Hydraulic Geometry of 

River Gross Sections— 

Theory of Minimum 

Variance

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1029

Hydraulic Geometry of 

River Gross Sections— 

Theory of Minimum 

Variance

By GARNETT P. WILLIAMS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1029

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1978
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

W. A. Radlinski, Acting Director

Library of Congress Cataloging in Publication Data

Williams, Garnett P

Hydraulic geometry of river cross sections. 

(Geological Survey professional paper ; 1029) 

Bibliography: p. 

Supt. of Docs, no.: I 19.16:1029

1. River channels. 2. Channels (Hydraulic engineering) I. Title. II. Series: United States. Geological Survey. 

Professional paper ; 1029.

GB561.W54 551.4'83 76-608396

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

Washington, D.C. 20402

Stock Number 024-001-03048-9
CONTENTS

Abstract ____________________________ 

Introduction and problem ________________________

Basic variables and relations __________________

Factors that can influence the hydraulic exponents___ 

Minimum variance ________________________

Theory ——___________________________________

Hydraulic exponents based on the concept of 

minimum adjustment ______________

Computation of minimum variance ______

Influence of choice of variables __________

Test of minimum-variance theory with field data 

Collection of data ________________________

Sources of error _____________________

Comparison of measured to theoretical 

hydraulic exponents ______________

Case A _________________

Case B ______________________________

Case C ______________________________

Case B/C ___________________________

Case D ______________________________

Page 

1

1

2

2

4

4

5

7

10

11

11

12

15

17

17

19

19

20

Hydraulic exponents of individual stations _____. 

Collection of special data ___________.

The width-versus-area relation _____. 

Maximum and minimum geometrical properties of a section ________________.

Bed-sediment sizes and estimates of bed 

roughness ____________________.

Slope ______________________.

Accuracy of exponents determined by minimum 

variance __________—______—————.

Empirical formulae for hydraulic exponents _ 

Hydraulic exponents from the Gauckler-Manning 

and Chezy equations _—______—___————.

Method ____—_—————_——————————-

Results __________________—_—.

Comparison of methods of computing hydraulic 

exponents __—___————_——_————_.

Summary and conclusions __——_—————————. 

References ______———————— — — ———————-

Statistical variance and a hydraulic exponent —__.

ILLUSTRATIONS

Page 

21 

21 

21

24

28

28

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30

31

31

32

33

34

34

39

Page

FIGURE 1. Diagram of hypothetical change in two dependent variables _____________________ 6

2-8. Graphs:

2. Hydraulic geometry plots for Colorado River near Grand Canyon, Ariz ___________ 13

3. Hydraulic geometry plots for Prairie Dog Town Fork Red River near Childress, Tex _— 14

4. Frequency distribution of exponent values, for different categories of stations _______ 16

5. Width versus flow area for a firm-bank station _________________________ 22

6. Width versus flow area for a loose-bank station _______________________——____ 23

7. Comparison of estimated to observed width-area relations ___________________ 25

8. Cross-sectional profile and hydraulic geometry, Humboldt River near Argenta, Nev ——— 26

9. Sketch showing concept of bank inclinations ____________________________——____ 27

10-13. Graphs:

10. Computed versus measured values of exponent 6 __________—__ ——___- —— _ 31

11. Computed versus measured values of exponent / _______________-_______ 32

12. Computed versus measured values of exponent m _________________—— ———— ——— 33

13. Power relationship between velocity and discharge where the exponent=0.5 _______——___ 41

TABLES

TABLE 1. Trial-and-error example of finding / for which the sum of variances is a minimum _________

2. Rate of change of dependent variables with increase in discharge, for different sets of variables __

3. Comparison of exponents determined by least squares with those fitted by eye _______________

4. Theoretical rates of change of dependent quantities with increase in discharge, for different sets 

of variables. (Case A: width and slope constant) ________________________________

Page

8

10

15

17

III
IV CONTENTS

Page

TABLE 5. Theoretical rates of change of dependent factors with increase in discharge, for groups of variables surviving case A and for different constraints (Case B: firm banks) _____——_—— 18

6. Theoretical rates of change of dependent quantities with increase in discharge, for different sets of 

variables (Case C: slope constant; loose, noncoherent banks allowing complete freedom for 

width to adjust) __________________________________________—__——- 19

7. Measured and predicted values of hydraulic exponents for stations having variable slope, with bed

and banks readily erodible ______________________-_____—_-—_——— 20

8. Comparison of average measured exponents to exponents predicted by the minimum variance

theory, using V, D, W, r, ff, S, and QS as the appropriate variables ——————————————— 21

9. Accuracy of methods of predicting hydraulic exponents _________—_____—_____—___—_—____ 34

10. Standard deviation of log V and of log Q _______________——_——————————————— 40

11. Summary of data __________________________________—__———————————— 42

SYMBOLS

61 

dso 

f

h 

ff

m

hydraulic exponent of channel width, in the proportionality between width and water discharge,

for example, W <x Q* 

exponent of width included in 61/(6i+/i), as estimated from a width-versus-area plot. 

grain size at which 50 percent of the distribution is

finer (dlfl=10 percent finer, and so forth.) 

hydraulic exponent of mean water depth, in the proportionality between depth and water discharge 

exponent of depth included in 6i/(6i+/i), as estimated from a width-versus-area plot 

Darcy-Weisbach friction factor for wide channels,

A 

A,

hydraulic exponent of mean velocity, in the proportionality between mean velocity and water discharge

resistance coefficient in Gauckler-Manning formula

correlation coefficient

distance

hydraulic exponent of friction factor, in the proportionality between friction factor and water discharge 

hydraulic exponent of slope, in the proportionality between slope and water discharge 

cross-sectional flow area

cross-sectional flow area at the upper 

hydraulic geometry power relation

end of the

Dmln

Q 

R

S

So

S.E.

V

W

X 

Y

cross-sectional flow area at the lower end of the 

hydraulic geometry power relation

Chezy resistance coefficient

mean flow depth=A/W

mean flow depth at the upper end of the hydraulic 

geometry power relation

depth at the lower end of the hydraulic geometry 

power relation

water discharge

hydraulic radius

slope or energy gradient

sorting coefficient=log doo—log d]0

standard error

mean flow velocity

water-surface width

water-surface width at the upper end of the hydraulic geometry power relation

water-surface width at the lower end of the hydraulic geometry power relation

horizontal distance from channel center toward bank

depth at distance X

maximum depth at channel center

difference between the logarithms of two quantities

standard deviation

bed shear stress

angle of bank inclination

average inclination of the two banks at a cross section

CONVERSION FACTORS

English

ft (feet)

ft (feet)

ft2 (square feet)

ft3 (cubic feet)

mia (square miles)

Multiply by

0.305

304.8

0.0929

0.0283

2.59

Metric

m (meters)

mm (millimeters)

m2 (square meters)

m3 (cubic meters)

km2 (square kilometers)
HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONSTHEORY OF MINIMUM VARIANCE

By GARNETT P. WILLIAMS

ABSTRACT

This study deals with the rates at which mean velocity, 

mean depth, and water-surface width increase with water 

discharge at a cross section on an alluvial stream. Such relations often follow power laws, the exponents in which are 

called hydraulic exponents. The Langbein (1964) minimum 

variance theory is examined in regard to its validity and its 

ability to predict observed hydraulic exponents.

A major part of the study is devoted to identifying the 

important variables to use with the theory. These variables 

are velocity, depth, width, bed shear stress, friction factor, 

slope (energy gradient), and stream power. If the slope at 

a particular station is constant, only the first five of these 

variables need be considered.

The second aspect of the study tests the theory against 

field data. The 165 cross sections used reflect the following 

ranges of hydraulic exponents: 0.00=^6^0.82 (width), 

0.10^/^0.78 (depth), and 0.03^m^=0.81 (velocity). Flow 

conditions range from 0.000283 cubic meters per second 

(0.01 cubic feet per second) to 1,980 cubic meters per second (70,000 cubic feet per second), widths from 0.31 meter 

(1 foot) to 579 meters (1,900 feet), mean depths from 

0.031 meter (0.1 foot) to 10.7 meters (35 feet), and median 

bed-material sizes from 0.06 millimeter to 100 millimeters. 

Most geographic regions of the contiguous United States 

are represented. The original theory was intended to produce 

only the average hydraulic exponents for a group of cross 

sections in a similar type of geologic or hydraulic environment. The present test shows that the theory does indeed 

predict these average exponents, with a reasonable degree 

of accuracy.

An attempt to forecast the exponents at any selected cross 

section was only moderately successful. Empirical equations 

are more accurate than the minimum variance, GaucklerManning, or Chezy methods. Predictions of the exponent of 

width are most reliable, the exponent of depth fair, and the 

exponent of mean velocity poor.

INTRODUCTION AND PROBLEM

Rivers have always been the arteries of civilizations. Because societies are so closely dependent 

upon the flow of water, people have for many years 

looked for methods of predicting the relations 

among the hydraulic features of a river—the water 

discharge, mean depth, width, mean velocity, and 

other variables. Such flow characteristics affect not

only man but also the plants and animals living in 

or along the river.

The subject of this study is the rates at which 

water-surface width, mean depth, and mean velocity 

change with water discharge at a given cross section or station on a stream. Only streams that have 

loose particles on the bed will be considered. For 

such alluvial streams, there are no reliable ways to 

predict the rates of change of the flow variables 

mentioned above. Accurate methods are elusive because of the irregular shape of the cross section 

and the changing roughnesses of the flow boundary. 

The flowing water molds the loose particles into 

various patterns and configurations, and these 

roughness changes can vary with discharge. Bed 

roughness can also vary with distance across the 

stream. Under certain circumstances, such changes 

in bed roughness have been associated with abrupt 

changes in mean water depth and mean velocity. 

A search for general relationships can be complicated further by the variability of bank roughness, 

which differs considerably with lithology, vegetation, and other factors.

This paper begins with a discussion of basic relations and of the minimum variance theory (Langbein, 1964). The rest of the paper has three main 

parts. The first part of the study examines the question of which variables to use with the minimumvariance theory. The second, treated concurrently 

with the first, tests the theory in regard to its ability 

to predict the average rates (considering a large 

group of rivers and cross sections) at which mean 

velocity, depth, and water-surface width change 

with discharge. Finally, the third part is an attempt to find an objective way to forecast the hydraulic relations at any given cross section on an 

alluvial channel.

I would like to thank Walter B. Langbein, Edward 

J. Gilroy, Marshall E. Moss, and William H. Kirby 

for their many helpful discussions on certain as-
HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

pects of the study. Others who have contributed useful comments include William W. Emmett, Kenneth 

L. Wahl, Carl F. Nordin, Jr., Dallas D. Rhodes, M. 

Gordon Wolman, Edward J. Pluhowski, Henry C. 

Riggs, and Lucien M. Brush, Jr. U.S. Geological 

Survey personnel, too numerous to mention, at district and subdistrict offices have been most helpful 

in providing streamflow records and other basic 

data.

BASIC VARIABLES AND RELATIONS

The multitude of variables that may be involved 

when water flows down an alluvial channel can 

be classified into flow properties (discharge, mean 

velocity, mean depth, etc.), water characteristics 

(temperature, specific weight, and viscosity), channel features (alinement or sinuosity, slope or energy 

gradient, shape of cross section), and sediment-related features (rates and grain sizes of sediment 

in transport, scour and fill, roughness of bed and 

banks, cohesiveness of bed and banks, sizes and 

shapes of boundary particles, and other aspects).

The flow variables of primary interest in this 

paper are the discharge or flow rate, Q; watersurface width, W; mean depth, D (defined as crosssectional flow area A divided by W) ; and mean 

velocity V (defined as Q/A). The continuity relation specifies that Q = VDW.

Changes in these variables commonly are studied 

for two situations. The first is at a given station 

or cross section on a river, and the second is for 

a series of stations proceeding downstream on a 

river (Leopold and Haddock, 1953). Only the ata-station case is treated in this paper.

A wealth of data (Leopold and Haddock, 1953; 

Stall and Yang, 1970; Fahnestock, 1963; and many 

others) shows that at cross sections on many rivers, 

canals, and laboratory flumes, the variables velocity, 

depth, width, slope S, and friction factor ff (defined 

later), all considered dependent, often change in 

proportion to some power of water discharge, considered independent. Thus

V oc Qm 

D « Qf 

W oc Q6 

S oc Q* 

ff « Qy

(D

(2)

(3)

(4)

(5)

where the exponents, called hydraulic exponents, 

represent the rate of change of the dependent variables with change in Q. Power relations of the sort

exemplified in equations 1-5 plot as straight lines 

on logarithmic paper. Leopold, Wolman, and Hiller 

(1964, p. 215-281) discuss these hydraulic relations in some detail.

Power functions do not necessarily hold for the 

complete range of flows at a given cross section 

(Richards, 1976). Some of the many factors that 

can interrupt or prevent consistent relations over 

a wide range of discharge are discussed later. Also, 

more than one power relation can exist for withinbank flows. For instance, very low flows may follow 

one power law, and flows approaching bankfull may 

follow another. Finally, Richards (1973) points out 

that for some cross sections simple power functions 

do not apply. He suggests quadratic or higher order 

curves for such sections.

Hydraulic relations change drastically when a 

river overflows its banks. Such overbank flow will 

not be considered in this paper.

The following discussion deals with equations 1 

through 5 and other equations of this form.

The general problem is to determine the hydraulic 

exponents wherever a power function relates the 

dependent and independent variables. Special attention will be devoted to the exponents of velocity, 

depth, and width (m, /, and b, respectively, in equations 1 through 3). This paper deals only with the 

exponents or rates of change. It does not consider 

methods of predicting absolute values of mean depth, 

width, and mean velocity.

Inserting the power equations 1 through 3 into 

the continuity relation Q = VDW gives Q^Q^QfQ*. 

The basic relation among exponents therefore, is

FACTORS THAT CAN INFLUENCE THE 

HYDRAULIC EXPONENTS

There is very little information on how water, 

channel, and sediment features affect hydraulic exponents. Intuitively, it can be reasoned that any 

water characteristic effects probably are small. For 

lack of evidence (admittedly not a proof of the assumption), these water features will be neglected 

in this analysis.

Of the channel features, channel alinement could 

be important because the cross-sectional flow pattern in a meander bend is different from that in a 

straight reach. Knighton (1975, p. 206), studying 

selected streams in England, found a lower rate of 

change of width on straight-reach sections than in 

meander sections. Channel alinement or sinuosity
FACTORS THAT CAN INFLUENCE HYDRAULIC EXPONENTS

will largely be eliminated as a factor in this study 

by dealing only with stations located on straight or 

slightly curving reaches.

Water-surface slope, or energy gradient of the 

flow, is an important variable in many formulae 

for calculating discharge and mean velocity. A channel's slope is determined primarily by the general 

topography of the landscape, but the water-surface 

slope may vary somewhat with discharge at a 

station. In some parts of the present study, the 

slope at a station will be assumed constant, even 

though some error may thereby be introduced. The 

slope for the reach that includes the cross section 

will be studied in a later part of this investigation 

as a possibly important variable.

There may never be an accurate way to account 

for all channel-shape irregularities in alluvial channels. Some simplifications are unavoidable. Channel 

shape in this study is described by width/depth 

ratios, by the approximate average angle of inclination of the banks, and by certain other geometrical 

attributes of the cross section. These features are 

defined and examined later in this paper.

The hydraulic exponents for wide, flat channels 

(large values of width/depth) should differ from 

the exponents for deep, narrow ones. In a deep, 

narrow channel the water depth increases more 

rapidly with given increases in Q. The exponent of 

discharge associated with depth (hereafter called 

the exponent of depth, with symbol /) therefore 

should be higher in such channels, and ra and b 

should be correspondingly lower.

The angle of inclination of the banks should also 

affect the exponents (Lewis, 1966; Knighton, 1974). 

If the banks are firm and vertical, the width remains constant with change in discharge, and the 

exponent of width (b in equation 3) is zero. At the 

other extreme, very flat banks would allow the width 

to increase considerably for a given increase in discharge, and the exponent of width would then be 

large.

The cohesiveness or the credibility of the bed and 

bank material varies with the degree of consolidation of the particles, the grain sizes and their sizefrequency distribution, the orientation, packing, and 

specific weight of the grains, the electrochemical 

bond between particles, the bulk density of the 

particles, antecedent moisture content, the age of 

the deposit (in many instances), and the water 

temperature (American Society of Civil Engineers, 

1968; Partheniades and Paaswell, 1970; Fisk, 1952; 

Schumm, 1960; Raudkivi, 1967). These factors

exert their influence in various ways, but their net 

result generally appears in the channel shape. For 

example, Friedkin (1945, p. 17) found that deep, 

narrow channels developed where banks were highly 

resistant to erosion. He noted that as bank resistance 

decreased the channels became progressively wider 

and shallower. Schumm (1960) plotted data for 69 

rivers of the western United States and found that, 

for his data, the channel width/depth ratio was about 

inversely proportional to the percentage of silt and 

clay in the bed and banks. Wolman and Brush 

(1961) found that the force required to move the 

grains that make up the banks was a chief determinant of the channel shape. Knighton (1974) concluded that the bank silt-clay content is strongly 

correlated with bank inclination for 12 rivers in 

England. Thus, the way in which bank credibility 

affects hydraulic exponents will be accounted for in 

this study mostly by an evaluation of the shape of 

the channel cross section.

Little is known as to whether the rate of sediment 

transport independently exerts an influence on the 

mean velocity, mean depth and width, and on their 

hydraulic exponents. Evaluation of any such effect 

is beyond the scope of this study. The sediment 

transport rate is generally associated with the channel shape. Channels that carry relatively large quantities of sediment, especially as bedload, tend to be 

wide and shallow. Those that carry small bed loads 

tend to be relatively narrow and deep. Consideration of the channel shape, therefore, may reflect any 

influence of the prevailing sediment transport rates 

on the hydraulic exponents. Also, the sizes of the 

moving particles help determine the kind of bed 

roughness for a given discharge.

The changeable bed roughness is associated with 

the mean flow depth. Because of this interrelation, 

an influence of bed roughness on hydraulic exponents cannot automatically be ruled out. Bed 

roughness depends on the sizes of the bed particles 

(grain roughness) and on the bedforms into which 

the flowing water molds these particles (form 

roughness).

The way in which the sizes of bed particles affect 

the hydraulic exponents is unknown. However, tKere 

is evidence (for example, Hack, 1957) that this 

factor may be important in stream behavior. Therefore, the sizes of the particles on the bed will be 

considered in this investigation.

The form roughness (ripples, dunes, and so forth) 

of alluvial channel beds often changes with dis-
HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

charge in a manner predictable only qualitatively. 

Some investigators believe that the change in water 

depth associated with the change in form roughness for certain limited flow and sediment conditions is a very important problem in alluvial 

streams. The hydraulic data plotted in this study 

cover many types of form roughness; however, the 

hydrographer rarely recorded the bedforms at the 

time of his discharge measurement. Therefore, this 

study will not analyze the role of changing bedform 

roughness on exponent values. The important point 

is that hydraulic exponents were readily definable, 

whatever the kinds of bed roughness. This suggests 

that if changes in depth did result from changes in 

bed-form roughness, such depth changes were not 

significant enough to disrupt the plotted power relations or could not be measured with sufficient accuracy.

A frustrating practical problem on some sandbed streams is the "discontinuous" stage-discharge 

relation that sometimes occurs (Colby, 1960). For 

a given discharge, the stage or elevation of the 

water surface (referenced to a fixed bench mark) 

can vary from day to day because of the sediment 

moving through the reach. Development of a sand 

bar, for example, can cause the water level to rise 

and can disrupt the previous relation of water level 

to discharge. Many of the 165 stations examined in 

the present study have sandy beds. A number of 

these are notorious "shifters"—stations where the 

discharge associated with a given water elevation is 

not constant with time. These stations nevertheless 

had consistent relations between mean depth and 

discharge. Therefore, even though the relation between water level and discharge may change with 

time, the relation between mean depth and discharge may not be significantly affected.

