WATER RESOURCES RESEARCH, VOL. 32, NO. 11, PAGES 3367-3374, NOVEMBER 1996

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OriljH-ack's lawRiccardo Rigon,1-2 Ignacio Rodriguez-Iturbe,1 Amos Maritan,3Achille Giacometti,4 David G. Tarboton,5 and Andrea Rinaldo6

Abstract. Hack's law is reviewed, emphasizing its implications for the elongation of riverbasins as well as its connections with their fractal characteristics. The relation betweenHack's law and the internal structure of river basins is investigated experimentally throughdigital elevation models. It is found that Hack's exponent, elongation, and some relevantfractal characters are closely related. The self-affine character of basin boundaries isshown to be connected to the power law decay of the probability of total contributingareas at any link and to Hack's law. An explanation for Hack's law is derived from scalingarguments. From the results we suggest that a statistical framework referring to the scalinginvariance of the entire basin structure should be used in the interpretation of Hack's law.

1. IntroductionHack [1957] demonstrated the applicability of a power func

tion relating length and area for streams of the ShenandoahValley and adjacent mountains in Virginia. He found the equation

L = 1 . 4 4 0 ' 6 ( 1 )

where L is the length of the longest stream in miles from theoutlet to the divide and A is the corresponding area in squaremiles. Hack also corroborated his equation through the measurements oi Langbein [1947], who had measured L and ,4 fornearly 400 sites in the northeastern United States. Gray [1961]later refined the analysis, finding a relationship L « /I0'568.Many other researchers have corroborated Hack's originalstudy, and, although the exponent in the power law may slightlyvary from region to region, it is generally accepted to beslightly below 0.6. Equation (1) rewritten as L t*Ah with h >0.5 is usually termed "Hack's law."

Midler [1973], on the basis of extensive data analysis ofseveral thousand basins, found that the exponent in Hack'sequation was not constant but that it changed from 0.6 forbasins less than 8,000 square miles (20,720 km2) to 0.5 forbasins between 8,000 and 105 square miles (20,720-259,000km2), and to 0.47 for basins larger than 105 square miles(259,000 km2).

As Mesa and Gupta [1987] point out, Midler's empiricalobservations are not consistent with the implications of therandom topology model of channel network structure first in-

'Department of Civil Engineering, Texas A&M University, CollegeStation.Permanently at Dipartimcnto di Ingcgneria Civile e Ambientale,Universita di Trento, Trento, Italy.3Istituto Nazionalc di Fisica della Materia, Scuola Internazionale

Superiore di Studi Avanzati and Istituto Nazionale di Fisica Nucleare,Trieste, Italy.

4Dipartimento di Scienze Ambientali, Universita di Venezia, Ven-ezia, Italy>^^

5Departmenf of Civil and Environmental Engineering, Utah StateUniversity, Logan. "^6Istituto di Idraulica "G. Poleni," Universita di Padova, Padova,Italy.Copyright 1996 by the American Geophysical Union.Paper number 96WR02397.0043-1397/96/96WR-O2397$O9.O0

traduced in the classic paper of Shreve [1966]. In fact, theytheoretically derived the value of. Hack's exponent, h, for therandom topology model of channel networks as

M») = 21 /ir+ (irlnY

rr - 1/n (2)

where n is the basin's magnitude. Equation (2) implies a continuously decreasing h(n) with an increasing n. For n =10,100, and 500 the exponent h(n) is 0.68, 0.530, and 0.513,respectively. When n tends to infinity, h tends to the asymptotic value of 0.5. This result makes clear the importance of themagnitude of the network in the exponent h under the premises of the random topology model. Further and more general resultS/Ojt__andom trees can also be found in work byDurret et K [1991}/

The classit^rexplanation for the exponent h being largerthan 0.5 was to conjecture that basins have anisotropic shapesand tend to become narrower as they enlarge or elongate. Thehypothesis of basin elongation was verified by Ijjasz-Vasquez etal. [1993] under the framework of optimal channel networks(OCNs), which are the result of the search of fluvial systemsfor a drainage configuration whose total energy expenditure isminimized [Rodriguez-Iturbe et al., 1992a; Rinaldo et al., 1992].Thus Hack's relationship may result from the competition andminimization of energy in river basins.

