£

York, .McGraw-Hill Book Co., v. 2,

1%3, Structural investigations in th..fountain Land, eastern Transvauiiuth Africa Trans., v. 66, p. 353 -in;'rogressive deformation in tectonicmerican Geophys. Union Trans., v106.olding and fracturing of rocks: NY...:aw-HUl Book Co., 568 p.965, The tectonics of the main goldrea of the Barberton Mountain Land., Econ. Geology Research Unit, L'niv.md, Johannesburg, 23 p.0. R.; Joubert, G. K.; Sohnt,,.,Van Zyl, J. S.; Roussouw, P. j.;J. J.; and Visser, D. J. L., 1956, Th,the Barberton area: Geol. Surveya Spec. Pub. 15, 253 p.J., 1964, The geology of the I.ihad portion of the Eureka Synclinr.> Consort Mine and Joe's Luck Siding,Mountain Land: Unpub. M.Sc. thesi>.i-atcrsrand, Johannesburg.1935, Bcitrag zur Schotteranaly?,-Mineralog. Petrog. Mitt., v. 15, p

^ ■ G ^ b p *

STREAM LENGTHS AND BASIN AREAS IN TOPOLOGICALLYRANDOM CHANNEL NETWORKS1-2

RONALD L. SHREVEUniversity of California, Los Angeles, California 90024

ABSTRACTIn order to comprehend the geometry of drainage basins and channel networks, which is prerequisite to

explaming their mechanics, it is necessary to understand the close connection between network topoloev andfuch planimetnc elements as stream lengths and basin areas. Topologically random channel networks constitute an important theoretical case. In an infinite topologically random network, (1) the expected mae-nuutlc of a randomly drawn link of order a is (2^ + l)/3,.(2) | of all links and . of the interior linkshead streams (3) complete subnetworks of any given order have the same distribution of magnitudes as allnetworks of hat order, which explains why it seems to make little difference whether or not basins chosen°rT^_Xfre c-?-T'mX' ,(4) t]?e .Probability that a randomly drawn stream of order « will consist of\ links 1S 2 * (1 -2 £ «)»i and g) the average stream of order Q will have 2Q-* - 1 tributaries enter-,nB from the sides, of which 20-—* Wlll on the average be of or(]er u Unk . h_ measured Qn ,.24000-scale maps of eastern Kentucky are approximated more closely by a gamma density with parameter v = 2than by either a log normal or an exponential. The mean length of the exterior links is almost twice that ofthe interior links, in agreement with the findings of others. In an infinite topologically random channel network whose interior link lengths are gamma distributed with „ = 2, (1) the densities of the logarithms of the,iream lengths arc slightly left-skewed and increase in dispersion with increasing order, and (2) the densitieso.| Te °8arllm"? of lhe. Schumm total length of all streams in a subbasin of given order are highly symmetrical, decrease slightly in dispersion with increasing order, and for orders 1 and 2 agree well with Schumm'sobservations at Perth Amboy, \ ew Jersey. If the link lengths are exponentially distributed, then the streamlengths will also be exponent* Iy distributed. In finite topologically random channel networks with specifiedstream numbers, the expected number of links per stream and per subnetwork of given order are given;.y closed but complicated formulas, from which the expected stream lengths and Schumm lengths can becalculated. As observed in natural networks, in a topologically random population, (1) the expected stream.cngths do not satisfy Horton's geometric-series law as well as the expected stream numbers, whereas theexpected Schumm lengths do, (2) on Horton diagrams the curves are straightest for the most probable networks, and are concave upward or downward for networks of order lower or higher, respectively, than themost probable and (3) the geometric-mean stream-length and Schumm-Iength ratios are about 2 and 4 5

ffKrtS; 1 CKrr^Pin$n! \°SUh f°r basLn areas are exactly the same •* for Schumm lengths. InhPerth) Amboy basin studied by Schumm, the average area draining directly overland into unit length£??£_ £ "_ ♦_ C'°nSvT mdePfcnd.ent °.f i])9 Particular position of the channel in the basin as he proposed,but is less for exterior links than for interior links. Substantial differences in the relative areas draining intosources and in other gcomorphic characteristics are not reflected in Horton diagrams because the Horton

Sore ^n,TZ\SS "" "^ '£72ned by f^"0* t0p°logy* 0f most fundamental signif icance,therefore, are not the Horton variables, but the more elementary quantities, such as link lengths and source

INTRODUCTIONThe way in which the planimetric ele

ments of a drainage basin are put togetheris intimately connected with the topologyof the channel network. Indeed, certain.xomorphological relationships, such as thelaws of drainage composition proposed byHorton (1945, p. 286-291), apparently are

1 -Manuscript received September 30, 1968; revised January 16, 1969.

^Publication 709, Institute of Geophysics andPlanetary Physics, University of California, LosAngeles, California 90024.

Jwrxal of Geology, 1969, Vol. 77, p. 397-414]0 1969. The University of Chicago. AU rights reserved.

in large part the result of randomness innetwork topology (Shreve 1966, 1967;Smart 1968). Thus, in order to comprehendthe geometry of drainage basins and channelnetworks, which is prerequisite to explaining their mechanics, it is necessary to understand the perhaps dominant topologicaleffects. The purpose of this paper is to investigate these effects in the importanttheoretical case of stream lengths and basinareas in topologically random channel networks.

DEFINITIONSAll of the specialized geomorphological

terms used in this paper have been denned397

398 RONALD L. SHREVE

in previous publications (Shreve 1966, p.20, 22, 27; 1967, p. 178-179). Strahlerstream orders will be used throughout(Strahler 1952, p. 1120; Shreve 1966, p.21-22). The term channel network will referto idealized networks with no lakes, noislands, and no junctions with more thantwo tributary channels. A complete networkis one that discharges into a stream of higherorder. Sources are the points farthest upstream in a network; their existence as objectively describable points in natural networks is a fundamental assumption of thetheory. Exterior links are the reaches ofchannel from the sources to the highestforks; and interior links are the reaches between the forks. The magnitude of a link isequal to the number of sources upstream ofit; and the magnitude of a channel networkor drainage basin is equal to the magnitudeof its outlet link. A link drawn at random isone selected in such a way that all links inthe specified target population are equallylikely to be drawn. An infinite topologicallyrandom channel network is a network ofinfinite extent in which all topologicallydistinct subnetworks of equal magnitudeoccur with equal probability. Finally, theexpectation, or expected value, of a randomvariable will, as usual (Feller 1957, p. 207;1966, p. 5), signify the mean, or first moment, of its probability density.

PROBABILITY DENSITIESLet p(fi, co) be the probability that a

link drawn at random from an infinitetopologically random network will havemagnitude n and order a>. Then p((i, c.)satisfies the recursive relationship

p - i/>(M, ") = _I_:i>(a,w- 1)

0=1

X p{n - a, to - 1)_ - l

+ 2/>(a,W)__/>(M - a,j8)] ,

p( \ , 1) - . , p (u , \ ) =0,

p(l,a) = 0 , M, w = 2, 3, . . .

