Horton's Laws Research Articles

Application of Horton's first and second laws of branching to fungi

A total of 183 colonies representing 14 species of fungi were examined for their agreement with Horton's laws of branch number and of branch length. Most conformed to the law of branch number, an inverse regression of logarithm of branch number per order on branch order. Only a minority of the colonies conformed to the law of branch length, a direct regression of logarithm of mean branch length per order on branch order. More of the colonies were well described by the regression of mean branch length on branch order, and more still by the regression of mean branch length on logarithm of branch order. Branch ratios for colonies were found to vary widely within an isolate and the value of single determinations is questioned. The calculation of length ratios for growing fungal colonies is considered to be valueless. The significance of these findings is discussed in relation to stage of colony development and differentiation.

Does Horton's law of branch length apply to open branching systems?

Horton analysis has been applied to colonies of mycelial fungi and the mean branch lengths of branches per branch order tend to form a direct arithmetical, not geometric, series with branch order. Data for two botanical trees also show the arithmetical relationship. It is argued, from simulation models, that for branching systems showing outward unconstrained growth an arithmetical relationship results when linear growth rate of the elements is constant, but that a geometric relationship may result when linear growth rate shows a progressive decline.

ROAD DRAINAGE AND EQUILIBRIUM IN SMALL STREAM BASINS∗

Roads and road drainage accompany the development of the physical landscape for cultural purposes. The addition of a road network to a small stream basin creates a situation of geomorphic disequilibrium by adding to the length of surface drainage channels. Effects of road drainage on the geomorphic equilibrium of small stream basins are analyzed using Horton's Law of Stream Lengths. The effects of different types of road networks are analyzed using an hypothetical stream basin and eight hypothetical road networks, suggesting a method of predicting the location and extent of disequilibrium in a stream basin system as a result of road drainage.

ON EFFECTS OF MAP SCALE ON CHARACTERISTICS OF CHANNEL NETWORK

In this paper, the authors propose a method of standardizing a number of first-order streams and a drainage density for different map scales. Measurements of channel network are done at 8 drainage basins in the Kanto District, using topographic maps with the scales of 1:25, 000 and 1:50, 000 in order to analyze effects of map scale on characteristics of channel network. There are some differences between channel network drawn from 1:25, 000 and 1:50, 000 as shown in Figs. 1 and 2. Results of measurement at 8 drainage basins are shown in Table 1. The ratio of the number of first-order streams between 1:25, 000 and 1:50, 000 maps is about 2.0, and the ratio of the drainage density is about 1.4. The number of the first-order streams and the drainage density are smaller in the smaller scale map. On the contrary, the ratio of the average length of the first-order streams is about 0.7 and the ratio of the average drainage area of the first-order streams is about 0.5. The average length and the average drainage area of the first-order streams are larger in the smaller scale map. In general, however, the parameters of Horton's law are not affected by map scale (Fig. 3). The ratios of the average length of the first-order streams, the average drainage area of the first-order streams and the drainage density between different map scales are derived as a function of the ratio of the number of the first-order streams (Eqs. (9), (10) and (12)). The calculated ratios and the measured ones are summarized in Table 3. For analyzing the channel networks, it is important to indicate the map scale used in the analysis, and if one needs the comparisons between the values from the different map scales, the conversion by Eq. (14) could be possible.

