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CfnAAe-S'̂ 2 HYDROLOGICAL PROCESSES, VOL. 8, 125-135 (1994)

\ \ & A A ■

[NVARIANCE AND SCALING PROPERTIES IN THEDISTRIBUTIONS OF CONTRIBUTING AREA AND ENERGY IN

DRAINAGE BASINS

4&-%:

(Quebec)', ^

/ Disposal

sediment',

s\ J. Geo-

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P. LA BARBERAInstitute of Hydraulics, University of Genoa, Montallegro 1, 16145 Genoa, Italy

ANDG. ROTH

Institute of Hydraulics, University of Perugia, S. Lucia, 06100 Perugia, Italylr\yUL 5^ •

ABSTRACT» * " * * ^

(A>°The cumulative probability distributions for stream order, stream length) contributing area, and energy dissipation perunit length of channel are derived, for an ordered drainage system, from Horton's laws of network composition. It isshown how these distributions can be related to the fractal nature of single rivers and river networks. Finally, it isshown that the structure proposed here for these probability distributions is able to fit the observed frequency distributions, and their deviations from straight lines in a log-log plot.

key words Geomorphology Fractals Drainage basins

INTRODUCTIONMandelbrot (1982) suggests that dissipative systems with many spatial degrees of freedom, such as rivernetworks, will follow power law distributions of mass and energy. The cumulative probability distributionof these characteristics can therefore be written as

P [ X > x ) o c x ~ 0 ( 1 )where oc indicates the presence of a proportionality factor. Rodriguez-Iturbe et al. (1992a) describe theinvariance properties of this probability distribution for mass and energy in river basins under widechanges of spatial scales. Assuming the total cumulative area draining into a given site as a surrogatevariable for discharge, the values of the exponent 0 are explained on the basis (1) of the empirical relations h i p p r o p o s e d b y G r a y ( 1 9 6 1 ) , t h a t i s . .

L o c A r k & v J r \ \ t v O ^ ' ( 2 )

which, with a = 0-568, provides a very good fit between main stream length L, and basin area, A, and (2)assuming that the Euclidean length from the outlet to the most distant point in the boundary of the basincan be described as the first collision time of two fractal trails (Mandelbrot, 1982; Feder, 1988; Takayasuet al, 1988). From this, a value of 0 approximately equal to 0-45 and 0-90 is predicted for mass and energy,respectively, in good agreement with the observed values measured in five basins in North America. In fact,the exponent 0 in the power law distribution of mass is 'statistically indistinguishable among the differentbasins and approximately equal to 0-43', Rodriguez-Iturbe et al. (1992a).

In this paper the values assumed in Equation (1) for both the exponent (3 and the proportionality factor

CCC 0885-6087/94/020125-11© 1994 by John Wiley & Sons, Ltd. Received 16 September 1992

Accepted 7 January 1993

I126 P. LA BARBERA AND G. ROTH

are explained on the basis of a quantitative analysis of river networks by means of Horton's laws of network composition (Horton, 1932; 1945; Strahler, 1952; Shumm, 1956). The structures proposed here forthe cumulative probability distributions for mass and energy are able to fit the observed distributionsand their deviation from straight lines at large values of area. It is also shown that the value of 0 can belinked to the fractal structure of single rivers, with fractal dimension d, and river networks, with fractaldimension D. These measures are derived, for an ordered drainage system, from Horton's laws of drainagecomposition, as proposed and discussed in Mandelbrot (1982), La Barbera and Rosso (1987; 1989; 1990),Tarboton et al. (1988; 1990), and Rosso et al. (1991). Finally, with reference to the exponent fitted in theempirical Equation (2), it is shown that the expected value of 0 is 0-432.

HORTON'S LAWS AND THE FRACTAL STRUCTURE OF RIVERS AND RIVER NETWORKSHorton's laws of network composition are stated here in terms of Strahler's ordering scheme. The structureof a river network is therefore described as a system of streams which recognizes, through stream order, ahierarchy among the different branches. Strahler's ordering scheme postulates that: (1) source streams areof order 1; (2) when two streams of equal order join, a stream of one order higher is formed; and (3) whentwo streams of different order join, the continuing stream retains the order of the higher order stream. Theempirical laws of stream numbers and stream lengths (Horton, 1945) state that the bifurcation ratio, Rb,and the stream length ratio, R\, are constant within a catchment; the empirical laws of stream areas andstream slopes (Shumm, 1956; Strahler, 1952) states that the stream area ratio, Ra, and the stream sloperatio, Rs, are also constant. Denoting with u the order of a stream segment, these ratios are defined as

«,.,Kb =

' U i

R\

R*A *

Rs Sw+l

(3)

(4)

(5)

(6)

where ww is the number of streams of order to; lu is the mean length of streams of order u\ A0 is the meantributary area of streams of order w; and S^ is the mean slope of streams of order u.

