The Distribution of 

Branches in River Networks

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-G
The Distribution of 

Branches in River Networks

By ENNIO V. GIUSTI and WILLIAM J. SCHNEIDER 

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-G

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1965
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 20 cents (paper cover)
CONTENTS

Abstract___ ___________________

Introduction. _________________

Sources of data_______________

Variability of bifurcation ratios. _ 

Distribution of bifurcation ratios,

Page 

Gl

1

1

1

3

Distribution of number of major tributaries__ 

Distribution of number of smaller tributaries. 

Number of branches in river networks___-__- 

Summary and conclusions.________________

References cited__________________________

Page 

G5

6

7

8

10

ILLUSTRATIONS

FIGURES 1-3. Graphs:

1. Relation between number of 

streams and stream order____

2. Relation between bifurcation ratio 

and order____--____--_______

3. Relation between area and bifurcation ratio_ _ ______________

4. Random-walk model._________________

5-11. Graphs:

5. Relation between number of coalescing tributaries and order,__

Page

G2

FIGURES 5-11. Graphs Continued 

6. Distributions of bifurcation

7. Distribution of bifurcation ratios 

of order 4_ _________________

8. Distributions of major tributaries. ___ __________________

9. Combined distribution ________

10. Distributions by order lag_____

11. Growth curves of stream branching.------.----------------

G6

7

8

9

10

TABLES

TABLE 1. Distribution of streams by order, within the 

Yellow River drainage system___________

2. Means and Standard deviations of bifurcation 

ratios for Yellow River drainage_________

Page

G2

TABLE 3. Selected statistical data for distributions shown 

in figures 6, 7, and 8_____________________

4. Probability of occurrence of number of major 

tributaries to a stream.__________________

5. Skewness of distributions for given order-lags.

m

G5

5

7

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

THE DISTRIBUTION OF BRANCHES IN RIVER NETWORKS

By ENNIO V. GIUSTI and WILLIAM J. SCHNEIDEB

ABSTRACT

Bifurcation ratios derived from streams ordered according to 

the Strahler system are not wholly independent of the stream 

orders from which they are computed and, within a basin, tend 

to decrease in a downstream direction. The bifurcation ratios 

computed from two successive constant orders of streams within 

equal-order basins increase with the area of the basin but tend 

to become constant where the basin reaches a certain size.

The distribution of the number of major tributaries (streams 

of one order-lag) is exponential with a maximum frequency of 

two and can be expressed as the probability function

The distribution of the number of smaller tributaries is both 

unimodal and skewed and there appears to be an orderly succession of distributions from exponential to normal from the major 

tributaries to the smallest fingertip branches.

INTRODUCTION

Many geomorphic studies make use of an ordering 

system of stream branches proposed by Horton (1945). 

A fundamental parameter of this ordering system is the 

bifurcation ratio, which is defined as the ratio of the 

number of stream branches of a given order to the number of stream branches of the next higher order. This 

ratio can be expressed by

Nu ~NU

+1 (1)

where 56 =bifurcation ratio,

Af«=number of streams of given order, 

and Nu+l =number of streams of next higher order.

According to Horton (1945, p. 290), the bifurcation 

ratio varies from a minimum of 2 in "flat or rolling 

drainage basins" to 3 or 4 in "mountainous or highly 

dissected drainage basins"; it is a parameter used in 

equations giving the number of streams in a basin. 

As expressed by Horton, the equation is

(2)

where s=order of main stream 

and /z/=given order.

Strahler (1957) expresses the equation as 

log Nu=a bu (3)

where the antilog of b is the bifurcation ratio. He 

further states that the bifurcation ratio is "highly 

stable and shows a small range of variation from region 

to region." The average (mean) is about 3.5.

This paper discusses the distribution of bifurcation 

ratios and further analyzes the distribution of the 

number of any order tributary in a basin.

SOURCES OF DATA

Several sources of data were used for this study. 