Mandelbrot [1983] suggested that an exponent larger than0.5 in L « Ah could arise from the fractal characters of riverchannels which cause the measured length to vary with thespatial scale of the object. Thus (1) would be a reflection of afractal dimension of river channels close to d, = 2 X 0.6 =1.2[e.g., Feder, 1988]. However, as explained in detail by Feder[1988], to derive the fractal dimension of a river from theprevious argument, one needs to assume that the shapes of theriver basins are self-similar. This assumption does not seem tobe verified in nature [Ijjasz-Vasquez et al. 1994; Nikora andSapozhnikov, 1993].

Peckham [1995] derived a relationship between Hack's exponent h and a fractal dimension, say DSST, of a self-similartree as h = 1/DSst- In tne caSG of a space-filling tree, DSST =2 and h = 0.5, while for DSST ̂ 2 Hack's exponent h can behigher. Similar arguments connecting Hack's exponent to theglobal fractal characters of the network have been proposed byNikora and Sapozhnikov [1993].

One goal of this paper is to suggest a new statistical uiter-

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RIGON ET AL.: ON HACK'S LAW 3369

<f,t smaller than 1.14 and 1.2, respectively, for elongation tooccur. Higher values of <p,_ will imply contraction of riverbasins when increasing with size.

The relationships derived above were first tested in riverchannel networks extracted from digital elevations models(DEMs) using standard procedures [e.g., Jenson andDomingue, 1988; Tarboton et al., 1991]. The basins used in theanalysis are given in Table 1.

Figure 2a shows the result of Hack's analysis. While the bestestimate of the slope is close to the commonly found value of0.6, wejoticejhat the detailed study of the morphology of thefluvialb^sirisaUowed_byJDE_^^ the analysis- of thestatistjjal^flucluaJlQJ^-jiejent in Hack's type~of diagrams.These fluctuations were neither considered in the originalwork by Hack nor in the experimental studies following it.From Figure 2b we have

L.cc.4' (15)which gives a Hurst exponent H = 0.92. The relationshipbetween L and Ln is presented in Figure 2c:

L « Ll (16)

implying a fractal scaling exponent for channel sinuosity ofcbL = 1.15. With cbL = 1.15 and H = 0.92 in (14), one obtainsh = 0.60 that matches the value obtained by direct fitting inFigure 2a.

Thus assuming self-similar shapes and neglecting elongationwould have resulted in h = <pJ2 = 0.575, and taking streamsas nonfractal would have resulted instead in h = 1/(0.92 + 1)= 0.52. In both cases one underestimates the value of h. Themain contribution to the exponent h comes from fractal sinuosity, with a smaller contribution coming from elongation.

The above analysis gives standard results and also clarifiesthe conditions which define geometrical elongation. Nevertheless, the requirement that the shape coefficient s is a constantis too restrictive to be fulfilled exactly. To account for the

Table 1. Geometrical Data Relative to the River BasinsAnalyzed

River Basin LocationA,km2

L,km km

GuyandottcLittle CoalTug Dry ForkJohnsBig CoalRacoonPingeonMoshannonBrushyRock CastleSturgeonIslandWolfSextonBigIndianB i g ^Schoarie Creek HeadwaterPoundTipton Run.Blackberry Fork

W.Va.W.Va.W.Va.Ky.W.Va.Pa.W.Va.Pa.Ala.Ky.Ky.W.Va.Ky.Ky.Ky.W.Va.IdahoN . Y.Ky.Pa.