Bank roughness could influence the exponents in 

that rough banks, for example, those with lots of 

vegetation, retard the water flow along the sides of 

the channel and, thereby influence the flow in the 

center. To eliminate this factor, the amount and 

kind of bank roughness should be constant for all 

stations. As this is impractical, some variation in 

observed exponents may be due to differences in 

bank roughness.

A number of features therefore could influence 

the hydraulic exponents. These relevant factors include width/depth ratios, average bank inclinations, 

other channel-shape aspects, channel slope, and the 

sizes and size distribution of bed particles. The 

channel-shape characteristics probably reflect the

cohesiveness of the bank sediments and any influence of prevailing sediment-transport rates and 

types.

MINIMUM VARIANCE

THEORY

Data collected for this study show a wide range 

of possible values for each of the three main exponents (m, /, and 6). What theories can explain 

the various observed exponents? Several different 

hypotheses would be desirable to insure greater impartiality when testing them against the data.

The literature contains at least seven theoretical 

or semitheoretical attempts to predict hydraulic exponents: the theory of minimum entropy production (Leopold and Langbein, 1962), the minimumvariance theory (Langbein, 1964), the minimal 

channel-mobility theory (Tou Kuo-Jen, 1964), the 

similarity principle (Engelund and Hansen, 1967), 

the minimum energy-degradation theory (Brebner 

and Wilson, 1967), the threshold-channel theory 

(Li, 1974), and the conservation-sediment transport 

theory (Smith, 1974). Only two of these—those by 

Langbein (1964) and Li (1974)—deal with the 

at-a-station case.

Li's theory is restricted to streams having gravel 

or bounder bed and banks in small watersheds (less 

than about 26 to 52 km2 or 10 to 20 mi2 ). For 

this limited situation, the exponents are fixed at 

6 = 0.24, /=0.46, and m = 0.30, according to Li's 

theory.

Langbein's approach (Langbein, 1964, 1965; 

Scheidegger and Langbein, 1966) is flexible in regard to the predicted values of the hydraulic exponents, and applies to a wide range of channel 

types. It applies to both the at-a-station and downstream cases. His theory takes a statistical or probabilistic viewpoint and tries to provide the average 

exponents for a group of stations of approximately 

comparable environments. As it is the only theory 

generally applicable to all at-a-station situations, 

Langbein's theory will be explored in detail in this 

paper. No new theories are introduced here.

The philosophy underlying a statistical or probabilitistic approach is that average or most probable relations are all that man can produce 

from theoretical considerations. This viewpoint 

holds that the particular complexities of any natural 

environment, such as the chance emplacement and 

distribution of rocks of various sizes on a stream-
THEORY OF MINIMUM VARIANCE

bed, are so numerous that the hydraulic exponents 

for any one spot can never be confidently and precisely calculated. For this reason, the minimumvariance theory was proposed as a method of forecasting the average relations for a group of rivers, 

with the understanding that any one case may not 

exactly agree with the expectation (see, for example, Langbein and Leopold, 1964). If such a hypothesis comes close to predicting observed relations, it may have the potential of becoming dependable enough to use in practical problems.

Many "laws" of science that have become established are based on the concept of the minimization 

of effects. Examples are Fermat's principle of lightray travel, the principle of least action or least work 

(de Maupertuis or Le Chatelier, respectively), the 

principle of least constraint (Gauss), the principle 

of the straightest path (Hertz), the law of the equipartition of energy (Maxwell and others), the law 

of the survival of the fittest, and, in the business 

world, the law of supply and demand. The basic 

concept is that physical effects in the operations of 

nature, once having reached an equilibrium condition, change as little as possible from then on. In 

other words, a system tends to react to an imposed 

stress so as to minimize the disturbance, or to restore or to keep the previous conditions.

Consider the classical theorem of the equipartition of energy (Resnick and Halliday, 1960, p. 506- 

510). This principle deals with the various kinds 

of energy of a gaseous system—mainly kinetic 

energy of translation of individual molecules, kinetic energy of rotation of individual molecules, 

kinetic energy of vibration of the atoms in a molecule, and potential energy of vibration of the atoms 

in a molecule. These four kinds of energy represent 

different and independent ways in which the total 

energy of the system can be apportioned. An increase in the total energy could be absorbed in 

varying proportions by the four different kinds of 

energy. The theorem of the equipartition of energy 

states that the total available energy of a system 

containing a large number of molecules distributes 

itself in equal shares to each of the various ways 

in which the molecules can absorb energy.

The Langbein (1964) theory applies such a concept to the changes that occur in the variables of 

a river system. In keeping with the minimal principle, the changes in the variables are such that 

the total effect, action, work, or adjustment is a 

minimum. For a cross section on a river, the theory 

would suggest that all variables strive to resist any

imposed change (maintain original equilibrium conditions), with the net result being that all of them 

change equally, insofar as possible. In other words, 

the dependent variables adjust by an equal percentage of their former values, su,bject to the restrictions of the situation. A typical restriction might 

be steep, cohesive banks that prevent the watersurface width from changing significantly as discharge increases.

A number of investigators have noted the applicability of such a minimization principle to alluvial 

channels. Examples are Velikanov (1947, p. 304), 

Mackin (1948, p. 492), Rubey (1952, p. 135), Leopold and Maddock (1953, p. 46), and Bretting 

(1958).

HYDRAULIC EXPONENTS BASED ON THE CONCEPT 

OF MINIMUM ADJUSTMENT

According to the minimum-variance theory just 

described, the problem is to find those particular 

hydraulic exponent values that, subject to any local 

physical restrictions, represent a minimum and 

equal adjustment to a change in the independent 

variable, usually water discharge. The thesis is that 

for these exponent values the sum of the squares 

of the exponents is a minimum. The logic behjnd 

this approach is best seen by analogies, presented 

below. Such a minimization approach is used in 

various branches of science and engineering to solve 

problems of indeterminate systems.

Consider a system with two dependent variables, 

say V and D. The question is how these variables 

will change in response to a new discharge. Select 

a discharge and plot the associated values of log 

V and log D on a graph (fig. 1) obtaining point X-L. 

(Power laws relate the variables, and log units are 

used here for the convenience of working with 

straight Ilines on arithmetic paper.) Now it rains 

upstream, and soon there is a new discharge at the 

station. This forces log V and log D to change by 

some magnitude, producing a new point x2 on figure 1. The changes are Alog V and Alog D. Assuming that neither depth nor velocity will decrease 

with the increase in discharge, point x2 will lie 

somewhere in the quadrant that is bounded by a 

vertical line from x± (zero change in V, entire 

change absorbed by D) and a horizontal line from 

»i (zero change in D, entire change absorbed by V).

Given that the new Q causes the variables to move 

to some new point x2 , the total adjustment can be 

represented graphically by the straight-line distance 

s between xt and x2 . This distance, being the hypote-
HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

logD

AlogD

Alog V

log V

FIGURE 1.—Hypothetical change in two dependent variables, with resultant change represented by resultant 

or hypotenuse s; s2=(Alog Z>) 2+ (Alog V) 2.

nuse of a right triangle, is related to Alog V and 

Alog D by

sz = (Alog F) 2 + (Alog D) 2 . (6)

The change in the independent variable, Alog Q, is

the same for both Alog V and Alog D. Dividing all

terms in equation 6 by the common constant (Alog

Q) 2 gives

(Alog F) 2 + (AlogD) 2

(Alog Q) 2 (Alog Q) 2 (Alog Q) 2 

/Alog F\ 2 /Alog D\2

) \Alog Q) .

(7)

\Alog Q \Alog Q

This equation is, therefore, another way of expressing the amount of adjustment of the system.

The terms in parentheses on the right-hand side 

of equation 7 are hydraulic exponents. For example, 

from Fa Qm we have

Alog V m = ———— .

Alog Q 

Equation 7 therefore can be written as

——-——— = m2 + /2 . (8) 

(Alog Q) 2

The left-hand side of this equation includes only the 

distance s and the constant Alog Q and therefore 

reflects s, the amount of adjustment. Thus, the magnitude of the adjustment, as indicated by the lefthand side of the equation, is proportional to the 

sum of the squares of the hydraulic exponents. Furthermore, least total adjustment occurs when s is a 

minimum, that is, when the sum of the squares of 

hydraulic exponents is the lowest number.

This type of relation can be extended to three or 

more dimensions to include any number of dependent

variables. The least total adjustment for the entire 

system would be that for which the sum of squares 

of the hydraulic exponents is the lowest possible 

number, consistent with any local restrictions (for 

example, vertical rock banks) that may apply. 

Knighton (1977) derives this same principle from 

an entirely different viewpoint, namely Euclidean 

space.

Another analogy is the determination of the center of gravity of a two- or three-dimensional group 

of points. The center of gravity is that point from 

which the squared distances to all other points add 

up to the lowest number.

Examples that Langbein (Scheidegger and Langbein, 1966, p. C7) gave wherein the minimum sum of 

squares of changes corresponds to least total work 

are the displacement of joints in a truss and the 

distribution of the QH products (H being head loss) 

in a network of pipes.

The analogies thus far have suggested that minimizing the sum of the squares of the hydraulic exponents corresponds to least total change for the 

system as a whole. In addition, minimizing the 

sum of squares of exponents corresponds to an equal 

division of any change in an independent variable. 

Consider, for example, the analogy used in figure 1, 

whereby the entire increase in discharge was absorbed by V and D. The total adjustment (length of 

hypotenuse s of the triangle whose other sides are 

Alog D and Alog V in fig. 1) is greatest when the 

hypotenuse becomes equal to either Alog D or Alog 

V, such that the other is zero. As Alog D becomes 

more equal to Alog V, the hypotenuse (magnitude 

of adjustment) decreases, and it reaches its minimum value when Alog D = Alog V. In other words, 

the smallest net adjustment occurs when the dependent variables divide the imposed change equally 

amongst themselves.

A second example of this principle involves the 

many possible ways in which velocity, depth, and 

width can adjust to an increase in discharge. Consider the basic relation of the exponents, m+f+b 

= 1.0. For the hypothetical situation where no other 

variables are involved and where all three dependent 

variables ( V, D, W) are unrestricted, the minimum 

sum of squares for all possible combinations of m, 

f, and b occurs when m=/=& = l/3. Mathematically, 

this sum of squares is (.33) 2 + (.33) 2 + (.33) 2 

= 0.3267. Trial calculations show that any other 

values of m, f, and b, where m+/+& = 1.0, produce 

a sum of squares that is greater than this minimum. 

For example, if m-0.72, /=0.21, and 6 = 0.07, the 

sum of squares is 0.5674, which is greater than than
THEORY OF MINIMUM VARIANCE

0.3267. Thus the variables absorb the change in Q 

equally when the squares of their exponents add up 

to the lowest number. Note also that the sum of 

squares of the exponents m, f, and b is a maximum 

(1.0000) when the change is wholly concentrated in 

one of the three dependent variables, such that the 

other two remain constant and their exponents are 

zero. A river would be least likely to adjust by 

changing in this manner, and it seems significant 

that the sum of squares of exponents is furthest 

from being a minimum for such a situation.

The concept of minimizing the sum of squares of 

changes or deviations is very similar to the wellknown and widely accepted least-squares method 

for finding the best-fit relation to a group of values. 

The least-squares method can be used to approximate the "best-fit" relation not only to two variables 

or dimensions but also to three or more variables. 

Multiple regression and trend surface analysis are 

two examples.

What is the meaning of the term "variance" as 

used in Langbein's theory? Any group of water 

depths, mean velocities, and other values for a station can be analyzed statistically; for example, in 

regard to the arithmetic average of the depths, the 

standard deviation of those depth values, the variance (the square of the standard deviation), and 

so forth. The section "Statistical Variance and a 

Hydraulic Exponent" at the end of this report shows 

that if we take all the mean velocities measured for 

a station, list the logarithms of these velocities and 

compute the variance of this group, such a variance 

will be proportional to the square of the hydraulic 

exponent m (the exponent of velocity). The same is 

true of the other dependent variables and their respective exponents. Thus the square of a hydraulic 

exponent is proportional to the variance of the logs 

of the dependent variable, where variance is defined 

as the square of a standard deviation, as in normal 

statistical usage. Because of this close relationship 

and because the word "variance" is a convenient 

term which is already established in connection with 

the present theory, "variance" is used as a replacement for the more accurate phrase "square of hydraulic exponent." Thus, from DxQf the square 

of the exponent (f2 ) is called the variance of depth. 

The process of finding those exponents whose 

squares add up to the lowest possible number is 

termed "minimizing the variances." This concept 

of minimum variance is similar to, but not exactly 

the same as, the minimum variance of conventional 

statistics.

COMPUTATION OF MINIMUM VARIANCE

In calculating the most probable exponents in the 

hydraulic geometry relations, the simple laws of 

exponents apply. When two quantities are multiplied, their exponents are added ; when one quantity 

is divided by another, the exponent of the latter is 

subtracted from that of the former. Shear stress 

in wide channels (T), for example, is proportional 

to depth times slope. Substituting equations 2 and 4 

into this expression, we have shear ccDSccQfQ* 

<xQf+s. The variance of shear ("var shear") therefore is expressed by squaring the sum of the exponent of depth (/) and the exponent of slope (z), 

that is, var shear =(/+z) 2 . The Darcy-Weisbach 

friction factor (ff) in wide channels equals 8gDS/V2 , 

where g is acceleration due to gravity. This friction 

factor is proportional to depth times slope divided 

by the square of velocity, or ff<x'QfQ*/Q2m, and var

Because m + /+6 = 1.0, the variance of width, 

62 , can be written in an alternate way as (1 -m-f) 2 . 

Similarly, f=l — m — b and^ m = I — b — f. Also, one exponent may be known to have a certain relation 

to another, such as 6 = 0.25/. In the latter case, the 

quantity 0.25/ would be substituted for b, and the 

variances can all be written in terms of /. Finally, 

one exponent may be known in advance, permitting 

all variances to be written in terms of just one unknown. For instance, if & = 0.10, then m + /=0.90 

and m = 0.90-/. Substituting 0.10 for b and 0.90 -/ 

for m allows all unknown variances to be written 

in terms of /. These many alternative ways of writing variances are used extensively to reduce the 

number of unknowns in the analysis.

Some factors may vary over a certain restricted 

range or are fixed so they do not change at all 

(remain constant). Such limitations are known as 

constraints. A constraint is merely a physical condition that governs the extent to which a variable 

can change. Examples are the fixed, constant width 

in flume experiments and the constant slope typical 

of certain reaches of some natural rivers and 

streams. The importance of a constraint is that it 

limits or otherwise influences the values that other 

variables can take. Constraints are usually present 

in nature, and in most cases they preclude a perfectly uniform distribution of any change in an 

independent variable. Partly for this reason, we 

rarely find a case where m = / = 6 = 0.33.

One way of determining those particular values 

of the exponents that provide the minimum sum of
8 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

variances is by trial and error. A case that Langbein (1965) used illustrates this method. Consider 

a river cross section where the banks are rigid and 

vertical so that the width is constant. The watersurface slope at some stream stations does not 

change significantly for a range of flow conditions 

(Leopold and Haddock, 1953, p. 36), so slope will 

also be considered constant for this example. Let 

the bed of the channel consist of unconsolidated 

sand, so that the channel resistance is adjustable. 

Now an increased water discharge arrives at the 

station. The question is how this and other increases in discharge will be reflected among the 

various pertinent dependent variables.

With the width constant, mean velocity and depth 

must absorb all of the new discharge. Changes in 

such factors as bed shear stress will also occur. 

Assume for illustrative purposes that this bed shear 

and the resistance to flow (friction factor), along 

with velocity and depth, are the only important dependent variables whose exponents are not already 

known. The problem is to find the values of the 

hydraulic exponents of these dependent variables.

The first step is to identify the variances of the 

dependent variables. In so doing, we must consider 

the constraints of constant width and constant slope. 

Because Q = VDW, or Q1 = QmQfQ° (width constant), 

this continuity expression specifies that w + /=1.0. 

The variance of velocity, m2 , can therefore be written in an alternate way as (I — /) 2 . Shear in wide 

channels is ordinarily proportional to depth times 

slope or QfQs, so that the variance of shear would 

normally be (f+z) 2 ; however with slope constant its 

exponent z = 0, shear stress varies only with depth, 

and the variance of shear is simply equal to /2 . The 

friction factor, proportional to DS/V2 , is proportional to D/V2 or Qf/Q2m when slope is held constant. 

The variance in friction factor, being by definition 

the square of the exponent relationship, is, therefore, (/-2m) 2 . Substituting w = l-/ in order to put 

this expression in terms of /, as the variances of the 

other variables are expressed, then gives the var 

ff as (3/-2) 2 . The variances of the selected dependent variables, when width and slope are constant, are therefore

velocity 

depth 

shear 

friction

(I-/) 2

P

f2

(3/-2) 2

(For this initial example, it has been possible to 

write all variances in terms of one unknown, /. Although this will not be possible in many other cases, 

the alternative ways of writing variances should be 

used whenever possible in order to reduce the number of unknowns involved in the computation.)

The goal is that value of / for which the sum of 

all variances is a minimum. The theory holds that 

only under such a condition is the change in Q 

distributed as uniformly as possible among the dependent quantities. Suppose we guess that the exponent /—the rate of change in depth with change 

in discharge—is 0.3. We then compute the variances 

of each of the four dependent variables. Var vel, for 

example, is m2 = (l-/) 2 = (1.0-0.3) 2 = 0.49. (See 

column 3 of table 1.) After the separate variances

TABLE 1.—Trial and error example of finding f for which 

the sum of variances is a minimum

Variable Variance

Computed variance for value of /

0.3 0.4 0.5 0.6 0.7 0.58

Velocity ___ (

Friction -.--.(

!-/)'

f

f2

3/-2;

' ----- 0.49

____ .09

____ .09

I 2 __ 1.21

0.36 

.16

.16

.64

0.25 

.25

.25

.25

0.16 

.36

.36

.04

0.09 

.49

.49

.02

0.176 

.336

.336

.068

Sum of variances — l.i 1.32 1.00 .92 1.09 .916

are each computed in this manner, they are summed. 

Table 1 shows that when /=0.3, the sum of the 

variances is 1.88. Is this the minimum sum attainable? Continuing the trial and error method, the 

sum of variances turns out to be the lowest number 

when /=0.58. In other words, the change in Q is 

accommodated as equally as possible by the four 

major dependent variables when /=0.58, that is, 

when the rate of increase in log depth with increase in log Q is 0.58. The exponent of velocity, m, 

is therefore m = l-/=0.42. Width and slope are 

already known to be constant, so b and z both equal 

zero. Shear increases directly with depth, so the exponent of shear is 0.58 in this case. The rate of 

change in friction factor with increase in Q is equal 

to Qf/Q2m or Q~° 26 . The friction factor thus varies 

with the —0.26 power of discharge.

Such trial and error computations are laborious. 

Fortunately there is a quick, easy method of arriving at the answer /=0.58. The goal is that value 

of / that takes all of the variances, when added together, to a minimum. Therefore, add the variances 

figuratively and designate that their sum goes to a 

minimum : 

var vel + var depth + var shear

+ var friction-»minimum

(3/-2) 2-»minimum 

Squaring as indicated gives

(1-2/+ /2 ) + /2 + /2 + (9/2 -12/+ 4) -^minimum.
THEORY OF MINIMUM VARIANCE

Collecting the like terms:

12/2 -14/+ 5->mi«imum.

The value of /, representing the slope of the depthdischarge relation, is obtained by taking the first 

derivative. Because the function goes to a minimum, 

the derivative according to the rules of basic calculus is set equal to zero. Setting the first derivative equal to zero gives

24/-14 = 0 

24/=14 

/ = 0.58.

(In fact, setting the first derivative equal to zero 

can give either a minimum or maximum value for 

a function, but in this case, the foregoing trial-anderror procedure has shown that /=0.58 is the minimum value. Taking the first derivative for all other 

minimum variance computations also gives a minimum value, as can be verified by taking the second 

derivative. If the second derivative of a function 

is positive, then the first derivative has produced 

the minimum value. The second derivative of all 

minimum variance relations is alway positive.)