(Shreve 1967, p. 180); and therefore thegenerating function (Feller 1957, p. 248)

(1)

M = l

must satisfyC J - l

Ru = __?_,__ + ./?_, / ]Ra(8=1

2__ = \r , « = 2, 3, .

(2a)

(21

The marginal probability (Feller 1957,201)

«(") = _Ci>0*, ») , (3i»i=i

which is the probability of drawing at ndom a link of order o> and unspecified ma„nitude, and the expectation (Feller 1957,207)

E_(y) = f> pip., «) /«(«) , (31

which is the expected value of the mafnitude /x for links of order o> drawn at ranidom, can be found from (2b) without firs!finding explicit formulas for it_ or p(n, o>)jbecause from (2a)

« ( _ > ) = R u ( l ) ( 3 c jand

where Ri = dR^/dr. Following convention(Mood and Graybill 1963, p. 45), y is bold-jface to indicate that it is a random variableand not, as in p(n, oj), the argument offunction.

Rewriting (2b) with the index o> decreaseby 1 and eliminating the sum betweenoriginal and the rewritten equations leacto

R_ \ A _ _ i /

X ( & #

(4a

By induction, using (3c) and the fact thai

•W = _C*G«, w) r"

fy

(2a

P = i ( 2 bi - ir , w = 2, 3, ... .

inal probability (Feller 1957, !>•

< i - i

ie probability of drawing at ran-of order co and unspecified mag-

1 the expectation (Feller 1957, p.

= _Lm Pin, »)/*(«) ,u = i / (31,

he expected value of the ma".>r links of order co drawn at ran-)e found from (2b) without firsi'licit fonnulas for i?_ or p(u, _• ,»m (2a)

u(u) = i.u(i) [dc

(3(

- dRu'dr. Following conventionGraybill 1963, p. 45), y is bold-

cate that it is a random variables in p(u, a), the argument of a

g (2b) with the index co decreasedliminating the sum between thed the rewritten equations lead;

x(fe + *»(4a I

STREAM LENGTHS AND BASIN AREAS

967, p. 180); and therefore _ncr, function (Feller 1957, p. 248)

/e,(D = I((i) = -_-from (2b),

«(_> —

and _._(!) = «(2) =

1)«(co)

2 = 0 (4b)

but

hence,hence,

b(») = 1/2- , u = l, 2, ... , (4c)

an important result that was derived by adifferent method in a previous publication(Shreve 1967, p. 181).

Solving (2b) for _?_, differentiating, substituting in (3d), and using (4c) leads to

£*(») = 1 ;

. = £ 1/2*̂ -1 = § .

399

(6c)

(fid)

«(«) = Z/Km, co) , (3a, E_ = 2E__. + J>-#^ ,

>n, using (3c) and the fact that

(5a)Ei = 1 , a «. 2, 3,

Rewriting this with the index co decreased by1 and eliminating the sum in the samemanner as before then gives

4E__, - E_ = 4EW_, - E___ . (5b)

By induction, using the fact that E, = 1 andE. = 3 from (5a),

Thus, 3 of all links and £ of the interiorlinks head streams. Finally, substituting(fid) in (6a) and (6b),

.(«) - 3/4" , a, = i. 2, . . . , (6c)a formula that was obtained by a less revealing method in a previous publication(Shreve 1967, p. 182).

Let c(n, a,) be the probability of drawingat random from an infinite topologicallyrandom network a link that is the outlet ofa complete network of magnitude /x andorder a. This is the same as the probabilityof drawing a link with one tributary of magnitude u and order oj and the other of unspecified magnitude and order co or greater.Thus,

hence,4E___ - E_ = 1 ;

E_(_0 = (2*-i + l)/3 ,co = 1, 2, ... .

(5c) c(*, w) = H2PQh ")__>03)J ; (7a)

(5d)

Let s(w) be the probability of drawing atrandom a stream of order co from the streams(not the links) comprising an infinite topologically random network; and let q be theprobability of drawing at random a linkthat heads a stream. Then, because s(u) isthe same as the probability of drawing atrandom a link of order co from the linksthat head streams, a _(_,) is the probabilityof drawing at random an interior link withtwo tributary links of order co - 1. Thus,using (4c),

aSfw) = |[M(_j _ i)p __ 1/2^-1co = 2, 3, . . .

'Shreve 1967, p. ISO) and

qs(l) = k(1) - |;

(6a)

(6b)

and, using (4c) and summing the series,

c(n, co) = pfa ia)/2r-* . (7b)The significance of this result is that complete networks of any given order in aninfinite topologically random network havethe same distribution of magnitudes as allnetworks of that order. This explains whyit seems to make little difference whetheror not basins chosen for geomorphologicalinvestigation are complete.

The probability that a link drawn atrandom will be the outlet of a completenetwork of order co is

__>(,_, co) = 1/22""!A« = l

« - 1, 2, . . . ,

(8)

which, as should be expected, is identicalto the probability of drawing a link thatheads a stream.

..

400 RONALD L. SHREVELet /(A; co) be the probability that a

stream of order co drawn at random from aninfinite topologically random network willconsist of X links; and let qu be the probability that a link of order co drawn at randomwill be the terminal link of a stream, that is,the outlet link of a complete network. Then,using the rule for conditional probabilities(Feller 1957, p. 105) and substituting (4c)and (8),

_« = T,c(ft,w)/u(co) = l/2r-i. (9a)

In topologically random networks the factthat a particular link is not a terminal linkin no way influences the probability thatthe next link downstream will be one;therefore,/(X; co) is geometric,

/"(X; co) = (7.(1 - <7_,)*-i

= 2-<«-»(l - 2-<«-«)x-1 , (9b)

X , co = 1 ,2 , . . . .

The expected number of links in streams oforder co is (Feller 1957, p. 210)

E_(:\) = 1 + (1 - qm)/qm = 2»-i . (9c)

Thus, the expected number of links perstream increases with order as a geometricseries with ratio 2, as found previously by adifferent argument (Shreve 1967, p. 184).

Let T(co; ft) be the probability that astream chosen at random from the streamsthat debouch into the sides (but not theheads) of the streams of order ft in an infinitetopologically random network will haveorder co; and let /(co, ft) be the probabilitythat a link drawn at random will have tributaries of order co and ft, where co < ft. Then

2X«;0) - /(co,ft)/__>(/3,ft) (10a)

but, using (4c),/(co, ft) = _[2„(co)"(ft)] = l/2n+"

co = 1, 2, . . . , ft - 1 (10b)

ft = 2, 3, ... ;

hence,

r(co; ft = 2°-«-y(2Q-1 - 1) ,co = 1, 2, ...,__- 1, (lo,

ft = 2, 3,

According to (9c) an average stream of ord«ft in an infinite topologically random nelwork will consist of 2n_1 links andtherefore have 2Q_1 — 1 tributaries dijcharging into it from the sides; and accoiing to (10c) 2n~<--1 of these tributarieson the average be of order co, as suggest*by Smart (personal communication).

