Statistical Properties 

of Dune Profiles

GEOLOGICAL SURVEY PROFESSIONAL PAPER 562-F
Statistical Properties 

of Dune Profiles

By CARL F. NORDIN, JR.

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 562-F

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1971
UNITED STATES DEPARTMENT OF THE INTERIOR

GEOLOGICAL SURVEY 

William T. Pecora, Director

Library of Congress catalog-card No. 77-609557

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

Washington, D.C. 20402 - Price 50 cents (paper cover)
CONTENTS

	Page Page

Abstract.......................................................................... Fl Data and analysis Continued

Introduction..................................................................... 1 Stationarity and equilibrium flow............................. F18

Background............................................................... 1 The Markov model for dune profiles......................... 20

Purpose and scope...................................................... 3 Dimensionless spectra........................................... 21

Acknowledgments....................................................... 3 Other properties of the spectra................................ 27

Some properties of a Gaussian process................................. 4 Cross correlation and cross-spectral analysis.............. 32

Zero and h-level crossings............................................ 4 Prediction................................................................. 35

Wave heights and amplitudes........................................ 5 Summary and conclusions.................................................. 37

Theory of time series analysis....................................... 6 Discussion of results.................................................... 37

Data and analysis.............................................................. 8 Conclusions............................................................... 38

Basic data................................................................. 8 Recommendations for future studies............................... 39

Zero and Wevel crossing analyses.................................. 10 References...................................................................... 39

Distribution of wave heights and amplitudes.................... 15 Planning of data requirements............................................. 40

Spectral analysis......................................................... 18

ILLUSTRATIONS

Page

FIGURE 1. Sketch of idealized dune shape........................................................................................................................... Fl

2. Definition sketch of bed profile........................................................................................................................... 2

3-37. Graphs showing 

3. Ratio of the expected number of /z-level crossings to the expected number of zero crossings, E{N/,}/E{N(>}, as a

function of h ................................................................................................................................. 5

4. Cumulative probability distribution functions for dimensionless maxima, TJ, as a function of 8........................... 6

5. Distributions of bed elevations................................................................................................................ 10

6. Comparison of the expected length or duration between zero crossings for a Gaussian process, E{In }, with the

observed values, /0 ......................................................................................................................... 11

7. Observed values of Nh /N0 as a function of h............................................................................................. 12

8. Average values of Nh/Nu as a function of h ................................................................................................ 12

9. Observed values of the ratio /^//,{ plotted against h....... ............................................................................. 12

10. Observed distributions of 1% for run 1...................................................................................................... 13

11. Observed distributions of 1$ for runs 1, 2, and 3........................................................................................ 14

12. Observed values of the coefficient of variation, Cr, for the distributions of /^ plotted as functions of h.................. 14

13. Comparison of observed 1% distributions with the theoretical class frequencies from a gamma distribution............ 14

14. Percent of particles in the bed above the level h....................................................................................... 15

15. Distribution of positive and negative a values for run 1............................................................................... 16

16. Distribution of positive and negative a values for run 2............................................................................... 16

17. Distribution of positive and negative a values for run 3............................................................................... 17

18. Approximate exponential distribution of positive a values for run 55.............................................................. 17

19. Distribution of wave heights, //, for run 55................................................................................................ 17

20. Relation between twice the average maximum ordinate between zero crossings, 2a+, and H.............................. 18

21. Relation between mean values of the distances between successive zero upcrossings of y and the mean dune length,

L..... ........................................................................................................................................... 18

22. Relation between average of maximum ordinates between zero crossing, a +, and standard deviation of bed elevation, (Ty. .................................................................................................................................... 18

23. Comparison of means and standard deviations for eight short segments of the record of run 3, Atrisco Lateral and

	for eight longitudinal profiles during equilibrium flow in the 8-foot flume, runs 8 through 15......................... 19

24. Spectra for indicated flume records......................................................................................................... 20

25. Relation of Co, Ci, and C2 to unit water discharge...................................................................................... 20

26. Comparison of estimated and observed average distance in feet between zero crossings.................................... 21

27. Spectra of the processes, y=y(x)... ........................................................................................................ 22

28. Spectra of the processes, y=y (t)........................................................................................................... 23

29. Comparison of ripple and dune spectra, 2-foot flume................................................................................... 24
IV CONTENTS

FIGURE 3-37. Graphs Showing Continued Page

30. Dimensionless spectra for the process y=y(x)... ....................................................................................... F25

31. Dimensionless spectra for the process y=y(t).................. ........................................................................ 26

32. Relation of maximum value of G'(x) to mean velocity and to e'.................................................................... 27

33. Dimensionless spectrum based on similitude criteria.................................................................................. 28

34. Spectra of longitudinal profiles, y=y(x), showing effect of channel size and bed configuration........................... 29

35. Autocovariance function and spectrum of longitudinal profile, y=y(x), for run 3, Atrisco Lateral....................... 30

36. Relation between average distance or time between successive zero upcrossings and the mean wavelength or period

of spectra from equation 8............................................................................................................... 31

37. Relation of wave celerity, c, to wave number, e, for various wave-number components of a single record, flow conditions constant, and for average wave numbers from the spectral moments of records obtained under different flow 

conditions.. ................................................................................................................................... 32

38. Cross correlograms for run 8 with runs 9 through 15............................................................................................... 34

39. Graphs showing distance from the origin of the maximum cross correlation as a function of time and change in maximum correlation with time.......................................................................................................................................... 34

40. Graphs showing coherence, y2 , and gain functions, A, for runs 8 and 9 and for runs 9 and 15.......................................... 35

41. Correlograms and spectra of y = y(x) for run 16, 2 feet left of centerline; for run 17, centerline; and for run 18, 2 feet right of

centerline.................................................................................................................................................. 35

42. Cross correlograms........................................................................................................................................... 36

43. Coherence diagrams.......................................................................................................................................... 36

44. Graphs showing relation of standard deviation of bed profiles, ay , to mean flow depth, D, and relation of standard deviation of

bed profiles, ay , to unit water discharge, Q... ......................................................................................................... 37

TABLES

Page

TABLE 1. Summary of basic data........................................................................................................................................... F9

2. Statistical properties of raw data.............................................................................................................................. 11

3. Miscellaneous properties from the zero-crossing and spectral analysis............................................................................. 16

4. Summary of flow characteristics for dimensionless spectra........................................................................................... 27

5. Comparison of wave celerities.................................................................................................................................. 31

6. Summary of wave properties................................................................................................................................... 33

7. Average values of observed variables........................................................................................................................ 37

SYMBOLS

Symbol Definition Units

A ((a)...... Gain function....................................................... 0

a............ Dune amplitude, the maximum ordinate of y between

zero crossings................................................... 0

c............ Wave celerity..................................................... Fpm

c(co) ...... Cospectrum......................................................... 0

D........... Mean flow depth................................................... Ft

E........... Expected value..................................................... 0

F........... Froude number..................................................... 0

f............ Frequency number......................... Cycles per unit time

f........... Dimensionless frequency........................................ 0

Gyy(ct>)..... Spectral density function............ (Cycles per unit time)"'

Gys(cn) ..... Cross spectrum........................ (Cycles per unit time)^ 1

G(x)........ Spectral density function for

the process y = y(x)........... (Cycles per unit distance)" 1

G(f) Spectral density function for

the process y=y(t)................ (Cycles per unit time)" 1

G'.......... Dimensionless spectra........................................ 0

Symbol Definition Units

g............ Acceleration due to gravity........................... Ft per sec 2

h............ A fixed leved of the bed, measured in units of standard

deviation of y from the mean bed level................ 0

H........... Dune or wave height, from crest to trough.............. Ft

H(ct>)...... Frequency response function................................... 0

/£.......... Duration or length of the upward excursion of the

process y above the fixed level A........................... or ft

L........... Dune length, from trough to trough.......................... Ft

mn .......... The nth moment of the spectrum.............................. 0

No.......... Number of zero crossings....................................... 0

Nh.......... Number of A-level crossings.................................... 0

P........... Probability distribution function............................... 0

p............ Proability density function...................................... 0

q............ Water discharge per foot of width............................ Cfs

q(<a)....... Quadrature spectrum............................................. 0

s............ A time or distance lag.................................... Min or ft

S............ Water-surface slope............................................... 0
CONTENTS

Symbol Definition Units

t............ Time................................................................. Min

T........... Wave period........................................................ Min

V........... Mean flow velocity................................................ Fps

x............ Distance along channel in flow direction.................... Ft

y............ Bed elevation, measured from y=0.......................... Ft

a........... Height of local maximum of y(t)............................... Ft

 y'yz(co)..... Coherence function............................................... 0

82 ........... A parameter........................................................ 0

e............ Wave number......................................... Cycles per ft

e'........... Dimensionless wave number................................... 0

Symbol Definition Units

TJ........... A dimensionless wave height................................... 0

6((i))....... Phase angle......................................................... 0

(Ji........... Mean number of upcrossings per unit time............ (Min)~'

Pyy(s)..... Autocorrelation function......................................... 0

Pyz(s)..... Cross-correlation function....................................... 0

o"y.......... Variance of the process y.................................... Sq ft

(j)yy(s)..... Covariance function.............................................. 0

(f>yz(s)...... Cross-covariance function....................................... 0

co........... Angular frequency, <a = 2irf............ Radians per unit time

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

STATISTICAL PROPERTIES OF DUNE PROFILES

BY CARL F. NORDIN, JR.

ABSTRACT

Properties of sand waves formed by subcritical unidirectional water 

currents are investigated by statistical analyses of records of streambed 

profiles. Records of bed elevation y as a function of distance x along the 

channel, y=y(x), and time records at a fixed point of the channel, 

y=yUh were collected in three laboratory flumes that were 8 inches, 

2 feet, and 8 feet wide and in a straight alluvial channel that was 55 feet 

wide. All bed material was fine sand. The continuous analog records 

were converted to discrete data points and were analyzed by digital 

computer.

The analyses show that both types of records, y(x) and y(t), can be 

approximately represented as stationary Gaussian processes. When the 

data are standardized and the lengths or distances are expressed as 

ratios of the mean duration between zero crossings of y, the statistical 

properties of all the flume data are similar, with no distinguishing characteristics that can be attributed to size of flume or to whether the bed 

forms were ripples or dunes. The field data, however, reflect the influence of large alternate bars that were not present in the flumes.

The Gaussian assumption, together with the spectral properties of the 

records as expressed by a dimensionless parameter, 8, permits predicting the distributions of maximum and minimum values of y between 

successive zeros of y. These distributions represent the probability 

distributions of the depth of local scour and fill due to the formation and 

migration of sand waves, and the parameters that specify the distributions relate approximately to flow velocity and depth.

Observed values of the number of zero and A-level crossings, the 

mean duration between zero crossings, and the mean duration of upward excursions of the process y(t) above the fixed level h compared 

reasonably well with theoretical values for the Gaussian model. The 

distribution of the duration of upward excursions is the conditional probability distribution of the rest period of a particle, given that it is deposited on the downstream face of a ripple or dune at the level h. 

Observed distributions of these durations can be approximated by a 

gamma distribution with parameters that relate to h, where h is measured in units of standard deviation from the mean bed level. These 

distributions and other probability distributions that enter into stochastic 

models of sediment transport can be determined either from the theoretical model or empirically from the observed data. The results of the 

study show that, even though the bed elevation deviates somewhat 

from the postulated normal distribution, reasonable estimates of many 

properties of the bed profiles can be derived from fairly simple statistical 

models.

INTRODUCTION

BACKGROUND

A distinguishing characteristic of sand waves formed 

by unidirectional subcritical water currents is their

tendency to form "en echelon" with gently sloping upstream faces and more steeply sloping downstream faces 

that meet the horizontal at approximately the natural 

repose angle of the sand. These features migrate slowly 

in the mean flow direction as material is eroded from their 

upstream faces and deposited on their downstream faces. 

Generally, these features are described as simple triangular forms, in profile, somewhat as sketched in figure 

1, with a mean length from crest to crest or trough to

FIGURE 1. Idealized dune shape.

trough, L, a mean height from crest to trough, H, and a 

constant angle of downstream face, j8. If the waves are 

long-crested, or two-dimensional, their geometric properties then are considered completely specified by L, H, 

j8. The ratio of mean length to mean height, L/H, called 

the ripple index, is a measure of the wave steepness.

In reality, the simple waveforms of figure 1 rarely exist. 

Long-crested sand waves occur apparently only under 

rather restricted flow conditions which will not be considered here. In the general case of flow in a wide channel 

with a bed of fine sand, the ripples and dunes that form 

are three-dimensional and highly irregular in size, shape, 

and spacing.

The three-dimensional properties of these features are 

completely described by a contour map of the bed, and 

with modern sounding, navigation, and computing equipment, large areas of a streambed can be mapped with 

ease and dispatch. However, the expense involved in 

obtaining detailed contour maps is generally prohibitive, 

and more generally, one has available only profiles of the 

streambed, obtained either by sounding along the channel 

from a boat or by sounding at some fixed point in the flow 

and recording changes in the bed elevation as the dunes 

and other bed features migrate past the sounding point.

Fl
F2 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

Although the profiles give only a two-dimensional 

picture of the streambed, they still provide a great amount 

of useful information. From longitudinal records, one can 

determine directly the distributions of lengths and heights 

associated with a particular ensemble of waveforms. From 

time records, the average wave period is easily found, 

which, together with mean wave height, provides a very 

good estimate of the bed load transport (Simons and 

others, 1965). The distribution of troughs and crests indicates the amount of local scour and fill associated with 

the migrating sand waves; this information may be important in such practical problems as designing and maintaining navigation channels or estimating the depth to 

which a structure such as a pipeline or siphon should be 

buried beneath the mean bed level to minimize the 

probability of local scour exposing the structure to the 

current.

From a comparison of the properties of different 

streambed profiles, it may be possible to establish 

whether or not there are any essential differences, other 

than scale, between ripples and dunes, and whether or 

not there are statistical properties of the dune profiles 

other than scale that can be attributed to the size of the 

channel. Both questions have important implications in 

modeling alluvial channel processes.

Perhaps the potentially most useful information to be 

derived from streambed profiles is information that relates 

to stochastic models of sediment transport. For example, 

in their two-dimensional stochastic model for the transport and dispersion of bed-material sediment particles, 

Sayre and Conover (1967) require the probability that a 

sediment particle will be deposited at a given level in the 

bed and the conditional probability for the length of time a 

particle will remain buried in the bed (that is, that it will 

experience a rest period of a certain duration), given that 

it is deposited at a particular level. These probabilities, 

together with some other distributions of interest, are 

easily found from the bed profiles.

To be more specific, consider the short segment of 

profile sketched in figure 2. Assume a straight uniform 

channel with equilibrium flow conditions, as defined by 

Simons and Richardson (1966, p. J3). If y is the bed elevah -level crossings

'Zero crossings' ~ U P crossings

*-Wavelength or wave period-* 

FIGURE 2. Definition sketch of bed profile.

tion, measured from the mean bed level so that y=0, 

and x is distance along the channel in the direction of 

flow, the bed profile then can be represented in the form 

y=y(x, t), xeX, teT. At any given position x = x0 along 

the channel, one may record the change with time of bed 

elevation y to produce the record, y= y(t). Similarly, at 

any given instant of time, t to, one can sound along the 

channel to obtain a record of the bed profile, y= y(x). 

In reality, of course, it is impossible to obtain instantaneously a longitudinal profile, but practically the time required to obtain a profile is small compared with the time 

required for a dune to shift appreciably downstream, so 

that the assumption y=y(x) is quite reasonable.

For either y== y(x) or y y(t), the bed elevation y, 

measured about the mean bed level, is a random variable 

that depends on a parameter (t or x) defined on an arbitrary parameter set (T or X, respectively). By definition, 

then y = y(x, t) is a stochastic process (Cramer, 1964, 

p. 137).

In all cases, y is a continuous function; obviously, there 

can be no discontinuities in the sand bed of a stream. 

Intuitively, one also would expect that, if the mean properties of the flow, of the sediment, and of the sediment 

transport do not change with time or with distance along 

the channel, then y will represent a stochastic process 

which meets the requirements both of stationarity and of 

ergodicity.

Consider next, in figure 2, some simple definitions that 

will be used later. The points where the processes y(x) 

or y(t) cross the zero axis are zero crossings, and the 

average distance between successive upcrossings 

(values of y going from negative to positive) is an average 

wavelength or wave period, somewhat analogous to L 

shown in figure 1. The maximum ordinate between zero 

crossings, in absolute values, is defined as the amplitude, 

a, and is roughly comparable to one-half of the wave 

height of figure 1.

Crossings of the level h are defined in a similar manner. 

Note that the average duration of the upward excursion of 

the process y(t) above the fixed level h is the average 

rest period experienced by a particle after it is deposited 

on the downstream faces of the sand waves at the level h.

The probability that a particle will be deposited at the 

level h also can be determined from the bed profiles for 

at least some simple postulated depositional patterns. 