The summarized steps in the minimum variance 

computation are:

1. Determine an independent variable and write 

all other pertinent variables (dependent or 

constant) as power functions of the independent variable. Thus, if Q is independent, 

then V<xQM, DxQf, slope <xQz, shear <xDS 

ocQ/Q*, etc.

2. Define the variances (the square of the exponent) of all variables, using the same unknowns insofar as possible. Thus, for example, 

with m + /+6 = 1.0 and with m and / necessarily involved, express var width as (1-m 

-/) 2 rather than b2 .

3. Write the variances in a group and designate 

that the sum of the variances goes to a minimum.

4. Square any "compound" variances (a variance 

consisting of a sum of letters and possibly 

numbers) and collect like terms for the complete group.

5. Set the first derivative equal to zero and solve. 

If more than one unknown is present after 

step 4 is completed, set the derivative of each 

unknown equal to zero in turn while holding 

any other unknowns constant. Then solve the 

resulting equations simultaneously for the 

values of the unknowns.

Taking a more complex example, again presented 

only to demonstrate the method of computation, consider a reach of stream where width and slope are 

free to vary rather than being constant. The bed 

is deformable, as with the last example. Suppose a 

change in Q, as the independent variable, will be reflected in the dependent variables velocity, depth, 

width, bed shear, and frictional resistance. According to the minimum variance theory, what are 

the most probable hydraulic exponents in the power 

equations that relate these factors to discharge?

Each of the steps just listed is applied, in turn, 

to obtain the expected hydraulic exponents. Writing 

the variables as power functions of Q (step 1) and 

defining the variances (step 2) :

Step 1 (relations)

SacQ*

Step 2 (variances) 

1

m2

(l-m-f) 2

z2

x Qf+z-*» (f+z-2m) 2

Writing the variances (of V, D, W, T and ff for this 

example) as a group and designating this sum to be 

a minimum (step 3) :

var vel + var depth + var width + var shear

+ var friction factor^minimum

m2 + f- + (I-m-f)* + (f+z) 2

+ (f+z-2m) 2 -^minimum

Squaring the compound variances and collecting like 

terms (step 4) gives

6m2 + 4/2 + 2z2 +1 - 2m-2f-2mf

+ 4/2 — 4mz-»minimum.

Next the first derivative with respect to m is set 

equal to zero, holding / and z temporarily constant 

(step 5). This gives

12m-2/-4z-2=0.

Then the first derivative with respect to / is set 

equal to zero, holding m and z constant:

-2w + 8/+4z-2 = 0.

Finally, holding m and / constant, setting the first 

derivative with respect to z equal to zero gives

In this manner, three equations are obtained 

(these last three, for this example) which are 

solved simultaneously for the three unknowns. For 

the present example, this solution produces m = 0.14,
10 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

/=0.43, and z = 0.29. Since m + /+5 = 1.0, 6 = 0.43. 

Velocity, therefore, varies as Q°- 14 ; mean flow depth, 

D, varies as Q°-43, and so forth

INFLUENCE OF CHOICE OF VARIABLES

The present theory, like many statistical theories, 

unfortunately does not tell a person what variables 

to use in the computation. And the variables included in (or excluded from) the analysis have a 

strong effect on the values of the predicted exponents. For example, with width constant, velocity and depth absorb a change in Q; whereas, 

if width also takes up some of the change, then 

velocity and depth will change by lesser amounts, 

and their exponents will be smaller. This same principle applies to other variables such as shear stress 

and stream power. If such variables absorb some of 

the change in flow conditions, then the predicted 

exponents are affected. So a very important problem is the question of which variables to include in 

the analysis.

A simple example will show the extent to which 

the predicted exponents can vary with different sets 

of variables. Suppose the variables likely to be important are the discharge (independent), velocity, 

depth, friction factor, and shear stress. (All other 

factors constant or not influential.) The relations 

and variances are:

velocity ocQm

depth ocQ'

shear «Q/Q°ocQ/

friction <xQfQ0/Q2m <xQf~ 2m

m2

f2 or (1-ra) 2

f2 or (1-ra) 2

(/-2ra) 2 = 9ra2 -6ra + l

Table 2 shows the predicted exponents for all possible combinations of variables. As far as the grouping of variables is concerned, this table is not based 

on sound hydrologic reasoning nor on any theory— 

it merely lists in a systematic manner the different 

possible ways of combining two or more of the four 

dependent variables.

Each exponent in table 2 has a considerable range 

of values, depending on which particular set of 

variables is involved in the minimization. The range 

of ra, for example, is from 0.30 to 1.0, and the exponent of the friction factor ranges from —2.00 to 

+ 0.10. Some values of exponents can be produced 

by more than one combination of variables (group 

nos. 1 and 2, 5 and 6, and 8 and 9), because slope 

is constant and hence shear stress varies directly 

with depth.

TABLE 2.—Rate of change of dependent variables with increase in discharge, for different sets of variables

Group 

No.

1

2

4

5

6

7

8

9

10

11

Dependent 

variables

V, D

V, T

V, ff

D, T

D, ff

T, ff

V, D. T

V, D, ff

V, r.ff

D, r, ff

V, D, T, ff

V 

Velocity

(m)

0.50

.50

.30

1.00

.40

.40

.67

.36

.36

.45

.42

Values

D 

Depth

(./)

0.50

.50

.70

.00

.60

.60

.33

.64

.64

.55

.58

of exponents

T

Shear

(/)

0.50

.50

.70

.00

.60

.60

.33

.64

.64

.5&

.58

ff 

Friction 

factor

(/— 2w)

— 0.50

— 0.50

.10

— 2.00

— .20

— .20

— 1.00

— .08

— .08

— .35

— .26

In view of the wide range of mathematically possible exponent values, how does one know what combination of variables to use in the minimum variance 

analysis ?

In his original paper (1964), Langbein dealt with 

the components of stream power, QS, or velocity, 

depth, width, and slope as functions of discharge. 

Subsequently, (Langbein, 1965; Scheidegger and 

Langbein, 1966), for at-a-station cases, he added 

shear and friction factor to the group. He stated 

(1965, p. 304) that "there might be questions as 

to the proper variables, but these are used for consistency in the several examples." In the 1966 paper 

(p. C8), he suggested that a large set of problems 

can be explained by using as dominant factors the 

width, hydraulic radius, velocity, shear, and friction 

factor. Actually, depth was used in place of hydraulic radius in all the examples given. Langbein 

(1965) chose the combination of V, D, W, T, and 

ff for at-a-station situations because minimizing the 

variances of these variables produced exponents 

closest to those of the few case samples he cited.

In both the 1965 and 1966 papers, Langbein 

treated the downstream cases—rivers and straight 

canals—as special or different from the at-a-station 

situation. For downstream cases, he minimized the 

variances of five different aspects of stream power.

The problem of selecting the correct combination 

of variables is very similar to that encountered in dimensional analysis, where you have a list of dimensionless terms and must decide which terms are insignificant. The variables used in earlier pages of this 

paper (shear, friction factor, and so forth) may or 

may not be the most important variables, and, unfortunately, the concept of least total adjustment by 

itself cannot suggest the most important factors. 

Two possible solutions are (1) introduce another
THEORY OF MINIMUM VARIANCE 11

theory to show which variables should be involved, 

and (or) (2) compare actual field data to the exponents predicted by various combinations of variables. Any "proof" of the minimum variance theory, 

in fact, must include the latter sort of comparison, 

and it is the second approach that will be followed 

in this paper. If the predictions for a given group 

of variables match the field observations, the correct 

or best combination of variables has been found and 

the theory acquires a certain degree of reliability. 

On the other hand, if no combination of variables 

produces exponents close to the field data, the minimum-variance theory could then be said to lack a 

firm factual basis.

An alternative to dealing with a standard set of 

variables is to use different combinations of variables for various hydraulic situations (Haddock, 

1969). The difficulty with this procedure is knowing 

which variables to use for any given situation 

(Dozier, 1976).

TEST OF MINIMUM-VARIANCE THEORY WITH 

FIELD DATA

COLLECTION OF DATA

The U.S. Geological Survey has for many years 

measured the water discharge at selected stream 

stations throughout the United States. In making 

such measurements, the hydrographer obtains data 

that yield V, D, W, Q, and the shape of the channel 

cross section. Records of this sort afforded an opportunity to test the theory against actual data. 

Although the Geological Survey monitors thousands 

of stations, the selection criteria dictated by the 

present study eliminated the vast majority of sites. 

These criteria were that the stream have (a) a 

movable bed (silt, sand, gravel, cobbles, and (or) 

boulders), (b) no artificial or natural control or 

apron on the bed at the measuring site, (c) negligible influence on flow variables from bridge piers, 

(d) no history of wide-scale dredging, (e) no dam 

immediately upstream or any other feature that 

caused observable net degradation or aggradation 

along the gaging station reach, (f) no extremely 

heavy bank vegetation significantly affecting the 

flow range interest, and (g) a range of discharge 

preferably encompassing at least one log cycle over 

which the plotted hydraulic data showed well-defined power relations.

When making a discharge measurement, the hydrographer commonly walks as much as several 

hundred feet upstream or downstream from the 

gage to find a cross section that is easy to wade. 

The records for a given station therefore, often

consist of measurements made at many cross sections along a reach. Unless the channel shape remains approximately constant along the entire 

reach, the plots will show a certain amount of 

scatter attributable to the variety of measuring sites 

(Wolman, 1955, p. 11; Lewis, 1965, p. 12-13). It 

was therefore necessary to inspect the original 

streamflow measurement notes and, for each gaging 

station, to select only data taken at the same cross 

section. In almost all cases, this task was accomplished by accepting data taken either at the same 

wading site or from cableways, as recorded in the 

hydrographer's field notes. In rare instances, the 

channel shape was sufficiently constant along the 

reach to permit the use of data taken anywhere 

along the reach. The requirement that measurements 

be made at the same section eliminated many of the 

stations that had passed the seven criteria listed 

in the previous paragraph.

Table 11 (appended to the end of this report) 

lists the 165 stations finally accepted for analysis. 

For each of these stations, I plotted values of W, D, 

and V versus Q on log paper and measured the hydraulic exponents graphically.

The 165 stations were chosen to represent many 

physiographic regions. Included are streams in different climates, soils, lithologies, and types of landscape; streams, with beds of very small grains and 

streams with large cobblestones and even boulders 

on the bed; streams a few feet wide and streams 

many hundred feet wide, with a correspondingly 

wide range of typical discharges, depths, and velocities ; and streams with banks ranging from firm and 

steep to rather flat and easily erodible. Some of the 

selected stations are on ephemeral reaches.

The observed exponents range from 0.00 to 0.82 

for b, 0.10 to 0.78 for /, and 0.03 to 0.81 for m. The 

channel widths, or rather the values on the hydraulic-geometry plots, range from 0.31 m (1 ft) 

(minimum width on W versus Q relation) to a maximum of about 579 m (1,900 ft). Depths range from 

about 0.031 m (0.1 ft) to 10.7 m (35 ft). The 

smallest bed-material size (median diameter) is 

0.06 mm, and the largest is about 100 mm (table 

11). The lowest discharge on a hydraulic-geometry 

power relation is 0.000283 m3 /s (0.01 ft3 /s) (Belle 

Fourche River below Moorcroft, Wyo.) and the 

highest is about 1,980 m3 /s (70,000 ft3 /s) (Skagit 

River near Mt. Vernon, Wash.)

The period of time covered by the various discharges on any one plot averaged about 3 to 5 yr 

and ranged from about 1 to 17 yr, depending mainly
12 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

on how often the more extreme discharges flowed 

and on how frequently the different flows were 

measured.

SOURCES OF ERROR

The basic data (Q, V, D, and W) for every station involve a certain amount of measurement error 

that could affect the hydraulic exponents. A discharge measurement is made by observing the total 

depth and the velocity at one or more intermediate 

depths, at each of many successive verticals across 

a stream (Buchanan and Somers, 1969). The mean 

velocity and an applied cross-sectional flow area at 

each vertical give the discharge for a subsection, 

and the various subsection discharges are summed 

over the entire stream width to get the total discharge. Carter and Anderson (1963) analyzed and 

discussed in detail some of the sources of error in 

such measurements. They concluded that with normal stream-gaging procedures, the errors due to the 

instrument and the general current-meter method 

would be about ± 2 percent or less in most cases and 

would be randomly distributed. However, in natural 

streams, such features as water waves and bed 

forms, especially in shallow depths and at extremely 

high and low discharges, add to the errors involved. 

The hydrographer estimates the accuracy of each 

discharge measurement by rating it excellent (2 

percent error), good (5 percent), fair (8 percent), 

or poor (over 8 percent), according to the flow-, 

channel-, and instrument characteristics at the time 

of the measurement. Nearly all of the data in the 

present study were rated good or fair. The error 

in the plotted Q-, V-, and D-values, in other words, 

probably is no more than about 8 percent in most 

cases. Water-surface widths could be measured quite 

accurately and should have a negligible error for 

the present purposes.

Trial calculations with typical best-fit lines indicated that errors of 8 percent in the measurement 

of Q, V, and D could cause a difference of about 20 

percent in the exponents of velocity and depth if 

many errors at each end of the best-fit line happened 

to be distributed so as to cause the maximum possible deviation in slope of the line. Similarly, 

the exponent of width could have a maximum of 

about 5 percent error. However, measurements of 

Q, V, D, and W have an equal chance of being off on 

either side of their true value, and, therefore, such 

measurement errors should tend to offset one another over a number of observations. Hence, any

errors in the exponent values due to errors in measuring Q, V, D, and W probably are not significant.

The number of points on any one plot ranged 

from 10 to 105 and averaged 30 per graph for the 

entire study.

The lines on the plots of velocity, depth, and 

width versus discharge were, in most cases, fitted 

by eye. Those for stations 15, 41, 48, and 69 were 

fitted by least squares.

Figures 2 and 3 show the plots for two of the 

stations. These examples were selected because they 

represent approximately the least scatter (fig. 2) 

and the most scatter (fig. 3) for the 165 stations 

studied.

The amount of scatter on all 495 graphs (3 graphs 

for each of the 165 stations) was measured and is 

expressed in table 11 as an approximate percentage 

of the best-fit-line value of the dependent variable 

for any given discharge. The percentages encompass 

about 90 percent of the total number of plotted 

points for each station and were determined as 

follows.

On each hydraulic-geometry graph, two lines parallel to the best-fit line were drawn. One line excluded the 5 percent of the points having the 

greatest positive or upward departure from the 

best-fit line, and the lower line excluded the 5 percent of the points having the greatest downward 

departure from the best-fit relation. These two 

parallel lines thus included about 90 percent of the 

total number of plotted points. (This figure was 

selected in order to exclude the occasional outliers.) 

For any given discharge, the value of the dependent 

variable, as indicated by the upper line, was read 

and expressed as the percentage of the best-fit-line 

value by which it exceeded the latter. The corresponding percentage for the lower line is the percent of the best-fit line value by which the lower 

line falls below the best-fit line. These two percentages are both listed for each graph. They indicate the approximate range of percentage within 

which nearly all (that is, about 90 percent) of the 

plotted points fell, relative to the best-fit line. For 

example, a listing of 82/51 for velocity means that 

90 percent of the plotted points fall between two 

parallel lines that are, respectively, 82 percent above 

and 51 percent below the best-fit line, at any given 

discharge. Incidentally, the popular correlation coefficient proved unsuitable for the present purpose 

because it varied with the slope of the line and 

the extent of the discharge range, as well as with 

the scatter. A line with a low slope gave a low
to WATER-SURFACE WIDTH,

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14 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

1.0

DISCHARGE, IN CUBIC FEET PER SECOND 

10.0 100.0

0.1 1.0 

DISCHARGE, IN CUBIC METERS PER SECOND

10.0

FIGURE 3.—Hydraulic-geometry plots for Prairie Dog Town Fork Red River near Childress, Tex. (station 60, this 

study). Dashed lines include about 90 percent of the plotted points.

correlation coefficient, which presumably implies a 

poor relation, even though all of the plotted points 

fell virtually right on the best-fit line.

The scatter on the log-velocity versus log-discharge graphs ranged from 2 to 110 percent with an 

arithmetic mean of 23 percent. For depth versus

discharge, the range was from 2 to 120 percent with 

a mean of 24 percent. And for width versus discharge, from 1 to 110 percent with a mean of 21 

percent.

The plots shown in figure 2 have scatter of about 

3/4 percent for velocity, 5/4 percent for depth and
THEORY OF MINIMUM VARIANCE 15

2/2 percent for width. On figure 3, the scatter is 

about 87/50 percent for velocity, 78/34 percent for 

depth, and 110/53 percent for width.

The final source of analysis error is the error due 

to fitting the lines by eye rather than by least 

squares (assuming least squares would be the best 

possible way). To evaluate this error, the exponents 

for 10 randomly selected stations were computed by 

least squares and compared to the corresponding 

eye-drawn exponents, the latter having been determined first in all cases.

For the 30 test plots (10 stations) examined, the 

maximum discrepancy between an exponent determined by least squares versus one fitted by eye was 

±0.03 exponent units (table 3). The average absolute error or discrepancy, computed without regard to the sign, was about 0.01 exponent units for 

each of the three exponents. Drawing lines of best 

fit by eye, therefore, was reasonably accurate and 

reliable.

TABLE 3.—Comparison of exponents determined by least 

squares with those fitted by eye

Station

Least-squares 

exponents

Eye-fitted 

exponents Discrepancy

1 „

16 _.

31 ..

47 —

64 _.

81 __

98 ..

115 ._

132 ._

149 ._

Average

absolute

error

m

0.51

.34

.52

.36

.42

.20

.25

.48

.35

.71

/

0.42

.59

.49

.42

.53

.16

.14

.50

.26

.24

6

0.07

.07

.01

.22

.06

.67

.60

.04

.39

.05

m

0.51

.35

.52

.36

.40

.18

.25

.47

.37

.72

/

0.43

.60

.48

.41

.56

.15

.13

.50

.23

.24

b

0.07

.08

.01

.21

.05

.65

.61

.02

.40

.04

m

0.00

.01

.00

.00

—.02

.02

.00

—.01

.02

.01

.009

/

0.01

.01

—.01

—.01

.03

.01

—.01

.00

—.03

.00

A,-! 9

*\JL£t

b

0.00

.01

.00

—.01

—.01

.02

.01 —.02

.01

—.01

.010

COMPARISON OF MEASURED TO THEORETICAL 

HYDRAULIC EXPONENTS

A number of variables, as discussed earlier, might 

have some influence on the hydraulic exponents. 

However, Langbein's papers suggest that most such 

factors cannot be taken into account individually in 

a minimum-variance analysis because their effects 

usually cannot be determined separately. He believes 

that in spite of the interaction and net influence of 

such variables, there will result in nature a statistical array of exponent vaues in which certain values 

(the averages) are more common. These most common values of m, /, and b represent a central tendency, and the correct combination of variables is 

that for which the minimization of variances yields 

the most common exponents.

Five cases, including four that Langbein (1965) 

specified, are tested here. The first case deals with 

stations where the water-surface width is approximately constant. The banks are firm and are steep 

enough to prevent the width from changing significantly with discharge. This category was arbitrarily 

defined as including all stations for which 6^0.03. 

These stations, a total of 22, are labeled "A" in 

table 11. For the second case, the banks again are 

firm but, in this case, they are not steep enough to 

keep the water-surface width approximately constant (76 stations). In table 11, these stations are 

labeled "B". Also, a few are labeled "R" for rock 

banks, or B/R for one rock bank and one firmsediment bank. The third case (labeled "C") includes channels where the entire flow boundary is 

loose and easily eroded. The present study has 16 

such stations. A fourth case (B/C, C/R) consists 

of the 51 stations having one firm bank and one 

noncohesive bank. Langbein (1965) did not examine 

this case.

In all four of the above cases, the energy gradient, 

or water-surface slope, is specified to remain essentially constant as discharge varies. (On some 

reaches, this may be only approximately true and 

may account for some unknown amount of discrepancy between predicted and observed results.)