20889845864844494464053933253102952602121861521481461081056453

145.190.563.768.856.253.649.149.752.445.946.330.530.333.330.241.021.518.523.518.818.0

75.857.541.645.740.735.235.333.829.933.527.123.021.723.421.128.316.915.817.812.513.3

Data were obtained by processing U.S." Geological Survey 7.5"D E M s . \ ° *

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Figure 2. (a) Hack's law for the basins in Table 1. (b) Elongation for the same set of data as in Figure 2a. An exponentgreater than 0.5 means that basins elongate, (c) Stream lengthversus diameter for the basins in Table 1.

natural variability of river network structures, it is necessary torelax the deterministic definitions in favor of statistical ones.This will be pursued in the following sections, together with ageneralization of Hack's law that preserves the validity of (14).

3. Hack's Law and Scaling ArgumentsHack's original work dealt with nonoverlapping basins. In

this section we will start with the assumption that a Hack-typerelationship is valid for the different subbasins embedded in alarger basin. Thus a statistical framework is developed for the

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3372 RIGON ET AL.: ON HACK'S LAW

Table 2. Values Relative to the Rivers in Table 1 WhoseArea Is Larger Than 200 km2River Basin <t>L h H (*_/*) - B y 0//i

Guyandotte 1.06 0.56 0.92 0.89 0.43 0.74 0.77Little Coal 1.06 0.56 0.92 0.89 0.41 0.74 0.73Tug Dry Fork 1.07 0.54 1.00 0.98 0.41 0.77 0.76Johns 1.06 0.60 0.75 0.77 0.40 0.67 0.67Big Coal 1.05 0.56 0.89 0.88 0.42 0.75 0.75Racoon 1.02 0.52 1.00 0.96 0.45 0.89 0.86Pingeon 1.06 0.55 0.92 0.93 0.45 0.75 0.82Moshannon 1.02 0.52 0.96 0.96 0.45 0.86 0.85Brushy 1.04 0.54 0.96 0.93 0.43 0.82 0.80Rock Castle 1.06 0.55 0.92 0.93 0.43 0.79 0.78Sturgeon 1.03 0.52 1.01 0.98 0.46 0.92 0.88Island 1.07 0.54 0.96 0.98 0.43 0.78 0.80Wolf 1.07 0.55 0.92 0.96 0.41 0.75 0.75

P[X > /] « p[Ah > /] = P[A > /'"•] (31)

Here <£_, sinuosity exponent; h, Hack's exponent; H, Hurst's exponent; B, power law exponent for areas; y, power law exponent forlengths. Data were obtained after extracting the drainage networksfrom DEMs using the threshold A, = 50 pixels as the maximumcontributing area necessary to maintain a channel. The exponent h wasestimated by fitting (_?)_. versus the contributing area in a log-log plot.The value of H was obtained from (__,)_ ~ _<I///>+1 where (_?„>_ isthe expected value of diameters with the contributing area, a. Incolumn 5, (14) is used to estimate H from the values of cf>L and hwhose values are reported in columns 2 and 3, respectively. In column8, y is estimated according to (31) being B and h as given in columns6 and 3, respectively.

This argument, (29), and (30) clearly show that the self-affine characteristics (e.g., Hurst exponent) of the basin boundaries are intimately related to Hack's exponent and the distribution law of contributing areas.

Now consider the random variable defined as the distance,i_, measured through the network from a randomly chosenpoint to the boundary of the basin. The exceedence probabilitydistribution of this distance may be derived from Hack's lawand the power law distribution of contributing areas.

Calling _6 the random stream length to the divide, we have

where /3 is defined in (28) and h is Hack's exponent, assumedhere to hold also for the internal structure of a basin. Thisholds for £ = 1 in (22). The power law of lengths defined in(31) has been verified in numerous basins with excellent fittingthroughout several logarithmic scales. It links the aggregationpattern of areas with the stream length structure of the network and Hack's law.

Table 2 shows a synthesis of the experimental analyses conducted on the DEMs of the basins in Table 1 with area largerthan 200 km2. The exponent y falls between 0.70 and 0.90 forvalues of /3 between 0.41_to 0.46 and h between 0.52 and 0.60with an average of 0.55. Tne average exponents of the probability distributions are 0.79/ for the lengths and 0.43 for totalcontributing areas.