LINK LENGTHS

Practically nothing is known aboutstatistical distribution of the lengthslinks in natural channel networks. Schui(1956, p. 607-608) concluded frommeasurements in badlands basins at PerAmboy that the distribution of first-ordestream lengths, that is, of exterior liilengths, is log normal. He did not repoiinterior link lengths. M. A. Melton (personscommunication), on the basis of unpub-jlished studies of various basins in the west-]ern United States, has reached the sameconclusion concerning both exterior ancinterior links. Smart (1968, p. 1011-1012)measured link lengths on maps of basins iiMissouri, Virginia, and Arizona. He founcdistributions qualitatively similar to thosof Schumm and Melton, but concludefrom a goodness-of-fit test that, except fo2the paucity of very short links, the distributions could be exponential, as suggested b)computer simulation (Smart et al. 1967}using the model proposed by Leopold ancLangbein (1962, p. A18). M. J. Kirby (per_sonal communication) has found that linlmeasured in the field fit an exponentialdistribution fairly well, but that thosemeasured on maps fit a distribution more]like the log normal.

\V. C. Krumbein and I, with the help ofstudents in a class, have studied map linllengths in thirty networks selected ajrandom from the drainage basins of magni-'j

__ 20-»-J/(2n-i - 1) ,

1,2, . . . , f t- 1 , (10cj9. = 2, 3, ... .

9c) an average stream of orderte topologically random nc-t-•isist of 2"-1 links and will•e 2"~l — 1 tributaries dis-it from the sides; and accord-;;;-_-i 0f these tributaries will:e be of order co, as suggestedsonal communication).

LINK LENGTHS

nothing is known about the;tribution of the lengths ofal channel networks. Schumm17-608) concluded from hisi in badlands basins at Perththe distribution of lirst-orck-riis, that is, of exterior linkg normal. He did not report•ngths. M. A. Melton (personalin), on the basis of unpub-of various basins in the west-

states, has reached the samejncerning both exterior and. Smart (1968, p. 1011-1012.); lengths on maps of basins inginia, and Arizona. He foundqualitatively similar to thoseand Melton, but concluded

less-of-fit test that, except forf very short links, the distribu-e exponential, as suggested bynulation (Smart et al. 1967)xlel proposed by Leopold and'62, p. A18). M. J. Kirby (per-.nication) has found that links

the field fit an exponentialfairly well, but that thosemaps fit a distribution more

lormai.imbein and I, with the help ofi class, have studied map linkthirty networks selected ati the drainage basins of magni-

STREAM LENGTHS AND BASIN AREAS 401tude 10 tributary to Rockhouse and Wolf.reeks, Martin County, eastern Kentucky.Local relief is a few hundred feet, anddrainage density is about 10 miles of channelper square mile. Bedrock is flat-lying relatively homogeneous coal-bearing Pennsyl-vanian sandstone. Modern 1:24,000-scaletopographic and geologic maps publishedby the U.S. Geological Survey completelycover the area. Sources and channels weredrawn on the maps by eye on the basis ofcontour curvature, and link lengths were

4 0

measured as the straight-line distances between the ends of the links. The resultinghistograms (fig. 1) are right-skewed likethe log normal, in agreement with thefindings of the previous investigators.

Interestingly, although the two histograms have the same general form, the meanlength of the exterior links is almost twicethat of the interior links. A similar differencehas been reported by Melton (personal communication) and Smart (1968, p. 1012) andis implied by the published data of Schumm

30<>_;.

_a 20

10

EXTERIOR LINKSMean = O. II ml.

o -*-oooJ l - J I I I I I L - L __1 I I I I I I I

005 0 10 0 1 5 0 . 2 0 0 . 2 5Link length (mi.)

0.30 0.35 0.40

4 0

30

_-<b 20

10

r̂ n.INTERIOR LINKS

Mean -0.056mi.

T.

-JIi i i i i i i i i i i L i 11111 ' ' ■ V ' I I ' ■

0 . 0 0 0 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5Link length (mi.)

0.30 0.35 0.40

ric. 1.—Histograms of map link lengths in drainage basins in Martin County, eastern Kentucky

1 " ■U

■■■ nr ■■—

402 RONALD L. SHREVE

(1956, p. 607-608) and Melton (1957,table 2, facing p. 88). Thus, the first-orderstreams, which consist of exterior links,must be considered separately from thehigher-order streams, which consist of interior links, and the conclusion from (4c)that in infinite topologically random networks the average length of streams increases with order as a geometric series

INTERIOR LINKS

4 0

3 0

DjI-Jb 20

10

I O b s e r v e dExpected iflog-normal

X0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5

Link length (mi.)0.20

Fig. 2.—Comparison of observed distribution oflink lengths from figure 1 with log normal fittedusing maximum-likelihood estimators of parameters(Aitchison and Brown 1957, p. 39).

with ratio 2 (Shreve 1967, p. 184) must berestricted to streams of second and higherorder.

The log normal density (fig. 2) is butone of many possible right-skewed densitiesthat might fit the histogram for interiorlinks in figure 1. Another possibility is oneof the family of gamma densities

_,._,(*.) - a',-/:--1 exp (-a,/,)/l» , .^v, a, > 0 ,

in which v, which is not necessarily aninteger, is a shape parameter and

a . = > / » ( / . ) , ( l i b )where E(f.) is the expected (Feller 1966, p.5) interior link length. Of the two densities,

the gamma with v = 2 (fig. 3) fits bettethan the log normal, although neither &[very well. Moreover, it is mathematicalconvenient and it reduces to the exponentiidensity in the special case v = 1 (fig. 4). F^these reasons, therefore, and not becausany particular geomorphological sijcance, (11a) with v = 2 and a,- independeof order will be assumed to be a reasonajiapproximation to the truth in the deirjtions that follow.

STREAM LENGTHS

The lengths Z_, of streams of ordergreater than 1 are the sums of randjnumbers _■. of links of random length'where *_•» and li have the densities/(X; co)

INTERIOR LINKS

4 0

30

o?20

10

J L _ O b s e r v e dExpected ifgamma, v =2

Ki i i 1 1 i 1 1 i iHfiL.1 i i i i Ft-1-

0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5Link length (mi.)

Fig. 3.—Comparison of observed distributionlink lengths from figure 1 with gamma with paraters v = 2 and a,- = 2/?,-, where i\- is the obsermean length. Maximum-likelihood estimatorsparameters (Mood and Graybill 1963, p. 181; CIman 1956, p. 499) are v = 2.02 and a. = v/h.

g,.aXli) given by (9b) and (11a). Theytherefore random variables with densities

*.- £/(*;«).?.*.. (12;

(Feller 1966, p. 53), where the superscrijX* signifies the X-fold convolution of g,\

Th v = 2 (fig. 3) fits betterirmal, although neither fitseover, it is mathematicallyit reduces to the exponential▶ecial case v = 1 (fig. 4). Fornerefore, and not because of

geomorphological signifi-.h v = 2 and a, independentassumed to be a reasonableto the truth in the deriva-

REAM LENGTHS

Z_ of streams of order _are the sums of random

links of random length /„nave the densities/(X; co) and

NTERIOR LINKS

! O b s e r v e dExpected ifgamma,v=2

\

=tq_i_L___l_J_ J__J_J_____;. 0 5 0 . 1 0 0 . 1 5 0 . 2 0Link length (mi.)

parison of observed distribution offigure 1 with gamma with parame-

n = 2 li. where /, is the observe!laximum-likelihood estimators oid and Graybill 1963, p. 181; Chap-) are v = 2.02 and a» = vfli.

by (9b) and (11a). They areom variables with densities

= E/(X,«)_,\*-, (12a)

p. 53), where the superscriptle X-fold conyolution of _»-e.