The fifth case (D) consists of stations where 

the entire flow boundary is loose and readily erodible and the water-surface slope varies with discharge. None of the present stations were measured 

for slope changes and are here considered to have 

virtually constant slopes. However, some flume data 

from other studies are available for case D.

Classification of general bank firmness for each 

station was done subjectively from U.S. Geological 

Survey unpublished descriptions of the channel and 

banks at each station and from verbal descriptions 

given by Geological Survey hydrographers.

It is first necessary to find out whether a central 

tendency of exponent values exists for each of these 

cases. The few data for case D are insufficient to 

determine any clustering or distribution trend. For 

the other four cases, figure 4 summarizes the field 

data collected. The exponents for most cases have 

quite a wide range of values. However, with the 

possible exception of the exponent b for B/C stations, the distribution of exponent values for a 

given case and exponent does show an observable 

peak. Thus, a central tendency does exist for each 

exponent.

Some of the distributions in figure 4 are not symmetrical. However, plotting the data on both logprobability and arithmetic-probability paper showed 

that the distributions tend to be much more sym-
PERCENT OF STATIONS HAVING INDICATED VALUE OF EXPONENT

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THEORY OF MINIMUM VARIANCE 17

metrical on arithmetic than on log scales. (The 

only exception was case B, B/R, and R for the 

exponent b, where all ^-values < 0.04 were arbitrarily eliminated beforehand for use in case A.)

In practice, the average exponent for a number of 

field stations may not represent the true central 

tendency, depending on the number of stations for 

which data have been acquired and on how close 

to the "ideal" case these stations are. As a first 

approximation, however, the average of a given exponent for the stations of each category should be 

reasonably representative. Two averages — the arithmetic mean and the mode — were used for all cases.

The next questions are, can the minimum-variance 

theory predict these average or most common exponents and, if so, what combination of variables 

provides those average values ?

CASE A

The simplest at-a-station situation consists of 

cross sections at which both width and slope are 

approximately constant. As far as V, D, and W are 

concerned, any change in discharge is absorbed only 

by V and D. The exponent b, therefore, is approximately zero (taken as ^ 0.03 for present purposes), 

and ra + /«1.0. The field data (table 11) have means 

of /=0.52 (modal / = 0.55) and ra = 0.47 (modal 

w = 0.46) for the 22 applicable stations. Standard 

deviations are 0.13 and 0.13 exponent units for m 

and /. What group of variables, if any, predicts 

these average exponents in a minimum-variance 

computation ?

Table 4 lists the theoretical exponents produced 

by minimizing the variances of various combinations 

of variables. The variables included in this table (V, 

D, T, ff and VS, with width and slope constant) are 

those which can vary and which are deemed most 

likely to be important — a subjective but unavoidable 

approach. (QS and QS/W, proportional to stream 

power and stream power per unit bed area, respecTABLE 4. — Theoretical rates of change of dependent quantities with increase in discharge, for different sets of variables

I'Case A: width and slope constant. Average field values: m = 0.47 (mode 

0.46); /=0.52 (mode 0.55)]

Value of exponents

NnmhpT- Dependent Number variables

1 

2 

3 

4 

5 

6 

7 

8

V, D 

V, D, T 

V, D, ff 

V, D. VS 

V, D, T, ff 

V, D, T, VS 

V, D, ff, VS 

V, D, T, ff, VS

V

Velocity 

(m)

0.50 

.67 

.36 

.33 

.43 

.50 

.33 

.38

D 

Depth 

(/)

0.50 

.33 

.64 

.67 

.57 

.50 

.67 

.62

r 

Shear 

(/)

0.50 

.33 

.64 

.67 

.57 

.50 

.67 

.62

ff Resistance 

(/-2m)

—0.50 

— 1.00 

— .08 

,00 

— .29 

— .50 

.00 

— .14

VS 

Power 

per 

unit 

weight 

(m)

0.50 

.67 

.36 

.33 

.43 

.50 

.33 

.38

tively, may also be important and will be considered 

later. For this case these two variables both vary 

directly with Q and do not influence the computed 

exponents.) VS is proportional to stream power per 

unit weight of water. Yang (1972) believes this 

variable governs the nature of a stream network, 

the formation and behavior of meanders, the river 

profile, the formation of riffles and pools, and the 

rates of sediment transport.

Inspection of the predicted exponents shows an 

expected range of values for each exponent. No 

combination in table 4 exactly corresponds to the 

field averages of m = 0.47 or 0.46 and /=0.52 or 0.55. 

Three groups (numbers 1, 5, and 6 in the table) 

produce exponents rather close to these values: (a) 

V and D, the minimization of which yields ra = 0.50 

and /=0.50; (b) V, D, T and ff, the minimization 

of which yields m = 0.43 and /=0.57; and (c) V, D, 

r and VS, for which m = 0.50 and /=0.50. The evidence for stations having constant width and slope 

therefore suggests that if the minimum-variance 

theory is valid, one of the three combinations just 

listed may be the basic or best group of variables. 

For more general cases, these groups could also include one or more of W, S, QS, and QS/W, factors 

that are constant or noninfluential for case A. In 

fact, since the width W always absorbs some of a 

change in discharge (except, of course, where the 

banks are firm and vertical), width must be included 

as an indispensable variable. Thus, if minimizing 

the variances of one group of variables applies to 

all at-a-station situations, field data for case A 

narrow the choice to three general groups: (a) 

V, D, and W, possibly with S, QS, and (or) QS/W] 

(b) V, D, W, T, and ff, possibly with S, QS, and (or) 

QS/W; or (c) V, D, W, T, and VS, possibly with 

S, QS and (or) QS/W.

CASE B

Case B consists of stations with cohesive but nonvertical banks with the water-surface slope again 

constant. The water-surface width, in other words, 

increases with increase in discharge but is controlled 

or constrained by the firm banks. The rate of change 

of width depends, at least in part, on the shape of 

the cross section or angle of the banks at various 

flow stages. Because of this control which the firm 

banks exert on the exponent of width b, Langbein 

(1964) felt that some relationship reflecting such 

control should be incorporated into the minimumvariance analysis. For this case, he introduced the 

constraint b = 0.55/, saying this relation obtains for 

the stable channel described by one of Nizery and
18 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

Braudeau's (1955) sine equations. (See also Chow, 

1959, p. 178, eq. 7-12). The equation supposedly 

describes a stable hydraulic section, such as the 

cross section of an erodible channel (canal) in which 

no erosion will occur at a minimum water area for 

a given discharge. The formula is

, „ /r tan0 "1 v .7. eosH ——|X (9)

where Y = the depth at a horizontal distance X from 

the channel center, Y0 = the maximum depth at the 

channel center, and 6 is the angle between the horizontal and the bank at bankfull stage and is taken 

to be the angle of repose of sand grains (about 33°).

For such a channel, the relation 6-0.55/ results 

from taking the bankfull-stage cross section as 

given by the above equation, choosing various 

water-surface widths for different stages, getting 

the cross-sectional flow area corresponding to each 

width, computing the mean depth associated with 

each water-surface width and flow area, and finally 

plotting water-surface width (on the ordinate) versus mean depth on logarithmic paper. The points 

thus plotted follow a power law and have a slope 

of 0.55, which means W <x D°- 55. Thus, b/f = 0.55 or 

b = 0.55/.

Kennedy, Richardson, and Sutera (1964, pp. 338- 

339) legitimately asked whether the above cosine 

equation—a theoretical one intended for canals— 

does, in fact, describe the shape of river channels 

having movable beds and relatively firm banks. 

Thus, for testing the minimum-variance theory, an 

important question centers on the validity of Langbein's assumed relation 6 = 0.55/. Data accumulated 

for the present study should be sufficient to determine whether this relation is accurate.

Table 11 shows 76 stations labeled B, R, or B/R; 

however, two of these (stations 20 and 25) possibly

should be classified as having at least one "loose" 

boundary. To be safe, these two stations were eliminated for this minimum variance analysis, leaving 

74 stations for testing case B.

Of these 74 stations, the value b//=0.55 is approximated or exceeded in only three instances. For 

the vast majority of stations, b is a relatively low 

percentage of /. The average value of b/f for all 

74 stations is 0.19, with two-thirds of the cases 

falling within the range 0.09 ^(b//)^0.28. The 

average value of b/f, therefore, is reasonably well 

defined, and the assumption 6// = 0.55 definitely is 

not justified for the stations studied here.

In keeping with the policy of dealing with average 

values, case B was tested using the contraint that 

b = 0.19/. This empirical relation presumably accounts for the control that the firm banks exert 

on the hydraulic exponents. However, a plot (not 

shown) of b versus / for the 74 stations shows that 

b tends to decrease with increase in /, rather than 

increasing as the equation 6 = 0.19/ suggests. An 

eye-fitted line on the graph yields the very approximate relation 6 = 0.12-0.06/. This may express the 

relation between b and / as well as the former equation and was used in an alternate test of case B. 

As a matter of fact, the value of b for all 74 firmbank stations tends to be low. The average is 0.08, 

and two-thirds of the cases fall within the range 

0.04^6^0.11. It is, therefore, not unreasonable to 

take b = 0.08 as an average, for these 74 stations. A 

third test of case B, accordingly, was made using the 

constraint that 6 = 0.08, so that /+m = 0.92.

Table 5 shows the exponent values predicted by 

minimizing the variances of those combinations of 

variables that survived case A. The variables S and 

QS are constant and noninfluential, respectively, for 

case B; for brevity, they are not included in the

TABLE 5.—Theoretical rates of change of dependent factors with increase in discharge for groups of variables surviving

case A and for different constraints

[Case B: firm banks. Average field values: m = 0.42 (mode 0.40); / = 0.50 (mode 0.48); 6 = 0.08 (mode 0.07), for 74 stations]

Values of exponents

Sets of 

dependent Constraint 

variables

V, D, W 6 = 0.19/

V, D, W, QS/W

V, D, W, T. ff

V, D, W, r, ff, QS/W

V, D. W, r, VS

V, D, W. r, VS, QS/W

V, D, W 6 = 0.12 — 0.06/

V, D, W, QS/W

V, D, W, r, ff

V, D, W, T, ff, QS/W

V, D, W, r, VS

V, D, W. r, VS, QS/W

V, D, (W) 6 = 0.08

V, D, (W), r.ff

V, D, (W), T, VS

V

Velocity 

(m)

0.42

.34

.36

.35

.42

.38

.47

.49

.38

.39

.47

.48

.46

.38

.46

D 

Depth

(/)

0.49

.55

.53

.55

.49

.52

.44

.41

.53

.52

.44

.43

.46

.54

.46

W 

Width 

(6)

0.09

.11

.11

.10

.09

.10

.09

.10

.09

.09

.09

.09

( .08)

( .08)

( .08)

T

Shear 

(/)

0.49

.55

.53

.55

.49

.52

.44

.41

.53

.52

.44

.43

.46

.54

.46

ff 

Friction 

factor 

(/-2m)

—0.35

— .13

— .19

— .15

— .35

— .24

— .50

— .57

— .23

— .26

— .50

— .53

— .46

— .22

— .46

QS/W 

Power per 

unit area 

(1-6)

0.91

.89

.89

.90

.91

.90

.91

.90

.91

.91

.91

.91

——

VS 

Power per 

unit weight 

(m)

0.42

.34

.36

.35

.42

.38

.47

.49

.38

.39

.47

.48

.46

.38

.46
THEORY OF MINIMUM VARIANCE 19

table, but the possibility that they are part of the 

best group of variables has not yet been ruled out. 

Also, for the situation where Z> = 0.08, QS/W always varies as 1 — b or 0.92. Hence where b = 0.08 the 

factor QS/W is not included in the minimizations nor 

in the table.

Consider first the effects of the three different 

constraints, for a given combination of flow variables. (Incidentally, note that in case B, it seems 

necessary to incorporate an empirical relation—the 

constraint involving b—along with the theory.) As 

far as predicted exponents are concerned, values of 

b are essentially the same for all three constraints 

and range from 0.08 to 0.11. The exponents m and / 

show some differences—mostly minor—depending 

on the constraint.

The average field values for the 74 stations are: 

for m, mean = 0.42, mode = 0.40; for /, mean = 0.50, 

mode = 0.48; and for b, mean = 0.08, mode = 0.07. 

Standard deviations are 0.13, 0.14, and 0.05 for m, 

f, and b, respectively. All predictions in table 5 are 

within a reasonable range of these average measured 

exponents. Thus the case B test is inconclusive. All 

groups of variables examined for this case, including S and QS, will therefore be considered in the 

next test (case C).

CASE C

Case C consists of streams in which the slope at 

the station remains constant but the entire flow 

boundary is loose and readily eroded. The entire 

channel, in other words, is developed in noncohesive 

material—usually sand or sandy gravel. The loose 

"banks" in such channels may be reshaped from 

one flow to the next, and a change in discharge often 

alters the channel shape by erosion or deposition or 

both. The channel width is completely free to adjust 

to each new discharge.

Sixteen of the 165 stations listed in table 11 

(appended to end of report) qualify for this case. 

Most of the 16 stations are sandy streams for which 

the flow data have been plotted for low-flow conditions. The entire flow boundary for these low flows 

would be defined as the bed during large discharges. 

Average exponents (arithmetic means) for the 16 

stations are m = 0.21, /=0.26, and & = 0.54. Modal 

values are m = 0.22, /=0.27, and & = 0.45. The standard deviations are 0.07, 0.07, and 0.09 exponent 

units for m, /, and b respectively.

Table 6 shows the exponents predicted by the 

various combinations of variables, with S and QS 

again being omitted because they have no effect on 

the exponents for this case. Three of the combinations (groups 1, 4, and 6) predict ^-values that are 

markedly discrepant from measured averages. Two 

groups are fairly close to the field values: (a) V, D, 

W, T and ff, which predicts m = 0.22, /=0.30 and 

& = 0.48, and (b) V, D, W, r and VS, which yields 

m = 0.25, / = 0.25 and & = 0.50. Both of these combinations deserve to remain in the competition. The 

sixth group (V, D, W and QS/W) predicts m rather 

accurately (forecasting w = 0.20) but is about one 

standard deviation away for both / and b. It will be 

included in the next test, although it was not as 

impressive as the two groups just mentioned for 

case C. Neither S nor QS nor both combined have 

yet been eliminated as possibly relevant.

CASE B/C

The fourth major test involves stations having 

one firm and one loose bank (labeled B/C and C/R 

in table 11), with slope still constant. Fifty one of 

the stations are in this category.

An expression for the constraining effect of the 

firm bank should be incorporated into the minimum 

variance analysis. For the 51 applicable stations, I 

obtained such an expression by plotting & as a function of /. A definite trend appeared although with 

some scatter. Two-thirds of the plotted points fall 

within ±0.11 exponent units of the value indicated 

by the eye-drawn best-fit line. The line through the 

plotted points has the equation & = 0.84 —1.45/. This 

relation was used in minimizing the variances of the 

three groups of variables remaining in the competition.

TABLE 6.—Theoretical rates of change of dependent quantities with increase in discharge, for different sets of variables

[Case C: slope constant; loose, noncoherent banks allowing complete freedom for width to adjust. Average values for 16 field sites: m = 0.2l mean,

0.22 mode; / = 0.26 mean, 0.27 mode; and 6 = 0.54 mean, 0.45 mode]

Value of exponents

Group ~ 

No.

1

2

3

4

5

6

Dependent 

variables

V, D, W

V, D, W, QS/W

V, D, W, T, ff

V, D. W, T, ff QS/W

V, D, W. T, VS

V. D, W, T, VS, QS/W

V 

Velocity 

(m)

0.33

.20

.22

.14

.25

.16

D 

Depth 

(/)

0.33

.20

.30

.20

.25

.17

W 

Width 

<&)

0.33

.60

.48

.66

.50

.67

T

Sheaor 

(/)

0.33

.20

.30

.20

.25

.17

ff 

Resistance 

(/-2m)

—0.33

— 20

— .14

— .08

— .25

— .15

QS/W 

Power per 

unit bed area 

(1-6)

0.67

.40

.52

.34

.50

.33

VS 

Power per 

unit weight 

(m)

0.33

20

.22

.14

.25

.16
20 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

Because slope is constant, the variables S and QS 

are not included in the minimization. Minimizing the 

variances of V, D, W, T, and ff yields ra = 0.28, 

/=0.27, and 6 = 0.45. For V, D, W, r, and VS the 

predictions are m = 0.27, /=0.24, and b = 0.49. Minimizing the variances of V, D, W, and QS/W produces m = 0.24, / = 0.17, and fc = 0.59.

The average field values for the 51 stations are 

ra = 0.30 (mode 0.27), /=0.31 (mode 0.25), and 

fc = 0.40 (mode 0.45, poorly defined). Standard deviations are 0.10, 0.14, and 0.19 exponent units for 

m, /, and b, respectively.

All three groups of variables are fairly close in 

predicting m. For / and b the combination V, D, W, 

and QS/W is not as close as the other two groups. 

This combination also was less accurate in the previous test (case C). The combinations V, D, W, T, 

and ff and V, D, W, T, and VS again are reasonably 

close to the field values. The results therefore show 

that, for the four cases examined thus far, these 

latter two combinations are the only ones that consistently yield exponents close to the average observed exponents. Also, because slope has been 

assumed constant in the four cases examined above, 

there has been no way of determining whether S 

and (or) QS should be included in the complete set 

of variables. One test remains.

CASE D

The fifth test includes channels in loose, readily 

erodible material, as with case C, but now the watersurface slope varies with discharge rather than remaining constant. Measurements for such stations 

are extremely scarce. Only three sets of data could 

be found for this case: two of them are the Wolman 

and Brush (1961) flume study of 0.67 mm and 2.0 

mm sand, respectively, and the third is Ackers' 

(1964) flume study with 0.16 mm and 0.34 mm sand.

Table 7 shows the measured hydraulic exponents 

for the three flume studies. Ackers' values are those 

published in his paper and were determined by least 

squares. His graph of slope versus discharge showed 

sufficient scatter that he decided no relation could be 

defined; however, the plot strongly suggests a negative value for the exponent z. To get the Wolman and 

Brush exponents, I plotted their experimental data 

and, except for the z-values (determined by least 

squares), drew lines of best fit by eye. The plots of 

slope versus discharge, as with the Ackers data, 

show a decidedly negative exponent, but the scatter 

is such that the values of z, while definable, are not

TABLE 7.—Measured and predicted values of hydraulic exponents for stations having variable slope, with bed and 

banks readily erodible

Measured values

Wolman and Brush (1961), 0.67 mm sand. 

Wolman and Brush (1961), 2.0 mm sand.. 

Ackers (1964), 0.16 mm and

0.19 

.11

15

0.39 0.48 —0.34 

.54 .38 — .74

.42 .43

Average values of the above data _ .15 .45 .43 — .54

Theoretical values (minimum variance)

V, 

V

V, 

V, 

V, 

V, 

V

V

D, 

D

D, 

D, 

D, 

D, 

D

D

W,

W

W, 

W, 

W. 

W, 

W

W

,ff ,ff, s

, ff, QS 

,ff, S, QS

, vs 

, vs, s

, VS, QS

. VS. S. OS

0.14 

.19 

.03 

.10 

.33 

.30 

.40 

.36

0.43 

.35 

.62 

.50 

.33 

.30 

.40 

.36

0.43 

.46 

.35 

.40 

.34 

.40 

.20 

.28

—0.29 

— .11 

— .73 

— .45 

— .33 

— .20 

— .60 

— .43

accurate. Values of m, f, and b, among the three 

studies, are reasonably consistent for experiments 

of this type. The average exponents (arithmetic 

means) for the three studies are m = 0.15, /=0.45, 

6 = 0.43, and z =-0.54.

Inspection of the theoretical values of table 7 

shows that none of the four groups having V, D, 

W, T, and VS comes very close for the exponent m. 

Two of the other four combinations are not particularly close, either, for one or more exponents: 

the group V, D, W, T, ff, and QS is 0.17 exponent 

units off for / and -0.12 units off for m, while the 

group V, D, W, T, ff, and S is 0.10 exponent units 

low for / and somewhat high for z.