ON THE CHARACTERISTICS OF THE CHANNEL NETWORKS IN JAPANESE DRAINAGE BASINS

This paper analyses the composition of channel network, especially parameters of Horton's laws, and discusses the characteristics of drainage basins in Japan. Basic data are collected from 180 partial drainage basins in the whole country and 19 full drainage basins in Hokkaido. Horton's laws of drainage net composition hold well in drainage basins of Japan as well as in drainage basins of other countries. Parameters of Horton's laws vary slightly from area to area and from geology to geology. Ranges of values of the bifurcation ratio, the drainage area ratio, and the stream fall ratio are about the same between the partial drainage basins and full drainage basins. There are some differences in the ranges of values of the stream length ratio and the stream slope ratio between the partial drainage basins and the full drainage basins. These values of the full drainage basins are greater than those of the partial drainage basins. It is considered that these differences are due to more frequent meandering of rivers in middle and lower parts of drainage system. The linear relation is found between the bifurcation ratio and the drainage area ratio, and the slope of its regerssion equation is nearly 1.0. It is interesting that the drainage area ratio is always about 0.4 greater than the bifurcation ratio. The average of the bifurcation ratio of drainage basins in Japan is 4.24, which is greater than a theoretical value of 4.0. The average of the stream fall ratio is about 1.0 for drainage basins in Japan. Drainage basins are considered, in general, to be in the stage of maturity in stream channel development in Japan. As the stream channel is liable to develop on an originally inclined land surface, the frequency of excess stream is greater in Japan than in other continental countries. Although the stream bed is nearly in dynamic equilibrium, it tends to be slightly aggraded, especially in fresh volcanic rocks areas.

The groundwater outcrop-erosion model; evolution of the stream network in The Netherlands

The stream network in the higher, pleistocene part of The Netherlands is genetically coupled with groundwater discharge systems of various extents. Streams of a given order can be explained as outcrops of groundwater flow systems of corresponding order. The drainage system is controlled by the precipitation surplus (climate), the resistance of the subsurface to groundwater flow (geology) as well as the previous relief and the incision depth (topography). This concept is defined as the groundwater outcrop-erosion model (GOEM). Stream nets can be synthesized theoretically from this model using geologic, geomorphologic and climatic data based on groundwater flow formulas and Horton's law on stream order vs. stream density.

GENERALIZATION OF THE FIRST HORTON LAW

GENERALIZATION OF THE FIRST HORTON LAW CHRISTIAN WERNER School of Social Sciences, University of California, Irvine THE CONCEPT of stream numbers, first introduced by Horton and later modified by Strahler, underlies one of the most widely used methods in the description of channel networks (Horton 1945, p. 281 ; Strahler 1952, p. 1 120). Not only does it provide a numerical (but incomplete) characterization of a network‘s topology, but it has also led to the observation and formulation of empirical regularities (e.g., the now famous “laws” established by Horton) and their formal explanation by means of theory construction (Shreve 1966, 1969; Smart 1968; Werner 1970). Nevertheless, the stream number concept is less than satisfactory for at least two reasons: 1 . The disaggregation of channel networks into sets of streams is not based on any geologic or morphologic reasoning, nor does the physiography of a drainage basin and its network lend itself to this type of analysis. Rather, the stream concept as defined by Strahler is a strictly mathematical notion from graph theory (Riordan 1958, p. 135). It is therefore not surprising that it is strongly related only to those concepts which are derived from it, i.e., to the length, slopes, and drainage areas of the streams. Conversely, it exhibits little relation to most of the geomorphic and hydrologic parameters traditionally used in drainage basin and network description (e.g., Anderson 1949; Morisawa 1962; Werner and Smart 1973, p. 293). 2. In determining the stream numbers of a given channel network, each individual stream has to be identified. From these raw data only the stream numbers are obtained, and the remaining information is not utilized. In particular, the stream numbers allow one only to conclude how many streams of a given order merge to form streams of the next higher order; most of the information contained in the data (e.g., how many streams of order i merge with streams of order j when i # j , and from which side in each case) is not used at all, although it holds the key to various generalizations of the Horton Laws and, possibly, to other topologic/morphometric relationships in drainage basins. This paper addresses the second of the two shortcomings of the stream number concept as described above. It offers a partial solution to the problem of data utiliza- tion and derives several interesting conclusions. More specifically: 1 . The use of the raw data is increased by establishing a more detailed description of the network under consideration. 2. Based on this detailed description, the paper gives additional insight into the mathematical structure of channel network topology and provides a new network classification scheme. 3. The First Horton Law is shown to be a special case of the realtionships charac- terizing the patterns of merging streams in topologically random channel networks. THE STREAM MERGER MATRIX OF A CHANNEL NETWORK For the purpose of this paper we define the stream merger matrix, M M , of a channel network of order m as the matrix ( n i j I i, j = 1,2, ..., m), where nfj is defined as the CANADIAN GEOGRAPHER, XIX, 4,1975