If a particle is equally likely to be deposited at any place 

on the bed, the distribution is simply the frequency 

distribution of the y values. If deposition occurs only on 

the downstream faces of the ripples or dunes, the distribution can be determined from the distribution of y values 

where the process y(x) has a negative slope or y(t) has 

a positive slope. If nothing is known of the previous 

history of a particle, that is, if it is equally likely to be
STATISTICAL PROPERTIES OF DUNE PROFILES F3

found any place in the bed above the lowest point of 

particle motion, then the probability that it will be found 

at the level h can be determined approximately as the 

ratio of the area bounded by the fixed level h and the upward excursions of the process y(x) above h to the total 

area of the bed profile above the minimum y value.

Many properties of the processes y(x) and y(t) that 

are of interest can be determined empirically if suitable 

records of streambed profiles are available. Usually, 

though, it is difficult to obtain both types of records. In 

the laboratory, where flows can be controlled, it is possible 

to obtain long records of y(t) , but longitudinal profiles of 

the process y(x) are limited by the effective length of the 

flume. On the other hand, in field studies, it may be 

possible to obtain suitable longitudinal profiles, but the 

sand waves generally move so slowly that satisfactory 

samples of the process y(t) cannot be obtained under 

constant flow conditions. It has not been established that 

the statistical properties of the processes y(x) and y(t) 

are comparable, although some similarities have been 

noted (Nordin and Algert, 1966). Thus, it is extremely 

important to determine in what respects the two types of 

records are similar and to develop methods of correlating 

the properties of the two types of records.

Ultimately, of course, one will wish to predict something 

about the streambed profiles, given only information on 

the characteristics of the flow and the bed sediment. In 

order to do this, it is necessary first to determine if there 

are any consistent or recognizable patterns in the properties of interest, and then to attempt to relate these 

properties to flow and sediment parameters.

There is little theoretical or empirical basis upon which 

to postulate the statistical properties of y(x) or y(t). 

However, during the course of preliminary studies of the 

bed profiles, several facts emerged that led to the approach adopted for this investigation. First, it was noted 

that the y values were distributed about their mean values 

approximately as normal distributions, which was to be 

expected as most natural processes that develop under 

the influence of many random factors exhibit approximate Gaussian distributions. Second, many of the properties of a Gaussian process with known covariance 

functions, particularly the mean values of duration between zero and /t-level crossings and the maximum 

between zero crossings, are well established from previous 

work on the statistical properties of random noise (Rice, 

1954) and on ocean waves (Longuet-Higgins, 1958, 1962, 

1963). Finally, it has been shown that properties of the 

covariance function near the origin relate to a simple 

flow parameter, at least for a limited range of flow conditions (Nordin and Algert, 1966), so that in considering the 

prediction problem, the assumption of a known covariance 

function may be rather simple to satisfy.

Therefore, the approach adopted for this study was to 

compare the observed properties of the bed profiles with 

the theoretical properties of a Gaussian process of known 

covariance function. In the following section, the scope 

and specific objectives of the study are given in more 

detail.

PURPOSE AND SCOPE

In broad terms, this study was designed to investigate 

the statistical properties of streambed profiles. Data were 

collected in both laboratory and field investigations, and 

attention was always restricted to equilibrium flow over 

a bed of fine sand in a straight uniform channel with either 

ripple or dune bed configuration. The classification of bed 

configurations as either ripples or dunes is according to 

Simons and Richardson (1966, p. J5-J7). Details of the 

hydraulic and sediment data are given in a later section.

Specifically, we are interested in the mean values and 

the distributions of the durations between zero and h -level 

crossings, of the durations of upward excursions of the 

process y(t) above the fixed level h, and of the positive 

and negative maximums of y(x) or y(t) between zero 

crossings. In addition, it is of interest to consider whether 

or not the statistical properties of y(x) and y(t) are 

similar, whether or not there are any significant differences other than scale in the statistical properties of 

ripple and dune profiles, and whether or not or to what 

extent the statistical properties of dune profiles depend 

on the scale of the flow system.

Particular attention is paid to the spectral representations of the process y because the distributions of the 

amplitudes a for a Gaussian process depend to a large 

extent on the properties of the spectra (Cartwright and 

Longuet-Higgins, 1956). Some applications of cross correlation and of cross-spectral analysis are examined briefly.

As indicated above, the approach used in this study is 

to compare the observed properties of the bed profiles 

to the theoretical properties of a Gaussian process of 

known covariance function. Similarities and differences 

between the observed and the theoretical processes are 

noted, and some of the statistical parameters that describe 

the observed processes are related empirically to properties of the flow.

In the following section, a review of the properties of a 

Gaussian process is given, and some of the mathematical 

relations for spectral analysis are listed. Then the data 

is described and the results of the analysis are presented. 

Finally, the paper discusses the implication of the results, 

summarizes the conclusions drawn, and lists some recommendations for future research along these same lines.

ACKNOWLEDGMENTS

This study was a part of the U.S. Geological Survey 

Water Resources Division research program on me405-448 O - 71 - 2
F4 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

chanics of flow and sediment transport in alluvial channels. This report was prepared in partial fulfillment of the 

requirements for the degree of Doctor of Philosophy in 

Civil Engineering from Colorado State University, Fort 

Collins, Colo. Dr. V. M. Yevjevich, Colorado State 

University, assisted in the initial planning of this work, 

and Mrs. Lois Niemann did the programing and 

computing.

The writer expresses his appreciation for the counsel 

and encouragement of his major professor, Dr. D. B. 

Simons, and of his committee members, Dr. E. V. 

Richardson, Dr. E. J. Plate, Dr. M. M. Siddiqui and 

Dr. Scott Creely.

Special thanks are due to J. K. Culbertson, R. E. 

Rathbun, and H. P. Guy, U.S. Geological Survey, and 

Tsung Yang, Colorado State University, for permitting 

use of their unpublished data, and to W. W. Sayre and 

[gnacio Rodriguez-Iturbe for many useful suggestions 

and stimulating discussions.

SOME PROPERTIES OF A GAUSSIAN PROCESS

ZERO AND fc-LEVEL CROSSINGS

Suppose that y(t) is a real stationary Gaussian random 

function of the continuous parameter ?, 0 < t < °o, with 

zero mean and covariance function (f> yy (s) and possessing 

a spectral density function Gy y (a)). The covariance function is the expected value of the lagged product of y(t) 

and y(t + s), or

<l>vv(s)=E{y(t)y(t + . (1)

The process y(t) is assumed to be ergodic, so that the 

covariance function is given by

=r\ y(t)y(t + s) (2)

which by stationarity is a function only of s. The covariance function is related to the spectral density function 

by the equation

(3)

where w is the angular frequency.

The spectral density and covariance are Fourier transform pairs:

=-f

w Jo cos o)sds. (4)

The variance of y(t) is

0-2 = var y(t) = <f> yy (0) = Gyy (a>)da>. (5)

The first moment of the spectrum about the origin is 

given by

foe

m\= I o)G yy (a))da) (6) 

Jo

and, in general, the rth moment, r=0, 1, 2, . . ., is 

defined as

» 

a)r Gyy((t))da>. (7)

(8)

The mean frequency of the spectrum is then

and the derivatives of the covariance function at the 

origin, if they exist, are

(9)

where r is even, and

where r is odd.

Apparently, there is no general solution for determining 

the probability distribution of the interval between zero 

crossings, p{/o}, but its mean value is the reciprocal of 

the average number of zero crossings per unit time, 

E{No}, which was given by Rice (1954), as

(10)

The expected number of h -level crossings is

(ID

The ratio of the average number of h -level crossings to 

the average number of zero crossings from equations 10 

and 11 is

(12)

The expected duration of an upward excursion above 

the level h is given by Cramer and Leadbetter (1967):

*} = /*-'pr{y(0)> (13)
STATISTICAL PROPERTIES OF DUNE PROFILES F5

where /A is the mean number of upcrossings per unit time 

and pr denotes probability. The mean number of upcrossings is one-half of the mean number of crossings per unit 

time, so that, combining equations 10 and 13, the ratio of 

expected duration of upward excursions above the level 

h to expected duration of upward excursions above the 

zero level is

0+ } = 2 pr{y(0) > h} (14)

An interesting alternative to Rice's approach is given 

by Tick and Shaman (1966) for a straight line interpolation of an underlying continuous Gaussian process determined by sampling the underlying process at equispaced 

intervals. Again, it is assumed that y(t) is a stationary 

Gaussian process with continuous parameter t, °° < t < °°, 

and with zero mean and covariance function (f) yy (s) , 

and possessing a spectral density function Gyy (ct)) defined by equation 3. Assume that 0(0) = 1 and that y(t) 

is sampled discretely at the time points . . . 2Af, Ai, 

0, A£, 2A£, .... The sampled process also is Gaussian 

with covariance sequence (/>(/?A£), n = 0, ±1,±2, . . .. 

Connecting successively the observed ordinates of the 

sampled sequence with straight line segments yields the 

interpolation process mentioned above.

The expected number of zero crossings in a record of 

length k&t is found to be

To determine the expected number of /i-level crossings, 

choose two adjacent values, y(/iAf) and y({rc+l}Af), 

and denote them by Y\ and Y2 . The joint distribution of 

Y\ and ¥2 is bivariate normal with correlation (/>(A?)- 

Then, the expected number of A-level crossing is

l >h,Y2 <h) + pr(F, < h, Y2 > h)] 

= 2A-[pr(F, >fl, Y2 >-a°)-pr(Y1 >h, Y2 >h)].

(16)

From equations 15 and 16, the value of the ratio of the 

expected number of h -level to the expected number of 

zero crossings is seen to depend on the covariance function at one lag, $(A£)- Values were computed for 

<£(AO=0.7 and 0.9, and the curve for ^(At)=0.7 is 

plotted in figure 3 along with equation 12 for the continuous process. There is so little difference in the two 

curves for values of </>(Af) greater than 0.7 that the 

simpler expression of equation 12 is to be preferred.

Although the probability distribution of the / values 

cannot be precisely determined in the general case,

FIGURE 3. Ratio of the expected number of /i-level crossings to the 

expected number of zero crossings, E{Nn }jE{N0}, as a function of h.

Longuet-Higgins (1962, 1963) gives some approximations 

for upper and lower bounds of p(/o), with particular attention to certain ideal forms of the spectra of y(t), and 

Cramer and Leadbetter (1967) give equations for the 

moments of the distribution functions of the duration of 

upward excursions above the level h. From a practical 

point of view, the mean values of 1% are of most interest, 

particularly the mean in an interval hi h\, which is the 

mean rest period of a particle deposited on the downstream face of a sand wave between the elevations h\ and 

h-2, and which can be determined easily by integrating 

equation 13 between appropriate limits.

WAVE HEIGHTS AND AMPLITUDES

The wave height H was defined as the difference in 

elevation between a crest (maximum) and the following 

trough (minimum) in figure 1. The statistical distribution 

of// generally is not known, but where y(t) has a narrow 

spectrum, it has been established that ///2 is distributed 

according to a Rayleigh distribution (see Cartwright and 

Longuet-Higgins, 1956):

P(HI2) =

H

(mo) 1/2 (17)

where mo, the variance of y(t), is determined by equation 

5 or equation 7. If the process y has unit variance, the 

equation simplifies to

(18)
F6 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

and the cumulative probability distribution of///2 is given 

by

Although the probability distribution of H is not established, the statistical distribution of the local maxima of 

y(t) is known. If y(t) is a strictly stationary process 

possessing a continuous sample derivative y' (t) , a local 

maximum or crest is said to occur at t = to if y'(t) has a 

downcrossing of zero at t (Cramer and Leadbetter, 1967, 

p. 242). Define a as the difference in height between the 

crest and the mean level of y(t). Then, for the Gaussian 

process considered here, the probability distribution of 

a depends only on (mo) 1/2 and on a parameter 8 that 

represents the relative width of the frequency spectrum,

(20)

where 0< 8< 1.

For 5 * 0, the spectrum becomes infinitely narrow and 

the dimensionless maxima, 17, tend to a Rayleigh distribution:

(21)

where 17 > 0, and

p(i?)=0,

where 17 < 0.

where 17 is defined by the relation

2 . (22)

When 8 approaches its maxima value of 1, the distribution 

of 17 is Gaussian,

PW = 7^I72 e ~" 2/2 ' - °° < >? < °°. (23)

(277) 1 ' 2

Derivations of the above distributions are given by Rice 

(1954) and are discussed in detail by Cartwright and 

Longuet-Higgins (1956). The cumulative probability 

distributions of 17 for various values of 8 are shown in 

figure 4.

In figure 2, a was defined as the maximum y value 

between zero crossings, and it can assume either positive 

or negative values. From a practical point of view, it 

may be advantageous to consider separately the distributions of positive a+ and negative a values, in the 

event that y(t) is not symmetric about its mean level. 

The theoretical Gaussian model is assumed symmetric, 

so that in this case it is permissible to consider only the 

the positive a values, keeping in mind the symmetry of

FIGURE 4. Cumulative probability distribution functions for dimensionless maxima, TJ, as a function of 8.

the distribution about zero when negative values are of 

concern.

Clearly, for narrow-band random noise, the distribution 

of a+ would approach the distribution of local maxima, 

the Rayleigh distribution. (See, for example, Bendat and 

Piersol, 1966, p. 17.) Generally, however, there apparently 

is no theoretical derivation for the probability distribution 

of the a values. For purposes of this study, therefore, the 

observed distributions of a simply will be compared with 

the theoretical distributions of local maxima shown in 

figure 4. At least intuitively, one would expect the distribution of a values for a process with a broad spectrum 

to approach a form of the normal distribution, but there 

is no reason to expect the distribution to depend only on 

8 and m.

THEORY OF TIME SERIES ANALYSIS

In the preceding section, the covariance function, $(s), 

or alternatively, the spectral density function, G(w), was 

assumed known. Values of these functions are required 

to predict mean durations between zero and A-level crossings and to estimate probability distributions of maxima 

for the Gaussian model. For the actual streambed profiles 

analyzed in this study, values of <$>(s} and G(co) were 

computed and the observed properties of the profiles 

were then compared with properties of a Gaussian process 

having the same covariance and spectral density functions.

Apart from the problem of predicting the zero and 

A-level crossings, the covariance functions and spectra 

together with the probability distributions of the y values 

provide a great deal of information about the process y(t) , 

and these functions will be found particularly useful in 

later sections when we consider the similarities and the differences of the streambed profiles that can be attributed 

to scale of the flow system or to channel size. Therefore, 

it is appropriate at this point to review briefly the theory of 

time series analysis and to mention some of the properties 

of the covariance functions and spectral density functions 

that were found useful in analyzing the dune profiles.
STATISTICAL PROPERTIES OF DUNE PROFILES F7

For this purpose, consider y(t] to be a real continuous 

random function of time t, o° < t < o°. The process y(t) 

is assumed to be stationary and ergodic. Stationarity 

implies time invariance of the general statistical properties of the process y(t) , and ergodicity insures that 

averages across an ensemble are equivalent to averages 

over time along a single sample function or realization of 

infinite extent. (For a more detailed discussion of these 

concepts, see Yaglom, 1962 or Cramer and Leadbetter, 

1967.)

By stationary, both the mean value, y, and the variance, 

cr*, are constants, so that it is always possible to form a 

new time series, [y(t) y]l(Ty, that has zero mean and 

unit variance. Unless otherwise specified, it will be 

assumed in the following discussion that this transformation has been made. Then the correlation function, 

defined as

(24)

is identical to the covariance function.

Both (j)(s) and G(OJ) are even functions of their respective arguments, and the usual equations for these functions are

=1 f"

TT Jo

= fC>

Jo

(25)

(26)

where negative values of both time and frequency are 

considered. Because y(t) is a real function, the correlation and spectral density functions often are given 

as a cosine transform pair, equations 3 and 4, where the 

G and <j> values are modified by the necessary constants 

to apply only to positive t and w values. Bendat and 

Piersol (1966, p. 77-84) give a complete discussion of the 

"one-sided" correlation and spectral density functions 

that generally are used in practical computations.

The autocovariance function measures the degree of 

dependence between the observed quantity y at one time 

and at another time s units later. It is especially useful in 

defining periodicities in the process y(t).

The power spectrum or spectral density function 

describes the general frequency composition of the data 

in terms of the density of its mean square value, that is, 

Gyy(aj)d(*) represents the contribution to the variance of 

the process from the frequencies between w and (&j+d&j). 

Thus, the spectra show directly similarities and differences in the streambed profiles and provide a quantitative basis for comparing the ripple bed forms with the

dune forms or for evaluating the effect of scale on the 

statistical properties of the streambed profiles.

The treatment is extended to two stochastic processes 

as follows. The cross correlation between two series 

y(t) and z(t) is given by

(27)

where <j>yz(s) is the cross covariance between the series 

and is defined as the expected value of the product 

y(t)z(t + s). Both y(t) and z(t) are ergodic, so that the 

cross covariance is

<f>y Z (s)=E{y(t)z(t +

I fT/2

= lim- y(t)z(t + s)dt.