The remaining two of the eight groups come 

closest to the observed values. If the measured z 

values are reliable, then the combination V, D, W, 

T, ff, S, and QS is closest (-.05, +.05, -.03, and 

+ .09 exponent units off, for m, f, b, and z, respectively) . If less weight is given to the z values which 

were not as well defined as the other exponents, then 

the group V, D, W, T, and ff is slightly closer. For 

some at-a-station cases, where slope is constant, the 

choice is irrelevant since S and QS drop out in the 

minimization calculations, leaving just V, D, W, T, 

and ff. Tentatively, however, the available data suggest that one group of variables, namely V, D, W, T, 

ff, s, and QS, applies to all at-a-station situations.

Table 8 shows the extent to which a minimum 

variance analysis, with V, D, W, T, ff, S, and QS as 

the appropriate group of variables, predicts the 

average measured exponents for the five cases.

To summarize the findings thus far: (1) the minimum-variance theory closely predicts the average 

hydraulic exponents for the five types of stream 

cross sections examined; and (2) the group of variables that consistently gives the most accurate results is V, D, W, T, ff, S, and QS.
HYDRAULIC EXPONENTS OF INDIVIDUAL STATIONS 21

TABLE 8.—Comparison of average measured exponents to exponents predicted by the minimum variance theory, using V,

D, W, r, ff, S, and QS as the appropriate variables

[Dashed entry following case means column does not apply]

Case Constraints

Number 

of

applicable 

•stations

Velocity 

exponent 

m

Data

Mean Mode

A _______ Width and slope

constant (bf&Q). 

B, B/R, R ..Banks firm but not 

vertical; slope constant 

6 = 0.19/ 

6 = 0.12 — 0.06/ 

6 = 0.08

B/C, C/R _ One bank firm; 

slope constant; 6 = 0.84 — 1.45/ 

D ..... ___ .None __________ ._ __.____..

22

74

16

51 

3 

(flumes)

0.47 0.46 

.42 .40

.21 .22

.30 .27 

.15 ( 2 )

0.43 

.36

00

.38

.22

.28 

.10

Depth 

exponent

Data

Mean Mode

0.52 0.55 

.50 .48

.26 .27

.31 .25 

.45 ( 2 )

Fheory -

0.57 

.53

.53

.54

.30

.27 

.50

Width 

exponent 

6

Slope 

exponent 

z

Data Data

Mean Mode Mean

0.08 0.07

.54 .45

.40 M5 

.43 (s)

0.11

no

.08

.48

.45 

.40 —0.54

Mode

(~2~)~ — 0~45

1 Poorly defined. 

2 Insufficient data.

HYDRAULIC EXPONENTS OF INDIVIDUAL 

STATIONS

The original intent of the minimum-variance approach was to find only the group averages. However, one of the eventual goals should be an accurate 

prediction of the exponents for any given station.

The hydraulic exponents at any given stream cross 

section probably depend to some extent on certain 

local features. Such features might reflect the channel shape, size, slope, boundary material, and other 

characteristics. Information of this sort was collected for all 165 stations to see if such special data 

can help provide the hydraulic exponents for any 

stream cross section.

Three approaches for predicting exponents were 

explored: (a) minimum variance, (b) empirical 

equations based on the data of this study, and (c) 

the Gauckler-Manning and Chezy relations.

COLLECTION OF SPECIAL DATA

THE WIDTH- VERSUS-AREA RELATION

An approximate 6-versus-/ relation applicable to 

each individual station can be obtained by plotting 

estimated water-surface widths and their associated 

estimated cross-sectional flow areas. Cross-sectional 

flow area A=DW. Thus, where power relations 

exist,

So, for the range over which hydraulic exponents 

are defined, a logarithmic graph of width-versusflow area will show a straight line having slope 

&/(& + /).

The value of &/(& + /) permits a definition of b 

in terms of /, or vice versa. For instance, if &/(& + /) 

= 0.20, then 6 = 0.25/.

Another advantage to such a width-versus-area 

graph is that the range of width and of area (and 

hence also of mean depth) over which the hydraulic 

exponents are valid will be readily shown by the 

range over which the straight-line relation holds.

Points plotted to define hydraulic geometry relations represent separate and individual flow measurements. A plot of W versus A, therefore, could 

also be defined from separate flow measurements. 

However, with stable channels, the gross channel 

dimensions and general shape are reasonably constant with time. Thus, the relation between watersurface widths and flow areas should be approximately the same for a series of separate flows as for 

various hypothetical flows within any one cross 

section profile, as long as the selected profile is representative or typical of the general channel shape and 

dimensions. The validity of this assumption is tested 

below.

Because the channel shape and hence the widths 

and areas change slightly from one flow to the next, 

more than one cross-sectional measurement is needed 

to get a reliable representation of a TF-versus-A 

relation. The experience of this study is that about 

three measured cross sections (taken at the same 

location at least several weeks apart) are needed for 

stations with firm, stable banks, but four to six 

cross sections are needed for stations where the 

entire flow boundary is cohesionless. Since widths 

and their respective areas are the only required 

data, surveys of the cross section serve as well as 

discharge measurements, as long as some flow has 

occurred between surveys. Surveys might even be 

better in that they can always extend up to and 

over the banks.
10.0

CROSS-SECTIONAL FLOW AREA, IN SQUARE FEET

100.0

Z 50

x" 

o

LJJ

o

LU

OC

20

10

EXPLANATION

A Cross-section of June 8, 1971 (Q = 802 ft 3/s) 

O Cross-section of January 19, 1971 (Q=2,450 ft 3 /s) 

X Cross-section of February 21, 1968 (Q=2,210 ft 3 /s) 

T Data for individual discharge measurements, 1966-71

O A

A

1.0 10.0

CROSS-SECTIONAL FLOW AREA, IN SQUARE METERS

FIGURE 5.—Width versus area for a station with firm banks—White River below Tygh Valley, Oreg. Hypothetical widths and are

three cross sections which were measured at the same section on different occasions. Lines of best fit were drawn for these d

separate flow measurements added afterward for comparison.
P 

b

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m

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p 

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3D

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FEET IN WIDTH, WATER-SURFACE 
24 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

Figures 5 and 6 show typical W-versus-A relations for a firm-bank and loose-bank station, respectively. The standard procedure in preparing these 

and similar graphs for every station was as follows: (a) plot a cross section for a very high measured flow; (b) draw horizontal lines (typically 6 to 

14) representing theoretical water surfaces; (c) 

measure with a planimeter the cross-sectional flow 

area associated with each water-surface width; (d) 

plot W versus A on log paper; (e) repeat for three or 

four other high-flow measurements at the same 

station, plotting all the data on the same graph; 

and (f) draw a single best-fit line by eye for the 

range over which a straight-line relation applies.

Two methods were tried for testing the reliability 

of the width-area relations. One method was to 

plot width against area for the many separate flows 

that made up the observed hydraulic-geometry relations and to compare the best-fit line with that obtained by the graphical planimeter method. This 

approach turned out to be infeasible because, for 

many stations, the range of areas for the real flows 

was too small. For stations having a reasonable 

range of measured flow areas, the widths and areas 

obtained by planimeter showed fair agreement with 

the data from real flows (figs. 5 and 6).

In the second test, the value of &i/(&i + /i), representing the slope of the line on the cross-section 

based width-versus-area graph, was compared to 

the "true" value of &/(&+/) as computed from the 

observed hydraulic exponents.

Figure 7 shows the results of this test. The data 

are not homoscedastic, and the points are more 

evenly distributed on log scales, as in figure 7. The 

graph indicates that, as a first approximation, the 

value of &/(&+/) can indeed be estimated by plotting widths and areas generated from cross-sectional surveys. On the other hand, the scatter undoubtedly leaves room for improvement. The biggest single discrepancy between &i/(&i+A) and the 

"true" &/(& + /) for a station is 0.35 exponent units. 

In a number of cases, however, the agreement is 

perfect. For two-thirds of the 165 cases, the estimated value &i/(&i + /i) is within ±25 percent of 

the true &/(& + /).

There is a slight tendency for the estimated \>J 

(&i + /i) to be low for stations having a high value 

of &/(& + /), and vice versa (fig. 7).

MAXIMUM AND MINIMUM GEOMETRICAL 

PROPERTIES OF A SECTION

The mathematical definition of hydraulic exponents suggests that their values might be influenced

by certain characteristics of the channel cross section. Take, for example, the exponent of width, 6, 

defined as

(10) ^ _ log W2 - log Wi _ Alog w 

logQ2 -logQj AlogQ 

where each subscript refers to a point on the bestfit line relating width to discharge.

A consistent way to define the two points on the 

best-fit line, for example, W, and W1} is to take them 

as the maximum and minimum values, respectively, 

for which the straight-line relations holds. These 

points can be readily seen on most plots of width 

versus area. Figure 5 is a good example. For that 

station, the maximum values of width (Wm&*) and 

area (Amax ) for which the power relation holds are 

29.3 m (96 ft) and 29.9 m2 (322 ft2 ), respectively. 

Therefore, the associated depth (Dmax ) is Amax/VPmax 

or 1.02 m (3.35 ft). Similarly, the lower end of the 

power relation is at a width Wmill of 24.4 m (80 ft) 

and an area Amin of 7.52 m2 (81 ft2 ). Thus, Dmiu 

= 0.31 m (1.01 ft). Alog W would then be log Wm&x 

-log Wmin = log 96-log 80 = 0.079. Alog A and Alog D 

were defined in the same way for all stations.

The denominator Alog Q in equation 10 is not 

directly available. Assume the hydraulic radius is 

approximately equal to mean depth D and that 

resistance and slope are constant. A number proportional to Qmax (discharge at the upper end of the 

power relation) can then be computed from the

2/3

Gauckler-Manning equation as DmaxAmax. Similarly,

2/3

Qmm would be proportional to DminAmin. The denomination Alog Q then is proportional to

log Qmax -log Qmln =0.667

Using the maximum and minimum widths, depths, 

and areas as just described, numbers were generated 

that hopefully would be proportional to the exponents b and /. The denominator in these numbers 

was the Alog Q of the previous paragraph, and the 

numerators were Alog W and Alog D for b and /, 

respectively.

For some of the loose-bank stations (such as fig. 

6), no break could be discerned at the lower end of 

the straight-line relation on the width-area plot. For 

these cases, the minimum values were taken as 

those corresponding to a depth of 0.03 m (0.1 ft), 

since the current-meter method probably would not 

measure shallower depths with any accuracy. In a 

few other cases, the available cross-sectional pro-
HYDRAULIC EXPONENTS OF INDIVIDUAL STATIONS 25

1.0

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Q

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-<T

0.01

I

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• /

0 /*>

/k v r^ ^^

/ o

V «X O

V XO ><

4-0 / i <t>

A / A

• 4 •txO A

4>», t >* A

— [J] -B- -B- D ^ y/ •

+

^ (fA^ 4/D/

_X ^

X ^

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XT

-B-

' /

• *A

T O^sf^

•Xo ^

f -©•

0 -B-

+

<l>

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^/ ?/

& -

_

EXPLANATION —

+ n-<»e vift $ 9Xv $ Symbol Stations

A D • /^ • ^< • V 1-4

• • A rtj A 5-7

/ 0

A f< D+tfOO + x

— ^/^ •$() *

y^ D

it D *$ / D ^

/^ T

/ 0

D ^^* $

X / m

/ / Line of perfect agreement +

x / / &

~~ ^ / X A

/// :

>/ X 4

X *

i i

0.01 0.1

-^- COMPUTED FROM HYDRAULIC EXPONEN

8-15

16-25

26-28

29-40

41-46

47-53

54-62

63-78

79-86

87-97

98-114

115-125

126-131

132-140

141-148

149-165

TS

States

N.Y. 

Del., Md.

Va., N.C.

S.C., Ga., Fla.,

Iowa

Nebr.

Kans.

Okla.

Tex.

N. Mex.

Colo.

Wyo.

Utah

Ariz.

Nev.

Calif.

Oreg.

Wash.

I

Miss.

—

1

FIGURE 7.—Width-area relations based on channel cross sections compared with true values as computed from hydraulic 

exponents. Two-thirds of the estimated values, encompassed by the dashed lines, fall within ± 25 percent of the 

true values.

files did not extend high enough up the banks to 

show the break at maximum values; in these cases, 

maximum values were taken at the maximum flow 

conditions recorded for the site.

The maximum and minimum properties as denned here were also used to form other channel 

descriptors. For instance, many analysts use the 

channel width/depth ratio as an indicator of channel shape. In the present study, the width/depth 

ratio at the upper end of the applicable power relation was denned as Wmax/Dmax, in which these 

values were measured as just described. Similarly, 

the lower end of the power relation is associated 

with a width/depth ratio of Wmin/Dmin .

The amount of error associated with the above 

method of estimating the maximum and minimum
w 

1

r

a:

n>

MEAN DEPTH, 

IN METERS

p 

b

o

b P

en _»

MEAN VELOCITY, IN 

METERS PER SECOND

WATER-SURFACE WIDTH, 

IN METERS

p

k>

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to 

b

on 

b

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MEAN DEPTH, 

IN FEET

MEAN VELOCITY, IN 

FEET PER SECOND

WATER-SURFACE WIDTH, 

IN FEET

DE

D

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m JJ

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— N3 

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D
HYDRAULIC EXPONENTS OF INDIVIDUAL STATIONS 27

geometrical properties of a channel section is difficult to assess. The accuracy of these values increases with the number of cross sections plotted.

Another use of maximum and minimum crosssection characteristics is to define "bank inclinations." The general steepness or flatness of the banks 

should affect the exponent b (Lewis, 1966; Knighton, 1974). The Humboldt River near Argenta, Nev. 

(station Nos. 126 and 127 of this study), shows 

how the inclination of the boundary, regardless of 

whether the latter be firm or loose, affects the hydraulic exponents. Figure SA is the channel cross 

section at the cable, and the plotted hydraulic-geomentry data are shown in figure SB. Inspection of 

the plotted data reveals two different sets of exponents, corresponding to discharges higher and 

lower than about 4.25 m3/s (150 ft3 /s), respectively. 

Why these different sets of exponents? From the 

plot of width versus discharge, the break in the 

exponent b is seen to occur at a water-surface width 

of about 26 m (85 ft). Transferring this width to 

the plotted profile (B-B' in fig. 8A), we see that this 

width just corresponds to the base of the banks, 

that is, to a noticeable change in the general inclination of the flow boundary. From the base of the 

banks up, the firm, regular boundary promotes a 

well-defined set of hydraulic-geometry relations, 

and the rather steep banks cause a relatively low 

value of the exponent b (0.06 in this case). At low 

discharges, on the other hand, the flattish and cohesionless bed is the flow boundary. Such a flattish 

boundary is associated with large changes in width 

for a given change in discharge, that is, a relatively 

high value of the exponent 6 (0.46) and more scatter 

on the hydraulic-geometry plots. Hence, the boundary inclination and regularity have a direct influence on the hydraulic exponents (Richards, 1976).

(This example, along with figure 4 and the data 

in table 11, suggests that there may be some risk 

or questionable significance associated with computing average hydraulic exponents for a physiographic region, as some investigators do. See also 

Rhodes, 1977, p. 83-84.)

Maximum and minimum areas and widths, defined 

as explained above, were used to define a bank angle 

B. The cross section was assumed trapezoidal, and 

the bank sections were taken as equal right triangles 

(fig. 9). The slope of these banks (0) was considered 

to be the average bank inclination of the natural 

channel.

The distances used to compute Twere the base 

and height of the bank sections, that is, tan 0 

= height/base. The average base of a bank section 

= 1/2(^max -^min). (See fig. 9.) The height EB, 

considering the two bank sections equal, is

\-**-max -"-mill)

wmin

Then for the bank inclination,

_ 4 ( Amax — Am jn 

tan 6= W* -W*

max miu

(ID

Since the tangent and other trigonometric functions 

are sometimes awkward to use mathematically (for 

example, the tangent goes to infinity as the banks 

become vertical), ^Twas expressed in degrees.

(Defining a bank angle 6 by plotting the cross 

section and drawing a straight line by eye for the 

general bank inclination was too subjective, partly 

because the transition from bank to bed was hard 

to recognize on some cross sections. The above sysUpper water surface has Wmax ,A

FIGURE 9.—Sketch showing concept of bank inclinations. In practice, the geometrical properties of maximum and minimum areas and widths are determined by logarithmic plotting of widths and corresponding areas measured from 

several cross-sectional profiles.
28 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

tern has the disadvantage that channels with various 

combinations of bank angles can have the same 6, 

so 0 is at best only an approximation. The method 

is consistent and objective, however.)

BED-SEDIMENT SIZES AND ESTIMATES OF BED 

ROUGHNESS

The bed particles for each station were sampled 

for size distribution at or near the cross section of 

interest. The median diameter, d50, of the size distribution was used to describe the bed sediments. 

For some streams, especially with sandy beds, this 

value was available in published literature. At most 

stations, however, the particles had to be sampled. 

In streams on which all the bed particles were less 

than coarse gravel in size, representative bed 

samples were taken at several points across the 

section and combined into a single composite sample 

for laboratory analysis. Coarse gravel and larger 

particles generally were measured directly in the 

field by the pebble-count method (Wolman, 1954). 

At these sites, the finer material also was sampled 

and, if equal to more than about 15 percent of the 

total size distribution, was analyzed by sieving. The 

pebble count is a surface-sample frequency by number, whereas the sand-sized material was a threedimensional sample analyzed by weight or volume 

frequency. On the basis of the work of Kellerhals 

and Bray (1971), the two distributions were combined. In making this computational union, each of 

the two distributions was weighted according to 

that percentage of stream-bed area it covered, as 

indicated by the pebble count.

Sieving was the usual method of size analysis for 

sand-sized material, though with a few samples a 

visual-accumulation (VA) tube was used. As the 

VA method is based on the principle of fall velocity 

rather than a direct measurement of the sieve diameter of the particle, the VA-tube data theoretically 

should be made comparable to sieve data by an appropriate adjustment factor (U.S. Interagency 

Committee, 1957, p. 37, fig. 7). However, assuming 

a grain-shape factor of 0.7 (naturally worn sediments), the difference in the results produced by 

the two methods is negligible for medium and fine 

sands.

Two additional size-frequency characteristics were 

determined for many stations: dS4 (the grain size for 

which 84 percent of the distribution is finer) and a 

sorting measure, So, defined as log d90 — log d10, 

where the subscripts indicate the percent finer in 

the size distribution. These data were not available 

for those sand-channel streams for which a median

grain size was published. For certain other stations, 

the dlo was not available.

Several sources of error are associated with the 

median bed-particle size. One question involves 

where to sample, when particles of different size 

groups form separate patches on the streambed. 

Furthermore, the sizes on the bed surface may not 

necessarily be the same as those just beneath the 

surface.

The sampling procedure itself involves some error. 

The variability in the results for measurements 

made at or near the same cross section, at least with 

the pebble counting, can amount to about 12 percent 

of the median diameter (Wolman, 1954, p. 954).

In most cases, there was a time lag averaging 

about 1 to 2 years and ranging from about 6 months 

to 17 years between the period of the hydraulic 

measurements and the occasion of the sediment 

sampling. I assumed that the bed sediments did 

not undergo any drastic changes in size distribution during this interval.

Finally, one particle-size distribution may not 

apply to the full range of plotted water discharges. 

The size distribution of the bed particles could 

change with flow conditions, at least for streams 

with a wide range of particle sizes.

Several relative-roughness variables were studied 

for possible influence on the hydraulic exponents. 

These variables applied only to grain roughness and 

did not include roughness due to bed forms and 

channel irregularities. Examined were Dmla/d50, 

-Dmax/^50, and the difference between these two, that

IS, (i/max — ^min ) /^50«

Another grain-size variable investigated was d1/6,

50

since according to the Strickler relation, this is proportional to the Gauckler-Manning resistance coefficient in streams having coarse bed material.

Finally, according to Henderson (1966, p. 98), 

the quantity (d50/-D) 1/3 is proportional to the DarcyWeisbach friction factor ff. Thus (d50/Dm&x y/3 would 

be proportional to the friction factor at maximum 

flow depth, (d50/Anin) 1/3 would be proportional to 

the friction factor at minimum depth, and the logarithmic change between these two extremes would 

be log f(Anin/AnaX ) 1/3]. These factors involve only 

the grain effects and do not include the influence of 

bed forms.

