Stochastic Analysis of 

Particle Movement 

over a Dune Bed

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1040

Stochastic Analysis of 

Particle Movement 

over a Dune Bed

By BAUM K. LEE and HARVEY E. JOBSON

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1040

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1977
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY

V. E. McKelvey, Director

Library of Congress Cataloging in Publication Data

Lee, Baum K.

Stochastic analysis of particle movement over a dune bed.

(Geological Survey Professional Paper 1040)

Bibliography: p. 41

1. Sediment transport Mathematical models. 2. Stochastic processes. 3. Sand-dunes.

I. Jobson, Harvey E., joint author. II. Title. III. Series:

United States Geological Survey Professional Paper 1040

GB562.L43 55l.3'7'5 77-16278

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

Washington, D.C. 20402 

Stock Number 024-001-03024-1
CONTENTS

Abstract ................................................ 1

Introduction ............................................ 1

Acknowledgment ........................................ 2

Background ............................................. 2

Theoretical models .................................. 2

Experimental studies ................................ 4

Remarks ............................................ 5

Development of theory ................................... 5

Characteristics of particle movement over a dune bed .. 5 

Estimation of the probability distributions of the elevations of deposition and erosion ...................... 6

Estimation of the probability distributions of the rest 

periods ........................................... 8

Estimation of the probability distributions of the step 

lengths ........................................... 10

Bed-material transport equations ..................... 14

Page

Development of theory Continued

General two-dimensional stochastic model for dispersion 

of bed-material sediment particles .................. 18

Analysis and discussion of results ......................... 22

Experiment and basic data ........................... 22

Probability distributions of the elevations of deposition

and erosion ....................................... 23

Rest period distributions ............................. 25

Step length distributions ............................. 30

Bed-material transport............................... 36

Variation of various statistics with flow conditions and

a relation between the step length and the rest period 39 

Two-dimensional stochastic model for dispersion of bedmaterial sediment particles ........................ 39

Summary and conclusions ................................ 40

References cited ......................................... 41

ILLUSTRATIONS

Page 

FIGURES 1-20. Graphs showing:

1. Typical yx(t) record illustrating the class marks for deposition and erosion ................................. 7

2. Typical yx(t) record illustrating the conditional rest periods of a particle ................................... 9

3. Statistic, {xt j >k } for the step length of a particle ........................................................ 11

4. Method for estimating the percentage of volume occupied by dunes between elevations i? -and TJJ+I ............ 16

5. Size distribution curve of bed material ................................................................ 22

6. Frequency histograms, triangular density function, and truncated Gaussian density function for the elevation

of deposition and erosion ........................................................................... 24

7 -9. Sample probability mass functions of the conditional rest periods with fitted two-parameter gamma functions for:

7. Run 4A ............................................................................................. 26

8. Run 16 .............................................................................................. 27

9. Run 17 .............................................................................................. 28

10. Variation of the conditional mean and variance of rest periods with bed elevation ............................... 29

11. Frequency histograms for the marginal rest period and exponential fits ......................................... 31

12 -14. Sample probability mass functions of the conditional step lengths given the elevation of erosion is 0.0 with Gamma fits 

for:

12. Run 4A ............................................................................................. 32

13. Run 16 .............................................................................................. 33

14. Run 17 .............................................................................................. 34

15. Variation of the conditional mean and variance of step lengths with bed elevation ............................... 35

16. Frequency histograms for the marginal step length with Gamma fits ........................................... 37

17. Effective volume ratio as a function of bed elevation .......................................................... 39

18. Variation of bed-load discharge with bed elevation ............................................................ 39

19. Mean transport speed of a bed-load particle as a function of bed elevation ...................................... 40

20. Ratio of the conditional variance of step lengths to the conditional variance of rest periods as a function of bed elevation ..................................................................................................... 40

in
IV CONTENTS

TABLES

[Tables 1 and 73 -75 are in text; all others follow references cited]

Page

TABLE 1. Basic data and computed parameters ......................................................................... 23

2. Sample probability mass functions of elevations of deposition and erosion ....................................... 44

3 -5. Sample conditional probability mass function of rest periods, p^ yD(£a\ yp for:

3. Run 4A ............................................................................................. 45

4. Run 16 .............................................................................................. 46

5. Run 17 .............................................................................................. 47

6. Variation of conditional mean and variance of rest periods with elevation of deposition; £[T\YD = y] and Var

[ T\ YD = y] ............................................................................................... 48

7. Estimates of parameters and the results of goodness of fit test for the conditional rest periods (two-parameter gamma)