T^>x 2 J-T/2

(28)

The cross spectrum is defined as the Fourier transform 

of the cross-covariance function

Gyz((*>) -f

TT J-

(29)

where w represents the angular frequency, c(oj) is the 

cospectrum, a measure of the in-phase covariance, and 

q(cti) is the quadrature spectrum, a measure of the out-ofphase covariance. The cospectrum measures the contributions of oscillations at the lag zero between two time 

series. The quadrature spectrum measures the contribution of the different harmonics to the total cross covariance between the series when all the harmonics of the 

series y(t) are delayed by a quarter period but the series 

z(t) remains unchanged.

The real quantity defined as coherence, 7y2 (w), is a 

direct measure of the square of the correlation of the 

amplitudes of frequency w of the processes y(t) and 

z(t), or

(30)

where Gy(oj) and G2 (&j) represent the spectra of y(t) 

and z(t), respectively.

Even if the amplitudes are fully correlated, it is possible 

that the corresponding frequency components will have 

different phases. The phase lag at each frequency is 

given by

=arctan c(o>) (31)

where 0(oj) is called the phase function.
F8 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

Another quantity sometimes useful in cross-spectral 

analysis is the frequency-response function, //(w), calculated from the relation

(32)

The quantity A (w) is the gain function or amplitude gain 

of the system and measures the ratio of the amplitude of 

the frequency components of the series y(t) and z(t) at 

each frequency a).

The cross covariance measures the dependence between two time series at the given lag, 5, and for this 

study, it was applied to investigate the three-dimensional 

properties of the sand waves, their mean rate of shifting, 

and the time or distance required for the waves effectively 

to lose their identity. The cross-spectral functions, equations 29-32, were used in conjunction with the cross 

covariance.

The above theory is presented for a continuous process 

with the parameter t representing time. The frequency 

f a)/2TT is then given in cycles per unit time. The parameter t is simply a member of an arbitrarily specified 

parameter set, and it can be any quantity that permits the 

set of y or z values to be ordered linearly. If time t is 

replaced by distance x, the frequency / is replaced by 

wave number e and the wave period T l/fis replaced 

by wavelength L= 1/e.

DATA AND ANALYSIS

BASIC DATA

Profiles from four different channels were selected for 

these analyses. Table 1 gives a summary of the records. 

Three of the channels were recirculating laboratory flumes 

located at the Research Center Hydraulics Laboratory, 

Colorado State University, Fort Collins, Colo. The other 

channel was Atrisco Lateral near Bernalillo, N. Mex., a 

conveyance channel with a sand bed and with banks 

stabilized by clay and vegetation. The dimensions of the 

flumes were: 0.67 foot wide by 30 feet long, 2 feet wide by 

60 feet long, and 8 feet wide by 200 feet long. Atrisco 

Lateral was approximately 55 feet wide, and the profiles 

were obtained about midway in a straight reach 12,000 

feet long. The median diameters of bed material and flow 

parameters are shown in table 1.

Fifty-four records representing six different flow conditions were selected for analysis. Runs 1, 2, and 3 were 

data collected from Atrisco Lateral on three different 

days, but with similar flows. Records 4 through 19 and 

20 through 39 are from the 8-foot flume for two different

flow conditions, and like the field data for the first three 

runs, the bed configuration was dunes. The records for 

runs 40-43 are for a ripple bed in the 2-foot flume, and 

runs 44-47 are for identical flow conditions. Runs 48-51 

are for a dune-bed flow in the 2-foot flume, and runs 

52-54 are for a ripple bed in the 0.67-foot flume, corresponding to the experiments reported by Rathbun and 

Guy (1967). Run 54 subsequently was discarded because 

of suspected errors in the basic data, so analysis eventually was carried out on 53 profiles from the six different 

flow conditions.

Profiles of the bed elevations were obtained with the 

sonic depth sounder described by Karaki, Gray, and 

Collins (1961) except for the smallest flume where the 

profiles were traced on a stripchart from the plastic sidewalls of the flume. All data were digitized with an analogto-digital converter at the intervals shown in table 1. The 

flume data, runs 4 through 54, were standardized with 

zero mean and unit variance after removing a straight 

line trend to account for the possibility that the sand bed 

in the flume was not parallel to the instrument carriage 

rails supporting the sonic sounder.

At first, the trend was not removed from the Atrisco 

Lateral data because the sounder was mounted on a boat 

at a constant depth below the water surface. However, 

initial analyses showed some long-term trends in the data, 

so that parts of records from run 2 were selected for trend 

removal. These shorter records from Atrisco Lateral are 

shown as runs 55 through 57 in table 1 and are discussed 

in detail in a later section.

Only longitudinal profiles were available for Atrisco 

Lateral and the smallest flume; for the 2- and 8-foot-wide 

flumes, time records y y(t) were available along with 

the longitudinal profiles. All the computations described 

in this and subsequent sections were accomplished on 

the CDC 6400 computer at Colorado State University.

For the discrete data used in this study, the values in 

equations 25 through 32 were approximated by the estimates presented by Granger and Hatanaka (1964), using 

the Blackman and Tukey spectral estimates with a 

Hanning window (Blackman and Tukey, 1958, p. 34). 

Formulas for the digital calculations of the covariance 

functions and spectra and some guidelines for estimating 

the length of record required for the various calculations 

are given in the appendix. A detailed discussion of the 

calculations and an excellent review of spectral theory 

are given in the above references and in a recent book by 

Bendat and Piersol (1966). Rodriguez-Iturbe (1967) investigated the application of cross-spectral analysis to hydrologic data and gave a thorough discussion of the computational procedures for discrete data. The procedures used 

in this study are identical to those listed by him (RodriguezIturbe, 1967, p. 5-7).
STATISTICAL PROPERTIES OF DUNE PROFILES

TABLE I. Summary of basic data

F9

[Transverse station: CL, centerline; L and R, left and right third points of the channel; 2, 4, and 6, stations from the left wall of the 8-ft flume. Lag interval: given in minutes for runs 4,19,33,

40, 44, and 48]

Channel 

and run

Atrisco Lateral: 

Run 1.......

2.......

3.......

8-ft flume: 

Run 4.......

5.......

6.......

7.......

8.......

9.......

10.....

11.....

12.....

13.....

14.....

15.....

16.....

17.....

18.....

19

20...... 

21.....

22.....

93

24.....

95

26.....

27.....

28.....

29.....

30.....

31.....

32.....

33.....

34...... 

35.....

36.....

37.....

38...... 

39.....

2-ft flume: 

Run 40......

41...... 

42...... 

43......

44......

45......

46...... 

47.....

48...... 

49...... 

50...... 

51.....

0.67-ft flume:

Run 52...... 

53.....

54...... 

Atrisco Lateral:

Run 55...... 

56.....

57.....

Date

1966 

June 23 

June 22 

June 21

Mar. 15

Mar. 17 

....do.....

....do.....

....do.....

....do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

July 12

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

...do.....

July 13

...do.....

...do.....

July 14 

...do.....

1967 

Feb. 23

Feb. 21 

Feb. 22 

Feb. 23 

Mar. 23

Mar. 21 

Mar. 22 

...do.....

Apr. 12 

Apr. 11 

Apr. 12

1965

Nov. 30

Dec. 1

1966 

June 22 

...do.....

Time

1115 

1115 

1115 

1255 

1300 

1305 

1310 

1315 

1320 

1325 

1330 

1530 

1536 

1540

1732 

2114 

0930 

0924 

0935 

0946 

0950 

0953 

1113 

1308 

1328 

1329 

1727 

1735 

0907 

0912 

1012 

1016 

0810 

1435

1110 

2350 

1412

1720 

0908 

2156 

1640 

1400 

1403 

1536

1000 

1000 

1000

Time or 

station

0-4000 

0-4000 

1000-4000

936 min 

60-180 

180-60 

60-180 

60-120 

60-120 

60-120 

60-120 

60-120 

60-120 

60-120 

60-120 

60-180 

60-180 

60-180 

888 min 

60-180 

60-180 

60-180 

60-180 

60-180 

60-180 

60-180 

60-180 

60-180 

60-180 

60-170 

170-60 

60-180 

505 min 

60-180 

60-180 

60-180 

60-180 

60-180 

60-180

395 min 

5-55 

5-55 

5-55 

532 min 

5-55 

5-55 

5-55 

1160 min 

5-55 

5-55 

5-55

0-7.2 

9-16 

0-7.2

0-2000 

0-200 

200-400

Transverse 

station

CL 

CL 

CL

CL 

2 

4 

6 

4 

4 

4 

4 

4 

4 

4 

4 

2 

4 

6 

CL 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4

CL 

CL 

L 

R 

CL 

CL 

CL 

L 

CL 

CL 

L 

L

L 

L 

L

CL 

CL 

CL

No. of 

data 

points

12,924 

13,116 

9,864

936 

924 

924 

912 

432 

444 

432 

432 

444 

444 

456 

444 

876 

900 

900 

888 

468 

736 

984 

720 

876 

612 

480 

528 

480 

468 

792 

969 

852 

672 

864 

480 

864 

480 

1,188 

1,176

395 

668 

667 

766 

532 

640 

696 

640 

776 

416 

407 

393

306 

348 

318

6,000 

600 

600

Lag Mean Mean Water 

interval depth velocity Slope temperature 

(ft or min) (ft) (fps) (° C)

0.333 

.333 

.333

1.00 

.130 

.130 

.132 

.139 

.135 

.139 

.139 

.135 

.135 

.131 

.135 

.137 

.133 

.137 

1.0 

.256 

.163 

.122 

.167 

.137 

.196 

.250 

.228 

.250 

.256 

.152 

.158 

.141 

.750 

.139 

.250 

.134 

.250 

.101 

.102

1.0 .075 ' 

.075 

.070. 

1.0 

.078 

.072 

.078 . 

1.50 .120" 

.123 

.127]

.0208 

.0208 

.0208

.333 

.333 

.333

2.20 2.16 0.00055 19 

2.30 2.11 .00055 20 

2.29 2.08 .00058 20

2.80 2.09 .00063 20

2.36 2.01 .00056 24

.00053 ' 

.00053 

.00053 

2.36 2.01 ' .00053 25 

.00045 

.00045

.518 1.10 .00088 20 

.522 1.07 .00088 20 

.521 1.62 .00212 20

.174 .508 .00148 20 

.174 .508 .00148 20 

.174 .508 .00148 20

2.30 2.11 .00055 20 

2.30 2.11 .00055 20 

2.30 2.11 .00055 20

Median 

diam Bed 

of bed configuramaterial tion 

(mm)

0.23 Dunes. 

.23 Do. 

.23 Do.

.24 Do.

.24 Do.

.24 Do.

.35 Ripples. 

.35 Do. 

.35 Dunes.

.30 Ripples. 

.30 Do. 

.30 Do.

.23 Dunes. 

.23 Do. 

.23 Do.
F10 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

ZERO AND Ji-LEVEL CROSSING ANALYSES

In this section, mean values of the durations between 

zero and A-level crossings, the mean durations of upward 

excursions above the level A, and some of the probability 

distributions of interest in the two-dimensional stochastic 

model of particle movement (Sayre and Conover, 1967) 

will be considered.

First, it is noted that the bed elevation y, measured 

from the mean bed level follows approximately a Gaussian distribution (fig. 5). Intuitively, an approximate 

normal distribution is expected, because physical phenomena governed by the complex interaction of many 

factors often exhibit such a distribution. Logically, the 

distribution can be only approximate; in a finite flow

depth, the variation of the bed elevation about the mean 

can never assume infinite values. In addition, the preferred orientation of the dune forms, with the characteristic steeply sloping downstream faces, suggests a pattern 

more regular than normal distribution. Figure 5 shows 

examples of the distributions from each of the four channels, and although the sets of values plot around the 

straight line of a normal distribution, each shows some 

departures from normality.

The cumulative distribution curves of figure 5 tend to 

smooth out irregularities of the data and are not really a 

good indication of the normality or lack of normality of 

the data. A better criterion, perhaps, is to compare the 

skewness of the data with the skewness of a normal diss> -1

(/)

8-inch flume

 T-I

2-foot flume

O

JZ 2 

<

LU

-1

-2

8-foot flume

I I I I I I

99 98 95 90 50 10 2 1

I 1 I

Atrisco Lateral

I I I I I I

99 98 95 90 50 10 2 1

PERCENTAGE EXCEEDING y 

FIGURE 5. Distributions of bed elevations.
STATISTICAL PROPERTIES OF DUNE PROFILES Fll

tribution, which is zero. Table 2 lists the properties of the 

raw data and shows that the skewness varied from 0.3 

to about 1.7, with a preponderance of values on the positive side. Both the skewness and kurtosis show a considerable range values with no recognizable pattern.

TABLE 2. Statistical properties of raw data

Run Variance Standard Skewness Excess of 

deviation kurtosis

1......... 6.70X10-2 2.59X10-

2......... 7.63X10-2 2.76X10-

3......... 7.28X10-2 2.70X10-

4......... 4.00X10-2 2.00X10-

5......... 2.09X10-2 1.44X10-

6......... 1.58X10-2 1.26X10-

7......... 1.77X10-2 1.33X10-

8......... 2.62X10-2 1.62X10-

9......... 1.90X10-2 1.38X10-

10....... 3.12X10- 2 1.77X10-

11....... 3.59X10-2 1.89X10-

12....... 3.38X10-2 1.84X10-

13....... 2.96X10-2 1.72X10-

14....... 3.80X10-2 1.95X10-

15....... 3.83X10-2 1.96X10-

16....... 3.22X10-2 1.79X10-

17....... 4.68X10-2 2.16X10-

18....... 3.37X10-2 1.84X10-

19....... 1.14X10- 2 1.07X10-

20........ 2.16X10-2 1.47X10-

21....... 1.23X10-2 1.11X10-

22....... 1.94X10-2 1.39X10-

23....... 1.88X10-2 1.37X10-

24....... 1.96X10-2 1.40X10-

25....... 2.61X10-2 1.62X10-

26....... 2.77X10-2 1.67X10-

27....... 2.98X10-2 1.73X10-

28....... 1.76X10-2 1.33X10-

29....... 1.44X10-2 1.20X10-

30....... 1.07X10-2 1.03X10-

31....... 1.17X10-2 1.08X10-

32....... 2.11X10-2 1.45X10-

33....... 1.45X10-2 1.20X10-

34....... 1.22X10-2 1.10X10-

35....... 1.23X10-2 1.11X10-

36....... 1.03X10-2 1.01X10-

37....... 1.04X10-2 1.02X10-

38....... 1.25X10-2 1.12X10-

39....... 8.52X10-3 9.23X10-

40....... 1.29X10-3 3.60X10-

41....... 1.91X10-3 4.37X10-

42....... 2.25X10- 3 4.74X10-

43....... 2.09X10-3 4.57X10-

44....... 1.20X10-3 3.46X10-

45....... 1.64X10-3 4.04X10-

46....... 1.50X10-3 3.87X10-

47....... 1.79X10-3 4.23X10-

48....... 5.62X10-3 7.50X10-

49....... 5.48X10-3 7.39X10-

50....... 5.54X10-3 7.44X10-

51....... 3.23X10-3 5.68X10-

52....... 2.16X10-3 4.64X10-

53....... 1.19X10-3 3.44X10-

55....... 8.09X10-2 2.84X10-

56....... 8.72X10-2 2.96X10-

57....... 3.95X10-2 1.99X10-

6.38X10- 

5.09X10- 

2.77X10- 

1.71X 10° 

4.04 X 10- 

4.92 X 10- 

1.48X10- 

6.06X 10- 

2.59X10- 

5.70X10- 

4.12 X 10- 

3.38X10- 

5.20X10- 

6.07X10- 

5.57X10- 

8.66 X 10- 

1.65X 10- 

2.06 X 10- 

1.29X10- 

4.50 X 10- 

2.28X10- 

3.17X10- 

1.47X10- 

5.09 X 10- 

8.33 X 10- 

6.97X10- 

6.91 X 10- 

4.06X10- 

5.37X10- 

9.84X10- 

-9.77X10- 

8.07 X 10- 

1.75X10- 

2.04X 10- 

1.16X10° 

1.60X10- 

-2.71X10- 

7.66 X 10- 

1.03X 10° 

6.78X10- 

2.81 X 10- 

-1.85X10- 

3.33X 10- 

-2.29X 10- 

7.92 X 10- 

2.15X10- 

2.40X10- 

9.27X10- 

1.03X10° 

3.60X 10- 

1.46X 10- 

4.44X10- 

2.84X10- 

5.41X10- 

1.16X10° 

4.81 X 10-

1.07X10° 

9.47 XlO- 1 

3.35 XlO- 1 

9.91 X 10° 

-8.12X10-3 

6.45X10-2 

-2. 87 XlO- 1 

-2.03 XlO- 1 

-5. 62 XlO- 1 

3.35 XlO- 1 

-3.15X10- 1 

-2.58 XlO- 1 

-1.49 XlO- 1 

-3. 23 XlO- 1 

-4.36 XlO- 1 

2.32 X 10° 

-5. 84 XlO- 1 

-1.53 XlO- 1 

-1.67 XlO- 1 

5.03 XlO- 1 

4.46 XlO- 1 

-2.21 XlO- 1 

-2.20X10- 1 

-8.21X10-2 

8.11 X 10- 1 

4.20 XlO- 1 

4.41 XlO- 1 

7.06 XlO- 1 

1.66X10° 

8.41 X lO- 3 

5.49 XlO- 1 

1.55X10° 

2.57 XlO- 1 

-1.79 XlO- 1 

9.00X10° 

2.91 XlO- 1 

-4.17X10-2 

1.12X10° 

1.71X10° 

4.76 XlO- 1 

1.98 XlO- 1 

-7.54X10-3 

-4. 25 XlO- 1 

-4.87 XlO- 1 

-5.28 XlO- 1 

-4.16X10- 1 

-1.78X10- 1 

6.70 XlO- 1 

1.64X10° 

2.68X10° 

-8.08 XlO- 1 

-4.99 XlO- 1 

-4.33 XlO- 1 

3.83X10° 

2.49 X 10° 

2 -5.73X10-2

NOTE. Run 54 not used.