SLOPE

Channel gradients in the vicinity of the cross 

section were determined either from field data or 

topographic maps. Most field data were from longi-
HYDRAULIC EXPONENTS OF INDIVIDUAL STATIONS 29

tudinal profiles of the channel or water-surface, 

otherwise from high-water marks on the banks or 

from two gages along the reach. The topographic 

map measurements were generally made on TVa-nrin 

quadrangle maps by measuring the horizontal distance between the contour above and the contour 

below 'the section.

The assumption that a single slope value applies 

to a given reach is valid only in a statistical sense. 

The approximate overall slope of a reach is directly 

related to the general topography of the area, and 

this average probably did not change significantly 

over the period of time for which the hydraulic data 

have been plotted. However, the actual slope, and 

especially the energy gradient of the flow, at any 

moment in time may vary with flow conditions and 

with channel scour and fill, at least for high discharges. Such potential variability may introduce an 

unknown amount of error into any relation involving slope.

There are no data available for assessing the error 

involved in measuring the slope with the various 

field methods.

Topographic map measurements introduce error 

both in the drawing of the map and in the measurement of the horizontal distance along the thalweg. 

Hack's data (1957, p. 91-93) for 64 rather steep 

streams in Maryland and Virginia show that slopes 

measured from topographic maps can differ from 

field-measured slopes (channel profiles measured 

over a distance of 61 to 152 m (200 to 500 ft)) by a 

factor ranging from 0.03 to nearly 15. The factors 

for two-thirds of his observations ranged from 0.64 

to 2.00.

ACCURACY OF EXPONENTS DETERMINED BY 

MINIMUM VARIANCE

By assuming that the theoretical predictions of 

the minimum variance theory are the true mean 

values for each case, a rough idea of the accuracy 

can be obtained by looking at the spread of measured 

exponents about the predicted exponent. The 22 stations for case A (width and slope constant) have 

a standard deviation of 0.14 exponent units for m 

and 0.15 exponent units for /. Thus, if the distribution were normal, about 68 percent of those stations 

would have an exponent m that is within ±0.14 

exponent units of the predicted minimum-variance 

value. For the 74 firm-bank stations (case B), 

standard deviations are 0.14 for m, 0.14 for /, and 

0.05 for b. The 16 loose-boundary stations (case C) 

have standard deviations of 0.07, 0.08, and 0.11 for 

m, f, and b, respectively. And for the 51 stations

having noncohesive boundaries except for one firm 

bank, the standard deviations are 0.10 for m, 0.14 

for /, and 0.20 for 6. Thus, additional refinements to 

the minimum-variance analysis, probably in terms 

of more specific expressions of the constraints, are 

desirable. Another improvement would be to eliminate the subjectivity in classifying banks as "firm" 

or "loose."

An attempt was made to use a more objectively 

determined constraint, namely b 1 /(b1 + fi), in minimum-variance computations (V, D, W, T, ff) for 

each station. This approach obviates the need to 

classify banks as firm or loose, and a function of / 

can be substituted for b in the calculations. For control and comparison, similar computations were also 

made using the true b/f as given by the measured 

hydraulic exponents. The predicted exponents for 

the 165 stations have the following accuracies:

Constraint

true b/f

Exponent

m

b 

m

f 

b

Standard error 

(Exponent units)

0.132

.102

.045

.131

.117

.082

Percent sums 

of squares 

explained

17

62

96

18

50

86

(The percent of the total sums of squares of the 

dependent variable explained is defined as

where S.E. = the standard error of the estimate of 

the dependent variable and a is the standard deviation of the dependent variable.)

The above calculations suggest that using b 1 /(bl 

+ /i) with minimum variance can provide a good 

estimate of b, a fair estimate of /, and a poor estimate of m.

Some reasons that might inhibit better agreement 

between predicted and observed exponents are:

1. A different group of variables possibly should 

be used in the minimization, rather than V, 

D, W, r, and ff.

2. Different groups of variables possibly should be 

used for different hydraulic and (or) geologic 

situations.

3. Too many simplifying assumptions, such as constant energy gradient over the flow range, 

may have been adopted.

4. The minimum-variance theory may apply to 

certain kinds of channels and (or) flow conditions better than to others.

5. A different constraint (other than b1 /(b1 + fi)) 

should be used.
30 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

6. The minimum variance calculations should be 

carried out with coefficients or weighting 

factors attached to each of the exponents, 

rather than with a single constraint applied 

to the entire minimization.

The first five of the above items were explored. 

These items deserve further study. However, no 

significant improvement in predicting the exponents 

emerged from the examination. A search for empirical coefficients (item 6) led to the surprising 

development that such coefficients, if known, estimate the hydraulic exponents directly, so that minimum variance apparently is not needed with this 

approach.

EMPIRICAL FORMULAE FOR HYDRAULIC 

EXPONENTS

The special data collected for this study, plus 

selected combinations of some variables, were tabulated for each station. The following independent 

variables, except when_their values were unavailable, were included: S, T, d50 , d84, &!/(&! + A), Amax,

•Amin> H'max, KK min, //max, Anin> "max/Dinax> Wmin/Dmin,

•^•max/Aniin, W max/ " min» Anax/Anin» Dm&yi/cL50f -DminA^Oj 

(Anax-AniJ/cU ( W/D ) max/ ( W/D ) min, d1/* , (d50/ 

Anax) 1/3 , (dao/Anin) 1/3 , (^O/Anax) VV (Ao/Anta) 1/3 , the

logs of all the variables just listed, So, 1/0, Amln/Amax,

in/ TF

A

6i

( W/D ) min/ ( W/D )

AlogQ

+ log

Lmln _

Alog TF/Alog Q and Alog D/Alog Q.

An equation for each hydraulic exponent was obtained by multiple regression, at a probability level 

of 0.05. Both the natural value and the logarithm of 

each exponent were tested. Each dependent variable was first regressed against the set of 59 independent variables listed in the previous paragraph. 

In many cases, at least one additional regression 

was made to reduce the equation to a more practical form of no more than two or three independent 

variables.

The most accurate empirical formulae are:

= 0.8 (12)

which has S.E. = 0.082 and explains 86 percent of 

the sums of squares of b (fig. 10) (data are not 

homoscedastic, but they are not on log scales either);

/=0.60-0.58 -O.OOlScL (13)

which has S.E. = 0.096 and explains 66 percent of 

of the sums of squares of / (fig. 11); and

ra = 0.24+ 0.16 -0.21 + 0.000021 AninX 

(14)

which has S.E. = 0.110 and explains 45 percent of 

the sums of squares of m (fig. 12). In these equations d50 is in millimeters and Dmin is in feet.

The coefficients in equations 13 and 14 carry appropriate units to make the equations dimensionless.

The three exponents for a cross section as given 

by equations 12-14 do not always add up to exactly 

1.0. Equations 12-14 with the present data produce 

sums of exponents ranging from about 0.93 to 1.07.

These general equations for the hydraulic exponents suggest that b, the exponent of width, is 

almost entirely a function of the channel geometry 

(&i/(&i + A), that is, widths and areas). The exponents / and especially m seem to depend partly on 

channel geometry, partly on roughness-related features, and possibly on additional features that were 

not studied or not measured well enough.

As with the minimum-variance test, the predictions are good for b, only fair for / and poor for m. 

Possible reasons why more accurate equations did 

not appear are:

1. The basic data, such as d50 and &i/ (&i +A)» P°Ssibly were not measured accurately enough.

2. The right variables were not included in the 

regression. For instance, maybe sediment 

transport rate should be involved. Or maybe 

d50 or d8 ± are not the best measurements to 

represent grain size. Also there is a question whether the true &/(& + /), rather than 

the measured &i/(&i+A)> should have been 

used in the regressions. (The measured & x / 

(&i+A), of course, would have to be used 

in practice.) The empirical equations, especially the one for &, all become slightly more 

accurate if &/(& + /) is used instead of bj 

(&i + A) in the regressions.

3. The true type of function was not found or 

considered.
HYDRAULIC EXPONENTS OF INDIVIDUAL STATIONS 31

0.90

0.80 -

Line of perfect agreement

0.10 0.20 0.30 0.40 0.50

Measured exponent b

0.60 0.70 0.80 0.90

FIGURE 10.—Computed versus measured values of exponent 6, where computed 6=0.8 (6i/(6]+/i)). Standard error=

0.082 exponent units, with 86 percent of sums of squares explained.

HYDRAULIC EXPONENTS FROM THE GAUCKLERMANNING AND CHEZY EQUATIONS

METHOD

The Gauckler-Manning and Chezy equations

(V=—- 

n

respectively, where R is hydraulic radius and n and 

C are roughness coefficients), can provide estimates 

of hydraulic exponents. Three assumptions are 

necessary: (a) the selected equation is valid for

every station to which it is applied, (b) the roughness coefficient (Manning n or Chezy C) is constant over the flow range of interest, and (c) the 

slope or energy gradient is constant for the flow 

range of interest. The validity of at least the first 

two of these assumptions is doubtful for many stations on alluvial channels. Thus, the present attempt is more exploratory in nature and is motivated 

by a curiosity to see how close the predicted exponents come to the observed exponents.
32 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

05

c 

o

Q. 

X 

05

-a

05 

Q.

E 

o 

o

0.70

0.60

0.50 -

0.40 -

0.30 -

0.20 -

0.10 -

Line of perfect agreement

0.10 0.20 0.30 0.40 0.50

Measured exponent/

0.60 0.70 0.80 0.90

FIGURE 11.—Computed versus measured values of exponent/, where computed f=0.60—0.58 (&i/(&i+/i)) 

Standard error=0.096 exponent units, with 66 percent of sums of squares explained.

-0.0018

At constant roughness and slope, the GaucklerManning formula specifies that V&R2/3 . Similarly, 

according to the Chezy equation, VocR1 /2 . Assuming 

D^R and substituting V=Q/A, we get Q« D2 /3A 

for Gauckler-Manning and Q<xDl/2A for Chezy. 

Thus, a number proportional to Q can be computed 

from an associated value of D and of A.

The estimated values of W and A generated from 

hypothetical water-surface widths drawn on plotted 

channel cross sections were used as the basic data. 

For each width and area, the associated mean depth 

D was computed as A/W. The corresponding "discharge" then is D2/3A and D^A for the GaucklerManning and Chezy equations, respectively. Finally, 

W versus "Q" and D versus "Q" were plotted on log 

paper, lines of best fit were drawn by eye, and the 

exponents measured graphically as usual. (The 

depth plot probably is partly spurious, in that D is 

plotted against D2/3A.) This procedure was followed

for all 165 stations. Only the exponents b and /, for 

width and depth, respectively, were studied in this 

way, and m was determined as l-b-f.

RESULTS

For all 165 stations the standard errors and percents of total sums of squares explained, for each 

exponent, are as follows:

Exponent Formula

m Gauckler-Manning ___ 

Chezy __—_—_

/ Gauckler-Manning ___ 

Chezy ____— —

6 Gauckler-Manning ___ 

Chezy _____—_

Percent of

Standard error total sums

(exponent units) of squares

explained

0.134

.164

.133

.159

.109

.105

15

0

35

7

75

77

Separating the results into firm-bank and loosebank categories did not bring about any significantly 

better accuracy in predicting the hydraulic exponents.
HYDRAULIC EXPONENTS OF INDIVIDUAL STATIONS 33

0.70 -

0.60 -

0.50 -

c

03 

C 

O 

Q. 

X 

05

Q.

E 

o 

o

0.40 -

0.30

0.20 -

0.10 -

0.10 0.20 0.30 0.40 0.50 

Measured exponent m

0.60 0.70 0.80 0.90

FIGURE 12.—Computed versus measured values of exponent m, where computed m=0.24+0.16d%—0.21 (&i/(&i+/j)) +0.00002

50

(•Dmin/dso). Standard error=0.110 exponent units, with 45 percent of sums of squares explained.

The Gauckler-Manning equation comes closer to 

estimating the exponents / and m than does the 

Chezy equation. (Standard errors for the GaucklerManning equation are lower, with a greater variance 

of the dependent variable explained.) The two flow 

equations are about equally accurate for the exponent b.

Both formulae, however, are somewhat less accurate in predicting hydraulic exponents than the 

minimum variance and empirical methods. The following section compares the results of these various 

methods.

COMPARISON OF METHODS OF COMPUTING 

HYDRAULIC EXPONENTS

Table 9 compares the statistical accuracy of the 

minimum variance (using &i/(6i + /i)), empirical 

and Gauckler-Manning methods of predicting the 

hydraulic exponents for the 165 cross sections of 

this study. The minimum-variance and empirical 

methods are about equally reliable for b. The empirical equations, while still in need of improvement, 

are the best of the three methods for / and m. This 

is not surprising, since the empirical equations were 

derived solely from the present data. However, the
34 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

TABLE 9.—Accuracy of methods of predicting hydraulic 

exponents

Method

Exponent

Minimum Variance (using 6i/(6i + /i) 

Standard error __________

Percent sums of squares explained

Empirical Equations:

Standard error __.._. — __________

Percent sums of squares explained

Gauckler-Manning:

Standard error _______________

Percent sums of squares explained

0.131 0.117 0.082

18 50 86

.110 .096 .082

45 66 86

.123 .133 .109

15 35 75

present data do cover a wide range of river conditions.

SUMMARY AND CONCLUSIONS

The minimum-variance theory assumes that in 

response to changes in water discharge, the adjustments in the major dependent variables tend to be 

as conservative as possible. These adjustments are 

reflected by the exponents in power relations between water discharge and each dependent variable.

Previous work done on this theory (Langbein, 

1964, 1965; Scheidegger and Langbein, 1966) has 

not explored the question of which variables are 

important—a question which must be resolved if 

the theory is to be applied. Also, no extensive testing 

of the theory has previously been carried out with 

field data. The 165 alluvial-channel cross sections of 

the present study provide tentative answers to these 

two problems. The data have the following ranges 

for hydraulic exponents: 0.00^&^0.82, O.IO^/ 

^0.78, and 0.03^m^0.81.

Results suggest that the major dependent variables in regard to flow adjustment at channel cross 

sections are mean velocity, water-surface width, 

mean depth, shear stress, friction factor, slope, and 

stream power. To the extent that slope remains constant at a channel cross section, the last two of this 

group can be dropped from consideration. For the 

five types of channels studied, minimum-variance 

calculations with these dependent variables produce 

values of the exponents m, f, b, and z that are reasonably close to the average exponents found for 

four natural-stream cases and for one flume case. 

The agreement suggests some promise for the 

theory.

Three methods—minimum variance, empirical relations derived from the present data, and the 

Gauckler-Manning formula—were examined for accuracy in predicting the hydraulic exponents at any 

given channel cross section. This phase of the study 

required the collection of such special data as bedsediment sizes, channel slope, and various geometrical properties of the channel cross section. The 

most accurate way to determine the at-a-station hydraulic exponents for the present data is with the 

empirical relations (equations 12-14). Using the 

equations, the predicted exponents agree with the 

measured exponents to the following extent: for m, 

the standard error S.E. is 0.110 exponent units with 

45 percent of the sums of squares of m explained; 

for /, S.E.-0.096 with 66 percent of the sums of 

squares of / explained; and for b, S.E. = 0.082 with 

86 percent of the sums of squares of b explained. 

These figures show that the equations provide only 

a rough approximation of observed hydraulic exponents. Many reasons could easily explain the differences between predicted and observed values.

To get the important channel characteristics, the 

investigator needs the median diameter of the bed 

material and the relation between water-surface 

width and cross-sectional flow area for all flows up 

to bankfull. The latter relation can be estimated 

from the data of at least three cross-sectional profiles. (As many as six profiles are recommended for 

stations in loose, sandy materials.) All profiles 

should be measured at the same section at time intervals of several weeks or months.

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1826.

STATISTICAL VARIANCE AND A HYDRAULIC EXPONENT

AND 

SUMMARY OF DATA

HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS 39

STATISTICAL VARIANCE AND A HYDRAULIC 

EXPONENT

This section explains the close relation between 

the square of a hydraulic exponent and the term 

"variance" of conventional statistics.

Consider a series of discharge measurements and 

the associated mean velocities, all taken at the same 

cross section on a stream. Change these measured 

values into logarithms and let <r = the standard deviation of the logarithms of each hydraulic property. 

For example, <riog v is the standard deviation of the 

log-velocity values, a measure of the spread of the 

log-velocity values about the mean of the log-velocity 

group. The following hypothetical example will show 

that the hydraulic exponent of velocity, m, can be 

defined as

m = '

flog V

\ flog Q I

where r is a correlation coefficient (Crow and others, 

1960, p. 158).

Suppose we have 10 different discharge measurements and a mean velocity for each discharge. It is 

specified that these two variables have a power relation, such as YocQ"1 or log V<xm (log Q). Figure 

13 is a graph of the data for this hypothetical example. The exponent m (the slope of the line) can 

be measured graphically and equals 0.5 for these 

data. The ratio of the standard deviations, <T\<>S V/ 

(nog Q, should also equal 0.5. Columns 2 and 5 of table 

10 give the measured values of log V and log Q, 

respectively. The standard deviations of the log V 

values (criogF) and of the log Q values (<ri0ge) are 

calculated in table 10 according to the usual procedure (Crow and others, 1960, p. 12). The values 

for this example come out to be ai0gF = 0.496 and 

fio«r o = 0.992. Then the ratio

flog V 0.496

= 0.5 =

alog Q 0.992

For this example, all the points lie on a straight 

line, so the correlation coefficient is 1.0 The correlation coefficients for velocity, depth, and width are 

rarely 1.0 in regression analyses but are usually 

higher than about 0.7. For example, the following 

correlation coefficients have been computed from 

Culbertson and Dawdy's (1964) data:

Rio Grande at Rio Grande near Rio Grande at 

San Felipe, N. Mex. BernalUlo, N. Mex. Cochiti, N. Mex.

Velocity __ 

Depth ____ 

Width __

1 Constant.

1.00 

.99

0.99 

.99 

O

0.98 

.98

.82

In some cases, it is therefore acceptable to assume 

that the correlation coefficients of the various hydraulic exponents are approximately equal or constant. Thus, just as m oc o-log F/o-iog c , as shown by 

this example, the same principle applies to each of 

the other dependent variables, so that

, O-log D flog W

/oc __, boc __, and so forth.

flog Q flog Q

All of the latter relations contain o-iog Q as a common factor. The proportionalities, therefore, are still 

valid if o-iog Q is deleted. This leaves /oc o-iog D, b<x o-log w, 

and so forth. In other words, each hydraulic exponent is proportional to the standard deviation 

of the logarithms of its respective hydraulic 

property. Now square both sides of these proportionalities: f°ca1; n,b2 K(r* w,and so forth. Since ' log D' log W

"variance" is the square of a standard deviation, the 

square of a hydraulic exponent is proportional to 

the variance of the logarithms of the associated hydraulic property. This is why "variance" is used as 

a shorthand term for "square of hydraulic exponent."

The above statistical definitions of the hydraulic 

exponents are not always valid. Some of the questionable procedures concerning the logarithms of the 

several hydraulic properties are the following:

1. They are treated as if they are distributed at 

random, that is, not predictable for any given 

case. Actually the hydraulic properties, rather 

than being randomly distributed, follow definite laws and so should be predictable on the 

basis of these laws. The values are not predictable at present due to insufficient knowledge. However, as explained on pages Cl and 

C2 of the Scheidegger and Langbein (1966) 

paper, the net result of many predictable actions often is the same as if the whole process 

were random. The process, therefore, lends 

itself to a statistical approach.

2. They are assumed to be mutually independent. 

The reason for this assumption is unclear and 

may depend on the definition of independence. 

Certainly the hydraulic properties (V, D, and 

W, for example) are related in that a change 

in one property usually is associated with 

changes in the other factors.