Special photogrammetrically prepared drainage maps 

of 130 square miles of the Yellow River basin in the 

Piedmont province of Georgia were analyzed to determine the distribution of streams in the drainage network 

of the basin. The maps, compiled at the scale of 

1:24,000 delineated all drainage courses visible on 

aerial photographs taken at a flight height of 7,200 

feet. Additional data were obtained from 108 standard 

topographic maps, ranging in scale from 1:24,000 to 

1:250,000, of the Piedmont province. Data published 

by Melton (1957), Coates (1958), and Leopold and 

Langbein (1962) were used also for analyses. Streams 

were classified according to a system of ordering proposed by Horton (1945) and modified by Strahler (1957).

A summary of the distribution of number of streams 

of a given order within the Yellow River drainage 

system is shown in table 1. The 130-square-mile area 

of the Yellow River basin consisted of two seventhorder streams. The table lists separately the distribution of streams in each seventh-order basin.

VARIABILITY OF BIFURCATION RATIOS

The number of streams of any order in each of the 

two seventh-order basins that comprise the part of the 

Yellow River drainage system analysed here are 

plotted against their order in figure 1. The figure is 

typical in that some curvature exists in the range of 

higher order subbasins. This effect is also shown in

Gl
G2

500i

BASIN B

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

6

*~\

i I

+

23456 78 

BASIN ORDER (u)

FIGURE 1. Relation between number of streams and stream order for the Yellow

River.

figure 2 where the bifurcation ratios computed as 

N1/N2) N2/N3 * * * Nt/N7 from the values of table 1 are 

plotted against the basin order. The position of the point 

suggests that the bifurcation ratio is highly variable. 

In order to clarify this variability, mean bifurcation 

ratios and their standard deviation were computed 

from 26 subbasins of fifth, sixth, and seventh order

BASIN A

234 567 

BASIN ORDER (</ + !)

FIGURE 2. Relation between bifurcation ratios and order for two seventh-order

basins.

within the Yellow River drainage system. These data 

are shown in table 2. Both figure 2 and table 2 suggest 

(1) that bifurcation ratios expressed as the number of 

next lower order tributaries or Ns_\ are smaller than 

bifurcation ratios computed from lower order tributaries and (2) that bifurcation ratios tend to become 

constant and less variable (smaller standard deviation) 

when expressed as ratios between the numbers of lower 

order branches.

Figure 3 suggests a relation between the bifurcation 

ratio and the drainage area. This relation, however, 

depends entirely upon the relation between order and 

area and can be expressed as follows: Basins of equal 

order but variable areas tend to have the smallest 

bifurcation ratios in the smallest areas; the ratios 

increase with increasing areas up to a certain size, 

beyond which the bifurcation ratios tend to become 

constant.

TABLE 1. Distribution of streams, by order, within the Yellow 

River drainage system

Subbasin

A 

B. ... ... .... - ---

Number of streams (Nn) of order 

1

4,026 

4,823

2

812 

1,060

3

200 

233

4

33

42

5

8 

11

6

2 

4

7

1 

1

TABLE 2. Means and standard deviations of bifurcation ratios for 

Yellow River drainage

Mean («») -.._ .. -

JV.-1 

N.

3.2

1.3

N,-t 

JV_,

4.5

1.5

N,-s 

JV_i

4.7

.75

Nr* 

JV.-I

4.8

.64

N^

JV_«

14.6

JV.-. 

JV_i

*4.75

Computed from 6 basins of 6th order and 2 basins of 7th order. 

1 Computed from 2 basins of 7th order.
DISTRIBUTION OF BRANCHES IN RIVER NETWORKS G3

20 r

10

a:

m

OO O..O'*O

OO OQ** O OO

Q S=2 

 S=3 

o S =4 

® 5=5

0.01 0.05 0.1 0.5 1.0

DRAINAGE AREA (A), IN SQUARE MILES

PIOUEE 3. Relation between drainage area and bifurcation ratios !Vi:2Vj for subbasins within the Yellow River basin.

The fact that bifurcation ratios become smaller as 

they are computed from higher order subbasins is 

basically due to the branching process. Consider 

figure 4 where a fourth-order basin derived by a randomwalk process is shown. For clarity, the same basin is 

shown successively in figure 4 with the number of 

streams of increasing order. It is apparent that as the 

stream order increases the percentage of streams that 

coalesce into a higher order tributary also increases 

and that this increase is due to the diminishing amount 

of area available. In figure 5, all the percentages of 

coalescing streams for the Yellow River and for the 

random-walk model developed by Leopold and Langbein (1962, p. A 18) have been plotted against their 

order. The increase in a downstream (with higher 

order) direction, particularly for the Yellow River, is 

evident.