.......................................................................................................... 48

8 -10. Sample joint probability mass function of rest periods and elevation of deposition, pT YD(ta ,yi) f°r:

8. Run 4A ..............................................................'............................... 49

9. Run 16 .............................................................................................. 50

10. Run 17 .............................................................................................. 51

1 1 -24. Sample conditional probability mass function of step lengths, px\ YF Yn(xp\ ^ty) (run 4A) ......................... 52-55

25 -42. Sample conditional probability mass function of step lengths, px\ YF Yn(xp\ yi>y^ (run 16) .......................... 55-59

43-56. Sample conditional probabilitymass function of step lengths, p_y\yy(aj3\>'p>'j) (run 17) .......................... 60-63

57-59. Conditional means and variances of step lengths; £[X\ YE = yit YD = yj] and

57. Run 4A ............................................................................................. 64

58. Run 16 .............................................................................................. 65

59. Run 17 .............................................................................................. 66

60 -62. Estimates of parameters describing two-parameter gamma distribution for conditional step lengths for:

60. Run 4A ............................................................................................. 67

61. Run 16 .............................................................................................. 68

62. Run 17 .............................................................................................. 69

63 -65. Results of goodness of fit test for conditional step lengths for:

63. Run 4A ............................................................................................. 70

64. Run 16 .............................................................................................. 70

65. Run 17 .............................................................................................. 70

66 -68. Sample conditional probability mass function of step lengths, p^ Y (XQ \ yj) for:

66. Run 4A ............................................. D. . ............................................. 71

67. Run 16 .............................................................................................. 71

68. Run 17 .............................................................................................. 71

69. Variation of conditional mean and variance of step lengths with elevation of deposition; £ [X\ YD = y] and Var

[X\ Y? = y] .............................................................................................. 71

70 -72. Sample joint probability mass function of step lengths and elevation of deposition, p% yD(*0 ,y,) for:

70. Run 4A ............................................................................................. 72

71. Run 16 .............................................................................................. 72

72. Run 17 .............................................................................................. 72

73. Variation of various statistics with stream power ............................................................. 38

74. Comparison of the effective volume ratios at elevation y^; £j, from yt(x) record and £j, from yx (t) record ............... 38

75. Comparison of measured and computed total bed-material transport rates ...................................... 38

METRIC-ENGLISH EQUIVALENTS

Metric unit English equivalent

Meter (m) = 3.28 feet

Centimeter (cm) = 3.28 x 10"2 foot

Tonne per day per meter (t/day «m) = .336 ton per day per foot

Dyne per square centimeter (d/cm2 ) = 2.09 x 10~3 pound per square foot

Dyne per centimeter per second (d/cm -s) = 6.85 x 10~5 pound per foot per second
CONTENTS

SYMBOLS

A,B 

C,D

EH,E

F

f('),F(>

(n) A-)

8 

h

*J,J

N(t)

Constants in estimating the conditional mean of the 

rest periods

Constants in estimating the conditional variance of 

the rest periods

Dimensionless Chezy discharge coefficient

Concentration of total bed-material discharge

Depth of flow

Median sieve diameter of bed material

Sieve diameter of bed material for which 84 percent 

is smaller

Sieve diameter of bed material for which 16 percent 

is smaller

Event that a particle eroded from elevation ^passes 

v dune crests before it is deposited at elevation ^

Event that a particle passes v dune crests before it is 

deposited

Mathematical expectation and its estimate, respectively

Froude number of the flow

Probability density and distribution functions, 

respectively

n-fold convolution of the probability density, f{ )

Gravitation acceleration

Average depth of the zone in which bed-material 

movement occurs determined from the yt(x) 

record

Scale parameter of the conditional step length distribution

Scale parameter of the conditional rest period distribution

Length of the yx(f) record

Length of the yt (x) record

Total number of bed forms contained in the yt(x) 

record for which the upstream side intersects the 

elevation yt and the downstream side intersects 

the elevation y^

Total number of possibilities of the event E^v contained in the yt (x) record

Total number of bed forms contained in the yx(t) 

record and which also contain some erosion in the 

class interval associated with the elevation yi

Total number of bed forms contained in the yx(t) 

record and which also contain some deposition in 

the class interval associated with the elevation ^

Total number of bed forms contained in the yx(t) 

record and which also contain both an up-crossing 

and a down-crossing at the elevation j

Number of steps taken by a particle or number of 

class intervals for the realization of YD and YE

Total number of particles per unit area deposited in 

time t

Total number of particles per unit area eroded in 

time t

Total number of particles per unit area deposited 

within the class interval (ra f ,ni+l] in time t

Total number of particles per unit area eroded within the class interval (ni ,ra,+1 ] in time t

Counting process describing number of steps taken 

by a particle in time t

P(-)

i\ Q '

y 

S.