STATISTICAL LAW OF STREAM NUMBERS FOR SUBNETWORKS IN INFINITE TOPOLOGICALLY RANDOM CHANNEL NETWORKS

Werner proved by the combinatorial theory that the relation of stream numbers and orders of infinite topologically random channel networks obeys statistically Horton's law with the bifurcation ratio 4 (Werner, 1972). A problem, however, still remains how to adapt the conclusion to natural channel networks, the orders of which are given. It is the purpose of this paper to lead an equation to describe the statistical relationship of stream numbers and orders of subnetworks in infinite topologically random channel networks, the orders of which are given, introducing parameters different from the bifurca-tion ratio. The combinatorial analysis shows that the mean mel of the number of the ‘excess’ (cf. Smart, 1967) streams of order l merging directly into a stream of order m is 2m-l-1 in infinite topologically random networks. Therefore, mem-1=1=e1 and _??_ 2 =K. Using the values of rl and K, the following equation is finally deduced. _??_ where mμl is the mean of the number of the streams of order Z merging into the streams of various orders higher than l in a subnetwork of order m, which is a portion of infinite topologically random channel networks. The above equation is convenient to describe the statistical law of stream numbers for finite channel networks and can explain the fact that Norton's diagrams of many of natural channel networks are concave upward. And the probability S (l) of drawing a stream of order l at random from the streams com-prising an infinite topologically random channel network is given as follows : _??_ This quite agrees with the equation by Shreve (1967, p. 182).

Patterns of Drainage Areas with Random Topology*

Geographical AnalysisVolume 4, Issue 2 p. 119-133 Free Access Patterns of Drainage Areas with Random Topology* Christian Werner, Christian Werner Christian Werner is associate professor of geography, University of California, IrvineSearch for more papers by this author Christian Werner, Christian Werner Christian Werner is associate professor of geography, University of California, IrvineSearch for more papers by this author First published: April 1972 https://doi.org/10.1111/j.1538-4632.1972.tb00464.xCitations: 6 * The support of the National Science Foundation, Grant GS-2989, is gratefully acknowledged. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat LITERATURE CITED 1 Dacey, M. F. Probability distribution of number of networks in topologically random drainage patterns,” Water Resources Res., submitted, 1971. 2 Hall, Marshall, Jr. Combinatorial Theory. Waltham, Mass. (1967). 3 James, W. R. and W. C. Krumbein, “Frequency Distribution of Stream Link Lengths Journal of Geology, 77, (1969), pp. 544– 565. 4 Leopold, L. B., M. G. Wolman, and J. P. Miller. Fluvial Process in Geomorphology. San Francisco: W. H. Freeman and Co., (1964). 5 Riordan, J. An Introduction to Combinatorial Mathematics. New York: J. Wiley, (1958). 6 Riordan, J. Combinatorial Identities. New York: J. Wiley, (1968). 7 Shreve, R. Statistical Law of Stream Numbers,” Journal of Geology, 74 (1966) 17– 37. 8 Shreve, R., “Infinite Topologically Random Channel Networks,” Journal of Geology. 75, (1967) 178– 186. 9 Werner, C. Topological Randomness in Line Patterns,” Proc. Assoc, Amer. Geographers, 1 (1969) 157– 62. 10 Werner, C. Expected Number and Magnitude of Stream Networks in Random Drainage Patterns,” Proc. Assoc. Amer. Geographers, 3, (1971) 181– 185. 11 Woldenberg, M. J. Horton's Law Justified in Terms of Allometric Growth and Steady State in Open Systems,” Bull. Geol. Soc. Amer., 77 (1966) 431– 434. 12 Woldenberg, M. J. Spatial Order in Fluvial Systems: Horton's Laws Derived from Mixed Hexagonal Hierarchies of Drainage Basin Areas,” Bull. Geol. Soc. Amer., 80 (1969) 97– 112. Citing Literature Volume4, Issue2April 1972Pages 119-133 ReferencesRelatedInformation