Thus, bifurcation ratios -^p- tend to become constant

/V $

for values of u > s 2, where s is the order of the main 

stem.

DISTRIBUTION OF BIFURCATION RATIOS

Frequency distributions of bifurcation ratios are 

shown in figures 6 and 7. The distributions are derived from data on all fourth- and higher order subbasins and random sampling of first-, second-, and 

third-order subbasins within the Yellow River drainage 

system. Data obtained by Melton (1957) fron aridto-humid mountainous basins in Arizona, New Mexico, 

Colorado, and Utah, and data obtained by Coates

(1958) for small basins in the humid Interior Low 

Plateaus province are included also.

According to Horton (1945, p. 296), equation 2 

becomes

£6 =AU (4)

if u=s 1. This relation indicates that the bifurcation 

ratio, Rh , is equal to the number of streams of the next to 

the highest order for a given drainage basin. Thus, the 

bifurcation ratio as defined by equation 4 can be equated 

to the number of maj or tributaries to a given stream. The 

major tributaries can also be defined as streams of one 

order-lag. Thus, it follows that a drainage system 

will have streams of one order-lag (major tributaries), 

two order-lags (smaller tributaries), and so on, and 

the number of tributaries may be defined as N,-i, 

Ns_2, * * * Ns_ n, the subscripts indicating the relative 

position within the drainage system.

Figures 4 and 5 show distributions of bifurcation 

ratios which are grouped according to their order-lag.

Thus ratios -^ in basins of order 4 can be written as

N3

ratios ^^ and can be compared to ratios -^ from

third-order basins which are also expressed as

The distributions of number of major tributaries 

(streams of one order-lag) are shown in figure 8. 

These distributions differ considerably from those 

of figures 6 and 7. Statistical parameters of the 

distributions of figures 6, 7, and 8 are tabulated in 

table 3.
G4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

BASIN SHOWING

FIRST ORDER 

BRANCHES ONLY

P = 18 

S =20

THIRD ORDER 

BRANCHES ONLY

SECOND ORDER 

BRANCHES ONLY

BASIN SHOWING 

ENTIRE NETWORK

NOTE:

P = Primary or coalescing branches 

S =Secondary 

Shading denotes drainage areas

FIGURE 4. Random-walk model of a fourth order basin (from Leopold and Langbein, 1962, fig. 8, p. A18).
DISTRIBUTION OF BRANCHES IN RIVER NETWORKS 

TABLE 3 Selected statistical data for distributions shown in figures 6, 7, and 8

G5

Statistic

Sample size ___ - -- - - 

Standard deviation of bifurcation ratios,

Skewness 1 of bifurcation ratios, &<»j>-

Source of data

Yellow River

270 2-7

AT.-, 

3.48

1.79 

.83

84 

3

AVl

AVi 

4.36

1.97 

.18

84

3.88

1.09 

-.11

84

AT.-3

4.72

1.05 

,21

Melton (1958)

132

3.04

1.43 

.73

45

4.04

1.17 

.034

45

AT.-J 

AVJ 

6.02

3.24 

.31

Coates (1959)

60 

3

AT.-i 

3.40

1.62 

.86

60 

3

AV2 

AT,-, 

4.38

1.17 

.32

Leopold 

and Langbein 

(1962)

92

Nr-i 

3.23

1.83 

.67

Topographic 

maps 

(blue 

lines)

269 2-6

3.27

1.66

.77

1 Skewness calculated by the approximate formula, skewness= "f d d vl tl 

0.7

< 0.6

DQ

o:

o

o:

LU

o_

0.5

0.4

0.3

0.2

0.1

3 4 

ORDER (S)

FIGTJBE 5. Relation between percentage of coalescing tributaries and order.