3. They are assumed to have approximately normal 

distributions. A computed standard deviation is 

most meaningful only if the distribution is approximately normal. There is no basis for assuming that the values are either normally or
40 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

TABLE 10.—Standard deviations of log V and of log Q (see fig. 13)

1

2 

3

4 

5 

6 

7 

8 

9 

10

Point on T _ T7 

graph Lo« V

0 4^

-_-_ —.30

- ___ __________ ______ +.05

- __ __ _ ____ _____ .25

. __ _ _ _ _ __ __ __ .40

- ___ _ __ — __ __ __ .55

. —— _ —— _______________ __ _ __ .70

- _ __ _ ______ __ __ .85

- _ __ _ _ _ _ ___ _ __ 1.00

Sums _ ______ 2.90

Means __ _ _ __ .29

i / S( deviations) 2

\ N-l 

]/ 2.2140

Deviation 

from 

mean

0.74

.59

.44

.24

.04

+.11 

.26 

.41 

.56 

.71

(Deviation) 2

0.5476

.3481 

.1936 

.0576 

.0016 

.0121 

.0676 

.1681 

.3136 

.5041 

2.2140

LogQ

0.10 

.40 

.70 

1.10 

1.50 

1.80 

2.10 

2.40 

2.70 

3.00 

15.80 

1.58

Deviation 

from 

mean

1.48

1.18

.88

.48

-.08 

+.22 

.52 

.82 

1.12 

1.42

——

(Deviation) 2

2.1904

1.3924 

.7744 

.2304 

.0064 

.0484 

.2704 

.6724 

1.2544 

2.0164 

8.8560

i / S( deviations) 2

\ N-l

] / 8.8560

=0.496

9

=0.992

symmetrically distributed, 

could resolve this question.

Measured data

One or more of the above three conditions probably is not satisfied in the field. The importance of 

this in regard to the statistical derivation of the 

exponents is uncertain. The statistical derivation relating the exponents to the variances of the respective hydraulic properties would, therefore, benefit 

from further study and clarification. However, such 

a derivation is not vital to the general minimumvariance theory. The theory merely proposes that 

the most probable exponents are those whose squares 

add up to the smallest number, because this situation corresponds to the most uniform distribution of 

the imposed change, as shown earlier. The association with the variance of conventional statistics is 

only of minor importance.

(For table 11 see p. 42.)
STATISTICAL VARIANCE AND A HYDRAULIC EXPONENT 41

CO

o

"> 05

«- o

I

I

.2* 'JS

IT)

c> q o

LO 

O

A 60|
42 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

TABLE 11.—

Station 

No. Station

Hydraulic exponents Minimum

variance

case 1

Log cycles

of Q 

on plot

Amount of scatter on 

hydraulic geometry plots

Velocity Depth 

(percent) (percent)

Width 

(percent)

1 Chenango River near Chenango Forks, N.Y _. 0.51 0.43 0.07 B

2 Genesee River near Mt. Morris, N.Y —————— .42 .55 .06 B

B Genesee River at Rochester, N.Y ————————— .61 .33 .05 B

4 Susquehanna River at Colliersville, N.Y ———— .42 .53 .05 B

5 White Clay Creek near Newark, Del —————— .65 .31 .04 B

6 Big Elk Creek at Elk Mills, Md ————————— .64 .32 .06 B

7 Murderkill River near Felton, Del ———————— .60 .35 .06 B

8 Rappahannock River at Remington, Va .——— .39 .47 .15 B

9 Neuse River at Kinston, N.C ..—————————. .61 .32 .08 B

10 Middle Creek near Clayton, N.C ..—-————— .33 .34 .30 B

11 Nahunta Swamp near Shine, N.C -——————— .53 .42 .04 B

12 Contentnea Creek at Hookerton, N.C —————— .27 .59 .13 B

13 Fishing Creek near Enfleld, N.C __________—— .41 .49 .07 B

14 Trent River near Trenton, N.C ________——— .59 .33 .09 B

15 Hyco River at McGehees Mill, N.C ———————— .14 .78 .OS B

16 Savannah River at Augusta, Ga. (Butler Cr.) — .35 .60 .08 B

17 Little River near Mt. Carmel, S.C __—————— .20 .76 .05 B

18 Little River near Adel, Ga ————————————— .33 .47 .20 B

19 Rocky Creek near Dudley, Ga __-_______—.___ .46 .44 .10 B

20 Suwannee River at Fargo, Ga __————_.—___ .20 .54 .26 B

21 Altamaha River at Doctortown, Ga __————— .49 .45 .07 B

22 Alapaha River at Statenville, Ga _________ .15 .76 .11 B

23 Spruce Creek near Samsula, Fl _.————_———_ .52 .38 .10 B

24 Econflna River near Perry, Fl ___________ .38 -49 .15 B

25 Bayou Pierre near Willows, Miss _________ .24 .43 .35 B

26 Nishnabotna River above Hamburg, Iowa ___ .35 .60 .05 B/C

27 Wapsipinicon River near DeWitt. Iowa ._—_. .29 .68 .03 A

28 Des Moines River near Tracy, Iowa ______ .37 .62 .01 A

29 Elkhorn River at Ewing, Nebr ___.______ .27 .35 .40 B/C

30 Elkhorn River near Waterloo, Nebr. .44 .50 .04 B 

	(upper cable).

31 Elkhorn River near Waterloo, Nebr. .52 .48 .01 A 

	(lower cable).

32 Elkhorn River at Neligh, Nebr ___________ .31 .65 .04 B

33 Elkhorn River near Norfolk, Nebr _.______ .35 .56 .09 B

34 Little Blue River near Deweese, Nebr __—__ .21 .50 .32 B/C

35 West Fork Big Blue River near .25 .60 .14 B 

	Dorchester, Nebr.

36 Niobrara River near Cody, Nebr _________ .46 .54 .02 A

37 Republican River at Stratton, Nebr ______ .19 .23 .62 B/C

38 Republican River below Harlan Co. Dam, Nebr .40 .58 .01 A

39 Republican River near Guide Rock, Nebr ___ .20 .72 .07 B

40 Plum Creek at Meadville, Nebr _________ .47 .52 .03 A

41 Smoky Hill River at Elkader, Kansas _____ .03 .22 .75 C

42 Beaver Creek at Cedar Bluffs, Kansas ____ .27 .55 .19 B/C

43 Arkansas River at Syracuse, Kansas— .23 .72 .04 B 

	300 ft above gage.

44 Arkansas River at Syracuse, Kansas— .20 .33 .49 B/C 

	400 ft above gage.

45 Sappa Creek near Oberiin, Kansas—low flows __ .16 .33 .51 C

46 Sappa Creek near Oberiin, Kansas— .18 .62 .20 B/C 

	medium flows.

47 Washita River near Durwood, Okla _______ .36 .41 .21 B/C

48 Wolf Creek near Fargo, Okla _________ .07 .13 .82 B/C

49 Canadian River near Noble, Okla ________ .34 .24 .43 C

50 Canadian River at Bridgeport, Okla _______ .24 .15 .63 B/C

51 North Canadian River near El Reno, Okla __ .21 .31 .50 C

52 Cimarron River near Buffalo, Okla _______ .18 .13 .67 C

53 Cimarron River near Guthrie, Okla ______ .24 .41 .34 B/C

54 Wichita River at Wichita Falls, Tex ______ .32 .59 .09 B

55 Wolf Creek at Lipscomb, Tex ______________ .31 .29 .42 C

56 Canadian River at Tascosa, Tex _________ .28 .22 .50 B/C

57 Canadian River near Amarillo, Tex ._______ -20 .17 .65 B/C

58 Red River near Quanah, Tex ___________ -21 .25 .53 B/C

59 Prairie Dog Town Fork Red River near -19 .24 .57 C 

	Lakeview, Tex.

60 Prairie Dog Town Fork Red River near -25 .24 .54 C 

	Childress, Tex.

61 Brazos River near South Bend, Tex __._.__ .37 .57 .08 B/C

62 Brazos River at Sevmour, Tex ___________ .34 .33 .33 B/C

63 Rio Chama near Chamita, N. Mex ________ .36 .43 .24 B/C

64 Rio Grande Floodway at San Marcial, N. Mex. .40 .56 .05 B

	—cable.

65 Rio Grande Floodway at San Marcial, N. Mex. .18 .21 .63 B/C

	—200-300 ft above gage.

66 Rio Grande at Otowi Bridge near .37 .55 .11 B 

	San Ildefonso, N. Mex.

67 Canadian River at Logan, N. Mex _________ -39 .57 .02 A

68 Revuelto Creek near Logan, N. Mex _____._ -26 .70 .07 B

69 Pecos River near Acme, N. Mex.— -38 .11 .51 B/C 

	low-intermediate flows.

70 Pecos River near Acme, N. Mex.—cable ____ -43 .56 .01 A

71 Pecos River near Artesia, N. Mex ________ .29 .65 .06 B

72 San Jlian River at Farmington, N. Mex ____ .35 .61 .07 B

73 San Juan River near Shiprock, N. Mex .-.___ .49 .49 .05 B/R

74 Gila River near Gila, N. Mex __________ .40 .52 .09 R

75 Gila River near Redrock, N. Mex.—low flows __ .26 .12 .64 C/R

76 Gila River near Redrock, N. Mex.—high flows _ .33 .59 .08 R

77 Rio Salado near San Acacia, N. Mex _____ .22 .17 .63 C

78 San Francisco River near AJma, N. Mex ___ .23 .22 .55 C

79 Cherry Creek near Melvin, Colo. (1940-60 site). .29 .20 .51 B/C

80 Cherry Creek near Melvin, Colo. (1960-68 site) _ .32 .38 .32 B/C

81 Cherry Creek near Franktown, Colo _______ .18 .15 .65 C

82 Kiowa Creek at Kiowa, Colo ____________ .42 .58 .02 A

See footnotes at end of table.

1.0 

1.4 

1.3 

1.0 

.8 

1.3 

1.2 

1.2 

1.2 

2.1 

1.2

1.0 

.8

1.1 

1.0

.7 

1.2 

1.9 

1.5 

1.3

.6

1.0

1.1 

1.8 

1.3 

2.0 

1.0 

1.4 

1.2 

1.2

1.2

1.0 

1.9 

1.2 

1.5

3.3

1.0

1.1

.7

1.9

1.6 

.6

1.7

2.3 

1.1

1.0 

1.2 

2.9 

1.9 

2.1 

2.5 

2.3 

.9 

2.3 

1.9 

1.9 

2.6 

4.1

2.5

1.4

2.1

1.4

.9

2.6

1.1 

2.0

1.2 

1.1 

1.1

.8 

1.3

.8 

3.4 

3.1 

2.9 

2.8 

2.0 

1.9

2/2 

12/10 

25/25 

12/5 

12/8 

42/24 

36/14 

30/18 

20/16 

18/29 

47/25 

9/16 

7/9 

5/4 

19/19 

15/10 

24/9 

74/32 

50/34 

26/22 

16/8 

14/10 

29/24 

42/30 

27/20 

14/14 

10/18 

11/6 

15/14 

20/18

12/18

15/15 

25/13 

28/26 

34/40

18/16

28/17 

5/10 

14/7 

15/22 

49/75 

10/9 

13/9

20/10

50/48 

13/15

13/12 

17/20 

20/25 

33/22 

70/25 

45/26 

31/27 

21/20 

40/20 

27/31 

23/28 

52/30 

53/47

87/50

32/18

47/36

41/25

8/13

42/27 

25/19

5/7 

22/13 

38/22

11/10 

55/18 

50/21 

27/20

3/4 

40/20

4/9 

31/27 

15/36 

32/40 

30/18 

42/25 

20/12

5/4 

10/10 

24/24 

12/12 

14/10 

41/32 

23/12 

15/14 

17/16 

27/28 

42/30 

16/10

8/6

9/6 

25/12 

10/13 

10/23 

34/44 

45/34 

25/20 

17/11

8/10 

40/28 

30/16 

27/15 

15/12 

23/7

8/9 

29/28 

21/17

20/12

26/12 

18/21 

41/18 

54/26

25/13

48/32

16/6

13/16

35/15

39/17

31/8

15/14

48/15

47/36 

16/13

20/14 

31/21 

53/20 

60/34 

60/39 

28/26 

47/15 

21/15 

46/22 

40/34 

64/32 

41/30 

95/28

78/34

30/27 

45/35 

31/26 

21/4

34/24 

27/20

8/4 

21/14 

60/48

13/11 

30/36 

25/27 

24/19

7/3 

30/20 

15/20 

40/18 

60/32 

56/41 

47/27 

47/28 

20/13

2/4

18/12

8/6

7/11

3/4

6/8

7/17

7/11

7/7

22/16

12/12

7/6

7/3

6/3

7/4

3/3

6/3

21/19

16/10

29/23

3/3

7/6

13/23

44/18

16/13

5/8

4/2

2/2

51/22

4/6

3/1

8/11 

12/18 

20/14 

14/8

4/3 

60/40

4/3 

19/10

9/6 

70/51 

10/10

3/17

24/37

65/24 

10/10

9/7 

47/35 

31/28 

55/46 

46 /35 

45/37 

41/30

6/12 

76/50 

65/36 

72/43 

60/46 

80/46

110/53

9/22

75/42

31/22

3/4

30/41 

10/19

3/2

8/5

70/32

11/4

12/14

10/5

9/5

3/4

33/24

13/16

46/30

38/36

100/42

56/42

50/35

10/16
SUMMARY OF DATA 43

Summary of data

Slope S 

(ft/ft)

0.0008

.0005

.0002

.0020

.0008

.0032

.0019

.0018

.0001

.0049

.0009

.0002

.0008

__

.0006

.0005

.0001

.0003

.0004

.0002

.0004

.0008

.0002

.0004

.0003

.0007

.0004

.0004

.0010

.0007

.0014

.0002

.0015

.0017

.0007

.0008

.0019

.0255

.0009

.0014

.0014

.0013

.0013

.0002

.0009

.0005

.0018

.0006

.0010

.0004

.0002

.0008

.0012

.0010

.0007

.0020

.0011

.0003

.0006

.0012

.0004

.0004

.0021

.0006

.0017

.0006

.0006

.0003

.0016

.0019

.0034

.0050

.0050

.0063

.0026

.0040

.0038

.0060

.0076

Average 

bank _

angle 9 

(degrees)

6.0

19.4

23.6

22.1

24.9

22.1

23.3

13.4

20.2

2.6

55.6

19.3

30.3

34.5

53.3

26.3

52.1

18.4

11.4

8.5

14.2

36.8

41.2

44.9

1.0

32.6

30.6

30.2

1.8

17.0

22.0

28.8

9.4

2.0

24.2

28.2

1.3

30.6

17.1

53.0

1.2

27.8

12.5

.9

7.7

28.1

10.5

.8

.6

.2

.8

.3

.5

40.3

.5

.1

.5

.4

.2

.1

15.4

1.2

.8

13.2

.4

36.1

18.6

37.9

.9

28.8

34.1

17.4

14.7

20.7

1.0

42.7

.2

.6

.5

.5

1.0

66.6

dso 

(mm)

47

21

85

50

35

2..7

7.6

1.4

.50

.83

.60

.58

1.1

1.6

1.8

.80

.80

..75

.35

.22

.60

.64

.17

.31

.38

,29

.64

.53

,30

.24

.24

.26

^25

1.1

.48

.27

.32

.56

.44

,75

1.2

.20

.58

.58

1.4

1.4

.17

.36

.19

.14

.21

.36

..35

.38

.38

.20

.22

J8

.20

.16

.31

.31

10.3

,14

.14

.53

.18

.22

.16

,16

J7

,33

.35

48

.70

.70

.14

.42

.40

.50

.95

.58

dst 

(mm)

77

12V"

93

140

14.5

30

5.5

.90

3.3

18

1.2

2.8

10

4.3

3.5

2.1

".53

.50

.90

.95

.26

.49

.55

.40

1.2

1.0

.50

.37

.50

.45

.43

~1~6~

.80

".4<T

1.0

12

 

____

 

.23

.53

.27

.25

.31

.53

.63

.60

.59

.28

.31

.25

.34

.30

.38

.45

31

___-

....

"".40

.22

.22

"".50

120"""

6.0

6.0

.38

.86

1.1

T.2~

Sorting 

So

0.875

"567

.967

1.360

1.553

1.427

1.301

.610

1.326

1.117

.686

1.233

1.865

1.137

1.599

1.152

".609

.699

.431

.447

.624

.699

.464

.322

.791

.909

.602

.693

.590

.565

.572

".667

.486

2.39§

1.029

2.000

 

___.

....

.431

.450

.364

.590

.418

.444

.637

.573

.496

.532

.568

.615

.699

.869

.312

.377

2.714

.---

____

____

".626

.427

.427

".555

l".735

1.430

1.430

1.258

.820

".985

"870

61

bi + h

0.16

.18

.09

.09

.13

.11

.12

.18

.20

.42

.06

.16

.12

.19

.06

.08

.06

.21

.20

.17

.11

.09

.16

.23

.42

.11

.05

.04

.48

.10

.04

.09

.30

.39

.22

.08

.52

.03

.09

.04

.63

.14

.05

.52

.30

.23

.17

.98

.60

.83

.58

.55

.59

.05

.58

.65

.69

.55

.60

.61

.14

.38

.43

.06

.70

.14

.04

.07

.68

.10

.08

.09

.09

.09

.84

.05

.67

.67

.55

.50

.55

.03

A max 

(ft2 )

4000

1800

2870

830

200

126

52

1730

2950

43

81

810

1470

760

525

9100

480

440

101

240

4300

1130

49

700

113

2750

1670

6200

95

2050

1970

162

3210

92

725

240

41.5

400

790

248

8.4

34.0

415

104

18.0

133

1320

23.5

3000

470

24.7

340

2200

825

3100

3700

215

14500

7000

4000

4100

360

125

500

90

2650

1040

2600

111

1070

940

1970

1220

267

87

430

277

46.5

137

400

31.0

545

Amln (ft2 )

94

202

225

275

51

15.8

6.8

94

202

.93

24

162

63

66

29.7

680

57

52

8.5

25.8

820

123

5.6

26.0

2.5

136

457

925

.8

580

173

53.0

605

1.68

118

41.5

.84

24.3

124

17.0

.53

1.14

132

1.1

.38

20.5

31.0

3.9

5.8

8.0

.94

2.2

7.4

69

.50

3.4

8.0

.94

4.8

.66

530

26

9.6

40

.50

238

157

495

.47

192

59

460

88

38.8

2.4

87

1.50

.45

1.26

2.0

.85

75

Wmax 

(ft)

404

168

242

160

64

51

30

182

191

47

39

127

123

72

67

420

73

80

48

95

415

122

18

50

122

170

245

475

80

282

296

66

281

80

90

74

63

127

173

58

30

17

212

116

19

35

168

58

760

520

63

335

660

127

780

1530

213

1900

1690

1780

315

220

151

170

147

150

262

216

118

145

113

275

210

86

99

106

333

92

168

295

63

81

Win in 

(ft)

222

114

192

143

54

41

24

107

113

9.3

37

98

84

44

56

339

64

51

29

65

349

100

13.4

23

25

122

228

441

8.0

248

266

60

170

17

60

64

8.4

117

148

52

5.3

10.8

200

11.0

6.2

23

86

10.0

18

17

9.4

22

23

113

5.0

17.5

22.5

9.4

21.0

8.5

233

81

51

147

3.7

106

242

191

2.8

123

90

240

169

72

4.7

99

10.0

4.0

12.6

20.0

8.5

76

ZVax 

(ft)

9.9

10.7

11.8

6.1

3.1

2.4

1.6

9.5

15.4

.9

2.0

6.3

11.9

10.5

7.8

21.6

6.5

5.5

2.1

2.5

10.3

9.2

2.6

14.0

.9

16.1

6.8

13.0

1.1

7.2

6.6

2.4

11.4

1.1

8.0

3.2

.6

3.1

4.5

4.2

.2

1.9

1.9

.8

.9

3.8

7.8

.4

3.9

.9

.4

1.0

3.3

6.5

4.0

2.4

1.0

7.6

4.1

2.2

13.0

1.6

.8

2.9

.6

17.7

4.0

12.0

.9

7.4

8.3

7.2

5.8

3.1

.9

4.1

.8

.5

.8

1.4

.5

6.7

Cm in (ft)

0.4

1.8

1.2

1.9

.9

.4

.3

.9

1.8

.1

.6

1.7

.8

1.5

.5

2.0

.9

1.0

.3

.4

2.3

1.2

.4

1.1

.1

1.1

2.0

2.1

.1

2.3

.7

.9

3.6

.1

2.0

.6

.1

.2

.8

.3

.1

.1

.7

.1

.06

.9

.4

.4

.3

.5

.1

.1

.3

.6

.1

.2

.4

.1

.2

.08

2.3

.3

.2

.3

.1

2.2

.6

2.6

.2

1.6

.7

1.9

.5

.5

.5

.9

.2

.1

.1

.1

.1

6.0
44 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

TABLE 11.—Summary

Amount of scatter on

Station «<.-«-,.