STREAM LENGTHS AND BASIN AREASwith itself (Feller 1966, p. 144). Althoughthey occur in a similar context, the densities/' and g are not the same as those definedby Smart (1968, p. 1005). Substituting (9b)and (11a) into (12a), using the fact that,?,,«*&.« = £,+„.„ (Feller 1966, p. 46) andletting /_ = 1 - 2-o-<> gives

//_ = PZ1+1"(1 -p„)m exp (-«.£_)

xZ(^a1i_)--1/r(x,), (12b>in which T is the gamma function (Dwight1961, p. 209). For v = 2 the infinite series isthat for the hyperbolic sine; hence, finally

I, (I ) = a." exp (--...__)»\ ■*>/ 2"*-'(l - 2"<"-'"))l/2

403The densities //_ are shown in figures 5

and 6. The broad, relatively flat distributions in figure 5, particularly for thehigher orders, explain the common observation (Smart 1968, p. 1007) that the lengthsof the single main streams in basins studiedusually seem abnormal compared to theaveraged lengths generally computed forthe lower orders.

If the link lengths are exponentially distributed, as proposed by Smart and Kirkby,then v = 1, and from (12b) (or Feller 1966^

X sinh {a,Z_[l - 2-(u-1']1/2}

/ - „ > ( ) , c o = 2 , 3 , . . . .

(12c)

£ 0.0_ 0

INTER/OR LINKS

[ _ , O b s e r v e dExpected ifexponential

' 2 3 4 5 6 7Stream length L^/Efl/J

Fig. o.—Densities of stream lengths in infinitetopologically random channel network if interiorlink lengths are gamma-distributed with v = 2 Thequantity E(/,) is the expected (or overall populationmean) interior link length.

0 0 0 0 0 5 0 1 0 0 . 1 5Link length (mi.)

0.20

Fig. 4.—Comparison of observed distribution oflink lengths from figure 1 with exponential fittedusing maximum-likelihood estimator of expectedlink length.

Alternatively, this result could be found byLaplace transforming (12a) (Feller 1966,P- 414), making a partial-fraction decomposition, and then using a standard table ofinverse transforms.

° - 5 I 2 4 8 1 6 3 2Stream length L^/Ed/J

Fig 6—The densities of figure 5 transformed sothat the horizontal coordinate is the logarithm ofstream length, in accordance with common practicein plotting observed lengths.

p. 54) the stream lengths will also be exponentially distributed (figs. 7 and 8). Unlikethe gamma, the exponential density for linklengths has some heuristic appeal because ofits "lack of memory." This property is perhaps best exemplified by radioactive decay.The remaining lifetime of a radioactive nucleus is a random variable that is indepen-

RONALD L. SHREVE

dent of the present age and has the samedistribution as the lifetime itself. Only theexponential has this property (Feller 1966,p. 8). In the case of links, knowledge thata link exceeds a certain length would giveno information as to how much it exceeds it.Put another way, if such a link is traversedin equal infinitesimal increments of length,the probability of termination is the samein each increment. Such a hypothesis isattractive because of its simplicity. Moreover, the computer experiments of Smartet al. (1967) show that the link lengths in

I 2 3 4 5 6 7 8Stream length L^/Eflj)

Fig. 7.—Densities of stream lengths in infinitetopologically random channel network if interiorlink lengths are exponentially distributed.

the discrete model of channel networksproposed by Leopold and Langbein (1962,p. A18) arc geometrically distributed (Smart1968, p. 1010-1011), which suggests thatin the continuous limit they might beexponentially distributed.

On the other hand, the hypothesis thatlink lengths are exponentially distributed isnot so closely connected with symmetry requirements >as the hypothesis that channelnetworks are topologically random, andtherefore is not so immediately plausible.Also, the model of Leopold and Langbeinis not geomorphologically very realistic.Real channel networks generally grow head-ward, and, unlike their counterparts inthe model, compete for drainage area,adjust to the competition (Schumm 1956,p. 617-622), and sometimes even gain orlose whole blocks of territory by capture.Tints, it is not obvious on a priori grounds

alone that the distribution of link lengthsshould be exponential, or indeed what itshould be. Nor, for that matter, is it obviousthat the distribution of interior link lengths;should be independent of order and mag-jnitude. Clearly, careful field measurements!in real channel networks are badly needed, i

SCHUMM LENGTHS

Unfortunately, //_ cannot be compared]with the histograms published by Schumm](1956, p. 607, 610) because the streailengths of given order as defined by him (p.;604) are not the mean lengths of individual!streams of that order but are instead the!mean total lengths of all the streams in sub-jbasins of that order. Thus, Schumm's stream!lengths include exterior links, so that finding]the densities corresponding to lua is con-]siderably more difficult.

0 . 5 I 2 4 8 1 6 3 2Stream length Lw/E/l/)

Fig. 8.—The densities of figure 7 transformedso that the horizontal coordinate is the logarithm ofstream length. The curves all have the same shape;the modal lengths increase with order as a geometricscries with ratio 2.

For subnetworks of magnitude n, the]Schumm length is the sum of ii exterior link]lengths and \j. — \ interior link lengths. In jan infinite topologically random network,therefore, its density is

hi = tp(u, *)&\*gt%*/<<*) > (Ua):co = 2, 3, ... ,

assuming, for lack of better information,that both exterior and interior link lengths

fstribution of link lengthsential, or indeed what it>r that matter, is it obvioustion of interior link lengthsindent of order and mag-careful field measurementsictworks are badly needed.