STATISTICAL LAWS OF DRAINAGE NETWORK COMPOSITION

SYMBOLS AND DEFINITIONS m, l Order of basin or stream channel. mel Mean value of numbers of the l-th order streams joining directly to one m-th order tream, but, when (l+1)=m, the two streams that join to form the m-th order are excluded. mel (n; _??_) Weighted mean value of mel of topologically distinct stream networks of order _??_ with n first order streams. (Denominator is the total number of the m-th order streams.) mμl Mean value of numbers of the l-th order streams in one m-th order stream network. _??_am Mean area of the m-th order basins in a large _??_-th order basin. _??_Lm Mean length of the m-th order stream segments in an _??_-th order stream network. k' Constant. Horton's law of stream numbers is a valid statistical law to describe approximate relationships beween stream orders and numbers in both actual and randomly-generated stream networks. In a more exact sense, however, many networks exhibit certain systematic deviations from this law in Strahler systems. Many students have called attention to the fact that, in many cases, the set of points number versus order plotted on the semilogalithmic graph shows distinct upconcavity. In a previous paper (Tokunaga, 1966), an equation was proposed to modify Horton's law of stream numbers in order to satisfy this tendency. The following assumptions were then needed to construct the equation. Assumptions: 1. mem-1=m-lem-2=………=m-λem-λ-1=………=2e1=e1 mem-2=m-1em-3=………=m-λem-λ-2=………=3e1=e2 _??_ mel=m-lel-1=………=em-λel-λ=………em-l 2. _??_ The equation is given by the following expression: _??_ (1) _??_ (2) _??_(2)' The above assumptions were checked by populations of topologically distinct networks with the first order streams to be the most probable for given orders. Calculated values of mel (n; 5) are tabulated in Table 3. Further, these assumptions lead to the following equation for the relationship between basin order and area: _??_ (3) Assuming the next relationship between basin area and stream length._??_ (4) a law of stream length is written _??_ (5) Now, let m→∞ in equations (1), (3), we get log Q, as gradients of symptotes of these equations on the semilogalithmic graph. Consequently, Q means the bifurcation ratio and the basin area ratio and _??_ the length ratio of infinite networks. Because the fifth order network is the most probable, when n=171, putting e1=2e1 (171; 5) and K=3e1(171; 5) /2e1(171; 5), (2) becomes Q=4.03 consequently _??_=2.01 These values are in good agreement with the bifurcation ratio or the basin area ratio and the length ratio in infinite topologically random stream network by R. E. Shreve (1967).

MAP SCALE EFFECT ON THE STREAM ORDER ANALYSIS

It can readily be imagined that the result of morphometry of a given region will be different according to the scale of topographical map used in morphometry. Stream order analysis is attempted on the drainage basin of the Oguri River, a tributary of the Tama River. Four kinds of topographical maps of different scale (1:3, 000, 1:10, 000, 1:25, 000, 1:50, 000) are used. Drainage system maps are drawn from these maps respectively. The Strahler system of stream ordering is adopted and the whole basin is subdivided into 26 subbasins. The drainage networks of these subbasins are shown in Figures 1, 2, 3 and 4. The result of the stream order analysis obtained from each map system is tabulated on Table 1. Comparison of these figures enables us to recognize the effect of map scale especially at the head of stream network where the number of first order streams decreases with the de crease in map scale. As can be seen from Fig. 7, plots of the logarithms of the number of first order streams against denominator of map scale show a linear relationship between them. The regression equation for this line is expressed approximately in the form of LogN=LogK-LogS where K denotes the vertical intercept of the line. Although the Strahler's stream number assigned to a particular stream segment depend on the scale of topographical map used in ordering the stream network, the Horton-Strahler straight line for the stream number is found to be independent of map scale as shown in Fig. 8. The result obtained supports the Yangs' conclusion that the Horton-Strahler relationship obtained from topographical maps of different scales for the same stream system have the same bifurcation ratio of stream number. Concerning stream length, most of the stream net works of subbasins do not conform to the law of stream length. Sometimes in some basins, the higher order stream segments are shorter than they should be. It may be that, in this case, lack of agreement with the Horton's law is a result of the Strahler method of ordering. Comparison of actual stream system and that of the largest scale map suggest that stream system obtained from larger scale maps includes intermittent streams, which is considered to be less effective as an erosive agents. Therefore, maps of larger scales are not necessarily indispensable for stream order analysis.