Even though the distributions of the data depart from 

normality, the relation of equation 10 provides a good 

estimate of the mean duration or the mean distance 

between zero crossings, /0 , as shown in figure 6. Although

50

10

O OO

10

>/2

 I , IN FEET OR MINUTES

50

FIGURE 6. Comparison of the expected length or duration between

zero crossings for a Gaussian process, E{lo} , with the observed values,

/o.

there is considerable scatter, the data group around the 

line of perfect agreement. No consistent trends in the 

scatter could be attributed to flume size or type of bed 

form, so that most of the scatter is assumed to be due to 

the shortness of the records of the flume data.

The ratio of the expected number of A-level crossings to 

the expected number of zero crossings was given for both 

the continuous process and the discrete approximation 

to the continuous process in figure 3. Because the two 

curves are so similar, the simpler expression of equation 

12 will be used. Observed values of the ratio Nh/No are 

plotted in figure 7, along with the curve representing 

equation 12. Values for 13 profiles were plotted. Six of the 

profiles, runs 4, 19, 33, 40, 44 and 48, are of the process 

y=y(t), and seven of the profiles, runs 16, 32, 41, 45, 

53 and 56, are of the process y=y(x). For positive values 

of A, the points scatter symmetrically about the curve 

of equation 12, but for negative values of h. most of the 

points fall above the curve for values of h from 0 to 1 

and below the curve for values of h from 1 to 2.

Figure 7 shows clearly that there are more crossings 

below the mean bed elevation than above, which would be 

expected from consideration of flow conditions over a 

dune. Below the mean bed elevation, the reverse flow in 

the trough and the flow impinging on the back of the dune 

result in lower velocities which promote the growth of 

small-scale features. Above the mean bed elevation, the 

converging flow up the back of the dune results in a

405-448 O - 71 - 3
F12 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

-2

FIGURE 7. Observed values of Nh/N0 as a function of h.

higher-than-average velocity and shear stress near the 

bed, and small-scale features cannot form.

Figure 8 shows the average of all observations, with the 

points connected by dashed lines to give an indication of 

the shape of the distribution of Nh/N0 values and with 

averages of y(t) and y(x) differentiated. The figure 

indicates that records of both y(t) and y(x) yield values 

of Nh/No that agree very well with the theoretical curve 

for h > 0, but that deviate appreciably from the curve 

of equation 12 for h ^ 0, with the values from the process 

y y(t) showing the greatest deviation. Figure 8 suggests 

that there may be some slight differences in the properties 

of y(x) and y(t). However, distributions of the raw data 

and the spectral analyses of the processes, discussed in a 

later section, indicate that there are no appreciable

Average of 13

o

Average ofj/(f)

A

Average

FIGURE 8. Average values of Nh/N0 as a function of h.

differences, so that the greater deviation of the values for 

the process y=y(t) probably is fortuitous.

No consistent differences in the deviations of the values 

from the curve in figure 7 could be attributed either to 

flume size or to bed forms, where both ripples and dunes 

are represented.

The mean duration of an upward excursion of the 

process y(t) above the level h, E{1%}, was given by equation 13, and equation 14 is the ratio of mean duration of 

upward excursions at the level h to mean duration of upward excursions at the zero level. Equation 14 is plotted 

in figure 9 along with observed values of the ratio 1^11$. 

Again, as in figure 7, there is systematic deviation from 

the curve for values of h from 0 to 1, and there is 

considerable scatter at the higher values of h. However, 

this is not too disturbing because the number of upward 

excursions that are observed in the relatively short flume 

records probably is too small to get reliable average values 

of^+ .

The average rest period of a particle deposited between 

the levels hi and hz can be computed from equation 14 or 

estimated graphically from figure 9. Theoretically, there 

are no limitations to equation 14; it is applicable between 

any two levels of the bed, °°<A< + °°. The ratio 

1%/lfi approaches zero as h assumes large positive values 

and approaches an infinite value as h assumes large negative values. From a practical point of view, these extremes

0.2

FIGURE 9. Observed values of the ratio plotted against h.
STATISTICAL PROPERTIES OF DUNE PROFILES F13

rarely would be of interest. The major transport of bed 

material occurs in the activity shifting part of the bed, and 

it is unlikely that one would need to consider anything 

beyond two standard deviations of the mean bed level to 

account for the bulk of the sediment movement. Within 

these limits, a reasonable estimate of an average rest 

period can be determined from figure 9 or equation 14.

Equation 13 gives the mean rest period, /£, of a particle 

deposited on the downstream face of a dune at the level 

h. We are interested in not only the mean value but also 

the distribution of /^, the conditional probability distribution of rest periods, given that a particle is deposited at 

the level h. As indicated previously, there is no theoretical 

basis for predicting a distribution of the /£ values, and, 

unfortunately, the records of y(t) were not long enough 

to establish the rest period distributions.

However, it may be useful to establish some of the 

properties of the distributions for l^ values of the process 

y(x), even though these values do not represent rest 

periods, because if the statistical properties of y(x) and 

y(t) are similar, the distribution of crossings and other 

features of interest should be the same. (For Gaussian 

processes with zero means, similar covariance functions 

insure similarity of all other properties.) For this purpose, 

the Atrisco Lateral records were selected because they 

were the longest available.

Figure 10, a bar graph of 1% distributions for run 1, 

shows that shapes of the distributions vary with h and 

suggests that at the level h = 0 the lengths of upward 

excursions follow an exponential distribution. Figure 11 

shows as solid lines exponential distributions with the 

same means as observed mean values of /J for runs 1, 2 

and 3. The plotted points represent the observed values. 

Obviously, the exponential distribution is only a rough 

approximation.

W. W. Sayre (oral commun., 1967) suggested that perhaps a gamma distribution would serve as a model for 

the distribution of It values over any practical range of 

bed elevations that are of interest. To investigate this 

possibility, consider the gamma distribution with parameters b > 0, X > 0,

(33)

where x > 0, and

where x < 0.

P(/)=0,

When 6=1, this is the negative exponential distribution 

shown in figure 11, with X=l//0 . The variance of the

50

40

30

20

10

ill

*§§;!

liiiii

.

h =1.2

II

lillliliil

-

-

to 5°

3o

40

30

20

h =

=-1.2

10 20 30

/*+

40 50

FIGURE 10. Observed distributions of If for run 1.

gamma distribution is ft/X2 , and its coefficient of variation 

C p , the standard deviation divided by the mean, is I/Vft.

Figure 12 shows observed values of C v for the distributions of It values plotted as functions of h. The trend 

line sketched through the plotted points is positioned with 

Cv = 1 at h = 0, corresponding to the exponential distribution of figure 11. Thus, the postulated gamma distributions 

of the lengths or durations of positive excursions above 

the level h can be determined directly from figures 12 

and 9, with X and b computed from the following equations:

6=1/C| (34)

(35)

Figure 13 shows bar graphs of the observed It distributions for run 1, with points plotted at the midpoint 

of each class interval representing the frequency for that 

class from a gamma distribution with parameters given 

by equations 34 and 35. The observed mean value of l\
F14 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

40 30 40 20 30

FIGURE 11. Observed distributions of / + for runs 1, 2, and 3. Solid lines are exponential distributions with the same mean values as the observed

means. The lengths are in multiples of the lag interval, 0.333 feet.

wa 

we 

thi 

] 

rur 

fro 

the 

ob 

rel

be

( 

(

i

s used to compute X for h = 0, and all other parameters 

re determined from figures 9 and 12. The results of 

s example certainly are encouraging but not conclusive, 

t should be noted that the data from one time record, 

i 19, are plotted in figure 12. These data deviate more 

m the trend line of the Cv values than do the data from 

longitudinal profiles, but the number of crossings 

served in the time records was too small to provide 

iable statistical estimates of the distribution of the 

values. Thus, the approach outlined above appears to 

equally applicable to records of y(x) ory(t), but confi-

) 

>

V D 2.0-

^V .A ^

AX. " O

f ^CL a 

S ^kj^ i

* 1.0-

1

k 

A Run 1, _/=_/ (x)

D Run 2, y y (x) 

O Run 3, y y U)

 Run 19, y-y (f)

<

I

2 )

»

50 

40 

30 

20 

10 

0

w 50 c/3

0 40

z

u 30

O

z 20

UJ

K 10

UJ 

0.

0

50 

40 

30 

20 

10 

0

.oi

^^ * «w» ^»» < 

i 7"^"1 r\ i i i

h=0 

0

'"Xn '

i i i i i

i i -

»»«-» j

_ ..L^jr""*"^"" ! " ' : | -

: " ' : l' : ^' : : O'''|-" O' ." '' 'fj"":"i :i*J : ^O^ t '-O'-: ' : ' : 'rt : ''i ' ' : '''"' : i

3 10 20 30 40 50

-2 -1

FIGURE 12. Observed values of the coefficient of variation, C«, for 

the distributions of If plotted as functions of h.

FIGURE 13. Comparison of observed 4+ distributions with the 

theoretical class frequencies from a gamma distribution. The 

plotted points are from the theoretical distribution.
STATISTICAL PROPERTIES OF DUNE PROFILES F15

dence levels for estimating the distributions of rest periods 

cannot be established until longer records of the process 

y~y(t) become available.

One other distribution of interest that will be considered in this section is the probability that a particle is 

residing in the bed at the level h. This is not the same as 

the probability that a particle is deposited at the level h, 

for deposition occurs on the downstream faces and in the 

troughs of the sand waves; this latter probability no doubt 

depends on flow conditions. The distributions derived 

empirically by investigating the area bounded by the curve 

y= y(t) °r y y(x) above a level h give the percent of 

particles found in the bed above the level h and represent 

the probability of finding a particle at the level h if nothing 

is known of its previous history. Figure 14 shows the computed values for y(x) and y(t) records from the 2-ft flume, 

with a smooth curve drawn by eye to indicate the trend 

of the distributions. An approximate equation forP(/i), 

the probability that a particle is residing in the bed above 

the level h, is given by

P(h) = 1 e-°- (36)

Equation 36 is a good approximation in the range 

0.1 < P < 0.9, but it is not applicable for extreme values 

ofh.

-2

o Run 40, ripples

o Run 44, ripples

 Run 48, dur>es

_/=/(*)

o Run 41, ripples 

o Run 45, ripples 

 Run 48, dunes

50 100 

PERCENTAGE OF PARTICLES ABOVE THE LEVEL h

FIGURE 14. Percentage of particles in the bed above the level h.

DISTRIBUTION OF WAVE HEIGHTS AND AMPLITUDES

The absolute maximum value of y between zero crossings was defined in figure 2 as the wave amplitude, a, and 

it was noted that the distribution of positive amplitudes, 

a+, represented the probability distribution of local 

deposition and that the distribution of negative values, 

a , represented the probability distribution of local 

depth of scour associated with the formation and migration of sand waves. If the processes y(x) and y(t) were 

symmetric about their mean values, the distribution of 

positive and negative a values would be identical, and for 

a Gaussian process, it was postulated that the distributions of a values would approximate the distribution of 

crest heights, TJ, shown in figure 4.

All of the records analyzed exhibited relatively broad 

spectra, with values of S2 from equation 20 varying from 

0.83 to 0.99 (see table 3) and with most of the values 

greater than 0.9. Figures 15 through 17 show distributions 

of positive and negative a values for runs 1 through 3. 

The solid curves on the figures represent the Gaussian 

distribution from figure 4 of crest heights for a process 

with a broad-band spectrum (82 =1.0). Two points of 

interest should be noted: (1) the distributions are not 

symmetric, and (2) the negative values, a , follow more 

closely the normal distribution than do the positive values.

The positive amplitudes, a+, followed approximately an 

exponential distribution, as shown in figure 18. In this 

figure, P(a) is the cumulative probability distribution of 

a, and for this form of plotting, a Rayleigh distribution 

(eq 21) would have a slope of two and an exponential 

distribution, a slope of one. Clearly, except for very small 

values of a, the slope is one, indicating an approximate 

exponential distribution.

As discussed previously, there is no basis for extimating 

the probability distribution of wave heights, H, the troughto-crest height, and there is no reason to expect similarities in the distribution of H and a values. Figure 19 

shows the distribution of wave heights to approach the 

Rayleigh distribution, rather than the exponential distribution. The same data were used to prepare figures 18 and 

19, but in figure 19 the raw data were not standardized 

and the wave heights are given in feet rather than in 

units of standard deviations.

Even though the distributions are different, the mean 

values of a+ and H relate reasonably well (fig. 20) as do 

the mean values of distances between successive upcrossings of y and mean dune lengths, L, (trough-totrough distance) (fig. 21). Note, however, that values of 

2a and of the distance between successive upcrossings 

are not strictly comparable to H and L because entire 

waveforms occur above or below the mean bed level, and 

their lengths and heights are not reflected in the average
F16 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

TABLE 3. Miscellaneous properties from the zero-crossing and spectral 

analysis

[Type of profile is y(x) except type for runs 4, 19, 33, 40, 44, and 48, which is y(t)]

Length: L0 is the average distance in feet, or time in minutes for runs 4, 19, 33, 40, 44, and 48, 

between successive upcrossings of y = 0. 

Amplitude: a + is the average of maximum y values between successive zeros. 

<r v : From eq 5. 

8 2 : From eq 20. 

I/mi: Mean wavelength in feet, or wave period in minutes for runs 4, 19, 33, 40, 44, and 48, from 

spectra, eq 8.

Length

Run

1........

2........

3........

4........

5........

6........

7........

8........

9

10......

11

12......

13......

14......

15......

16......

17

18......

19......

20......

92

23......

24......

25......

26......

27

28......

29......

30......

32......

33......

34,

35......

36......

37......

38......

39......

40......

41......

42......

43

44......

45.......

46....... 

47....... 

48....... 

49....... 

50...... 

51....... 

52....... 

53.......

Lo

8.96 

9.22 

7.79 

57.3 

4.97 

4.94 

5.37 

5.24 

6.13 

5.78 

6.59 

3.39 

6.39 

4.74 

7.38 

6.21 

8.45 

6.96 

29.3 

4.68 

4.34 

4.21 

4.19 

5.04 

5.61 

5.62 

5.79 

3.70 

4.04 

3.98 

5.69 

4.78 

16.6 

4.67 

5.78 

3.20 

4.12 

4.20 

2.79 

14.3 

.968 

1.16 

1.36 

27.4 

1.14 

1.01 

1.13 

25.5 

2.56 

2.77 

3.16 

.636 

.782

C,

0.577 

.589 

.560 

.504 

.599 

.681 

.618 

.631 

.379 

.511 

.596 

.877 

.481 

.789 

.186 

.512 

.384 

.358 

.591 

.448 

.648 

.575 

.623 

.416 

.412 

.598 

.530 

.635 

.706 

.620 

.576 

.644 

.817 

.662 

.453 

.709 

.756 

.552 

.757 

.989 

.635 

.666 

.542 

.630 

.546 

.515 

.679 

.562 

.472 

.610 

.665 

.534 

.460

Amplitude

(Ft)

0.277 

.287 

.302 

.386 

.164 

.158 

.156 

.180 

.225 

.221 

.223 

.119 

.243 

.192 

.306 

.256 

.274 

.226 

.099 

.169 

.138 

.189 

.167 

.190 

.201 

.212 

.227 

.137 

.144 

.091 

.129 

.157 

.103 

.132 

.179 

.093 

.106 

.072 

.088 

.0237 

.0432 

.0483 

.0512 

.0279 

.0493 

.0430 

.0431 

.0885 

.0952 

.0734 

.0653 

.0384 

.0416

cr

0.850 

.913 

.782 

1.46 

.868 

.886 

.673 

.712 

.573 

.687 

.684 

1.21 

.690 

1.02 

.467 

1.09 

.747 

.555 

.885 

.800 

.834 

.648 

.708 

.705 

.843 

.836 

.702 

.928 

1.00 

.899 

.716 

.853 

.815 

.732 

1.06 

.836 

.755 

1.01 

1.23 

1.16 

.857 

.666 

.680 

.963 

.619 

.652 

.820 

.817 

.901 

1.03 

.687 

.920 

.449

(Ft)

0.259 .

.276 .