VB(j)

VT ,V'T

VarN.VarH 

W

i,j,v,k

X

X(n)

X(t)

y,y'

Sample probability mass function

Probability

Estimate of the mean bed-load discharge per unit

width associated with the elevation yj 

Estimate of the mean total bed-material discharge

per unit width

Estimates of the mean bed-load discharge 

Estimates of the mean suspended-load discharge 

Estimates of the mean total bed-material discharge 

Water discharge

Number of class intervals for the realization of T 

Shape parameters for the conditional distribution of

the step lengths and the rest periods, respectively 

Number of class intervals for the realization of X 

Standard deviation of bed elevation 

Slope of energy-grade line 

Measure of time 

Statistic which measures the conditional rest

periods 

Class mark for the statistic, tj k

Random variable describing the rest periods of a

particle 

Random variable describing the duration of tth rest

period 

Stochastic process describing the sum of n rest

periods

Mean flow velocity 

Mean shear velocity

Estimates of the mean transport speed of a bed-load 

particle and a suspended-load particle, respectively

Estimate of the mean transport speed of a bed-load 

particle associated with the elevation y^

Estimates of the mean transport speed of a bedmaterial particle

Estimate of the mean transport speed of a bedmaterial particle associated with the elevation yj

Variance and its estimate, respectively

Width of channel

Measure of longitudinal distance

Average distance traveled by bed material in time t

Class mark for the conditional step lengths

Statistic which measures the conditional step 

lengths of a bed-load particle

Statistic which measures the conditional step 

lengths of a bed-material particle

Random variable describing the step lengths of a 

particle

Random variable describing the length of ith step of 

a particle

Stochastic process describing the longitudinal position of a particle after n steps

Stochastic process describing the longitudinal position of a particle at time t

Measure of vertical distance

Class marks for YE and YD
VI CONTENTS

>Vax> > 

yt(x)

yx(t)

YD , YE 

YD(n) 

y~ (£)

AyHighest and lowest elevations, respectively, at 

which particles are deposited or eroded

Estimates of ;ymax and ymin, respectively

Elevation of the bed, y, as a function of the 

longitudinal coordinates, x, at a given time, t

Elevation of the bed, y, as a function of time, t, at a 

fixed point, x

Random variables describing the elevation of particle deposition and erosion, respectively

Stochastic process describing the vertical position of 

a particle after n steps

Stochastic process describing the vertical position of 

a particle at time t

Class width associated with y:

Vertical rise of the bed in the class interval associated with yj for the kih deposition period of the

y (t) record

y, 

no

P

°J

T b

T a' T a+l

*c

fl

Vertical fall of the bed in the class interval associated with yt for the *th erosion period of the yx(t) 

record

Specific weight of bed material 

Gamma function 

Effective volume ratios

Lower and upper class limits for yj, respectively 

Bulk porosity of the bed material in place 

Lower and upper class limits for Xp, respectively 

Estimate of correlation coefficient 

Geometric standard deviation of particle size 

Mean bed shear stress

Lower and upper class limits for ta , respectively 

Critical value of chi-square statistic 

Number of particles per unit volume of the bed 

Number of particles per unit volume of the bed 

associated with ^
STOCHASTIC ANALYSIS OF PARTICLE 

MOVEMENT OVER A DUNE BED

By BAUM K. LEE and HARVEY E. JOBSON

ABSTRACT

Stochastic models are available that can be used to predict the 

transport and dispersion of bed-material sediment particles in an 

alluvial channel. These models are based on the proposition that the 

movement of a single bed-material sediment particle consists of a 

series of steps of random length separated by rest periods of random 

duration and, therefore, application of the models requires a 

knowledge of the probability distributions of the step lengths, the 

rest periods, the elevation of particle deposition, and the elevation of 

particle erosion. In the past, it has proven impossible to estimate 

these distributions except by use of tedious and time-consuming 

single particle experiments.

By considering a dune bed configuration which is composed of 

uniformly sized particles, the probability distributions of the rest 

period, the elevation of particle deposition, and the elevation of particle erosion are obtained from a record of the bed elevation at a fixed 

point as a continuous function of time. By restricting attention to a 

coarse sand, where the suspended load is negligible, the probability 

distribution of the step length is obtained from a series of "instantaneous" longitudinal bed profiles in addition to the above information. Using these probability distributions, three bed-material 

transport equations and a two-dimensional stochastic model for dispersion of bed-sediment particles are developed.