Note on the Map Scale Effect in the Study of Stream Morphology

The Strahler stream order assigned to a particular stream segment depends on the scale of the topographic map used in ordering the stream network. However, the Horton‐Strahler bifurcation ratio of stream number, stream length ratio, and stream concavity are found to be independent of map scale. The coefficients in Horton's laws obtained from a map system of one scale can be transferred to a map system of another scale if the stream order difference between the two map systems is given. Yang's theoretical and equilibrium longitudinal stream bed profiles are shown to be independent of map scale and can be compared favorably with an actual profile obtained from a topographic map of any scale.

Potential Energy and Stream Morphology

Use of the analogy of entropy in thermodynamics reveals two basic laws which govern the formation of all stream systems. The first law is the law of average stream fall, which states that under the dynamic equilibrium condition the ratio of average fall between any two different order streams in the same river basin is unity. The second law is the law of least rate of energy expenditure, which states that during the evolution toward its equilibrium condition a natural stream chooses its course of flow in such a manner that the rate of potential energy expenditure per unit mass of water along this course is a minimum. This minimum value depends on the external constraints applied to the stream. The concavity of a river basin is shown to be the determinative factor in the formation of a stream system. On the basis of Horton's law and the law of average stream fall, longitudinal stream profiles can be calculated. The agreement between observed data and the theories found in this study is excellent.

DISCHARGE AND HYDROLOGICAL SIMILARITY OF DRAINAGE BASINS

Discharge is closely related not only to geological characteristics and vegetation of a basin but also depends upon geomorphic factors such as drainage area, stream length, stream number, stream order, basin shape, and ground slope. Hack plotted average discharge on a logarithmic graph for all gaging stations in the Pot-omac River basin and fitted a regression line with an exponent of 1.0. In general, an empirical equation relating stream discharge (Q) to basin area has shown to be Q=jAm……………………………………………………………… (1) where j and in are constants. The exponent in generally falls in the range from 0.5 to 1.0. For instance, in the case of annual mean discharge on the Potomac River basin, in is 1.0; it is 0.70 for flood discharge on twelve streams of New Mexico. On the other hand, the specific discharge of annual mean decreases exponentially with an increase of drainage area in Japanese river basins. This result agrees with the case of m≠1.0 in equation (1) and is different from the example of the Potomac River basin shown by Hack. It is supposed that non-linear relation of specific discharge to basin area is caused not only by areal localization of precipitation but also depends on irregularity of stream network. The purpose of this paper is to show the relationship between discharge and geomorphic characters of a basin such as stream order, stream length, and stream number, in terms of the concept of similarity. In general, assuming that the discharge is constant during the period of observation, the discharge (Qu) of drainage with order U is expressed as follows:_??_.……………………………………………………………… (2) where u is the stream order defined by 5trahler's ordering using a map on a scale of 1 50, 000. _??_u and Nu are mean length and number of stream with order u respectively. qGu and qsu are mean discharge of stream of order zc due to ground and surface water flows respecti-vely and these have the dimension of (m2/sec). Using Horton's first and second laws: _??_u+1/_??_u=γl, Nu+1/Nu=γb and, further assuming that the following ratios are the same in the whole basin :_??_ the equation (1) is transformed as follows: _??_+………………+_??_………………(3) Moreover, in order to generalize, if we substitute the characteristics of u-th order, l1, N1, and qau, for l1, N1, and qG1, in equation (3), the dischorge is rewritten as follows: _??_………………………………………………………………… (4) _??_ where _??_u is a proportional constant. This theory is supported by the observations of discharge carried out on two basins, the Kanna River basin and the Koshin River basin. The discharge was measured by the floating method at seven points along the Kanna River on May 2021, 1967 and at seven points along the Koshin River on June 16, 1969. As the weather had no precipitation during the observations, the discussions in this paper are the case of without the surface water flow in equation (4). Figs. 2 and 4 show the relation discharge to (_??_uNu) and the slopes of regr-ession lines are 1.0 in all cases.