UMM LENGTHS

, //_ cannot be comparedams published by Schumm610) because the streamorder as defined by him (p.mean lengths of individualorder but are instead thehs of all the streams in sub-ler. Thus, Schumm's streamxtcrior links, so that findingirresponding to //co is con-difficult.

co = 3

1 2 4 8 1 6 3 2earn length L^/Ed,-)densities of figure 7 transformednta-l coordinate is the logarithm ofe curves all have the same shape;increase with order as a geometric

yorks of magnitude u, thei is the sum of /x exterior link— 1 interior link lengths. In^logically random network,.ensity is

o)A*^rA(w) ' (13a)« = 2, 3, . . . ,lack of better information,

jrior and interior link lengths

STREAM LENGTHS AND BASIN AREAS

are gamma-distributed with the same v andexpectations v/ae and v/a,. Not only is (13a)more complicated than (12a), but also nogeneral explicit formula for p(n, co) or itsbivariate generating function is known, although univariate formulas for successivevalues of io can be calculated from (2b). Forw = 2, the density p(ii, co)/«(co) is particularly simple, namely, from (1) and (4c),

p(ji, 2)/u(2) = l/2*-» ,ix = 2,2,, ... ,

which is a geometric density. Substituting(13b) into (13a), letting v = 2, and Laplacetransforming the result gives

405right-hand side of (13c) can be decomposedinto partial fractions,

L(Aft - (s + aey ""*" s + atC \ . c 2

S + rx s+.ft

C 3 , C i

(131)

s + rs s + r4 '

where, unless ae/ai satisfies (13e),

a = -al , b = 0 , (13g)c-i = ialaV[(ae - n)2(r2 - r_)

^ - Z M j r d ' X ( r , - r 0 ( r 4 - r , ) ] ,

X a,- 2(>-l)

and Co, <-.-,, and e_ are given by cyclic permutation of the numerical subscripts in the

, (13c) last formula. Thus, finally, using a standardtable of inverse transforms,

= frlal/Ks + aS-Ks + a,

(13d)

X (s + at)2 - hcCa-]} ,in which, following custom, s denotes thevariable in the frequency domain. Thequartic in square brackets in the denominator is the difference of two squares; hence,it can be factored into the product of twoquadratics, which in turn can be factoredby means of the quadratic formula. Theresult is(s + rk) = s + Ua* + ai)

± |[(o. - a,-)2 ± 2\/2afa,]1/2,where the four factors corresponding tothe subscript k are given by the four possiblecombinations of plus and minus signs. Notethat r„. will be complex for certain values ofk« and a,-. For a., a,- > 0 these factors aredistinct from each other and from s -f- a,except when"«/a. = (1 + y/2)

(13e)± V2(l + v/2)1-'2 ,that is, when ae/ai = 0.22 or 4.6. Hence, the

tfOfi - -alLf exp (-«_£?.4

+ go exp (-,__.) , (13h)

£2, a_, a,- > 0 ,a_/a, ^ (1 + V2)

± V2(i + V2Y'2.The expectation of this density is

E(lf) = 3E(/„) + 2E(/,) , (13i)in agreement with (5d).

By the same process, using (2b), thedensity of third-order Schumm lengths is

Jh(lJ) = -ojll exp (-a.L3S)•i

+ Z>i exp (-r*__f)* - i

(14a)_

+ ___tf_ exp (-tkLf) ,

£3, a-e, a; > 0 ,

and c«, r.-i, r4, <7_, __, . . . , d-, and d8 are givenby cyclic permutation of the numerical sttb-

Schumm's histograms in figure 11. Tlagreement between the calculated andobserved histograms is highly encouraginiEspecially interesting is the lower dispersicand higher symmetry of the higher-ord«densities compared to the lower-order on^in figures 10 and 11; it suggests the conj<

8 1 6 3 2Schumm length L.%/E(l,)

Fig. 10.—The densities of figure 9 transformcso that the horizontal coordinate is the logarithm _Schumm total stream length. Increasing the ratk]E('«)/E(/,) merely shifts curves to right.

— Observed — Theoretical

Second-order

E__l

Schumm length Li, /E(lf)

FiG. 9.—Densities of Schumm total streamlengths in infinite topologically random channel network if exterior and interior link lengths are gamma-distributed with v = 2 and expectations E(le) andE(Z.), where E(._)/E(Z.) = 1 (dark curves) andE(Z_)/E(/.) = 2 (light curves).

scripts in the last two formulas. The expectation of density (14a) is

E(Z-f) = HE(*g + 10E(/,-). (14d)The densities £.,_„ //_-', and k§ are shown

in figures 9 and 10, and are compared with

1 0 2 . 0 3 . 0Log Schumm length (in ft.)

Fig. 11.—Comparison of theoretical and ob-1served histograms of Schumm total stream length-for the Perth Amboy, New Jersey, badlands basinstudied by Schumm (1956, p. 607). First-order]theoretical density assumed gamma with v = 2;second-order calculated for infinite topological^random channel network assuming interior linllengths also gamma-distributed with v - 2. E:pected exterior and interior link lengths were takeequal to the observed means.

ture that, as a consequence of the central-*]limit theorem (Mood and Graybill 1963, p.149-150), the densities of the higher-orderlengths may be approximately log normalindependently of the specific densities oflink lengths and subbasin magnitudes.

The observed mean interior link length/,- needed as the estimator of E(/,) in com-

ms in figure 11. Thethe calculated and the3 is highly encouraging.g is the lower dispersiontry of the higher-orderto the lower-order ones; it suggests the conjec-

co = 2 oj = 3

2 4 8 1 6 3 27 length L%/E(l,)

sities of figure 9 transformedcoordinate is the logarithm oflength. Increasing the ratio

ts curves to right.

— Theoretical

0 2 0 3 . 0>umm length (in ft.)

rison of theoretical and ob-Schumm total stream length

,-, New Jersey, badlands basin\ (1956, p. 607). First-orderassumed gamma with v = 2;tted for infinite topological!)twork assuming interior link.-distributed with v = 2. Ex-interior link lengths were taken•d means.

onsequence of the central-lood and Graybill 1963, p.nsities of the higher-orderapproximately log normalf the specific densities ofsubbasin magnitudes,mean interior link lengthestimator of E(i\) in coni-

STREAM LENGTHS AND BASIN AREAS 407

pitting the theoretical histograms of figure11 was calculated from the data of Schumm11956, p. 606, table 2) using the equation

L = Ule+ (M- 1)&I (15)

where 7. and L are the mean first-order andthe total stream lengths in a basin of magnitude u (tables 1 and 2). Using these estimators in (13i) gives an expected second-order Schumm length of 43.2 feet, whichcompares well with the average (for forty-

1966, p. 21-22). In order to avoid theambiguity inherent in Horton's system,Strahler (1952, p. 1120) proposed the slightly different system used in this paper, whichis now the one most commonly used. Unfortunately, stream lengths defined byStrahler's system do not fit Horton's lawvery well (Strahler 1952, p. 1137; 1957, p.915; Broscoe 1959, p. 5; Maxwell 1960, p.23; Bowden and Wallis 1964, p. 769-770);hence, modification of the content of the law

TABLE 1

Measured Lengths and Areas*

Basin n L(Ft)

/*_(Ft)

A(Ft') (Ft*)

Chileno Canyon, CalifMill Dam Run, Md

214296150

3.53X1032.54X10*3.97X10*

10.1482.

1420.

3.10X10'8.61X10**1.54X10"

85.01.67X10*7.81X10*

* Data from Schumm (1956, p. 606, table 2).

TABLE 2Derived Lengths, Areas, and Ratios

Basin (Ft)j .