?70

.200 

.144 

.126 

.133 

.162 

.138 

.177 

.189 

.184 

.172 

.195 

.196 

.176 

.216 

.184 

.107 

.147 

.111 

.139 

.137 

.140 

.162 

.167 

.173 

.133 

.120 

.103 

.108 

.145 

.120 

.110 

.111 

.101 

.102 

.112 

.0923 

.0360 

.0437 

.0474 

.0457 

.0346 

.0404 

.0387 

.0423 

.0750 

.0739 

.0744 

.0568 

.0464 

.0344

8a

0.933 

.978 

.968 

.973 

.964 

.952 

.966 

.972 

.975 

.970 

.982 

.977 

.960 

.983 

.979 

.974 

.931 

.946 

.974 

.946 

.966 

.952 

.938 

.944 

.909 

.876 

.968 

.966 

.970 

.961 

.965 

.830 

.961 

.923 

.983 

.962 

.958 

.886 

.895 

.921 

.970 

.874 

.881 

.904 

.927 

.948 

.927 

.956 

.980 

.964

I/TO!

26.0 

6.38 

4.91 

5.75 

4.46 

3.85 

5.11 

5.56 

5.31 

4.91 

6.37 

6.24 

4.52 

7.59 

6.48 

42.5 

5.35 

4.01 

4.47 

4.07 

4.43 

5.15 

5.85 

5.68 

4.72 

3.48 

5.37 

5.86 

5.44 

26.8 

4.95 

3.02 

4.31 

5.25 

6.21 

3.32 

29.7 

1.36 

1.61 

1.75 

38.8 

1.33 

1.24 

1.58 

32.4 

3.00 

2.37 

3.56 

1.08 

.81

100

FIGURE 15. Distribution of positive and negative a values for run 1.

100

VI

H50

HI

o 

cc

HI 

Q_

(a-)-

Normal 

distribution

Run 2

FIGURE 16. Distribution of positive and negative a values for run 2.

distance or average of maximum ordinates between zero 

crossings.

An interesting correlation was found to exist between 

the average of maximum ordinates between zero crossings and the standard deviation of the bed elevation

(fig. 22). For the smaller features, a direct linear relation 

applies, with the standard deviation of the bed elevation 

approximately equal to the average amplitude of the sand 

waves. This is precisely the relation predicted for the 

mean value of local maxima given by equation 23 and
100

VI

I- 50

UJ

O 

o:

UJ

a.

STATISTICAL PROPERTIES OF DUNE PROFILES

5.0

F17

(a-)-

Normal 

distribution

(a + )

Run 3

FIGURE 17. Distribution of positive and negative a values for run 3.

shown in figure 4 as the curve corresponding to 82 = 1.0. 

The relation of figure 22, then, supports the assumption 

that the distribution of a values should be similar to the 

distribution of the local maxima. The deviation from the 

line of agreement at the larger a values might be attributed to the fact that some of the details of the record

i.o

J, o.i

0.01

-Exponential 

distribution

0.1 1.0 5.0

FIGURE 18. Approximate exponential distribution of positive a 

values for run 55.

1.0

0.1

RayleighExponential

0.1 1.0

H, IN FEET 

FIGURE 19. Distribution of wave heights, H, for run 55.

are lost in digitizing the continuous trace of bed elevation. 

The larger dunes are not more regular than the smaller 

features; in fact, the reverse is true. If the bed profile is 

represented as a succession of identical triangles, similar 

to the idealized dune form of figure 1, with ripple index, 

L/H, of 15 and tan /3 = 0.6, the relation of mean amplitude 

to standard deviation of y is given by a = 0.64 o> If this 

condition were to hold for the larger dunes represented 

in figure 22, the plotted points would fall to the left of 

the line of agreement.

Summarizing, the average amplitude a is approximately 

equal to the standard deviation of the bed elevation, and 

the distributions of a values predicted by the relations in 

figure 4 appear to give reasonable agreement with the 

observed distributions. However, the bed profile is not 

symmetric about its mean level, and the distribution of 

negative values a agrees more closely with the predicted 

normal distribution than do the positive values a+. For 

the positive values, an exponential distribution can be 

shown to apply (fig. 18).
F18 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

1.0

0.04

i f |

0.04 0.1 1.0

TWICE THE AVERAGE MAXIMUM ORDINATE BETWEEN ZERO 

CROSSINGS, IN FEET

FIGURE 20. Relation between twice the average maximum ordinate 

between zero crossings, 2a +, and H.

10.0

1.0

0.4

I I I

I I I I I I I

0.4 1.0 10.0

AVERAGE DISTANCE, IN FEET, BETWEEN SUCCESSIVE

UP-CROSSINGS

FIGURE 21. Relation between mean values of the distances between 

successive zero upcrossings of y and the mean dune length, L.

1.0

0.01

0.01 0.1

AVERAGE OF MAXIMUM ORDINATES BETWEEN ZERO 

CROSSINGS (J+), IN FEET

1.0

FIGURE 22. Relation between average of maximum ordinates between 

zero crossings, a+, and standard deviation of bed elevation, a- v .

SPECTRAL ANALYSIS

In the previous discussion, the autocovariance functions or the spectral density functions for the processes 

y(x) and y(t) were assumed known. These functions 

were required to predict the number of zero and h-level 

crossings for the model Gaussian process, and for the 

experimental data the functions were known because 

they were computed. In the following sections, the properties of the covariance functions and spectra will be 

considered in somewhat more detail.

STATIONARITY AND EQUILIBRIUM FLOW 

A critical assumption in spectral analysis is that the 

processes under consideration are stationary, at least to 

the second order. At the initiation of this study, it seemed 

intuitively obvious that stationarity of the processes y(t) 

and y(x) would be a direct consequence of equilibrium 

flow conditions. If the mean characteristics of the flow, 

the sediment, and the transport do not change with time 

or with distance along the channel, then surely the statistical properties of records of the bed profile should be 

invariant with respect to shifts in the origin of the records. 

For the longer field records, it can be demonstrated that 

the assumption of weak stationarity is justified. Figure 

23A shows mean values and standard deviations for eight 

short segments from the record of run 3 plotted with their 

respective 90 percent significance levels. The standard
STATISTICAL PROPERTIES OF DUNE PROFILES F19

i.O

1.4

0.4 

0.3 

0.2,

 * __ __ __ Mean + _

4

-

Standard deviation

i i i i i i i

1.6

1.5 

0.20

0.18

0.16

) 1 2 3 4 5 6 7 8 ~" '<

A

Mean

4

__ -

_J^___^.__ f __ _,_ _^, __ r ^ .

Standard deviation '

9 

312345678 

B

FIGURE 23. Comparisons of means and standard deviations, in feet: A, for eight short segments of the record of run 3, Atrisco Lateral; B, for 

eight longitudinal profiles during equilibrium flow in the 8-foot flume, runs 8 through 15. The solid line represents the average value and the 

dashed lines indicate a 90-percent significance level.

deviations do not vary significantly. The rather large 

variations in the mean values resulted from large alternate 

bars and a meandering thalweg that existed in the channel 

and that introduced apparent trends in the short segments 

of the record. The bed profiles were obtained by sounding 

from a boat with the water surface as a datum, and the 

depth of flow varied somewhat systematically along the 

channels. However, when the bed elevations were measured from the mean bed level established by a linear 

trend line through short segments of the reach, the 

assumption of second-order stationarity for these short 

segments of record was satisfactory. For longer segments 

of the record, on the order of 40 to 50 times the mean 

channel width, no significant differences were noted in 

the means and variances.

The flume records presented a somewhat different 

problem. It was assumed that, for equilibrium flow conditions, any two records of the bed profile would show 

approximately the same statistical properties. However, 

this was not found to be true. Figure 23B shows that the 

standard deviations for runs 8 through 15, all of which 

were taken during apparent equilibrium flow conditions 

over an identical reach of the flume (see table 1), vary 

significantly.

Two factors appear responsible for the large variations 

in the flume records. First, the flume records fory(.r) are 

short relative to the number of dunes that are observed, 

and the short records introduce inherently large variations 

in the statistics from one observation to the next. Second, 

the concept of equilibrium flow implies a time-averaged 

stability that may not necessarily apply to any single 

observation. Simons and Richardson (1966, p. J3) indicate 

that equilibrium flow obtains when the time-averaged

water-surface slope and bed slope are parallel and constant and the time-averaged sediment discharge is 

constant. Rathbun and Guy (1967, p. Ill) have shown 

that extremely large variations in sediment transport 

rates are to be expected in recirculating flumes, and 

probably these variations are reflected in changes in the 

properties of the bed profiles from one observation to the 

next.

Despite the rather large variations in the flume records 

due to the fact that equilibrium flow conditions do not 

prevail over short time intervals, the spectral properties 

of the individual records are remarkably similar for a 

given flow condition, provided the data are standardized 

to zero mean and unit variance. Figure 24 shows the 

spectral ordinates for runs 8, 10, 12 and 14 as points 

plotted against a dimensionless wave number, e/emax , 

where e max is the maximum wave number for which the 

computations were carried out; the solid line represents 

an average curve for the spectra and the dashed lines 

represent a 90-percent confidence band based on the 

procedure given by Blackman and Tukey (1958, p. 21 23). 

Only two of the 80 points fall outside the confidence band; 

the general shapes of the spectra are therefore quite 

reproducible from one observation to the next. The approximation used in this study, therefore, was to treat 

each individual record as if it were weakly stationary. 

The moments of the spectra for the individual records 

could then be used to estimate the number of zero crossings by equation 10 to compare with observed values. 

However, in attempting to relate the statistical properties, 

especially the variance, of the bed profiles to flow parameters, it will be found convenient to work with average 

values over a consistent set of flow conditions.
F20 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

FIGURE 24. Spectra for indicated flume records.

THE MARKOV MODEL FOR DUNE PROFILES 

A major question with regard to spectral analysis is 

whether or not the properties of the covariance functions 

or the spectra can be predicted from only the characteristics of the flow and the sediment. Some work already 

has been accomplished along these lines. Algert (1965) 

observed that the general shapes of the autocorrelation 

functions and of the spectra for the process y = y(x) were 

similar to that of a second-order Markov process, and he 

developed a model for the spectra based on previous 

work of Siddiqui (1962). Nordin and Algert (1966) presented virtually the same development, based on Algert's 

1965 study, but added some observations on the process 

y=y(t). Ashida and Tanaka (1967) noted that a secondorder Markov process fits the observed spectra only for 

dunes and postulated a higher-order linear regressive 

scheme for other bed forms but did not develop a model or 

apply the higher order scheme to any of their data.

In the studies by Algert (1965) and Nordin and Algert 

(1966), the parameters upon which the model of the 

spectra is based are the values of the covariance function 

at zero, one, and two lag intervals, and these values were 

shown to relate roughly to the flow parameter, unit 

discharge (fig. 25). We can use figure 25 and equation 15 

to predict the average length between zero crossings of 

the process y(x) by noting that the spacing of the discrete 

data points used by Nordin and Algert (1966) in figure 25 

was approximately given by

(37)

Here AJC is equivalent to A? in equation 15, Ci/Co is <£ (A?), 

and C, is cry $(iAt). Substituting these relations in equation 15 and reading values of C0 and Ci from figure 25 

for values of unit discharge given in table 1, the average 

distances between zero crossings were computed and are 

compared with observed values in figure 26. For the dune 

bed configuration, the observed values of /0 fall between 

plus and minus 40 percent of the expected value, £{/o}. 

For the ripple bed configuration, the variations are considerably greater, probably because the size of ripples is 

more dependent on grain size than flow conditions, so 

that figure 25 perhaps is not applicable. In view of the

2.5

2.0

^ 1.5

y i.o

0.5

2 46 8 10

UNIT DISCHARGE IN CUBIC FEET PER SECOND 

PER FOOT

12

FIGURE 25. Relation of C0 , C,, and C2 to unit water discharge (from 

Nordin and Algert, 1966).
STATISTICAL PROPERTIES OF DUNE PROFILES F21

10

~ 5

 /*y

10

/0 ,IN FEET

FIGURE 26. Comparison of estimated and observed average distance 

in feet between zero crossings.

gross assumptions that went into estimating £{/o}, the 

results of figure 26 are considered quite good.

There are a number of problems involved in using the 

second-order Markov model and the relations of figure 25 

for predictive purposes. Both the model and the relations 

of figure 25 are strongly dependent on the lag interval at 

which the continuous records are digitized. In addition, 

spectral analyses by the writer and by D. R. Dawdy and 

N. C. Matalas (personal commun., 1966) have shown that 

the Markov model does not apply for many of the dune 

profiles examined. Finally, Plate (1967) has raised some 

serious questions with regard to the lack of physical basis 

of this model. It is desirable, therefore, to consider an 

alternative approach to describe the properties of the 

spectra and to relate their properties to characteristics 

of the flow.

DIMENSIONLESS SPECTRA

Autocovariance functions and spectra were computed 

for all the data listed in table 1. Figure 27 shows spectra 

for the process y=y(x) for runs 17, 32, 43, 46 and 49. 

Spectra of the process y = y(t) for identical flow conditions, runs 19, 33, 40, 44 and 48, are shown in figure 28. 

The data include dune flows in the 8-ft flume and both 

ripples and dunes in the 2-ft flume. The abscissa in both 

figures is dimensionless, with each value of frequency or 

wave number divided by the maximum value. All the 

spectra are remarkably similar in general shape, and there 

is little in the two figures to differentiate the processes

y(t) and y(x); this fact suggests that the statistical properties of the two types of records are similar.

Figure 29 shows a comparison of spectra for ripples 

and dunes for the 2-ft flume, with a 90-percent confidence 

band estimated according to the method given by Blackman and Tukey (1958). Here, as in figures 27 and 28, the 

similarity in the spectra is apparent. For both ripples 

and dunes, the major part of the variance is contributed 

by the longer wavelength components, greater than 2 feet. 

Both spectra show peaks at about 0.3 cycle per foot and 

again at about 0.8 cycle per foot. Both show additional 

peaks in the range from 2 to 4 cycles per foot, but within 

the confidence limits the peaks on the dune spectra in 

this range probably are not significant, whereas those for 

the ripples are. The rather wide band of the confidence 

limits leaves open to question the significance of most of 

the peaks at the lower wave numbers, but there is no 

question as to the general shape of the spectra.

The similarity of shape of the spectra in figures 27 and 

28 suggests that a model incorporating flow parameters 

might be developed from dimensional considerations. For 

a first approximation, consider the major part of the 

spectra, that part which contributes ah1 but a small percent of the variance, to be a function (</;) of only four 

variables:

G(x) = i//(e, g, D, V) (39)

where D is mean flow depth, V is mean flow velocity, 

g is the acceleration due to gravity, and other symbols 

are as defined previously. Dimensionless spectra G' 

are then given in terms of a dimensionless frequency /' 

or wave number e' and the square of the Froude number, 

F, as foUows:

fhere

G'(x) = *'( ',

G'(t)=GglV, 

f'=fVlg,

(40)

(41)

(42)

(43)

(44)

(45)

(46)

Neither fluid nor sediment properties enter into equations 38 and 39 on the assumption that water temperature 

and particle size for the data analyzed were approximately

, and
F22 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

100.0

10.0

1.0

0.1

0.01

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

FIGURE 27. Spectra of the process y y(x).
STATISTICAL PROPERTIES OF DUNE PROFILES F23

100.0

10.0

0.01 -

0.001 0.1 0.2 0.3 0.4 0.5 0.6 

RELATIVE FREQUENCY f/fmax

0.7 0.8 0.9 1.0

FIGURE 28. Spectra of the process y=y(0-
F24 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

20.0

10.0

O

P 1.0

0.1

0.01

0.002

V\V\ ,90-percent confidence band

j_____i_____I_____I j____ i

0

,90-percent confidence band

WAVE NUMBER, IN CYCLES PER FOOT 

FIGURE 29. Comparison of ripple (left) and dune (right) spectra, 2-foot flume.

constant. Then, the sediment transport rate and the bed 

forms are primarily functions of velocity and depth 

(Colby, 1964, Nordin and others, 1965). The mean flow 

depth enters equations 40 and 41 only through the Froude 

number. It is assumed that the equilibrium height of the 

bed configurations depends on flow depth, and it was 

shown in the previous section that the mean height of the 

bed irregularities and the variance of the process y(x) 

are closely related. Thus, because all the records were 

previously standardized in terms of the variance, the 

Froude number should be a parameter of only minor 

importance in the spectra, and for the first approximation, 

G' should be a function only of the dimensionless frequency or wave number.

Figure 30 shows the relation between G' and e' for the 

same data given in figure 27. For values of dimensionless 

wave numbers greater than 0.03, all data follow the 

relation

G' = 0.0001 (e') (47)

For lower wave numbers, each set of data follows a 

separate curve without a consistent pattern. It does not 

appear possible to collapse the family of curves into a 

single curve on the basis of Froude number. The dashed 

curve on the figure is intended only to show the general 

trend of the data, which seems always to show a maximum 

for values of e' less than 0.025.