Figures 6 and 7 show.examples of P[X £_ /] for the riverbasins whose DEM features have been described in Table 1. Inparticular, Figure 6 shows the detailed results obtained for theBrushy Creek basin where w_\have used a support area of 50pixels to define the drainage networks and the lengths aremeasured in kilometers. In this case, the exponent of the powerlaw of lengths is 0.8, the value of /3 is 0.45, and the value of hresulting from (31) is 0.56. Figure 7 shows the same results forthe Little Coal River basin.

. o4

500.1000.

5 1 0 . 2 0 . 5 0 . 1 0 0 .

Stream Length [km]

Figure 6. (a) Brushy Creek with support area 50; (b) P[A > a ] versus a (0 ~ 0.45); (c) P[S£ > L] versus L(y-0.82).

RIGON ET AL.: ON HACK'S LAW 3373

1000.

0 . 5 1 5 1 0 .

Stream Length [km]

50. 100.

Figure 7. (a) Little Coal River with support area 50; (b) P[A > a] versus a (0 0-41); (c) />[__ > L]versus L (y ~ 0.74).

The consistency of the measured exponents suggests that thescaling theory developed in this paper is a comprehensivedescription of the planar organization of basins.

5. ConclusionsThe results of this paper suggest the following conclusions:

1. Hack's law appears to hold, in a statistical sense, forany point inside a basin.

2. The statistical framework developed here predictsthat the ratio of consecutive moments of the distribution ofstream lengths to the divide from points with the same contributing area should follow a power law as a function of areawith the exponent being close to Hack's h. This prediction isconfirmed in natural basins.

3. Hack's exponent, the self-affinity of the river basin,and the fractal sinuosity of its channels are related by H =4>Jh - 1. For elongation to occur, H needs to be smaller than1. This imposes constraints between Hack's exponent and thefractal characters of rivers. The values of h and <\>L obtained innatural river basins yield// < 1 and thus confirm the tendencyto elongate.

4. Hack's relationship and the self-affine character ofbasin boundaries are related. The values obtained for H and</>_ in real basins yield excellent agreement with the commonlyaccepted>alues of h.

5. The exponents /3 and h are found to be dependentvariables through /3 -f h. = 1. Estimations of /3 and h innatural basins tend to confirm this dependency.

6. The stream length from any point to the divide followsa power law distribution whose exponent, y, is the ratio of theexponent of the power law describing the contributing area at

any randomly chosen point, /3, and Hack's exponent, h. Analyses of DEMs yield an excellent fit to the power law for thestream lengths and moreover confirm the relation between y,/3, and h.

Acknowledgments. This work was supported by C.N.R. "GruppoNazionale per la protezione dalle catastrofi idrogeologiche," Linea 3,Progetti MIEP-METEORAIN, by funds provided by the R. P. GregoryChair in Civil Engineering at Texas A&M University, and byM.U.R.S.T. 40% "Processi Idrologici fondamentali." Their support isgratefully acknowledged.

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A. Giacometti, Dipartimento di Scienze Ambientali, Universita diVenezia, D. D. 2137 Venezia, Italy.

A. Maritan, International School for Advanced Studies and IstitutoNazionale di Fisica Nucleare, Trieste, Italy. \

R. Rigon and R. Rodriguez-Iturbe, Department-of Civil Engineering, Texas A&M University, CETTI Building, Rm. 201, College Station, TX 77843. (e-mail: rigon@csrp.tamu.edu; m9280@zeus.tamu.ed_)

A. Rinaldo, Istituto di Idraulica "G. Poleni," Universita di Padova,1-35121 Padova, Italy, (e-mail: rinaldo@idra.unipd.it)

D. G. Tarboton, Department of Civil and Environmental Engineering, Utah State University, Logan, UT84322-4110.

(Received March 4, 1996; revised July 30, 1996;accepted August 2, 1996.)