(Ft*) (Ft') le/U i j / d i a./is*

Perth Amboy, N.JChileno Canyon, CalifMill Dam Run, Md

6.433.77X1081.24X10*

60.31.24X10*2.47X10*

- 9 . 78.42X1034.97X10*

1.571.281.15

1.411.353.16

-0 .110.050.64

■« Calculated assuming c in (21) is independent of position of channel in basin.

five subbasins) of 40.4 feet found bySchumm.

horton's lawHorton's law of stream lengths states

that "the average lengths of streams ofeach of the different orders in a drainagebasin tend closely to approximate a directgeometric series in which the first term isthe average length of streams of the 1storder" (Horton 1945, p. 291). The "streamsof each of the different orders" are thosedefined by Horton's system of ordering,which involves subjective classification ofone of the streams entering each fork astrunk and the other as tributary (Shreve

or of the definition of its terms has beenproposed by a number of investigators.Strahler (1957, p. 915), for example, suggested changing the law to state that thetotal length of channel of each order variesinversely as some power of the order.Broscoe (1959, p. 5) and Bowden and Wallis(1964, p. 770) suggested that the geometric-series progression with order could be preserved by substituting the sum of the average stream lengths from the first through agiven order, which they termed cumulativemean length, for the average stream lengthof that order. The cumulative lengths areapproximations to the Horton lengths whenthe Strahler system of ordering is used. In a

40S RONALD L. SHREVE

similar vein, Schumm (1956, p. 604) suggested using the average total length ofchannels in the subbasins of the given order.

At first sight it is surprising that suchmodifications should seem necessary, because(Mood and Graybill 1963, p. 147) the average stream lengths should cluster aroundthe expected lengths (in the sense that, asthe number of measurements increases, themean observed lengths should approacharbitrarily close to the expected lengths withprobability 1), and in infinite topologicallyrandom networks, according to (9c), the expected lengths will increase with order as ageometric series just as required by Horton'slaw. The explanation is that the populationwhose expectation is given by (9c) is not theone to which Horton's law refers. First,natural networks probably are not strictlytopologically random. Such evidence asexists, however, suggests that in many, ifnot most, areas free of geologic controlsthey are not far from it (Shreve 1966, p.31-36; 1967, p. 184-185; Smart, personalcommunication). Second, as already mentioned, the mean length of the exterior linksin a basin generally is significantly greaterthan that of the interior links. This factprobably explains some of the concave-upward curvature in Horton diagrams oflogarithm of stream length versus orderreported by Broscoe (1959, p. 5) and Maxwell (1960, p. 62, fig. 11; note that text onp. 23 contradicts the figure). Third, andmost important, regardless of which systemof ordering is used, Horton's law applies toaverage stream lengths in finite networks,whereas (9c) refers to average streamlengths in infinite networks. Thus, comparison with (9c) is not necessarily proper,except as an indicator of general behavior(as in Shreve 1967, p. 184).

Expected average Strahler stream lengthsin finite topologically random channel networks with given order and magnitude canbe calculated in terms of the expected interior and exterior link lengths E(/,-) andE(/t.) independently of the specific distribu

tions of the link lengths, assuming as beforthat the distribution of interior link lengths]is independent of order, magnitude, or an}other characteristic. With this assumption]the expected average length of streams oxorder 1 is E(/_), and that of streams of ordeia > 1 is equal to the product of E(j\) anfthe expected average number of links pestream of order oj. The expected averagnumber of links per stream in turn is tiaverage of E(v_) over all of the possible setof stream numbers, weighted accordingthe probability of occurrence of each se\where

E(v_) = I IaVi - 1) / (2«/ i - 1) ,0 = 2 /

W = 2, 3, a a a , fl ,

(1

(Smart 1968, p. 1007) is the expected nurber of links per stream of order a> in networks with stream numbers «_, w2, . . .jMq_i, 1. For a topologically random poptihtion of networks of given magnitude an^order, the sets of stream numbers and theiiprobabilities can be computed by means oi-the algorithm and formulas given in a pre-jvious paper (Shreve 1966, p. 29, 31).

The expected average total channclengths of Schumm can be calculatedsimilar fashion from the weighted averag^of E(£_) over the possible sets of strearnumbers, whereE(U = 1

+ E I [ 2 ( n . - i - l ) / ( 2 » _ - l ) , ( 1 7 ]0 = 2 o - 2 '

u = 2, 3, ..., a,

is the expected number of links per sulnetwork of order a> in networks with stre*numbers »_, »_, . . . , Wq__, 1. The expectetotal length per subnetwork of order cothen _E(0[E(.») + 1] + WM& ~from Melton's relationships (Melton 195*5p. 345; Shreve 1966, p. 27).

The cumulative mean lengths of Broscc

T.ngths, assuming as beforeion of interior link lengthsorder, magnitude, or any

lie. With this assumption,-rage length of streams olnd that of streams of orderi the product of E(i,) and.rage number of links perw. The expected averageper stream in turn is the

over all of the possible setsers, weighted according toof occurrence of each set.

_ - l ) / ( 2 « 0 - D , (16)

2 , 6 ,

1007) is the expected num-• stream of order w in net-•eam numbers »_, nt,jpologically random populates of given magnitude andif stream numbers and theirn be computed by means ofmd formulas given in a pre-ireve 1966, p. 29, 31).ed average total channeliumm can be calculated in

from the weighted averagethe possible sets of stream

/___ — l)/(2//_ — 1), (17)

- 2 , 3 O ,

•d number of links per sub-ler co in networks with stream-,, . . . , «__._, 1. The expected)er subnetwork of order co is.u„) + i] + .Eao[E(^)-^5 relationships (Melton 190*3,e 1966, p. 27).ative mean lengths of Broscoe

can be computed simply by summation ofthe expected average Strahler streamlengths.

To derive (16) and (17), consider theprocess of constructing topologically random networks with given stream numbers;/b tit, • ■ • , «q-1j 1 by starting with thesingle main stream and in cycles addingthe streams of successively lower order asdone by Shreve [1966, p. 29] and in theoriginal derivation of (16) by Smart [1968,p. 1005-1007]. If after the ath cycle the expected number of links in streams of somegiven order co is E0(v_), then after the nextcycle it will be

Ea+i(v_) = Ea(v_)i?a ,

Ra = (»V.-a ~ l)/(2,.r,-a+l - 1) ,

a = 9. - co + 1 , (18a)

Q - a + 2, .... a - 1 ,

co = 2, 3, . . . , 9 ,

counting addition of the streams of order12—1 as the second cycle. These equationsexpress the fact that in the construction oftopologically random networks, in which alltopologically distinct arrangements areequally likely, the expected number of linksin individual streams of a given order increases in direct proportion to the totalnumber of links of that order and higher.Streams of order co will consist of a singlelink when a = Q — w + 1; hence, by induction,

STREAM LENGTHS AND BASIN AREAS

E*H(__) - EQ(U + 2Ea(UX (ttn-a — 2w„__+i)/(2»q_«+_ — 1)

n - iE(v_) = H Ra, (18b)

= 92 3 9 .

from which (16) follows by letting /3 =V. — a + 1 and reversing the order of multiplication.