The dimensionless spectra for the process y y(t) 

are shown in figure 31, where the general relation for 

dimensionless frequencies less than 0.001 is given by

G' = 0.000015 (/' (48)

In figure 31 as in figure 30, there is considerable scatter; each set of data defines a general trend which tends 

to parallel equation 48. Again, it was not possible to sort 

the curves in terms of the third parameter, Froude number, which fact supports the assumption that depth should 

be a variable of minor importance in this analysis.

Equation 47 by no means characterizes the entire 

spectrum, but if the maximum values of G' and the dimensionless wave numbers at which they occur were 

either constant or functions of the flow and sediment, 

the spectra would be reasonably well described. Table 4 

summarizes pertinent data for such an analysis for the 

process y=y(x). The maximum spectral ordinate relates roughly to flow velocity, as shown in figure 32 (upper). 

The same weak relation appears to hold between the 

maximum G'(x) and the dimensionless wave number, 

e', at which the maximum occurs, as shown in figure 32 

(lower). No apparent relation was found between values 

of e' corresponding to maximum values of G' (x) and any 

of the flow or sediment characteristics, but it simply
STATISTICAL PROPERTIES OF DUNE PROFILES F25

30

10

1.0

0.1

0.3

I i I I I

Froude number= 0.220

Froude number = 0.230

Froude number=0.270

D

Run 46 

Froude number=0.262

d>

Run 49 

Froude number=0.396

i i i i i I I i I

0.01 0.1 

FIGURE 30. Dimensionless spectra for the process y=y(.r).

may be that there was an insufficient number of observations to establish any trends.

From equations 47 and 48, the spectral density function G is approximately proportional to the square of the

wave period, 7"= I//, or to the third power of the wave 

length, L=l/e. The wave celerity c is defined as L/71=//e, 

so that from the equations for the dimensionless spectra, 

we can conclude that wave celerity is directly proportional
F26 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

100,000

10,000

1000

100

Run 19 

Froude number=0.220

o

Run 33 

Froude number=0.230

A

Run 40 

Froude number=0.270

D

Run 44 

Froude number = 0.262

*

Run 48 

Froude number=0.396

0.00001 0.0001

FIGURE 31. Dimensionless spectra for the process y=y(t).

to e1/2. Thus, for constant flow conditions, the small waves 

move faster than the large waves. This conclusion is 

supported by Simons, Richardson, and Albertson (1961), 

who state, "Smaller dunes with their higher velocities 

overtake the larger dunes."

One arrives at the same conclusion by considering a 

simple model for sediment transport based on continuity 

principles, such as the one given by Simons, Richardson, 

and Nordin (1965), which states that, for ripples or dunes, 

the transport rate of bedload, <?&, is directly proportional
STATISTICAL PROPERTIES OF DUNE PROFILES F27

TABLE

Run

53.......

46......

43......

49......

32.......

17.......

4. Summary of flow characteristics for dimensionless spectra

Mean

velocity 

(fps)

0.593 

1.07 

1.10 

1.62 

2.01 

2.09

Mean 

depth 

(ft)

0.174 

.522 

.518 

.521 

2.36 

2.80

Median 

particle 

diameter 

(mm)

0.30 

.35 

.35 

.35

.24 

.24

F

0.246 

.262 

.270 

.396 

.230 

.220

Maximum G'

29.0

15.5 

22.8 

17.8 

17.3 

21.8

e' 

for 

maximum G'

0.0086 

.012 

.004 

.023 

.015 

.009

to wave height, H, and wave celerity, c. Suppose that the 

bed forms are dunes with ripples superposed, and that 

the dune movement is due entirely to the ripples overtaking the dune crest and depositing material on its 

downstream face. Then, continuity requires that 

Qb ~ c\H\ ~ Cztizi which states simply that the larger 

features move at a smaller velocity.

Note, however, that equations 47 and 48 say nothing 

about how the mean wave celerity might vary with varying flow conditions. This matter is considered in the next

50

10

0.4 1.0 

V, IN FEET PER SECOND

4.0

DU

0

D

X

S

in

i i i i i i i

-

i i i i i I 1

0.004 0.01 0.04 

«'CORRESPONDING TO MAXIMUM G (x)

FIGURE 32. Relation of maximum value of G' (x) to mean 

velocity and to e'.

section in connection with mean frequencies and wave 

numbers of the spectra computed from equation 8. 

Equations 47 and 48 describe only the high frequency 

or high wave number parts of the spectra. The generality 

of the results are supported by Hino's (1968) study of 

sand wave spectra, in which he showed from dimensional 

considerations that the wave-number spectra at high 

wave numbers should follow the wave number, e, to the 

minus third power and that the frequency spectra should 

vary as the minus second power of the frequency.

In considering sand wave properties in relation to 

channel roughness, however, it is the low frequency 

part of the spectrum that is important, and the dimensionless spectra of figure 30 do not reduce the low frequency part to a single relation. Engelund (1969) derived a 

universal dimensionless wave-number spectrum from 

similitude criteria that is of the form

2gDS G(x)

J72 (T 2gDS (49)

where e is wave number in cycles per foot, G(x) is in 

(cycles per foot)" 1 , i/; is an arbitrary function, S is the 

water-surface slope, and the quantity (2gDS)/V2 is the 

friction factor.

Engelund computed the relation of equation 49 for 

selected data from this report and plotted the results 

to define the general shape of the function as shown in 

figure 33. The results are encouraging, but additional 

data for greater flow depths probably need to be collected 

before it will be possible to determine whether or not 

the relation of figure 33 is superior to the relation of 

figure 30.

OTHER PROPERTIES OF THE SPECTRA

The probability distributions of the y values, the characteristics of the zero and h -level crossings, and the consistent shapes of the dimensionless spectra all point to 

some remarkable similarities of the bed profiles, irrespective of the size of the channel or of whether the bed forms 

were ripples or dunes. Indeed, the analysis to this point 

has been directed primarily toward exposing such similarities, and the procedures for standardizing the raw data 

to zero mean and unit variance and for nondimensionalizing the spectra really are nothing more than procedures 

for applying appropriate scale factors to the data to facilitate comparison. Quite often, however, one is more 

interested in the differences than in the similarities of 

the bed profiles, and if this is the case, it is more appropriate to compare the spectra of the records directly 

in terms of the actual frequency or wave number compo-
F28 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

o.i

0.01

0.001

0.1

 Run 17

 Run 32 

x Run 56 

o Run 3

1.0

gel/ 2 

2gDS

10.0

FIGURE 33. Dimensionless spectrum based on similitude criteria (modified from Engelund, 1969).
STATISTICAL PROPERTIES OF DUNE PROFILES F29

nents. Figure 34 is an example of such a comparison. 

Spectra of longitudinal profiles, y=y(.r), from each of 

the four channels, are plotted against wave numbers in 

cycles per foot. The one-sided spectra were computed 

from the standardized data, so that the area under each 

curve is equal to 0.5. The standard deviation, cry , for each 

record is shown in the figure. The effects of channel size 

on the distribution of the variance over the various wave 

number components is obvious.

Several features of figure 34 are particularly interesting. First, note that the flow conditions for run 56 and run 

24 are quite similar (table 1), yet there are appreciable 

differences in the spectra of the dune profiles. Run 56 

is a 200-ft reach of Atrisco Lateral, and the major differences in fluid and flow characteristics between the 

field data and flume data are given below. In the field 

the velocity is about 5 percent higher and the temperature 

is lower by 4°C. In addition, a fine sediment load was 

associated with the field data that was not present in the

Atrisco Lateral 

run 56, dunes 

= 0.296

8-foot flume

run 24, dunes

=0.140

2-foot flume 

run 49, dunes 

av =0.0739

2-foot flume

run 46, ripples

°> =0.0387 8-inch flume

^- run 53. ripples 

' ~~--wl <rv =0.0344

1 2 

t, CYCLES PER FOOT

00 10

WAVELENGTH, IN

0.5 

FEET

0.33

FIGURE 34. Spectra of longitudinal profiles, y=j(.v), showing effect 

of channel size and bed configuration.

flume. For Atrisco Lateral, the observed sediment concentration from suspended sediment samples was 

1440 mg/1 (milligrams per liter) of which 220 mg/1 were 

in the sand sizes, greater than 0.062 mm. For the flume 

data, the sediment concentration was 164 mg/1, all of 

which was sand. It is rather surprising that the dune 

profiles would be so markedly different with only the 

minor differences in mean flow properties. However, 

the effects of channel width, of the 4°C difference in 

temperature, of the 1,200 mg/1 fine suspended sediment, 

and of the minor velocity variations are all undefined. 

Their combined effect on the bed forms, though, is appreciable, and comparison of the spectra indicates 

that, even if the mean velocity, depth, and slope are 

similar in the laboratory and the field, there may exist 

important differences in bed forms.

A second point of interest concerns the comparison 

of run 46 and run 49. Here, the difference in the distribution of variance is due strictly to bed configuration and 

not to channel size. The only difference in the flows for 

these two runs is the increase in velocity, slope, and sediment transport rate associated with the dune bed configuration; all other factors depth, width, water 

temperature, and particle size of bed meaterial were 

constant. For the dune bed configuration, values of the 

standard deviation of the bed elevation, the mean wave 

amplitude, and the mean length between zero crossings 

are approximately double the values observed for the 

ripples. Surprisingly, the flow resistance as measured by 

the dimensionless Chezy coefficient is very similar for 

these two runs.

Finally, a third interesting feature of figure 34 is the 

indication that, for all the dune records, the variance is 

distributed generally over components of wavelength 

greater than 2 feet, whereas for the ripple bed configuration, the variance is distributed fairly uniformly over a 

greater range of wave numbers. The higher wave-number, 

or shorter wavelength, components contribute an appreciable part to the total variance of the profiles of the ripple 

bed configurations, but their contribution to the variance 

of the dune profiles is almost negligible. The question 

of whether the transition from ripples to dunes is gradual 

or abrupt cannot be answered from these data, but if 

additional experimentation were to show the transition 

to be abrupt, then the distribution of variance over the 

wave-number components would serve as a reliable 

criterion for distinguishing between ripples and dunes.

Spectral analyses of the longer records from Atrisco 

Lateral also are of particular interest because they permit 

better definition of the spectra at the lower wave numbers.
F30 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

An example of these analyses is shown for run 3 in figure 

35. In part A, the autocovariance function shows an apparent cyclic trend with a wavelength approximating the 

maximum lag of 600 or 200 feet. Inspection of cross sections of the channel every 50 feet along the length of the 

reach revealed the existence of a meandering thalweg 

and of alternate bars spaced approximately three to five 

times the channel width through the entire 4,000-ft reach. 

The trend in the autocovariance function (fig. 35,4) is 

almost certainly due to the presence of these alternate 

bars. The spectra for these longer records show the generally decreasing values of G(x) with increasing wave

numbers (fig. 35Z?) and are very similar in shape to the 

spectra shown in figures 27 and 28, but afford much 

greater detail at the lower wave numbers. Figure 35C 

shows the spectrum for wave numbers less than 0.1, 

corresponding to wavelengths of 10 feet or greater, with 

the wavelengths of the spectral peaks noted. The chisquare test with 20 degrees of freedom to establish a 

90-percent confidence band on the spectral estimates 

indicates that the average ordinate should fall between 

1.85 G(x) and 0.66 G(x). The peak at 200 feet may or 

may not be related to the meandering thalweg, but there 

is clearly a significant peak in the spectrum at wave num-

-0.2

600

42

36

30

24

18

12

i I I i i i i

0.2 0.4 0.6 0 0.02 0.04 

WAVE NUMBER (O, IN CYCLES PER FOOT

0.06 0.08 0.1

B C

FIGURE 35. Autocovariance function (4) and spectrum (B and C) of longitudinal profile, y=y(.v), for run 3, Atrisco 

Lateral. Cis an expanded version of part of B to show greater detail for the low wave numbers.
STATISTICAL PROPERTIES OF DUNE PROFILES F31

bers less than 0.01. All other peaks shown on the spectrum 

of figure 35C are considered significant with the exception 

of the one at 13 feet, but the wavelengths at which these 

significant peaks occur were not found to correlate with 

any recognizable features of the bed profile. There is, of 

course, no reason to suspect that these peaks in the spectrum should relate to the lengths of the physical waves 

on which the analysis is being performed, although it 

has been observed that for dune records there sometimes 

is a correlation between the wavelength corresponding 

to the peak in the spectra and the average dune length 

(Nordin and Algert, 1966; Ashida and Tanaka, 1967).

As a final consideration of the properties of the spectra, 

it might be worthwhile to explore further the relation 

between the characteristics of the spectra and the physical characteristics of sand waves. It was previously shown 

(fig. 22) that a correlation exists between the average 

amplitude, a, and the standard deviation of the bed elevation, cry , which is related to the spectrum of y through 

equation 5. The reciprocal of the mean frequency or wave 

number from equation 8 is the mean period or wavelength 

of the spectrum, and this value relates reasonably well 

to the mean period or wavelength defined from the zerocrossing analysis, as shown in figure 36. Thus, it is possible to relate empirically the spectral properties and the 

observed properties of the sand waves.

Ashida and Tanaka (1967) used spectral analyses for 

determining the propagation velocity of sand waves by 

plotting the wavelength of the maximum spectral ordinate

100

LU CO 

00 C3

10

10

MEAN WAVELENGTH OR WAVE PERIOD OF SPECTRA

100

for the process y(x) against the wave period for the 

maximum spectral ordinate of the process y(t). For the 

records analyzed in this study, some of the corresponding 

records of y(x) and y(t) possessed maximum ordinate 

values of their spectra at the origin; therefore, this method 

could not be used. In addition, no consistent relations 

were found between the peaks in the spectra and the 

average wavelengths or periods of the ripples and dunes. 

Consequently, the procedure used by Ashida and Tanaka 

(1967) is not recommended. However, from the relation 

in figure 36, it should be possible to determine the mean 

wave celerity either from the mean length and period 

found in the zero-crossing analysis or from the mean 

wave number or frequency of the spectra by using equation 8. Table 5, which shows a comparison of mean wave 

celerities computed by these two methods, indicates 

that values of c obtained by the two methods are 

comparable.

TABLE 5. Comparison of wave celerities

Run

vU)

17....

32....

43....

46....

49....

v(0

.. 19..

.. 33..

.. 40..

.. 44..

.. 48..

From zero crossing

L

(ft)

8.45

4.78

1.36

1.01

2.56

T

(min)

29.3

16.6

14.3

27.4

25.5

c=L/T

(fpm)

0.288

.288

.0952

.0368

.100

From equation 8

Bed

£=!/ T=llf c=fjf configuration

(ft)

7.62

5.45

1.75

1.24

3.00

(min)

42.5

26.8

29.7

38.8

32.4

(fpm)

0.179

.204

.0590

.0320

.0926

Dunes.

Do.

Ripples.

Do.

Dunes.

FIGURE 36. Relation between average distance or time between successive zero up crossings and the mean wavelength or wave period 

of spectra from equation 8. Dimensions are in units of lag intervals.

The data in table 5 also give an indication of how the 

mean wave celerity varies with mean wave number for 

different flow conditions. In the previous section, it was 

established that for a constant flow condition, the wave 

celerity of the different wave-number components varied 

directly as the square root of the wave numbers. This 

relation is shown schematically in figure 37/4. In figure 

375, the mean wave celerities computed from equation 8 

are plotted against mean wave number from the spectral 

moments. Here we see that, with increasing flow velocity, 

the average wave celerity varies inversely with wave 

number or directly with wavelength. This implies that 

the mean wave celerity is independent of frequency; or 

on the average, for these data it took just as long for a 

ripple to move past a given point as it did for a dune.

The importance of the relations in figure 37 is in the 

fact that these relations permit determining the properties 

of the process, y(t), from the properties of only the longitudinal profiles, y(x). If the spectral properties of y(x) 

are known, then the spectral properties of y(t) can be 

determined from the relation in figure 37/4 and equations 

47 and 48. If the mean wave number for the process 

y(x) is known, then the mean frequency or wave period 

of y(t) can be established by the relation in figure 375,
F32 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

WAVE NUMBER dt IIL)

A

0.3

0.1

0.03

I I I I I

c=0.03(«H

0.1 1.0

AVERAGE WAVE NUMBER <F), IN CYCLES PER FOOT

B

FIGURE 37. Relation of wave celerity, C, to wave numer, e. A, for 

various wave-number components of a single record, flow conditions 

constant. B. for average wave numbers from the spectral moments of 

records obtained under different flow conditions.

and the mean particle rest period at any level h and 

the conditional probability distributions for the rest 

periods at any level h can be determined directly from the 

relations in figure 12 and equations 14 and 33. Note, 

however, that the relation in figure 37B is no doubt a 

consequence of the sediment transport rates associated 

with the particular flow conditions examined here, and 

it is not known if the relation is generally applicable for 

greater flow depths.