The procedure was tested by determining these distributions from 

bed profiles formed in a large laboratory flume with a coarse sand as 

the bed material. The elevation of particle deposition and the elevation of particle erosion can be considered to be identically distributed, and their distribution can be described by either a "truncated Gaussian" or a "triangular" density function. The conditional 

probability distribution of the rest period given the elevation of particle deposition closely followed the two-parameter gamma distribution. The conditional probability distribution of the step length given 

the elevation of particle erosion and the elevation of particle deposition also closely followed the two-parameter gamma density function. 

For a given flow, the scale and shape parameters describing the gamma probability distributions can be expressed as functions of bed 

elevation.

The bed-material transport equations were tested for three flow 

conditions. The errors in the predicted mean total bed-material 

transport rates were -3.0, +3.5, and 80.1 percent for equation 55, 

and -1.7, +26.9, and +64.1 percent for equation 63. For the run with 

the large error, the mean total load concentration was small 

(8.9 milligrams per liter), and flow conditions were somewhat out of 

equilibrium.

INTRODUCTION

The movement of sediment in alluvial streams is so 

complex a process that it may never be subjected completely to a deterministic solution. It represents, in fact, 

an extreme degree of unsteady, nonuniform flow, since 

the streambed as well as the water surface may be continuously changing with time and position.

Numerous formulas and equations have been 

developed to predict sediment transport rates. Most of 

these developments ignore the actual nature of sediment movement and have assumed that the sediment 

transport rate can be described by a deterministic function of certain flow parameters. Unfortunately, after 

decades of searching, no universally accepted sediment 

transport equation has been found. The theories of 

probability, statistics, and stochastic processes have 

been used to describe the kinematics of a single bedsediment particle in an alluvial channel flow and to predict the dispersion characteristics of a group of such 

particles. These theories have clearly demonstrated a 

great potential for development of stochastic models of 

sediment transport and dispersion.

Most of the stochastic models (Shen and Todorovic, 

1971; Grigg, 1969; Yang, 1968; Sayre and Conover, 

1967; Hubbell and Sayre, 1964; Crickmore and Lean, 

1962; Einstein, 1937) are based on the proposition that 

the movement of bed-sediment particles consists of a 

series of steps separated by rest periods, so that determination of the probability distributions for the step 

lengths and the rest periods of a bed-sediment particle 

plays the major role in quantifying the bed-sediment 

transport. While this movement concept can easily be 

verified through laboratory observations, Einstein 

(1937) was the first to use it. He developed a one-dimensional probabilistic model for bedload transport. More 

recently, Sayre and Conover (1967) derived a twodimensional stochastic model by introducing the prob-
STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

ability distribution of the elevation at which a bed-sediment particle is deposited.

The probability distributions of the step lengths and 

the rest periods of a bed-sediment particle have been 

estimated from single particle experiments (Grigg, 

1969) or by using a group of tracer particles (Yang, 

1968; Hubbell and Sayre, 1964; Crickmore and Lean, 

1962). Because of the considerable effort required to 

conduct such experiments, it seems clear that some way 

must be found to estimate the probability distributions 

from more readily accessible data if significant further 

progress is to be expected. To apply the Sayre-Conover 

(1967) two-dimensional stochastic model, the probability distribution of the elevation at which a bed-sediment particle is deposited must be known. A method for 

estimating this distribution is developed in this report.

The objectives of this study are:

1. To present a method of estimating the following 

probability distributions for dune-bed conditions using 

only sounding records of the bed elevation (a) probability distributions (note that there are two separate 

distributions) of the elevation at which a bed-sediment 

particle is eroded and deposited, and (b) conditional 

probability distributions of the step lengths of a bedsediment particle given the elevations at which the particle is eroded and deposited. A method estimating the 

conditional probability distribution of the rest periods 

of a bed-sediment particle given the elevation at which 

the particle is deposited has been presented by Sayre 

and Conover (1967).

2. To develop bed-material transport equations based 

on the above probability distributions and to compare 

the results with the experimentally measured values.

3. To derive a two-dimensional stochastic model for 

dispersion of bed-sediment particles as a function of the 

above probability distributions.