Similarly, if after the ath cycle the expected number of links in subnetworks oforder co is Ea(£_), then after the next cycleit will be

+ Ea( _.) + 1 ,

a = 9 - co + 1 ,

fl - c o + 2 , . . . , fl - l ,

co = 2, 3, . . . , 9 .

The second term on the right-hand side isthe increase in the expected number of linksdue to addition of new tributary linksalong the sides of the already existingstreams; and the sum of the last two termsis, from Melton's relationships (Melton1959, p. 345; Shreve 1966, p. 27), the increase due to addition of two new tributarylinks at the head of each stream. As before,subnetworks that will ultimately be of orderco will consist of a single link when a = fl —co + 1; hence, by induction, combining theterms in (19a),

E(U = 1 + 2i?„_.{l + 2i?„__

X [1 + . . . + 22?r>_(_+2

X (1 + 2i?n-_+i) ...]}.

409

(19a)

i '

(19b)

co = 2, 3, . . . , Q ,

from which (17) follows by successivelyeliminating the parenthetical expressions.

Figures 12, 13, and 14 show Horton diagrams of the expected stream lengths andnumbers for topologically random populations of networks of various magnitudes andorders. Certain generalizations are immediately apparent. First, as in natural networks, the Strahler lengths do not satisfyHorton's law nearly so well as do theStrahler numbers, whereas the Broscoe andSchumm lengths do. Second, the curves arestraightest for the most probable networks,as observed in natural networks by Smart(1968, p. 1007), and, except for the first-order lengths, are concave upward for networks whose order is less than the most

410 RONALD L. SHREVEprobable, that is, whose geometric-meanbifurcation ratio is less than about 4 (Shreve1966, p. 31), and the converse. Finally, forn given and E(/_)/E(/,-) in the range from1 to 2, the geometric-mean Strahler lengthratio, which, incidentally, is preferable tothe arithmetic-mean ratio sometimes used,is close to 2, and the Broscoe and Schummratios are close to 2.5 and 4.5, respectively,in approximate agreement with observedvalues (Schumm 1956, p. 604-605).

BASIN AREASAlthough Horton did not specifically in

clude basin areas in his laws of drainagecomposition, he implied that they shouldsatisfy a geometric-series law like streamnumbers and lengths (Horton 1945, p 294-Schumm 1956, p. 606). This law of basin

areas, as it was subsequently formulated b_Schumm (p. 606) in the style of Hortorstates that "the mean drainage-basin areaof streams of each order tend to approximatclosely a direct geometric series in which thfirst-order term is the mean area of the first!order basins." This law not only is identica,to the law of stream lengths but also can Mtreated theoretically in exactly the saiway.

Let ay and a, be, respectively, the first;order areas and the double-triangular areqdraining directly overland into individusinterior links. Like link lengths, theareas are random variables. Judgii,,from the data of Schumm (p. 607, 609)they are distributed according to rightskewed densities very similar to those fclink lengths. In the absence of better ir

3 4 5 5 7

200

ICO r: 550

2 -

7

6433.07xl0-3

2 3 4 5 6 7

200

2 3 4 5 6 7

2 -

of _££ oi2z^^:^x:^^s^^ ft ***** «*» frU*-lengths shown by solid circles, BroscoeS_S bv c re ^with 1^ "5 "^ "X^ Strahler Streamstream numbers bv triangles. Unit of stre-m leLthSTvl *_?•?* T)™ 'engths by °Pen circles. andstream order. The three numbers in ach iltrfj r *?* t "^T hl>k len8th- Horizontal coordinate iso r d e r , a n d p r o b a b U i t y t h a t a ^ 1 ^ ^ n e t w o r k

■ ■■■i*

•opiu qj3u3| uEatu-DU,3iuoD§ aqi si unnStnp ijdbd ui joqmnu qunoj •_] ajnSy ui sb aiuus sjoquivCs put? saimnpJOOQ -fonte wtsqjSuaj quq jouojxd pui; jotj„:j_i pa.oodxa ipiq.w ui no I pu*c 'c/ 'nc 'cc apit)tu3uui jo SJjJO.uiau jo suoi4_mdoduiopuBJ AHBDiSo[odo} joj sjaquinu pur sqi3u.i| ut.ai}. jappuis papadxa jo sumriteip uo,jo__—-fl "oij

-:

" i " :

MP6'9OOO' A£ AOOI i

00 66OOO2OOI

1 1 ■ y-62 £ZS£'£ 1

OO'trSOOO'

sa

'21 ojnSq ui si: .uitlS S|oquiXs pin; S3iBUipJ00Q•sq}3u3] quq jouaiui papodxa ddiavj ojb sqiSunj jp_f| jouaixa pajoodxo qaiq.w in j-o opnquSuui jo s^jo.upu josuoi.Bjndod uiopuBJ .qpjoiSojodo, joj sj.iqumu pun sqi3uo| uiroj-js popodxo |0 sUTBiSmp uouoj i—-£\ -oij2 9 9 t > 2 Z ,- »E

_• 901

-

A S £ - 0 l * 9 0 i |t>9

09'- 001

002

r:

•japjo pnupod? aqi ssjoaMSU '.pminStnu qJOAMOU <un»151 O.BUipJOOD [B .uozuojq -q.Su.-q *WipOT 'sapjio uado Xq sqtfuai uiuinq;

•suontqndod uropuBJ „fli»l8oiodoi jo

-ui aa.taq jo 93U9sq*c_3ipjoj _soip oi anions ajda-l-qgu oi §uipjo33*c pain'(609 '£09 'd) WWWPS -1SuiSpnf -S3iqT3U-CA uioiasatp 'sqiSuoi ^uij 93fl'prnpiAipra oiui puxqjaAOs_a__ jT.mSu^uvaiqnop 3l-isju aip, 'XpAtpsdsaJ '3<

.-hubs sqi £*p3W» ui a\1aq UT30 os\u inq «$$&[ ulponuapi si A.uo }OU avtj' s-isiy aq; jo -caaB a_ain aqoqi qoiq.w ill S9U3S DUpUia..iiuixoidd-c oi puoi *ra_M'Sl!3--B UlS.,q-3a*>\Hlu:.ip ltf-3'uoyoH jo S[&& olP u!.\'q pa-j.qnuuoj Aiiuonba?

412 RONALD L. SHREVE

formation, therefore, it seems justifiable asa first approximation to assume the sametype of densities, namely, gamma densitieswith v = 2 and expectations E(_y) = 2/afand E(o_) = 2/a,. With this assumption allof the previous results for Schumm totalstream lengths are directly applicable tobasin areas.

The theoretical densities of basin areasin infinite topologically random channelnetworks are shown in figures 9 and 10, andare compared with Schumm's histograms infigure 15. Again, the agreement between

0.3

••$0.2

Observed — TheoreticalSecond-order,

1 0 2 . 0 3 0Log basin area (inft.8)

Fio. 15.—Comparison of theoretical and observed histograms of subbasin area for the PerthAmboy, New Jersey, badlands basin studied bySchumm (1956, p. 607). Assumptions same as forfigure 9.

the calculated and the observed histogramsis highly encouraging.