O'Loughlin and Squarer (1967) and Squarer (1968) 

have urged that the standard deviation of the bed elevation and some characteristic wave length from the spectrum of the bed profiles be used to describe the geometric 

properties of sand waves, rather than simply the mean 

lengths and heights. Because no consistent terminology 

for describing bed configurations or methods for computing average wavelengths and wave heights exists in 

the literature, it is very difficult to evaluate the results 

of different investigators. Certainly, then, it is desirable 

to develop some standard and rigorous method for describing the properties of the bed profiles that will permit 

comparison of different studies, and perhaps the standard 

deviation of the bed elevation and the reciprocal of the 

mean wave number of the spectrum from equation 8 

would serve that purpose. In addition, it might be useful 

to specify the value of 8 (eq. 20), which generally may be 

interpreted as a measure of the root-mean-square width 

of the variance spectrum (Cartwright and LonguetHiggins, 1956, p. 216). Values of these parameters and of 

other measures of wavelengths and heights are given in 

tables 6 and 3.

CROSS CORRELATION AND CROSS-SPECTRAL ANALYSIS

Several interesting properties of the sand waves can be 

investigated using the techniques of cross correlation 

and cross-spectral analysis. Cross correlation is the 

simpler to use and admits to direct physical interpretations. Figure 38 shows the cross correlograms for correlations of run 8 with each of the runs 9 through 15. 

The profiles are all for the same reach of flume (see table 

1) and were obtained 5 minutes apart. If the distance of 

the peak of the correlogram from the origin is plotted 

against time (fig. 3ft4), the slope of the trend line, 7 feet 

per hour, may be taken as the mean speed of movement 

of the dunes. An independent check on this figure is 

provided by the zero-crossing analysis, where the mean 

dune length for runs 8 through 15 is 5.7 feet and the mean 

period of runs 4 and 19 is 0.72 hours; these data give a 

dune velocity of about 8 feet per hour.

The general trend for the attenuation of the peak value 

of the correlation with time (fig. 39B) gives an indication 

of the rate of change in shape of the dune profiles. The 

dune profiles are assumed to be uncorrelated when the 

maximum value of the correlation function is 0.3 or less. 

The value of 0.3 is chosen because cross correlation of any 

two profiles taken at random from a group collected during approximate equilibrium flow conditions shows a 

maximum correlation generally less than 0.3. Then, 

extrapolating figure 39B, it is inferred that the profile 

virtually loses its identity in about 90 minutes, or in about 

the same time for the average-size dune to migrate twice 

its own length.

From figure 38, it is seen that the maximum value of 

the cross-correlation function for runs 8 through 15 is 

approximately the same as the maximum value for runs
STATISTICAL PROPERTIES OF DUNE PROFILES F33

TABLE 6. Summary of wave properties

[Average length is trough-to-trough distance; height is difference in elevation, crest to trough]

Run

1.. ...............

2.................

3.................

4.................

5.................

6.................

7.................

8.................

9.................

10................

11.... .......... ..

12................

13................

14................

15................

16................

17................

18................

19................

20................

21................

22................

23................

24................

25................

26................

27................

28................

29................

30................

31................

32................

33................

34................

35................

36................

37................

38................

39................

40................

41.. .._.... ......

42................

43................

44................

45................

46................

47................

48................

49................

50................

51.. ........... ...

52................

53................

Average

length, in

feet

6.06

6.14

5.28

5.07

3.73

4.24

5.16

4.66

3.89

5.13

3.89

f\ 44

4.49

6.18

5.06

4.84

4 54

4.31

3.55

3.49

3.67

4.40

4.31

4.34

4.89

4.55

4.77

3.85

4.12

4.72

4.36

4.59

3.27

4.64

4.28

3.78

1.20

1.15

1.24

1.24

i 34,

1.28

2.73

2.75

3.15

.498

.440

Wave characteristics

Average Coefficient

wave period, of

in minutes variation

.................. 0.533

.................. .535

.................. .535

25.9 .763

.................. .658

.................. .700

.................. .604

.................. .408

.................. .553

.................. .742

.................. .524

.................. .757

.................. .340

.................. .812

.................. .462

.................. .575

.................. .779

.................. .698

25.6 .664

.................. .475

.................. .502

.................. .564

574

.................. .471

.................. .595

.................. .532

.................. .489

.................. .515

.................. .528

.................. .649

.................. .754

.................. .445

16.3 .941

.................. .521

.................. .562

.................. .538

.................. .428

.................. .510

.................. .545

30.2 .426

.................. .484

.................. .568

.................. .606

40.4 .744

.................. .534

549

.................. .518

17.7 .514

.................. .408

494

.................. .630

.................. .353

485

Height

Average

0.370

.368

.408

.214

.225

.184

.201

.341

.269

.268

.305

.251

.384

.264

.364

.315

.265

997

.149

.261

.221

.265

.257

.296

.316

.300

.367

.276

.253

.165

.161

.260

.133

.200

.255

.173

.204

.175

.185

.0783

.0922

.0902

.0913

.0750

.0856

.0827

0874

.111

.167

.149

.0609

.0520

in feet

Coefficient

of

variation

0.643

.655

.597

1.43

.571

.377

.388

.514

.535

.509

.482

.477

.468

.616

.483

.915

.547

483

.229

.473

.506

.476

.524

.578

.484

.546

.506

.538

.494

.347

.450

.506

.216

.411

.645

.381

.446

.380

.457

.277

.328

288

307

.353

.268

.216

.318

.491

.454

.482

.369

.752

.752

Ripple

Average

22.8

22.7

17.2

163.

25.7

21.6

22.2

22.3

21.7

17.5

20.5

18.8

19.4

21.7

18.3

23.7

25.0

17.8

175.

18.6

18.4

16.2

16.0

19.2

15.8

19.0

16.5

18.7

23.9

25.6

27.5

21.3

127.

27.4

25.5

21.6

26.7

26.5

21.6

399.

14.7

13.4

14 3.

577.

14.9

15.9

15.7

183.

18.7

94 4

97 4

13.7

14.3

index

Coefficient

of

variation

0.857

.815

.764

.786

.678

.733

.575

.870

.772

1.05

.722

.916

.590

.979

.513

.739

1.23

.821

.674

.429

.671

.826

.584

.770

.627

.897

.668

.536

.704

.727

.868

.577

.929

.735

.783

.647

.576

.559

.549

.426

.648

579

.679

.802

.545

.422

.583

.574

.642

.836

58Q

1.08

.697

8 and 9. Even though the maximum cross correlations 

are about the same, it is assumed that the distributions 

of variance over the wave-number components should 

be appreciably different for the two series of runs because

the waves change shape somewhat as they shift downstream. In other words, the general shapes of the cross 

correlograms are not the same even though their maximums are equal. Plots of the coherence function, y2 ,
F34 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

Runs 8 and 9

-0.6 

0.4 

0.21

0 

-0.2

i_____I

Runs 8 and 10

Runs 8 and 11 -0.6 

0.4

\

Runs 8 and 12

-0.8 

0.6 

0.4 

0.2

 -0.4

Runs 8 and 13

-0.4 

0.2

-0.2

-0.4

Runs 8 and 15

-0.6

-40 -30 -20 -10 10 20 30 40

FIGURE 38. Cross correlograms for run 8 with runs 9 through 15.

and the gain function, A, in figure 40 support this assumption. The general decrease in both coherence and gain 

from the cross-spectral analysis of runs 8 and 15, as 

opposed to the values for runs 8 and 9, shows that the

wave-number components of the two records 8 and 15 

are virtually uncorrelated.

The cross correlation and cross-spectral analysis also 

proved useful in investigating the three-dimensional 

properties of the dunes. Run 17 was a profile y(x) taken 

down the centerline of the 8-ft flume. Runs 16 and 18 

were profiles taken 2 feet on either side of the centerline. 

The correlograms and the spectra for the individual 

profiles (fig. 41) do not suggest any appreciable differences in the profiles. A major part of the variance in all 

profiles is contributed by the shorter wave-number components corresponding to wavelengths generally greater 

than 4 feet.

The cross correlogram for runs 16 and 17 (fig. 42) shows 

that the two profiles, taken 2 feet apart along the same 

reach of flume, are practically uncorrelated, indicating 

that the sand waves certainly are not long crested. However, the cross correlogram for runs 17 and 18 (fig. 42)

i.o,

o:

O 0.5

0 10 20 30 40

TIME, IN MINUTES

B

FIGURE 39. A. Distance from the origin of the maximum cross correlation as a function of time. B, Change in maximum correlation with 

time.
STATISTICAL PROPERTIES OF DUNE PROFILES F35

0.4

1 2 30 1 2 

WAVE NUMBER ( e ), IN CYCLES PER FOOT

FIGURE 40. Coherence, y2 . and gain functions, A. for (left) runs 8 and 9. (right) runs 9 and 15.

1.0

0.5

-0.5

0 10 20 30 40 50 60

H

Z 20

15

</> ^

LJ "?

" 10

10 20 30 40 50 60 0 10 20 30 40 5U 60 

LAG

0 0.5 1.0 0 0.5 1.0 0 0.5 1.0

ABC

DIMENSIONLESS WAVE NUMBER ( / max) 

FIGURE 41. Correlograms and spectra of \ = v(.v). A. Run 16. 2 feet left of centerline. B. Run 17. centerline. C, Run 18. 2 feet

right of centerline.

shows a somewhat better correlation, suggesting that the 

dunes along the centerline of the flume extend to or at 

least influence the dunes along the right side of the flume. 

The coherence diagrams for the spectral analyses of the 

two sets of data (fig. 43) show the same effect; values of 

the coherence function y2 for runs 17 and 18 are approximately twice those for runs 16 and 17.

PREDICTION

In previous sections, the problem of predicting the 

properties of the bed profile from only the characteristics 

of the flow and the sediment was considered briefly in

terms of the Markov model and the dimensionless spectra. 

Neither of these approaches was found to be completely 

satisfactory. At this point, some additional aspects of 

predicting the properties of the bed profiles will be 

considered.

Under the assumption that the bed profiles can be approximated by a random Gaussian process, only three 

factors are needed to predict the mean values of durations 

between zero and h-\evel crossings and the distribution of 

maximum and minimum y values between zero crossings. 

These factors are the variance of the bed profile, erf,, the 

second derivative of the covariance function at the origin,
F36 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

--0.2

LAG

FIGURE 42. Cross correlograms.

DIMENSIONLESS WAVE NUMBER (e/Cmax) 

Fl(;URE 43. Coherence diagrams.

</>"(0), and the parameter S2. The parameter </>"(0) 

enters only in calculations of the mean duration between 

zero crossings, so that it could be replaced by E{lo}.

Values of S2 approached unity, and therefore as a first approximation it can be considered equal to one, and the 

prediction problem reduces to determining cry and </>"(0)

or£{/o}.

Definition of the dimensionless spectra in terms of flow 

parameters would permit determination of </>"(0) and 

£"{/<)}, and this approach is favored by the writer. Some 

simple alternatives are to relate /o directly to flow parameters or to find a relation between /0 and the standard deviation of the bed profile; such a relation would be analogous 

to the ripple index commonly used in geologic literature. 

In either event, it would still be necessary to determine cry , 

because the dimensionless spectra are based on the 

standardized data.

As we might expect, the standard deviation of the bed 

elevation, cry , relates roughly to flow depth (fig. 444) or to 

the unit water discharge (fig. 445). Data for these figures 

are summarized in table 7 and include average values 

from this study and values reported by Nordin and Algert 

(1966). It was not possible to reduce the scatter in these 

relations by consideration of other hydraulic or fluid variables. The point falling farthest to the left in the figures 

may be unrepresentative, because the bed profiles (runs 

52 and 53) were collected after cessation of sediment 

motion and probably represent residual features from 

higher velocity flow. (See Rathbun and Guy, 1967, for 

details of this experiment.)

All things considered, the scatter of the points in figure 

44 is neither unexpected nor discouraging. The data represent observations from five different flumes and two 

field channels. Two of the flumes used a sand feed system 

and three were of the recirculating type. In all the experiments except those performed by Algert (1965) in the 

0.4-foot flume, information on the bed configurations was 

a consideration peripheral to the main objectives of the 

study. A number of unknown factors may contribute to 

the scatter in the data, such as flume entrance and exit 

characteristics, operating procedures in the experiment, 

imposed fine-sediment load (in the field case), and particlesize distribution of the bed material.

The one other property of the bed profiles that is of 

most interest is the probability distribution of the durations of upward excursions of the process y(t) above the 

fixed level h, which is the conditional probability distribution of the rest period of a particle, given that it is deposited on the downstream face of a dune at the level h. 

The gamma distribution with parameters that relate to h, 

as shown in figures 12 and 13, is an attractive possibility. 

However, before this approach can be recommended, it is
STATISTICAL PROPERTIES OF DUNE PROFILES F37

i.o

0.1

0.01

0.1 1.0

MEAN DEPTH (D), IN FEET 

A

10.0 0.1 1.0 10.0 

UNIT DISCHARGE (9), IN CUBIC FEET PER SECOND PER FOOT

B

FIGURE 44. A. Relation of standard deviation of bed profiles, tr,,, to mean flow depth, D. B, Relation of standard deviation of bed profile. tr;/ . to unit

water discharge. Q.

necessary to verify both the gamma distribution and the 

relation of figure 12 with additional observations of the 

process y = y(t}.

In summary, the problem of predicting the properties of 

interest of the bed profiles is still not solved, but the results of this study have suggested several promising approaches. Some crude empirical relations, such as those 

of figures 12, 13, 26, 37 and 44, permit approximations to 

to be made of most of the properties of interest, but the 

need for both refined theoretical models and additional 

experimental data is obvious.

SUMMARY AND CONCLUSIONS

DISCUSSION OF RESULTS

The application of statistical techniques to analyze the 

properties of sand waves is a relatively recent undertaking, and therefore it is perhaps difficult to evaluate 

objectively some of the ramifications of this investigation. Nonetheless, three implications of the results seem 

to the writer to be of particular importance and to merit 

further discussion.

TABLE 1. Average values of observed variables

Channel

Atrisco..................

Atrisco..................

Atrisco..................

8-ft flume..............

8-ft flume..............

2-ft flume..............

2-ft flume..............

2-ft flume..............

0.67-ft flume...........

Bernardo*..............

Bernardo*..............

8-ft flume*.............

8-ft flume*.............

0.4-ft flume*...........

0.4-ft flume*...........

0.4-ft flume*...........

V

(fps)

2.16

2.11

2.08

2.09

2.01

1.62

1.10

1.07

.80

3.62

2.48

2.11

1.91

1.81

1.78

1.74

D

(ft

2.20

2.30

2.29

2.80

2.36

.521

.518

.522

.197

2.60

4.15

1.05

.670

.580

.485

.400

w

(ft)

55

55

55

8

8

2

2

2

.67

70

70

8

8

.4

.4

.4

s

0.00057

.00055

.00058

.00063

.00056

.00212

.00088

.00088

.00148

.00058

.00058

.00134

.00136

.0044

.0038

.0037

T

ro

19

20

20

20

24

20

20

20

20

8.34

8.34

16.7

17.8

18.4

15.6

17.8

d-M

(mm)

0.23

.23

.23

.24

.24

.25

.35

.35

.30

.23

.23

.28

.28

.34

.34

.34

<TU

(ft)

0.259

.276

.270

.169

.127

.0700

.0432

.0390

.0404

.647 .

.745 .

.175 .

.115 .

.0478 .

.0330 .

.0270

AvS a +

(ft)

0.277

.287

.302

.214

.146

.0806

.0416

.0408

.0400

/(,

(ft)

4.48

4.61

3.90

2.95

2.28

1.42

.58

.54

.36

Remarks

Dunes.

Do.

Do.

Dunes; recirculating

Do.

Do.

Ripples; recirculatin

Do.

Ripples; sand feed s Dunes.

Do.

Dunes; recirculating

Do.

Sand feed system.

Do.

Do.

system.

g system.

system.

system.

*Data are from Nordin and Albert (1966).
F38 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

The first important conclusion drawn from these investigations is that the properties of the profiles obtained 

in the laboratory flumes are all very similar, regardless 

of the size of the flume or of whether the bed configurations are ripples or dunes, provided that scale effects 

are properly taken into consideration. The scale effects 

seem to be completely accounted for by standardizing 

the raw data to zero mean and unit variance and by expressing the length of h -level crossings as a ratio of the 

mean length between zero crossings, or by forming dimensionless parameters in terms of the flow properties, as 

in the case of the spectra. This means that certain simple 

properties, such as the average values of the conditional 

probability-density functions of rest periods, can be 

modeled in a ripple bed configuration or in a very small 

flow system, and the results can be extrapolated to dune 

configurations of much larger flow systems.

Similarity of the properties of the ripple and dune 

profiles does not imply that there are no differences in 

these features but only implies that their general shape 

and method of movement are similar, as noted by Taylor 

and Brooks (1961). In fact, the comparisons of distributions of variance over the wave-number components 

shown in figure 34 support very strongly the notion that 

there are appreciable differences. The major differences 

between ripples and dunes are described in detail by 

Simons and Richardson (1966) and need not be considered here.

In addition, there were important differences between 

the flume and field data for dune bed configurations. The 

longer records from Atrisco Lateral reflected the influence 

of a meandering thalweg and of accompanying large 

alternate bars that generally were not present in the 

flumes. Simons and Richardson (1966) have noted that 

alternate bars do form in flumes, but for the flume data 

analyzed here, the width-to-depth ratio was not great 

enough to permit the bars to develop significantly.