The probability distributions of the elevation at 

which a bed-sediment particle is eroded and deposited, 

and the probability distribution of the rest periods, conditioned on the elevation of deposition, will be obtained 

from a continuous record of the bed elevation at a particular point as a function of time. The probability distribution of the step lengths, conditioned on the elevation of erosion and the elevation of deposition, will be 

obtained from a series of "instantaneous" longitudinal 

bed profiles. With these distributions obtained, various 

related probability distributions of vital interest will be 

estimated, and a relation between the rest periods and 

the step lengths of a bed-sediment particle will be investigated.

Three experimental runs are analyzed and the relations between the statistics describing the postulated 

probability distributions and the hydraulic conditions 

are investigated. All data were obtained from a tilting

recirculating flume of rectangular cross section 61 m 

long, 2.4 m wide, and 1.2 m deep. The bed material used 

in these experiments was screened river sand with a 

median sieve diameter equal to 1.13 mm and a 

geometric standard deviation equal to 1.51.

ACKNOWLEDGMENT

The data contained herein are essentially the same 

as those contained in a dissertation by Lee (1973). The 

data were collected under the general supervision of the 

second author. Special thanks are due E. V. Richardson, 

D. B. Simons, C. F. Nordin, D. C. Boes, and R. P. 

Osborne.

BACKGROUND

THEORETICAL MODELS

Einstein (1937) treated the movement of a single 

sediment particle over an alluvial bed as a stochastic 

process described by an alternating sequence of two independent random variables, namely, step lengths and 

rest periods. Considering the particle movement in the 

distance-time plane on a Galton's board (Parzen, 1960), 

Einstein derived exponential probability density functions for the step lengths and the rest periods,

and

fx (x) = k.

fT (t) =

,

x>0

-kt

e , t > 0

(1)

(2)

respectively, where

X,T = random variables describing the step 

lengths and rest periods of a particle, respectively;

x,t = distance and time, respectively; 

fx (x),fT (t) = common probability density functions 

of the step lengths and rest periods, 

respectively; and 

&i» ^2 = positive constants.

For a sediment particle introduced into the stream at 

distance x = 0 in such a way that it takes its first step at 

time t = 0, Einstein obtained the probability density 

function of the total distance traveled by the particle at 

time t to be

f(x; t) =

-k x - k~t n - 1(kzt) n - 1

x> 0, t> 0 , (3)

in which F(«) denotes the gamma function. Equation 3 

also represents the concentration distribution of a 

group of identical sediment particles with respect to
BACKGROUND

longitudinal position, x, as a function of time, t.

The probability density function for the case when 

the particle is initially (t = 0) at rest at x = 0 also was 

obtained by a similar procedure,

f(x; t) =

-k,x -

E

n= 1

T(n + 1)

x> 0, t> 0 (4)

It should be noted that equation 4 applies only to the 

particle that has taken at least one step.

Einstein (1950) also developed his well-known bedload equation by considering the dynamic lift force as a 

random variable. The idea is that the probability of a 

sediment particle being eroded from the bed surface is 

equal to the probability that the lift force exerted on the 

particle exceeds its submerged weight. He obtained

B V - 

*-" it it Y1

r ^o

p = z dz = (5)

where

p = probability of a sediment particle being

eroded; 

Tjo^j-B* = constants;

^ * = intensity of shear for an individual particle

size; and

<£* = intensity of transport for an individual particle size.

Solving equation 5 for <!> ,, which is a function of the bedload transport rate, one obtains the bed-load discharge 

for individual particle sizes from hydraulic parameters 

and sediment properties.

Hubbell and Sayre (1964) presented a one-dimensional stochastic model for the longitudinal dispersion 

of bed-material particles in an alluvial channel. The 

results are identical to Einstein's (eqs. 1-4). The 

assumptions are: (1) the flow is in equilibrium (Simons 

and Richardson, 1966); (2) the particle always moves 

in a downstream direction with a series of alternate 

steps and rests; (3) the duration of movement is insignificant compared to the rest periods; and (4) the 

stochastic processes describing the number of steps 

taken by a particle in a distance interval and a time interval are independent of each other and both are 

homogeneous Poisson processes (Parzen, 1967). These 

assumptions are essentially the same as those of Einstein's (1937) although stated in a different way.

Based on the concept of continuity, Hubbell and 

Sayre (1964) proposed the transport equation for the 

bed material of a certain characteristic,

(6)

where

QT = bed-material discharge in weight per unit time; 

ic = ratio of the volume of particles possessing the 

characteristic size to the volume of bedmaterial particles in the zone of particle movement;

ys = specific weight of the bed material; 

9 = bulk porosity of the bed in place; 

W= width of channel; 

h = average depth of the zone in which particle

movement occurs; 

x = average distance traveled by bed material in

time t;

t = measure of time; and

c = subscript that denotes terms associated with 

particles possessing a certain characteristic 

size.