DISTRIBUTION OF NUMBER OF MAJOR TRIBUTARIES

Because the distributions for the various data shown 

in figure 8 did not differ significantly from each other, 

the variations among the moments of these distributions 

were assumed to be due to sampling errors, and all data 

of figure 8 were combined. The resulting distribution 

is shown in figure 9 as a semilogarithmic function of 

the form

j=ke-k'N(*-D (5)

where /=the frequency of occurrence, in percent, 

N(S-n =number of streams of one order-lag, 

and e=base of Naperian logarithms.

The computed regression equation is

(6)

which can be simplified to

(6a)

Because equation 6a represents a distribution, it may 

become a probability function as

n==e -^ff , (7) jf & .1 1 \' i

from which the probability, p, can be computed for any 

given number of major tributaries.

Probabilities of occurrences for values of Ns-i up to 

10 are shown in table 4. This table shows that more 

than one out of three streams similar to those studied 

here will have two major tributaries, and about one 

out of five three major tributaries. A second-order 

stream will most frequently have two first-order streams 

as branches; a third-order stream two second-order 

branches, and so on.

TABLE 4 Probability of occurrence of number of major tributaries

to a stream

Number Probof major ability of 

tributaries occur- 

(N,,-i) rence (p)

2.___.__.__-_------------------------- 0.368

3._____-_-----.-_--------------------- .233

4_ _____ ___-______ -- .135

5. __.____.___ _.___ _ - - .082

6___.___ -050

7_ ________ __ -030

8___ -018

9_._ -Oil

10...___--_--------------------------- -007

Melton (1958) defines a "conservative drainage 

system" as "one having the minimum number of 

channel segments necessary for the highest order of the 

system." His measure of conservancy for any order 

u is

Su=-Rl>tt:ti+l -i

and maximum conservancy corresponds to $w=0.

(8)
G6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

£ 40

z

LU

£30 

LU

a.

z-20

C"

^_

§ 10

LU

3 

CX ,

LU ^ CY- 0 ' u ' ' ' ' ' ' ' ' '

-

' N

- / \

- / \\

/

/ \

40

30

20

10 /

 '' , , ,

'

x« 

1 n

-

x«x

/ \

/ \

/ V\

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i i i i i i T"~ T

u-23 4567 23456 789 10

BIFURCATION RATIOS (Ni:N2) FROM BIFURCATION RATIOS (N^N^ FROM MELTON COATES (1958), THIRD " ~ ~~

BASINS

LU

0 40

z

LU

O __

cc 30

LU

a.

z 

T. 20

»^_

> ^

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LU 

3 ,

0 ' 

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-

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ORDER (1957), THIRD ORDER BASINS

40

30

20

-/

S ^

i i i T"~""~T f n

r 40 /*\

/ \

' ^ ^n 

- / \ 30

/ \

/ \

/ \

t \ 10

*V '

1 1 1 1 v» n

--

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- / \

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/ \

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 ^-* i i. j i T

3456789

BIFURCATION RATIOS (Ni:N2) FROM YELLOW RIVER, THIRD ORDER BASINS

234567

BIFURCATION RATIOS (N2-'N3) 

FROM YELLOW RIVER, FOURTH 

ORDER BASINS

34567 

BIFURCATION RATIOS (N2:N3) 

FROM MELTON (1957), FOURTH 

ORDER BASINS

4U

LU

0

< hz LU 30W

o

a:

LU

D.

z

^ 20

' '

0

LU8- 10

LU

o:

Ll_

o'

 X^""*^s

 s

/ \

/ \

/ \

/ \ DATA FROM

/ \ YELLOW RIVER

/ \

/ \

/ \ 

/ \

/ \

// x«

- / \ / \

J

J* ^

 ^ 1 1 1 1 >

234567

FIGURE 6. Distributions of bifurcation ratios defined as N,-z+Ne-i. 

40 r30 -

20 - /< 

/

0 f I I 

2 3 A

BIFURCATION RATIOS (N^N2) 

FIGURE 7. Distribution of bifurcation ratios N,-3+N,-z.

