that the solution lies more in the hands of a theoretical statistician than a geomorphologist. An important conclusion to be drawn from the random walk studies is that the Law of Stream Numbers is simply a statistical probability function that automatically follows the definition of order. It is basically not a geomorphic law, but an abstract law that must apply equally well to any branching system. The writer has in fact found that equation f 1 ) can be fitted to data from a plum tree, with a standard error of estimate of only 0.15 log units, which is similar to that found in the Miller Basin. Fig. 3A A Net having Bifurcation Ratio of 2. Fig. 3B A Net having Bifurcation Ratio of Infinity. 53

4 The third question is that of the value of the bifurcation ratio, Rb- It follows from Strahler's definition of stream order (1953) that the minimum possible value of Rb is 2, but theoretically there is no maximum (see fig. 3). Nevertheless, the value of Rb found in natural basins usually seems to be somewhere 2 and 5. Horton (1945, p. 290) claimed that the bifurcation ratio has a relatively low value in flat or rolling country, and a higher value in mountainous or highly dissected terrain, which suggests that the value of Rb is related to geomorphic factors such as relief ratio, drainage density, etc. Random walk nets are designed with constant drainage density and eliminate all geomorphic factors. The writer determined bifurcation ratios for five such nets, and found a range from 3.9 up to 5.9. This supports the hypothesis that in random walk nets the value of the bifurcation ratio is determined by probability factors alone. There must be an upper "limit" to Rt, in terms of very low probability that a certain arrangement of streams giving a high bifurcation ratio will appear during construction of a random walk net. If many random walk nets were prepared, then the distribution of values of Rb to be expected is that shown in figure 4. Fig. 4 Hypothetical Probability Curve for the Bifurcation Ratio in Random Walk Nets. Abscissa scale unknown. Fig. 5A A Dendritic Net Pattern. Fig. 5B A Trellis Net Pattern. In natural basins the values of Rb tend to be somewhat lower than in random walk nets. The effect of geomorphic factors, then, is to bring about a more frequent union of stream channels. In random walk nets the streams can flow in any direction, twist, and dodge around other streams, but in natural basins there are certain conditions that restrict such vagaries and so force more frequent union. The most obvious of these restrictive conditions is the net pattern. For any one net pattern the value of Rb is set by the angular relationships of the streams and the drainage density. For example, in the dendritic pattern illustrated in figure 5A, the 54

5 angular relationships of the streams are such that the value of Rt, is fairly low, and furthermore, Rt, cannot be increased unless either the stream orientation or the drainage density is altered. The net shown in figure 5B is typical of a steepsided valley or washout gully. Because the gradients of the tributaries incised in the valley walls are high and similar, these streams rarely unite and so the bifurcation ratio is high. Stream orientation and drainage density in turn appear to be controlled by many geomorphic factors such as runoff intensity, soil types, relative relief, and geological structure. Although there is much work in progress on drainage density, the question of stream orientation has not received much attention to date, from a quantitative point of view. THE LAW OF MEAN STREAM LENGTHS This law was stated by Horton (1945, p. 291): "The average length of streams of each of the different orders in a drainage basin tend closely to approximate a direct geometric series in which the first term is the average length of streams of first order. " This law is expressed mathematically by the equation: log 10 L k, = / 0 + / 1 l/ (2) The ratio of mean stream lengths, RL = log^1 h. This law has been teste_d by many workers and found satisfactory. In the Miller Basin equation ( 2 ) fits the data Li, L% Lz with a correlation coefficient of and a standard error of estimate (S y, x ) of 0.10 log units. However, the regression line fitted to the data L\ through L% has a correlation coefficient of only and a standard error of estimate of 0,285 log units. Leopold and Langbein (1962, p. A 19) found that the law holds for random walk net data, and this has been verified by the writer in five different random walk nets, with a maximum standard error of estimate of 0.23 log units. Following the same arguments as those presented for the Law of Stream Numbers, the conclusion is that the Law of Mean Stream Lengths is basically an abstract statistical probability function that applies to any bifurcating system. This explains the form of the equation, and the next problem is the value of RL. There is a minimum possible value of 2 for Rt,, but no analogous minimum exists for RL except that it cannot be zero. In random walk nets the length of a stream of order Uis a function of the probability that it will meet another stream of order U or higher as it grows longer. This explanation accounts for the form of the frequency distribution curve for stream lengths in a random walk net. In a dichotomous net (fig. 3A) there must be a very high probability that Lu+i, the length of a stream of order U+1, will be at least double L u because there are only half as many streams of order U+1. Over the whole net, then, RL^I- In nets having a higher Rt,, the fact that a stream of order U can terminate at junctions with streams of order U and greater reduces the most probable value of L u below that expected in a dichotomous net. Since for a first order stream there are more streams of higher order available than there are for second order streams, it follows that the most probable value of L\ is proportionally reduced more than that of L<>, and so RL is higher in complex than in dichotomous nets. The value of RL in random walk nets is thus related to the overall stream frequency of the net and to the amount by which the bifurcation ratio exceeds the dichotomy value of 2. In random walk nets there is no restraint on the direction of stream flow, and basins are free to interfinger in a complex manner. In the case of real nets, however, there are certain conditions such as the overall slope of the basin towards its mouth and the net pattern that restrict such vagaries and bring about more frequent union of streams and a correspondingly lower range of values of L u. It follows from geometric consideration that the more nearly perpendicular a stream of order U approaches one of order U or higher, the shorter will be L u. Therefore, the ratio of mean junction angles of first order streams with first and higher order streams (Zi i+ M ) to mean junction angles of second order streams influences the value of RL. According to Lubowe (1964) the junction angles of first order streams with streams of 55

6 order U increase with U because of the increasing ratio of gradients. A similar relationship holds for second, third order streams etc. Since first order streams have one more order of streams that they can enter than do second order streams, it follows that on the whole Zt i+«must be greater than Za i+ n, and hence RL must be > 1. The value of RL in natural basins is thus related to the overall frequency of streams, the bifurcation ratio, and the angular relationships between streams of different orders. SUMMARY Horton's Laws of Stream Numbers and Mean Stream Lengths represent the geomorphic application of abstract mathematical relationships that follow automatically from the definition of order and must apply equally well to any branching system. The value of the coefficients Rb and RL are controlled by geomorphic factors, and the influence of some of these factors has been explained in this article. REFERENCES FARVOLDEN, R.N. (1963), Geological Controls on Groundwater Storage and Base Flow, Journal of Hydrology, 1, pp HORTON, R.E. (1945), Erosional Development of streams and Their Drainage Basins; Hydrophysical Approach to Quantitative Morphology, Geol. Soc. Am. Bull., 56, pp LEOPOLD, L. B. and LANGBEIN, W. B. (1962), The Concept of Entropy in Landscape Evolution, U. S. Geol. Surv. Prof. Paper, 500-A. LUBOWE, J. K. (1964), Junction Angles in Dendritic Drainage Patterns, Am. Jour. Sci., 262, pp MAXWELL, J.C. (1955), The Bifurcation Ratio in Horton's Law of Stream Numbers, Trans. Am. Geophys. V., 34, p MILTON, L.E. (1965), M. Se. Thesis, Geology Department, University of Melbourne. MORISAWA, M.E. (1962), Quantitative Geomorphology of Some Watersheds in the Appalachian Plateau, Geol. Soc. Am. Bull., 73, pp STRAHLER, A.N. (1953), Revisions of Horons' Quantitative Factors in Erosional Terrain, Trans. Am. Geophys. U-, 34, p