Estimates of Rh, R\, i?a and Rs for a river network can be obtained from the slopes of the straight linesresulting from plots of the logarithmically transformed values of nUJ /w, Au and Su versus order u, for wranging from 1 to fi, the order of the basin.

Horton's laws are geometric scaling relationships which yield the self-similarity of the catchment streamsystem within a certain range of scales (Nikora, 1989); the following derivations should therefore beregarded as a mathematical descriptions which, applies at the range of scales associated with Horton slaws. Unfortunately, this range is not yet well assessed; a possible route in this direction could be fromthe description of the erosional development of drainage networks, on the basis of the results obtainedfrom field observations (Montgomery and Dietrich, 1988; 1989), laboratory experiments (Sawai et «..1986; Shumm et al., 1987), mathematical models (Roth et al., 1989; Willgoose et al., 1991) and stabilityanalysis (Smith and Bretherton, 1972; Loewenherz, 1991).

On the basis of the self-similarity described by laws of stream lengths and stream areas, Rosso et at.(1991) reported that rivers are fractal with a fractal dimension

1 log/?, (7)

The estimates of d obtained from Equation (7) fit the measured values satisfactorily and are close to

CONTRIBUTING AREA AND ENERGY IN DRAINAGE BASINS 127

(3)

(4)

(5)

(6)

value of 1-136 hypothesized by Mandelbrot (1982) under the assumption thatd = lex ^ P o < (8)

where a is the fitted exponent in Equation (2) between L and A.On the basis of the self-similarity described by the laws of stream numbers and stream lengths, La

Barbera and Rosso (1987) reported that the fractal dimension of river networks is given by>gRD = mm

/ log/\.b2'maT'bi7* (9)

The fractal dimension of a stream network, as estimated from Equation (9), can take values from two tounity for the combined ranges of Rb and R\ values observed in nature. Although it has been observed thatriver networks display varying values of fractal dimension D, this generally lies between 1-5 and 2, with anaverage of approximately 1-7.

The scaling properties of the river network as a whole can be viewed as the product of the structuralcomposition of the drainage system, reflected by D, and the fractal nature of river length, described byd. By introducing this source of fractal behaviour of individual streams in Equation (9), Tarboton et al.(1990) obtain the fractal measure

S = D d ( 1 0 )

and, by combining Equations (7), (9) and (10), Rosso et al. (1991) obtainlog/VD = min 2,2

'log*,(H)

At scales greater then the fundamental length scale given by the drainage density, it can be assumed thatthe network drains the whole river basin. This constrains it to be space filling with a fractal dimension£ = 2 (Mandelbrot, 1982; Tarboton et al., 1990). Assuming d = 2a = 1-136, the value D = "b/d- 1-761is predicted from Equation (10).

DISTRIBUTION OF CONTRIBUTING AREA

The cumulative probability distribution for contributing area (i.e. mass) is here derived from a generalapproach based on a knowledge of the distribution of stream order. This is obtained from the cumulativeprobability distribution of a site having an upstream total stream length larger than a given value. The totallength Zu of streams in a single subnetwork of order to < il is given by Eagleson (1970) as

zuj —1\ Rb- l

# - l (12)

in which /, is the mean length of first order streams. With u> = Q, Equation (12) gives the total length ofstreams for the whole catchment. The total length of streams in the subnetworks of order u) within a basinof order Q is therefore obtained as

Z l = Z w R % ~ u ( 1 3 )The cumulative probability distribution of a site having an upstream total stream length Z1 larger than

Zu> can be consequently written as

Z t i fi t , " 1if 1-1 1 -(«r AY(n-w*-i (14)

The total length Zl of streams in the subnetworks of a given order within a catchment of order Cl is

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CONTRIBUTING AREA AND ENERGY IN DRAINAGE BASINS 131

Table I Characteristics of the basins under analysis as evaluated from a digital elevation model. Areas are expressed inKm!; 0 and 0C are the calculated exponents in the cumulative probability distributions for contributing area andenergy, respectively

River

MagraEntella

Area n1600400

Rv

3-973-88

Rx

2102-11

R»4-254-15

0-470-49

1031-05

D

1-861-82

e1-92203

0-0-44-0-43

f t

-0-92-0-84

straight line with slope equal to

value of

mge ofto theited indimen-

1 to thewe canom themaliza-

(21)

sof thebed by

(22)

> - .B

0 - 1 -0-864 (24)

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Fig. 4. Drainage network for (a) Magra river basin and (b) Entella river basin

I132

4.5

P. LA BARBERA AND G. ROTH

5.0 5.5 6.0 6 . 5 7 . 0 7 . 5 8 . 0

Log (a) (a [sqm])

8.5 9.0 9.5

f .•■

Fie 5 Magra river basin: cumulative probability distribution and cumulative frequency distribution for (a) contributing area and (b)energy