A second important implication arises in the developments leading to figures 30 through 33 of the dimensionless spectra. These figures show that the spectral 

representations of the processes can be determined as 

unique functions of the flow, fluid, and sediment properties. It was fortunate for these studies that the water 

temperatures and sediment sizes varied over rather 

narrow ranges, because this permitted expressing the 

dimensionless spectra in extremely simple form. On the 

other hand, because of this limited range of conditions, 

it was not possible to define adequately the shapes of the 

spectra. Nevertheless, the results are encouraging, so 

far as they were carried, and they point the way to future 

experimental studies that will permit a better definition 

of the spectra.

However, the most important result by far is the demonstration in figures 6-9 and 15-17 that the properties of 

the profiles can be estimated by theoretical considerations of fairly simple models. Certainly, the assumption 

that the processes y(x) and y(t) are Gaussian is a crude 

approximation, but the results obtained using this model 

are reasonable and consistent.

In one important aspect, this study was unsuccessful. 

It was not possible to relate uniquely the statistical 

properties of the bed profiles to the characteristics of 

the flow. However, some simple empirical relations were 

considered that will permit predicting approximately 

many of the properties of interest. The most important 

relations in this regard are shown in figure 44, which 

gives ay as a function of either depth or unit water discharge, and figure 25, from Nordin and Algert (1966, p. 

109), which shows values of the correlation function at 

zero, one, and two lag intervals as functions of unit water 

discharge. These values, along with the length of profile 

record, permit calculations of most of the factors considered in the zero and /i-level crossing analysis and the 

approximations of //J7/<j, as given by equations 15 and 16. 

However, the relations of figure 25 apply only if the lag 

interval is selected so that it bears a constant ratio to 

the standard deviation, ay.

CONCLUSIONS

In this study statistical properties of streambed profiles from four different size channels are compared by 

the techniques of time-series analyses and by a consideration of the mean values and distributions of zero and hlevel crossings, the durations of upward excursions of 

the records y(x) and y(t) above a fixed level h, and the 

distribution of maximum y values between successive 

zero crossings. The principal conclusions drawn from the 

study are the following:

1. No appreciable differences in the statistical properties 

of the profiles from the flumes could be attributed 

to flume size or to whether the bed forms were 

ripples or dunes, provided that the raw data were 

standardized to zero mean and unit variance and 

that the length scales were expressed as ratios of 

the mean length between zero crossings. In addition, 

only minor differences were noted between the 

properties of the longitudinal profiles, y = y(x), 

and the properties of the profiles, y = y(t), obtained 

by sounding continuously in time at a fixed point. 

Longer records of both types, particularly the time 

records, are required to establish if the differences 

are real or apparent.

2. Spectra of the longer records from Atrisco Lateral 

appeared to reflect the influences of a meandering
STATISTICAL PROPERTIES OF DUNE PROFILES F39

thalweg and large alternate bars that were not 

present in the flume data.

3. By dimensional analyses, forms of dimensionless 

spectra were derived which describe reasonably 

well the observed forms in the higher frequency 

and wave number regions. From the equations of 

the dimensionless spectra, it was established that, 

for constant flow conditions, the celerities of the 

individual wavelength components vary directly 

as the square root of the wave numbers.

4. From the mean spectral moments, it was shown that 

the mean wave celerity varied inversely as the mean 

wave number for the different flow conditions considered. This relation, together with the relation 

for celerity of wave-number components under constant flow conditions, permits determining the properties of the process, y(t}, from the properties of the 

longitudinal profiles, y(x).

5. The techniques of cross-spectral analysis were found 

useful for defining the mean rate of shifting of 

ripples and dunes and investigating their threedimensional properties.

6. Values of the bed elevation follow an approximate 

Gaussian distribution. For a Gaussian process of 

known covariance function, the expected number 

of zero or /i-level crossings, the expected length 

between crossings, and the mean duration of upward 

excursions of the process y(t) above the fixed level h 

can be computed. The comparison of observed and 

computed values shows good agreement for positive 

values of h and indicates systematic deviations for 

values of h below the mean bed elevation. The mean 

duration of upward excurions of the process y(t) is 

the mean rest period of a particle at the level h.

7. The distributions of the distances between zero crossings, /o, are approximately exponential. Values of 

the durations of upward excursions of y above the 

level h follow a gamma distribution with parameters 

that relate to /i, as shown in figures 10 through 13.

8. The distributions of maximum values of y between 

zero crossings represent the distributions of scour 

and fill associated with the formation and migration 

of sand waves. The sand waves are not symmetric 

about the mean bed elevation. The positive maximums, a+, are distributed exponentially and the 

negative maximums, a , are distributed according 

to a Gaussian distribution. Both distributions are 

functions of the standard deviation of the bed 

elevation, o^, which relates approximately to mean 

flow depth or unit water discharge (fig. 44).

9. Finally, the results of this study show that some of the 

distributions entering the two-dimensional model of 

sediment transport (Sayre and Conover, 1967) and

most properties of the dune profiles that are of interest can be determined from theoretical considerations of fairly simple models.

RECOMMENDATIONS FOR FUTURE STUDIES

The results of this study suggest a number of areas for 

future investigations. Four, in particular, appear especially 

promising. These are:

1. Studies should be undertaken to include fluid and sediment properties in the dimensionless spectra and to 

determine more accurately the dimensionless wave 

numbers at which the peak spectral ordinates occur.

2. An investigation should be made of the possibility of 

incorporating the distribution of wave height and 

wavelength and the relation of wave celerity to wavelength components (fig. 37) into a continuity-type 

bedload transport relation based on size and rate of 

shifting of the sand waves.

3. Experiments should be designed to obtain longer records of y = y(t) so that the probability distributions 

of particle rest periods (the durations of upward excursions) can be adequately defined.

4. Models other than the simple Gaussian model should 

be investigated as possible representations of the 

processes y(x) and y(t) to see if more accurate predictions of the properties of the bed profiles can be 

obtained theoretically. In addition, it should be possible to investigate some models by simulation techniques. One such possibility would be a process of 

the form y(t) =p cos (Ay + 6), where p, X, and 0 are 

all random variables of some specified distribution. 

In addition to the above studies, it is necessary to obtain reliable field records of bed profiles to permit extending these studies to a greater range of flow conditions. 

Records of the process y = y(t) are especially lacking, 

and many existing longitudinal profiles suffer from inadequate horizontal control.

As noted in the section "Data and analysis," reliable 

relations between the statistical properties of the profiles and the characteristics of the flow still remain to be 

developed. Empirical relations exist, but they are far 

from satisfactory for predictive purposes.

Finally, detailed descriptions of bed profiles are of 

value only if they lead ultimately to a better understanding of sediment transport processes. It is hoped that the 

results of this study and of future studies along the lines 

outlined above will contribute to such an understanding.

REFERENCES

Algert, J. H., 1965, A statistical study of bed forms in alluvial channels: 

Colorado State Univ. Kept. CER65JHA26, 31 p., Fort Collins, Colo.

Ashida, Kazuo, and Tanaka, S., 1967, A statistical study of sand waves: 

Internal. Assoc. Hydraulic Research Con<j., 12th, Fort Collins, 

Colo., Sept. 11-14, 1967, Proc., v. 2, p. 103-110.
F40 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

Bendat, J. S., and Piersol, A. G., 1966, Measurement and analysis of 

random data: New York, John Wiley & Sons, Inc., 390 p.

Blackman, R. B., and Tukey, J. W., 1958, The measurement of power 

spectra: New York, Dover, 190 p.

Carlwright, D. E., and Longuet-Higgins, M. S., 1956, The statistical 

distribution of the maxima of a random function: Royal Soc. 

[London] Proc., ser. A. v. 237, p. 212-232.

Colby, B. R., 1964, Discharge of sands and mean-velocity relationships 

in sand-bed streams: U.S. Geol. Survey Prof. Paper 462 A, 47 p.

Cramer, Harald, 1964, Model building with the aid of stochastic processes: Technometrics, v. 6, no. 2, p. 133-159.

Cramer, Harald and Leadbetter, M. R., 1967, Stationary and related 

stochastic processes: New York, John Wiley & Sons, Inc., 348 p.

Engelund, F. A., 1969, On the possibility of formulating a universal 

spectrum for dunes: Basic Research Prog. Rept. 18, p. 1-4, Hydraulic Laboratory, Technical University of Denmark, Copenhagen.

Granger, C. W. J., and Hatanaka, Michio, 1964, Spectral analysis of 

economic time series: Princeton, N.J., Princeton Univ. Press, 299 p.

Hino, Mikio, 1968, Equilibrium-range spectra of sand waves formed by 

flowing water: Jour. Fluid Mech., v. 34, pt. 3, p. 565-573.

Karaki, S. S., Gray, E. E., and Collins, Jack, 1961, Dual channel stream 

monitor: Am. Soc. Civil Engineers Proc., v. 87, no. HY6, p. 1-16.

Longuet-Higgins, M. S., 1958, On the intervals between successive 

zeros of a random function; Royal Soc. [London] Proc., ser. A, v. 

246, p. 99-118.

 1962, The distribution of intervals between zeros of a stationary 

random function: Royal Soc. [London] Philos. Trans., ser. A, v. 254, 

p. 557-599.

 1963, Bounding approximations to the distribution of intervals

between zeros of a stationary Gaussian process, in Rosenblatt, 

Murray, ed.. Time series analysis: New York, John Wiley & Sons, 

p. 63-88.

Nordin, C. F., and Algerl, J. H., 1966, Spectral analysis of sand waves: 

Am. Soc. Civil Engineers Proc., v. 92, no. HY5, p. 95-114.

Nordin, C. F., Simons, D. B.. and Richardson, E. V.. 1965, Interpreting 

depositional environments of sedimentary structures: Colorado 

State Univ. Civil Eng. Rept. CER65CFN-DBS-EVR15, 17 P., 

Fort Collins, Colo.

O'Loughlin, E. M., and Squarer, David, 1967, Areal variations of bedform characteristics in meandering streams: Internal. Assoc. 

Hydraulic Research Cong., 12th, Fort Collins. Colo., Sept. 11-14, 

1967, Proc., v. 2, p. 118-127.

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C. F., Nordin, Jr. and J. H. Algert: Am. Soc. Civil Engineers Proc. 

v. 93, no. HY4, p. 310-316.

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sediment transport variables in a small recirculating flume: Water 

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PLANNING OF DATA REQUIREMENTS

This appendix gives some approximate guidelines for planning the 

lenglh of records required for the various analyses considered in the 

texl. The slalistical basis for determining record lenglh and ihe compulational procedures are given by Bendat and Piersol' and will nol be repealed here. The compuler programs used in this study are available 

from ihe wriler or from Mrs. Lois Niemann, Civil Engineering Department, Colorado Stale Universily.

The compulalional procedures used in this study can be applied lo 

any conlinuous record of slreambed elevalion, provided lhal ihe record 

meels ihe necessary condilions for stationarity. From a practical point of 

view, this means lhal, for longiludinal profiles, ihe channel cross seclion 

should nol vary appreciably along ihe channel and that, for either longitudinal or lime records, ihe flow and sedimenl Iransporl rales should be 

approximalely conslanl during ihe period of observalion.

We consider here only flow condilions which occur where welldefined sand waves are know to be presenl al ihe slreambed, lhal is, 

only lower regime flow. The bed configuralions for lower regime flow are 

ripples, dunes, bars, and combinalions of ihese features (Simons and 

Richardson -). In the following discussion the computalional procedures for speclral analysis of a single record firsl are oullined. Next are 

given some rough guidelines for delermining ihe lenglh of record and 

ihe spacing of observalions for converting a conlinuous record lo discrele dala poinls for speclral analysis. Finally, ihe record lenglh and 

spacing of dala points for olher types of analysis, such as estimating the 

mean lengths between zero and //-level crossings, are considered.

COMPUTATIONAL PROCEDURES

A continuous record of bed profile, y = y(x), of lenglh L, is converted 

lo discrele dala poinls by sampling ihe conlinuous records al intervals 

AJC = /J ft, so lhal ihe sampling rale is l/h samples per ft. The enlire 

record is converted lo N discrete data poinls, y,, i = l, 2, . . ., /V. We 

consider ihe enlire sequence of y\ values lo have zero mean and unit 

variance. The covariance function corresponding lo equalion 2 of ihe 

lexl is computed by the formula

1

(D

for 5 = 0, 1, m, where m is ihe maximum number of lags.

1 Bendat, J. S., and Piersol, A. (.,.. 1966, Measurement and analysis of random data: New York, 

John Wiley & Sons, Inc., p. 278-320.

- Simons, D. B., and Richardson, E. V., 1966, Resistance to flow in alluvial channels: I'.S. 

(,eol. Survey Prof. Paper 422-J, 61 p.
STATISTICAL PROPERTIES OF DUNE PROFILES F41

Next, the finite cosine series transform function of the autocovariances is computed from

(2)

for s = 0, 1, 2, . m. In the above,

i = l, 2, m 1, and (3)

The spectrum is computed from the equations

G(0)=0.5C(0)+0.5C(1) 

G(»)=0.25G(»-l)+0.5C(i)+0.25G(i + l),

C(m)=0.5G(m-l)+0.5G(m).

== ^ 2 , -l (4)

These computations yield m+ 1 values for the spectrum at each of the 

lags s = 0, 1, . . ., m, corresponding to the wave numbers ex = s/2/raA 

or to the wavelengths L-> = 2mhls.

LENGTH OF RECORD AND SPACING OF DATA POINTS

The computational procedures outlined above show that, in selecting 

a length of continuous record for spectral analysis by digital techniques, 

three quantities must be considered: the record length, Lr, the sampling 

(digitizing) rate, h, and the maximum lag, m. The smallest wavelength 

for which spectral estimates are computed is 2h and the largest wavelength for which the spectral estimates are computed is 2mh.

As a general rule, we should select a sampling rate such that 2h =£ Lm\n, 

where Lmin is the smallest wavelength component of interest, and a 

maximum lag m such that 2mh Lmax, where Lmax is the largest wavelength component of interest. The record length, Lr, should be no less 

than lOmh if the percentage error of the estimated spectrum is to be of 

reasonable size (Parzen 3).

For a given bed form, such as dunes, quite satisfactory results have 

been obtained by selecting mh approximately equal to the mean dune 

length, by sampling at an interval h so that the average dune length is 

represented by 20 to 30 data points, and by selecting a length of record 

10 to 20 times the mean dune length. If ripples are superposed on the 

backs of dunes, a sampling interval h of about one-fourth of the mean

"I'arzen, Emanuel, 1967, Time series analysis papers: San Francisco, Holden-Day, 565 p.

ripple length is recommended. Generally, the contribution to the total 

variance of wave-number components greater than four cycles per foot 

was negligible in the case of ripples. For dunes, the contribution to 

the variance from components greater than one cycle per foot was 

negligible.

Where large alternate bars with smaller bed forms superposed are 

known to exist in a channel, a somewhat different procedure is called 

for. The record should be sufficiently long to cover ten or a dozen of the 

bars if the properties of the bars are of interest. However, to investigate 

the properties of the smaller features, it is suggested that short segments of the longer record be analyzed after trend removal. Methods for 

removing trends are given in Bendat and Piersol 4 and Parzen. 5

RECORDS FOR ZERO AND /i-LEVEL CROSSING ANALYSIS

It has been shown by Tick and Shaman 6 that the expected number of 

maximums and minimums of y and the expected number of zero and 

A-level crossings determined from the discrete approximation to a continuous Gaussian process in a given length of record are always less than 

the expected number determined from the continuous record. If the 

sampling interval h is selected such that 4A =£ Lmin, the estimated number of crossings from the discrete process will be at least 90 percent of 

the number in the continuous process for any level h within two standard 

deviations of the mean. Therefore, a sampling rate at least twice that 

recommended for spectral analysis is required for determining the 

average number of zero or A-level crossings.

Much longer records are required to determine the probability distributions of maximum and minimum y values between zero crossings, 

of the lengths between zero crossings, and of the durations of the positive excursions of y above the level h than are required for the spectral 

analysis. The writer recommends at least one hundred observations of 

these values to obtain reasonable estimates of the distributions. As a 

rule of thumb, the records should be ten times as long as the records for 

spectral analysis, and if consideration is to be given to A-level crossings 

at levels beyond one and one-half times the standard deviation from the 

mean, for ripples or dunes, the record should be of the order of 300 

times the mean ripple or dune length. Because the ripples and dunes are 

not symmetric about the mean bed elevation and because the y values 

are not strictly Gaussian in distribution, no hard-and-fast rules can be 

established for the record lengths to determine the distributions of the 

A-level crossings. The probability distributions of maximum and minimum y values between zero crossings, as shown in figures 15 17 of the 

text, are the logical starting place to estimate record length requirements for a given number of observations of these extreme events.

4 Bendat, op. rit., 390 p. 

"' Parzen, loc. cit.

"Tick, I.. J., and Shaman, Paul, 1966, Sampling rates and appearance of stationary Gaussian 

processes: Technometrics, v. 8, no. 1, p. 91 106.

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