DATA FROM 

MELTON (1957)

This concept applied to the major tributaries only 

gives

Su=^-l (8a)

Zi

where maximum conservancy refers to streams having 

only two major tributaries; this relation, according to 

the probabilities computed, has a one-in-three occurrence. The manner and the number of joining tributaries no doubt greatly affects the hydrology of the 

drainage system, and the number of major tributaries 

probably bears some relation to the shape of the basin.

DISTRIBUTION OF NUMBER OF SMALLER 

TRIBUTARIES

The subbasins of the Yellow River were further investigated in terms of the smaller tributaries present
DISTRIBUTION OF BRANCHES IN RIVER NETWORKS G7

Ul

<

Ul

O 

o:

Ul

Q.

z

_

!±>

0

Ul

13

0

Ul

cr 

u_

50

40

30

20

10

n

-

*

\ DATA FROM COATES (1958)

\

~ \

\

\

\.\*

\

s

- ^

x«v.^

III! 1

50

40

30

20

10

\DATA FROM MELTON (1957) 

\

\

50

40

30

20

10

_ \

\

\

DATA FROM LEOPOLD (1962)

\

-T-* ?

79 11 2468 10 2 

NUMBER OF STREAMS OF ORDER ONE LESS THAN BASIN ORDER (Ns -i)

10

40

- 20

10

13 

O

DATA FROM YELLOW RIVER

50 i40

30

20

10

0

_ \ DATA FROM STANDARD 

TOPOGRAPHIC MAPS 

, AT VARIOUS SCALES 

\

I I

579 11 23579 

NUMBER OF STREAMS OF ORDER ONE LESS THAN BASIN ORDER (Ns -i)

11

FIGURE 8. Distributions of number of major tributaries or bifurcation ratios defined as N,-i (data from various sources as shown).

therein by combining the number of branches of two 

order-lags (such as first-order branches in third-order 

basins, and second-order branches in fourth-order 

basins) into one distribution. Similarly, branches of 

three order-lags (such as first-order branches in fourthorder basin and second in fifth) and four order-lags 

(such as first in fifth and second in sixth) were combined 

into one distribution for each. Figure 10 shows the 

distributions for various order-lags; the progression 

from exponential to unimodal skewed distribution with 

increasing order-lag is apparent. As a measure of the 

symmetry of the distributions, skewness was calculated 

for each distribution in figure 10. Results are shown 

in table 5. Skewness appears to decrease with increasing order-lag, and only the number of the smallest 

fingertip branches in higher order basins may be normally distributed.

TABLE 5. Skewness of distributions for given order lags

Order lag 

8-1...

s-2__

Skewness 

_ 0.83 

_ .93 

_ .88 

_ .67

NUMBER OP BRANCHES IN RIVER NETWORK

The minimum number of branchings of a stream is 

given by Melton (1957) in the equation

This equation states that a second order stream will 

have at least two first-order branches, a third-order 

stream at least two second-order branches and four 

first-order branches, and so on. A graph of this 

equation is shown in figure 11 together with a straight 

line fitted through the modal values of the distributions 

of figure 10. The stream branches of the Yellow
G8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

50 i10

0.5

I I I I

8 10 11 12 13

NUMBER OF STREAMS OF ORDER ONE LESS THAN 

BASIN ORDER (Ns_i)

FIGURE 9. Combined distribution of number of major tributaries, from figure 6.

River obviously multiply much more rapidly than the 

minimum growth. In a previous study, Giusti (1963) 

showed that the number of subbasins, Na , of any given 

area, a, within a region or basin of a given area, A, 

could be expressed by the equation

- 

CL

(10)

Similarly, from equation 2 or 3, the number of stream 

branches can be expressed in terms of the order of the 

subbasins with respect to the order of the main stem. 

The fact that the relation based on area is hyperbolic 

indicates a geometric progression. However, by definition, the number of branches follow a geometric progression and the order (or order-lags) an arithmetic 

progression. Consequently the relation between number of branches and their order is exponential.

A further equation of Melton (1958) gives the number of channel segments in a basin as

7V=0.8147

L l- n

(ID

where L is the total length of channels and R is the 

relative relief. There are then several equations from 

which the number of channel segments in a basin can 

be computed, and usage of one or another will depend 

on the preference of the user. However, because of 

the variability of the bifurcation ratios, the time 

required for arranging the stream segments of a drainage system into orders, and the lack of true portrayal 

of the drainage systems on topographic maps, equations 

10 and 11 will possibly be more applicable in practice.