ANALYSIS OF NATURAL RIVER NETWORKS

The cumulative frequency distributions for contributing area and energy can be obtained fromi the: analysisof natural river networks throughout the use of DEMs. Two different basins in Italy are analyses ^obtained from the interpretation of a DEM in a square grid with 225 m to a side. The drainage si ^the two basins is described in Figure 4, whereas Table I reports the relevant characteristics oI in ^works as obtained from the analysis of the network structure of the basins with no filtering proc • ^is equivalent to assuming that the channel maintenance area is equal to the pixel area V"jL 5 and 6.observed cumulative frequency distributions for contributing area and energy are given in i ig asdcrive<j;In the same figures the corresponding theoretical cumulative probability distributions are^ara , f .from Equations (17) and (23), with reference to the characteristics of the two basins reportea

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I134 P. LA BARBERA AND G. ROTH

distribution of energy can be linked to the fractal dimension of single rivers, river networks and to the ratiolog RJ logi?s.

ACKNOWLEDGMENTS

We gratefully acknowledge Franco Siccardi, University of Genoa, for his helpful advice during researchwork and for his comments on drafts of the manuscript, and Ignacio Rodriguez-Iturbe, Institute) Inter-nacional de Estudios Avanzados, Caracas, for his comments on some parts of the research. This workwas supported by Italian National Research Council grant 91.02600. PF42 under the framework of theNational Group for Prevention from Hydrogeological Disasters, GNDCI, and by the project MURST60% Misure di organizzazione delle reti di drenaggio.

! J

!

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Ro*!g'' ^ofi!lnd3sLardi, F. 1989. 'Hydrodynamic description of the erosional development of drainage patterns'. Wat. Re-

Sawal'K.! AsWda3, K^and Imamoto, H. 1986. 'Stream network evolution and sediment yield on a bare slope', 3rd International Symp o s i u m o n R i v e r S e d i m e n t a t i o n , J a c k s o n 3 1 M a r c h - 4 A p r i l 1 9 8 6 . - , , 6 7

ShVm. S- A. 1956. 'Evolution of drainage systems and slopes in badlands at Perth Amboy, New Jersey , Geol. Soc. Am. ou

Shumm64S6' A., Mosley, P. M., and Weaver, W. E. 1987. Experimental Fluvial ^^^^[^jj^^i1]^. Res.. 8.Smith, T. R. and Bretherton, F. P. 1972. 'Stability and the conservation of mass m drainage basin evolution , Wat. nuou

Strahler A. N. 1952. 'Hypsometric (area-altitude) analysis of erosional topography', Geol. Soc. Am. Bull., 63, lll7J.142, RerATakayasu, H., Nishikawa, I., and Tasaki, H. 1988. 'Power-law mass distribution of aggregation systems with inject.on ,r . ■

G e i T p h y s . , 3 7 , 3 1 1 0 - 3 1 1 7 . , _ - 2 4 1 3 1 7 - 1 3 2 2 .Tarboton, D. G., Bras, R. L., and Rodriguez-Iturbe, I. 1988. The fractal nature of river networks , Wat. ^sour^s..t . -Tarboton, D. G., Bras, R. L., and Rodriguez-Iturbe, I. 1990. 'Comment on: On the fractal dimension or stream nci

B a r l 5 e r a , P. , a n d R o s s o , R . ' , Wa t . R e s o u r . R e s . , 2 6 , 2 2 4 3 - 2 2 4 4 . p u n i „ , i 0 n m o d e l I . U * *Willggafe, G., Bras, R. L., and Rodriguez-Iturbe, I. 1991. 'A coupled channel network growth and hillslope evolution m

ory', Wat. Resour. Res., 27, 1671-1684.

APPENDIX

Distribution of contributing areaFrom the definition of contributing area ratio, Ra, we obtain

.:

CONTRIBUTING AREA AND ENERGY IN DRAINAGE BASINS135

Au. D-(n-w.) (ai:

and, consequently,- {n -u j . ) =

\og(A^/An)\ogRa

(A2)

It follows thati°EM.-./-<n>

Rl o g ( / l w. A l n ) l o g ^

logKs

= exp , (A,. , \ log*. ( . logflA

Ub -= exp

= exp l0gl^j "~log/?a

>-'>l0<i/ A n

-i(D-\

(A3)

and, finally, we obtain- ( fi - w. ) 4,,. -f(Z)-l) (A4)

■

Distribution of energyFrom Equation (Al) and from the definition of the slope ratio, Rs, we obtain

^ ^ = ( R M - { n - ^AnSn

(A5)

and, consequently,. t o g j A ^ S ^ / A a S n ) ( A 6 )-(n-w.) = — log(^s)

Following the same procedure used to derive Equation (A3), with 9 defined in Equation (22), we obtain, r A - ( « - > _ ( A u J ± X * ( D - l ) & ( A 7 )

AnSnJn