No.

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129 

130 

131 

132 

133

134

135

136

137

138 

139

140

141

142

143

144

145

146 

147

148

149

150

151

152

153

154 

155

See

HJM»b!Wll ————

Huerfano River below Huerfano Valley Dam,

Colo. — low flows.

Huerfano River below Huerfano Valley Dam,

Colo. — high flows.

Arkansas River near Coolidge, Kansas —

low flows.

Arkansas River near Coolidge, Kansas —

high flows.

Blacks Fork near Little America, Wyo _ __ _

Blacks Fork near Lyman, Wyo _ __ __ _ _

Blacks Fork near Millburne, Wyo ____________

Wind River near Crowheat, Wyo _._ _ _ _____

Muddy Creek near Shoshoni, Wyo ——— _ __ ___

Powder River at Arvada, Wyo _ _ ___ _ _ _

South Fork Powder River near Kaycee, Wyo _

Sweetwater River near Alcova, Wyo _______

Cheyenne River near Spencer, Wyo _ ____ _

Lance Creek at Spencer, Wyo __ — _ _ __ _

Belle Fourche River below Moorcroft, Wyo ___

Virgin River at Virgin, Utah ._ ____ ____

Summit Creek near Summit, Utah __ _ _ _

Coal Creek near Cedar City, Utah _ _ _

Santa Clara River near Santa Clara, Utah ____

Sevier River near Circleville, Utah _ ____ _

Sevier River near Lynndyl, Utah —

300-400 ft below gage.

Sevier River near Lynndyl, Utah —

200-300 ft above gage.

Escalante River near Escalante, Utah ___ _ _

Pine Creek near Escalante, Utah .-._ __ ___

San Juan River near Bluff, Utah .. _ _

Dirty Devil River above Poison Spring Wash

near Hanksville, Utah.

Huntington Creek near Huntington, Utah ______

Saleratus Wash at Green River, Utah ____ _

Green River at Green River, Utah ____ ___

White River below Tabbyune Creek near

Soldiers Summit, Utah.

Green River near Ouray, Utah _ _

Duchesne River at Myton, Utah . __

Tonto Creek above Gun Creek near

Roosevelt, Ariz.

San Pedro River at Winkelman, Ariz _ __ _ ... 

Verde River below Tangle Creek, Ariz

Gila River at Kelvin, Ariz __ _ _ ______ __

Gila River at head of Safford Valley near 

So'omon, Ariz — all flows.

Gila River at Safford Valley near Solomon,

Ariz. — intermediate flows.

Gila River at Calva, Ariz. — intermediate flows _

Gila River at Ca?va, Ariz. — high flows __ _ _

Gila River near Clifton, Ariz __ __ _. __ ___

Colorado River near Grand Canyon, Ariz _ .

Virgin River at Littiefield, Ariz ____ _______

Humboldt River near Argenta, Nev. — 

low flows.

Humboldt River near Argenta, Nev. —

high flows.

Humboldt River at Comus, Nev. — low flows __

Humboldt River at Comus, Nev. — high flows __ 

Humboldt River at Palisade, Nev _ _______ 

Humboldt River near Elko, Nev _ _ __ _ ____ 

Cajon Creek near Keenbrook, Calif __________ 

Santa Margarita River at Ysidora, Calif. — 

0-200 ft above gage.

Santa Margarita River at Ysidore, Calif. —

300-500 ft below gage.

San Onofre Creek near San Onofre, Calif ..__

San Luis Rey River near Bonsall. Calif ______

San Luis Rey River at Monserate Narrows near 

Pala, Calif.

Warm Creek near San Bernardino, Calif _____ 

Santa Ana River near Mentone, Calif. — 

low flows.

Santa Ana River near Mentone, Calif. —

medium flows.

Wilson River near Tillamook, Oreg __________

White River below Tygh Valley, Oreg ________

White River near Government Camp, Oreg ____

Flynn Creek near Salado, Oreg ______________

Deer Creek near Salado. Oreg _ _

Mo'alla River near Canby, Oreg _ ___ _ _____ 

Nehalem River near Foss, Oreg __ ____ _ 

Nestucca River near Beaver, Oreg _ ___ __

Stehekin River at Stehekin, Wash __ .. _.

Spokane River above Liberty Bridge near

Otis Orchards, Wash.

Okanogan River at Malott, Wash __ __________

Little Scokane River at Dartford, Wash _____

Kettle River near Ferry, Wash

White River near Plain, Wash _. 

Naches River below Tieton River near

Naches, Wash.

footnotes at end of table.

Hydraulic exponents

TO

.30

.40

.15

.30

.67

.30

.45

.44

.36

.46

.38

.36

.27

.16

.17

.25

.68

.49

.47

.47

.41

.48

.30

.47

.43

.23

.60

.24

.58

.46

.36

.57

.47

.43

.57

.25

.38

.54

.42

.22

.38

.52

.38

.22

.32

.16

.28 

.28 

.26 

.37 

.12

.26

.30

.22

.18

.34

.14

.42

.47

.38

.46

.34

.45

.68 

.43 

.49

.59

.62

.44

.36

.29 

.59

f

.29

.61

.18

.67

.28

.41

.23

.54

.61

.50

.15

.58

.28

.31

.37

.13

.22

.25

.14

.41

.56

.40

.32

.37

.55

.30

.31

.29

.39

.33

.46

.42

.50

.46 

.36

.28

.38

.44

.53

.67

.56

.44

.22

.32

.64

.29

.64 

.37 

.69 

.23 

.33

.29

.59

.25

.13

.60 

.10

.38

.52

.55

.47

.45

.40

.29 

.48

.47

'.1824

.33

.45

.60

.71 

.34

b

.41

.00

.69

.03

.05

.29

.31

.03

.05

.07

.47

.07

.45

.55

.46

.61

.12

.24

.41

.12

.05

.13

.41

.19

.04

.46

.08

.50

.03

.21

.19

.03

.02

.12 

.07

.49

.24

.01

.05

.13

.09

.04

.42

.46

.06

.55

.08 

.35 

.09 

.40 

.56

.48

.11

.56

.68

.04

.74

.23

.04

.05

.06

.24

.16

.03 

.10 

.06

.04

.23

.05

.11

.04

.01 

.07

Minimum

case 1

B/C

A

B/C

A

B

B/C

B/C

A

B/R

B

B/C

B/C

B/C

B/C

C

B/C

B

B/C

B/C

B

B

B/C

C

B/C

B

B/C

B

C

A

B/C

B/C

A

A

B/C 

B/R

B/C

B/C

A

B

B

B/R

R

B/C

C

B

C

B 

B/C 

B 

B/C 

B/C

B/C

B

B/C

B/C

B 

B/C

B/C

B

B

B

B/C

B

A 

B 

B

B

B

B

B

A 

B

Log cycles of Q

on plot

2.4

.8

1.9

1.4

1.9

1.7

1.6

.7

1.1

1.4

2.0

1.2

2.4

3.8

4.1

1.1

1.2

1.9

1.6

1.3

1.5

1.3

2.0

1.7

1.4

2.9

1.0

3.4

1.0

1.6

1.4

.9

1.5

1.7 

.9

.8

2.6

.9

1.7

.6

1.4

r.o

1.6

2.7

1.2"

2.8

1.6 

1.5 

1.4 

1.5 

3.0

2.3

1.8

3.3

2.0

1.9 

1.9

1.8

1.8

1.0

.9

2.1

2.1

1.2 

2.0 

1.0

.9

1.2

.7

1.0

1.0 

1.1

hydraulic geometry plots

Velocity

(percent)

14/23

7/8

31/15

17/10

15/12

15/13

26/32

10/10

23/21

20/16

22/25

14/14

82/30

88/42

80/32

28/27

20/21

35/23

22/25

13/9

11/10

35/26

26/29

32/39

40/26

26/33

9/10

30/42

6/7

20/26

21/21

15/12

18/16

48/18 

26/10

10/12

23/15

20/12

13/18

15/8

19/18

3/4

31/24

80/40

12/10

110/53

13/10 

28/31 

20/24 

29/28 

80/50

25/18

22/18

46/44

63/35

19/12 

57/38

82/28

32/12

15/17

22/7

62/47

82/43

13/20 

15/19 

5/6

8 a / 11t

5/5

20/11

4/4

5/4

9/15 

10/12

Depth

(percent)

48/14

10/8

37/27

30/14

10/13

30/26

19/20

10/11

25/22

21/15

62/39

21/16

27/34

67/30

120/44

38/30

12/16

22/25

30/20

6/7

13/16

38/30

40/32

42/24

22/24

25/24

14/9

32/24

10/7

40/26

37/20

19/12

24/25

23/30 

11/20

20/17

18/25

21/12

18/11

10/12

20/24

5/4

82/26

47/28

13/15

48/49

16/20 

38/18 

25/26 

45/33 

78/36

40/30

28/13

50/20

110/42

14/22 

32/21

58/34

10/28

25/9

9/16

57/27

71/35

25/10 

22/14 

6/7

2/4

2/7

8/18

5/7

6/5

13/10 

12/9

Width

(percent)

32/24

11/8

50/44

10/32

5/6

25/28

44/22

4/4

7/6

23/23

58/43

6/6

84/57

70/42

62/34

34/26

15/8

55/23

27/17

5/7

14/4

19/10

47/30

37/34

3/6

40/47

2/7

83/32

1/4

15/24

15/13

6/7

16/6

44/20 

7/6

32/25

30/19

15/13

17/22

3/6

21/22

2/2

32/32

62/42

12/7

90/45

9/13 

18/15 

12/7 

26/23 

40/34

52/41

10/16

60/40

75/43

13/8 

60/40

28/33

7/7

5/4

4/6

42/30

24/46

5/6

5/7 

1/5

6/4

2/6

4/5

3/6

2/3

2/2 

2/4
SUMMARY OF DATA 45

of data—Continued

Slope S 

(ft /ft)

.0113

.0113

.0013

.0013

.0008

.0008

.0140

.0033

.0041

.0009

.0019

.0011

.0018

.0014

.0006

.0080

.0450

.0170

.0076

.0006

.0006

.0006

.0080

.0213

.0019

.0024

.0160

.0020

.0009

.0060

.0002

.0029

.0035

.0030

.0047

.0015

.0019

.0019

.0017

.0017

.0017

.0001

.0021

.0006

.0006

.0004

.0004

.0012,

.0015

.0246

.0048

.0048

.0063

.0037

.0030

.0038

.0189

.0189

.0027

.0096

.0142

.0300

.0217

.0019

.0008

.0019

.0060

.0020

.0001

.0019

.0009

.0007

.0049

Average 

bank_ 

angle 8 

( degrees )

.6

.0

.7

23.1

17.5

1.5

2.6

25.4

41.8

9.3

.5

32.9

.7

.5

11.1

.9

34.3

4.5

1.9

22.7

24.2

17.4

2.3

23.4

25.6

1.0

14.7

5.9

30.8

4.4

2.6

46.4

17.0

11.0

31.0

1.5

3.3

16.8

30.2

39.1

36.3

50.4

.4

2.7

29.4

.7

24.8

2.5

36.4

3.5

1.7

1.1

17.9

1.2

.5

29.5

1.9

8.3

27.7

17.4

20.6

14.3

31.0

31.9

13.1

23.8

21.0

1.8

24.9

38.0

24.4

44.6

7.9

(JoO

(mm)

.35

,35

,68

,68

18

J5

46

35

.25

-.18

.55

.97

.80

,30

3J

.33

24

12

.73

.41

,22

.22

14

J6

J)83

6,2

.062

.074

2.7

.20

1.05

.66

23

w42

,48

.48

"TJii

.27

1.8

6.6

.66

.90

.53

.60

1.6

.27

.16

.58

.20

..36

.78

43

15

100is""""

2.8 32"

36

45

23

77

33

59

63

56

13

86

ds4 

(mm)

.68

.70

_-.-

30

.53

70

43

~2~.6~

~7~6~

53

38

2.1

1.0

10.0

9.7

21.5

.25

12.1

.75

.16

8.9

IB"""

8~7~~~

.---

-___

15~"~

20

4.8

5.4

1.2

2.2

12.7

.71

.38

4.5

.44

.77

3.3

149

98

240

62"""

11.7

88

75

110

65

244

115

150

153

157

41

147

Sorting 

So

1.100

1.111

.---

..--

.975

.470

.573

.301

..__

____

-._

.914

__-_

1~566

.820

1.248

1.041

.699

2.244

2.243

.447

1.107

1.845

1.875

.824

2.497

2. 391

1~398

——

-_-_

2~ 42 7

1.865

1.590

1.642

.925

1.491

1.793

1.086

1.146

1.942

.911

.806

1.371

1.754

2.727

1.121

2~.125

1.706

1.164

.981

.910

1.220

1.477

1.288

1.869

1.301

1.000

2.136

.608

bi

bi + h

.71

.00

.65

.05

.15

.57

.53

.09

.06

.14

.71

.06

.69

.68

.32

.47

.15

.54

.66

.21

.10

.15

.54

.32

.09

.52

.17

.37

.06

.35

.45

.10

.06

.21

.13

.57

.43

.08

.08

.13

.11

.07

.74

.49

.10

.75

.12

.41

.11

.42

.50

.65

.17

.61

.73

.09

.71

.31

.11

.12

.14

.37

.20

.09

.20

.08

.21

.56

.10

.08

.09

.07

.18

/I max 

(fta)

83

261

252

700

670

142

220

1330

51

113

26.0

200

74

31.5

387

175

10.5

26.0

3.35

180

317

190

6.4

81

1500

77

85

1400

3500

48

4100

545

460

600

440

96

2600

480

180

705

760

11000

101

36.2

660

19.0

500

147

940

160

600

104

166

270

31.0

202

5.1

75

2400

322

177

96

59

1220

4200

1120

1250

2200

3950

302

2200

1210

540

A i

(ft2 )"

7.1

83

8.0

252

95

1.24

.93

301

10.7

43.7

.39

52.5

.92

.48

1.15

1.79

1.11

.62

.29

13

52

58

.46

2.2

222

1.0

7.6

1.9

1000

.91

31

172

126

98

40

.77

6.2

113

56

180

84

880

3.7

.93

62

.10

19

1.2

157

.76

3.8

.50

7.3

.21

.17

17.7

.14

5.1

213

81

44

.81

2.0

290

72

68

390

3.0

257

29.0

340

112

67

Wmax (ft)

123

123

200

210

117

106

100

188

30

84

78

76

110

84,

68

157

9.8

27

14

43

84

70

22

23

190

99

42

172

335

40

445

79

182

125

70

88

310

151

71

86

92

315

160

41

101

54

80

87

106

76

206

102

49

160

81

59

17

41

185

96

64

30

19

157

254

149

144

370

260

63

227

117

147

Wmin (ft)

21.0

123

21.0

200

86

7.0

5.3

165

27

74

3.9

70

5.2

4.8

10.4

18.0

7.0

3.5

2.9

24

70

58

4.6

7.2

162

10.0

28

15

310

9.9

49

70

170

84

51

5.7

22

135

65

71

72

262

14

6.7

80

1.0

54

12

86

7.6

16

3.1

29

2.1

1.7

48

1.4

18

140

80

53

5.2

9.8

138

115

117

113

9.0

199

52

191

98

101

Umax (ft)

.7

2.1

1.3

3.3

5.7

1.3

2.2

7.1

1.7

1.3

.3

2.6

.7

.4

5.7

1.1

1.1

1.0

.2

4.2

3.8

2.7

.4

3.5

7.9

.8

2.0

8.1

10.4

1.2

9.2

6.9

2.5

4.8

6.3

1.1

8.4

3.2

2.5

8.2

8.3

35.0

.6

.9

6.5

.4

6.3

1.7

8.9

2.1

2.9

1.0

3.4

1.7

.4

3.4.

.3

1.8

13.0

3.4

2.8

3.2

3.0 7.8'

16.5

7.5

8.7

5.9

15.2

4.8

9.7

10.3

3.7

Dmln

(ft)

.3

.7

.4

1.3

1.1

.2

.2

1.8

.4

.6

.1

.8

.2

.1

.1

.1

.2

.2

.1

.5

.8

1.0

.1

.3

1.4

.1

.3

.1

3.2

.09

.6

2.5

.7

1.2

.8

.1

.3

.8

.9

2.5

1.2

3.4

.3

.1

.8

.1

.4

.1

1.8

.1

.2

.2

.3

.1

.1

.4

.1

.3

1.5

1.0

.8

.2

.2

2.1

.6

.6

3.5

.3

1.3

.6

1.8

1.1

.7
46 HYDRAULIC GEOMETRY OF RIVER CROSS SECTIONS

TABLE 11.—Summary

Station ot t._ JT Station

156 

157 

158 

159 

160 

161 

162 

163 

164 

165

Yakima River at Cle Elum, Wash .. . ____ .

Hydraulic exponents

m

ft 07

.53

.49

.51

.35

.42 

.54

.67

.81

/

0.60 

.43 

.60 

.44 

.38 

.61 

.54 

.41 

.31 

.15

6

0.02 

.03 

.07 

.05 

.10 

.04 

.04 

.05 

.02 

.02

Minimum 

variance 

case 1

A 

A 

B 

B 

B 

B 

B 

B 

A 

A

Log cycles 

of 0 

on plot

0.7 

1.0 

.9 

1.2 

2.0 

l.t 

1.3 

1.5 

1.0 

1.2

Amount of scatter on 

hydraulic geometry plots

Velocity 

(percent)

3/5 

9/9 

10/7 

3/7 

10/5 

5/6 

3/8 

30/10 

3/4 

7/7

Depth 

(percent)

7/5 

9/10 

7/6 

5/4 

6/12 

8/7 

6/5 

13/15 

7/4 

5/10

Width 

(percent)

1/2 

3/7 

5/7 

2/2 

4/6 

2/3 

1/4 

5/6 

3/4 

1/2

A = approximately vertical banks;

B = banks firm but not vertical;

C = loose, noncohesive banks;

R = rock banks;

B/C, B/R, C/R = one bank of each type indicated (for example, B/C = one bank firm but not vertical, the other bank noneohesive).
SUMMARY OF DATA 47

of data—Continued

Slope S

(ft /ft)

0.0030

.0059

.0064

.0036

.0240

.0002

.0005

.0120

.0004

.0011

Average

bank

angle 6

(degrees)

22.0

15.5

37.2

30.3

15.2

23.1

30.0

35.2

37.5

31.8

dso

(mm)

37

100

24

58

88

.65

100

90

104

95

ds*

(mm)

66

170

37

172

217

2.6

140

130

160

----

Sorting

So

1.393

.723

.544

1.653

1.033

1.286

1.176

1.029

.893

----

61

. ,

1

0.04

.12

.12

.08

.25

.06

.07

.05

.08

.09

A. x

(ft2 )

1190

370

435

2200

258

14600

1570

475

2950

3600

Am i

(W)

270

23.5

53

245

36

940

108

86

635

770

Wmax

(ft)

267

96

67

202

65

640

174

110

241

260

Wmin

(ft)

250

70

52

169

40

540

145

100

216

225

Dmax

(ft)

4.5

3.9

6.5

10.9

4.0

22.8

9.0

4.3

12.2

13.8

Dm In

(ft)

1.1

.3

1.0

1.4

.9

1.7

.7

.9

2.9

3.4