Incidentally, in the table given bySchumm (p. 608, table 4), the frequenciesfor the first-order areas are one column toto the left of their correct positions.

Just as in the case of link lengths, theobserved mean interior area u,- needed asthe estimator of E(a,) was calculated usingthe equation

A = udf + (m — l)di , (20)where df and A are the mean first-order andtotal areas in a basin of magnitude n (tables1 and 2). Using these estimators in the equation analogous to (13i) gives an expectedsecond-order area of 376 ft2, which may becompared with the average (for forty-fivesubbasins) of 343 ft2 found by Schumm (p.606, table 2).

Horton diagrams of the expected areas infinite topologically random channel net

works are shown in figures 12, 13, and 16.Like expected lengths, expected areas donot depend upon the specific distributions]of the first-order and interior areas, but only!upon their expectations, provided as before}that the distribution of interior areas isindependent of order, magnitude, or anjother characteristic. The diagrams shotthat the expected areas satisfy Horton's la\rather well. The curves are straightest foithe most probable networks, that is, fo|those whose geometric-mean bifurcatioiratio is about 4 (Shreve 1966, p. 31), and aiconcave upward for networks whose order;less than the most probable, and the coiverse. Finally, the geometric-mean arratio for the expected areas in the mosjprobable networks is about 4.5, in approjmate agreement with observed value(Schumm 1956, p. 604-605).

CONSTANT OF CHANNEL MAINTENANCESchumm (1956, p. 606-607) pointed out

the nearly linear relationship between areaand Schumm total stream length in drainagebasins, and proposed (p^60)_} that theaverage area c draining directly overlandinto unit length of channel is a constantcharacteristic of the bedrock and environment but independent of the particularposition of the channel in the basin.

Although c seems to be tne average arearequired to maintain unit length of channel!as meant by Schumm (p. 607), it is not theconstant of channel maintenance as definedby him, which is instead the slope in a simplelinear regression model relating the averagedrainage area to the average total channellength in the subbasins of the various ordersin a single drainage basin. Mathematically,neither c nor, contrary to the assertion bySchumm (p. 607), the constant of channelmaintenance is the reciprocal of drainagedensity, that is, is equal to A/L, because ofthe contribution of the areas draining directly into the sources in the first case and thepresence of the nonzero constant term inthe model in the second. Practically, however, the constant of channel maintenancewill be approximately equal to the reciprocal

Tigures 12, 13, and 16.s, expected areas dospecific distributions

nterior areas, but onlyns, provided as beforei of interior areas is•, magnitude, or anyThe diagrams show

is satisfy Horton's law,-es are straightest forictworks, that is, fortrie-mean bifurcation-e 1966, p. 31), and areetworks whose order isrobable, and the con-geometric-mean area

ed areas in the mostabout 4.5, in approxi-ith observed values.04-605).\NEL MAINTENANCE

606-607) pointed out.tionship between area•earn length in drainaged (p. 608) that theling directly overlandchannel is a constantbedrock and environ-

ent of the particularid in the basin,to be tne average areaunit length of channel

n (p. 607), it is not themaintenance as defined.•ad the slope in a simplelei relating the average: average total channelns of the various ordersbasin. Mathematically,try to the assertion byhe constant of channelreciprocal of drainage

juaJ to A/L, because ofie areas draining direct-n the first case and thelzero constant term incond. Practically, how-)f channel maintenancey equal to the reciprocal

STREAM LENGTHS AND BASIN AREAS 413

of drainage density whenever the basin areais large compared to the constant term inthe model.

Incidentally, the model is valid onlyfor the discrete points corresponding tothe orders, so the meaning of the constantterm is complex. In particular, it is not theaverage area draining into each source, asthe improper procedure of setting thelength equal to zero would suggest.

(tables 1 and 2). Thus, c cannot be independent of the particular position of thechannel in the basin, but instead must beless for exterior links than for interior links.

The same calculation, also based on thedata of Schumm, gives d,/df = 0.05 forChileno Canyon, California, and 0.64 forMill Dam Run, Maryland. These differences are readily apparent in the basinsthemselves. In both the Perth Amboy basin

252

.OOOD 37.00M49.00

j i i i i

5 02

.OOOC 74.50m 99.00

_ i i i i _

752

.OOOb 11'2.00■ 149.00

i i i i i

t oo2

.OOOUI49.50ml 99.00

_ i i i i i

Fig. 16.—Horton diagrams of expected subbasin areas (or Schumm total stream lengths) for topologicallyrandom populations of networks of magnitude 25, 50, 75, and 100 in which the ratio of exterior to interiorexpected drainage areas (or link lengths) is 1 (solid squares) and 2 (open squares). Coordinates and symbolssame as in figure 12. Fourth and fifth numbers in each diagram arc geometric-mean area (or Schumm-length)ratios.

The proposal that c is constant is notsupported by Schtimm's own data from thePerth Amboy, New Jersey, badlands. Thisis shown by solving the identity

A = c L + L i d , ( 2 1 )

for the average source area a, assumingthat c = di/li as required if it is constant.The quantities I. and _"_ can be found bymeans of (15) and (20). Substitution of thenumerical values reported by Schumm thengives the impossible result d,/as = —0.11

and Chileno Canyon, slopes are near theangle of repose, ridges are sharp, and manyfirst-order basins arejlong, narrow chuteswith tiny source areas at their heads. In thebasin of Mill Dam Run, on the other hand,slopes are gentle, interfluves are broad, andfirst-order basins are more ovoid in planwith large source areas draining flat uplands.

These striking differences are not reflected in the Horton diagrams for the threebasins. The stream numbers, lengths, andareas conform to Horton's laws and the

r -<_

414 RONALD L. SHREVE

bifurcation, length, and area ratios are comparable in all three basins (Schumm 1956, p.603, 604, 605). The calculations in this andprevious papers (Shreve 1966, 1967; Smart1968) show that this lack of sensitivity tosubstantial differences in geomorphic character is primarily due to two factors. First,channel networks developed in the absenceof geologic controls are to a considerabledegree topologically random. Second, thestream lengths and areas in a Horton analysis are sums of many link lengths or theirassociated areas. The Horton variables, inother words, behave so well because the

deviations have been averaged out. Of mostfundamental significance, therefore, are notthe Horton variables but the more elementary quantities, such as link lengthsand source areas.

Acknowledgments.—Essential unpublishedinformation on theoretical developments wasfreely given by J. S. Smart, and on link lengthsby Smart, M. A. Melton, W. C. Krumbein, andM. J. Kirkby. Financial support was providedby the National Science Foundation (grantGA-1137) and the University of California.The numerical computations were carried outon the IBM 360/75 and associated equipmentof the UCLA Campus Computing Network.

REFERENCES CITED

Aitchison, J., and Brown, J. A. C, 1957, Thelognormal distribution: New York, CambridgeUniv. Press, 176 p.

Bowden, K. L., and Wallis, J. R., 1964, Effect ofstream-ordering technique on Horton's laws ofdrainage composition: Geol. Soc. America Bull.,v. 75, p. 767-774.

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