A preliminary report on an empirical analysis of drainage network adjustment to precipitation input

In an analysis of the effect of precipitation on drainage network adjustment and on cumulative stream discharge, seven 4th-order drainage basins in each of two areas were investigated. Both areas are covered by glacial till and have similar precipitation intensities and thunderstorm frequencies, but one area receives approximately 30 percent greater average annual precipitation than the other. Values of certain geomorphic parameters were determined for each basin, and these values were compared statistically. As might be expected from considerations based on Horton's law, the number of segments per order per unit area of a 4th-order basin in the wetter area is systematically related to that in the drier area. The analysis indicates that a basin of a given order in the drier area contains more segments of every order than does a basin of the same order in the wetter area. But on a unit area basis more segments per order per unit area of 4th-order basins occur in the wetter area. Runoff data indicate that cumulative stream discharge per segment per order per unit area should be appreciably greater in the wetter area than in the drier area. On the basis of data collected from the two areas it appears that some variable other than precipitation, materials, evapotranspiration, and precipitation intensity is operative in the adjustment process. This variable may be initial relief.

Spatial Order in Fluvial Systems: Horton's Laws Derived from Mixed Hexagonal Hierarchies of Drainage Basin Areas

The number of stream basin areas (that is, the number of Strahler stream segments) is predicted on the basis of an hexagonal model for basin areas. It is suggested that the number of orders and the number of basins per order balance opposing tendencies for minimum overland work for streams flowing in small basins and maximum work savings in large, as opposed to small, channels. In addition, the entropy of the system approaches the maximum possible. The model agrees with empirical data in cases where the land surface is reasonably uniform with regard to structure and lithology. The necessity for approximate geometric progressions of basin areas with order creates geometric progressions of other basin parameters, leading directly to the power function relationships between variables, known as allometric growth. This model is identical to one proposed to explain the numbers of market areas per order in a system of cities. In addition, the model predicts the structure of bile ducts in one bovine liver cited in this paper.

MICROCANONICAL ENSEMBLES OF RIVER NETS

Abstract It is shown that a thermodynamic analogy can be set up for structurally cyclic as well as for noncyclic river nets that satisfy Horton's law of stream numbers. In the latter case, the microcanonical formalism of statistical thermodynamics has to be used.

Horton's Laws of Stream Lengths and Drainage Areas

Most customary explanations for Horton's metric laws of stream lengths and drainage areas are based on the idea that in river nets one can speak of cycles or generations of rivers. However, this customary explanation presupposes certain topological features in river basins that do not seem to occur in nature. Hence, the theory in which river nets are assumed to be simply random connections of channels is further applied. Upon this basis, a rational explanation of the two metric laws is obtained which is free from any specific structural assumption regarding a river net.

Statistical Properties of Stream Lengths

Two basic assumptions are employed in this treatment of the statistics of stream lengths: (1) All topologically distinct networks with a given number of sources are equally likely (after Shreve). (2) Lengths of interior links in a given network are independent random variables drawn from the same population. The mathematical development leads to an approximate expression for that contains no adjustable parameters and that depends only on the stream numbers and the mean link length. This expression gives somewhat better agreement with data on actual stream systems than does Horton's law of stream lengths; other advantages over Horton's law are cited. Quantitatively, our procedure appears to account for about 65% of the variance in mean stream length data for third‐ and fourth‐order basins. If exact values of the link numbers are introduced into the calculation, the unexplained variance is reduced to about 15%. For a complete statistical description of stream lengths, it is necessary to know the distribution of interior link lengths. Results of computer simulation studies suggest that this distribution is negative exponential. Data taken on two small watersheds (140 links) show reasonable agreement with this hypothesis.

Horton's Law of Stream Numbers

The possibilities for obtaining a rational explanation of Horton's law of stream numbers are reviewed. The previous explanations of the stream‐number law by (1) a growth model referring to generations of rivers and by (2) statistical graph‐theory are compared. The graph‐theoretical explanation seems superior to the other model of Horton's law, inasmuch as it produces not only the form of the law, but even, numerically, the naturally observed bifurcation ratio. It is also independent of the structural properties of the river net.