SUMMARY AND CONCLUSIONS

River networks have been analyzed in terms of the 

number of branches ordered according to a system 

proposed by Horton (1945) and modified by Strahler 

(1957). All networks were dendritic and in different 

climatic and geologic environments. The bifurcation 

ratios computed from two successive constant orders in 

any stream network vary according to the two stream 

orders from which they are computed, and decrease in 

a downstream direction. Bifurcation ratios for equalorder basins increase somewhat with the area of the 

basins but tend to become constant for basins beyond a 

certain size. These two factors the order used for 

computation and the size of the area also affect the 

parameters of the distributions of bifurcation ratios. 

Bifurcation ratios tend to become constant for ratios 

made between number of branches which are two or 

more orders removed from the order of the main stem 

(two order-lags).

The distribution of the number of one order-lag 

branches (considered the major tributaries to a given 

stream) was found to be exponential, with a maximum 

frequency at two. On an average, one out of three 

rivers of any given order will have two major tributaries, 

and one out of five rivers will have three. The distribution of number of higher order-lag branches or number 

of smaller tributaries was found to be both unimodal 

and skewed, and only the number of smallest fingertip 

branches may approach a normal distribution.
FREQUENCY (f), IN PERCENTAGE FREQUENCY (f), IN PERCENTA

c

z 

c

CDm

0 fi

-n

-1

m oo

>ij O 

S -0 

§ 0 

S m 

n 33

5= r- M

8 168

AG TRIBUTARIES 

 Distributions ofnui

<y Z is) 

S <-? o

i- N) CO !- N) 

D O O OO O O 

I 1 1 1 1 1 z 

 -_ _ _ _ c

% m

- ^^^ o w

*--"""'" o

 ^-'"" o

x'^ O ^ rn 

/ 33

 / >

~ / -1

/

 33 

- / 2 vo 

/m 

GO

1

1 2

 1 ' 7

-I ^ H1 1 1 1 1 

^^-^ ""

^^

^^

 /x

/

1

s c

CDm

33

o

!_ O

O 

33

CD a
G10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

200 i100 -

O

cc

LU

Q 

CC 

O

LU 

O

o:

00

o:

u_

o 

cc

LU 

GO

10 -

5 -

3456 

ORDER LAG (S -u)

FIGURE 11. Growth curves of stream branching. Lower curve is minimum growth, 

upper curve from model values of distributions of figure 10.

The variability of bifurcation ratios found in this 

study indicates that bifurcation ratios must be carefully 

defined in terms of the two successive orders from 

which they are computed and the order of the main 

stem. Comparisons of undefined bifurcation ratios 

may lead to erroneous conclusions.

REFERENCES CITED

Coates, D. R., 1958, Quantitative geomorphology of small 

drainage basins of Southern Indiana: New York, Columbia 

Univ. Dept. Geology Tech. Rept. 10.

Giusti, E. V., 1963, Distribution of river basin areas in the conterminous United States: Bull. Internat. Assoc. Scientific 

Hydrology, v. 8, no. 3.

Horton, R. E., 1945, Erosional development of streams and their 

drainage basins hydrophysical approach to quantitative 

morphology: Geol. Soc. America Bull., v. 56, no. 3, p. 

275-370.

Leopold, L. B., and Langbein, W. B., 1962, The concept of 

entropy in landscape evolution: U.S. Geol. Survey Prof. 

Paper 500-A, 20 p.

Melton, M. A., 1957, An analysis of the relations among elements 

of climate, surface properties, and geomorphology: New 

York, Columbia Univ. Dept. Geology, Tech. Rept. 11.

Melton, M. A., 1958, Geometric properties of mature drainage 

systems and their representation in an E4 phase space: 

Journ. Geology v. 66, no. 1.

Strahler, A. N., 1957, Quantitative analysis of watershed geomorphology: Am. Geophys. Union Trans., v. 38, no. 6, 

p. 913-920.

O