Combining equation 6 with the result from the HubbellSayre one-dimensional stochastic model gives the total 

bed-material discharge for all particle sizes,

(7)

in which ^ and k2 are defined in equations 1 and 2, 

respectively.

Sayre and Conover (1967) extended the one-dimensional stochastic model derived by Hubbell and Sayre 

(1964) to two dimensions by introducing the vertical 

level at which particles are deposited. Their analysis led 

to the joint probability density function for the event 

that a particle has, at time t, traveled a distance equal 

to AC and is located at an elevation equal to y,

f(x,y,t)

(n) t (n)

, n = 1

t -

, (8)

where

fy D(y) = probability density function for the 

(n} (n) elevation of particle deposition; 

fx (x),fT (t) = rc-fold convolutions of fx(x) and fT (t),

respectively;

fr\Y (* \v) = conditional probability density function 

for the rest periods given the elevation at which the particle is 

deposited; and

t' = sum of the first n rest periods.

If a group of identical sediment particles are released 

simultaneously at jc = 0, v = v0, and t = 0, equation 8 

gives the concentration of the particles, which were initially at rest and have moved from their respective initial positions, with respect to longitudinal position, jc, 

and vertical position, v, as a function of time, t.
STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

In order to apply equation 8, the density functions, 

fyD(y)> fr\ YD(t \^)> a11^ fx (*) must be specified. The unconditional density function, fT (£) is related to 

fr\ Y (t \y) and fy (y) by the relation

max

min D D

(y)dy (9)

in which ymax and ^min are the highest and lowest elevations at which particles can be deposited, respectively. 

The marginal case of equation 8 is

f(x; t) =

(n)

n = 1

(n)

dt

(n)

fx (x) P[N(t) =n], (10)

n=

in which P[N(t) = n] denotes the probability that a particle takes n steps in a time interval t. Equation 10 is a 

general one -dimensional stochastic model where only 

longitudinal dispersion is considered. One may note 

that the substitution of equations 1 and 2 into equation 

10 reduces to equation 4.

Yang (1968) assumed the step lengths are gamma 

distributed with a shape parameter, r, and the common 

density function,

r- 1 L

~ kx

x>0 (11)

and the rest periods are exponentially distributed with 

the common density function given in equation 2. 

Substituting equations 2 and 11 into equation 10, he obtained

f(x;t) =

~ k lx ~ k2 l

p ^

L* i

nr ~ l (k2 t) n

n = 1

T(nr) F(n + 1) '

x > 0, t > 0 . (12)

Since the gamma distribution reduces to the exponential distribution when r = 1, equation 4 is actually a 

special case of equation 12.

Shen and Todorovic (1971) generalized the HubbellSayre one-dimensional model given in equations 1, 2, 

and 4. The essential difference between the two models 

is that the former was based on the nonhomogeneous 

Poisson processes (Parzen, 1967), while the latter was 

based on the homogeneous Poisson processes. In the 

Shen-Todorovic model, the probability density functions of the step lengths and the rest periods are, 

respectively,

-f fc,(s)ds l

fx (x) = fc 1 (x) e , x > 0 , (13)

and

-f k2 (s)ds 

fT (t) =k2 (t) e ° , t>0 , (14)

where

k^x), k2 (t) = functions of xand t, respectively; and 

XQ, tQ = initial position and time, respectively. 

The probability density function of the total travel distance of a particle, which was initially at rest and has 

moved from its initial position, XQ, was found to be

x t 

- J k l (s)ds - J k2 (s)ds

x.

f(x; t) = k (x) e

[

x -in - 1 r t

f k l (s)ds\ J fc2 (s)ds|

xn J L *Q J

(n) T(n+ 1)

(15)

, x > 0, t > 0 .

n=

It is seen from equations 13 and 14 that the mean number of steps taken by a particle in (XQ , x] and (tQ , t] are

fxk,(s)dsand f k2 (s)ds,

JXn "In

respectively, whereas those of Hubbell-Sayre's model 

are ^(X-XQ) and k2 (t-tQ), respectively. The HubbellSayre (1964) one-dimensional model is a special case of 

the Shen-Todorovic model.

EXPERIMENTAL STUDIES

Hubbell and Sayre (1964) conducted concentration 

distribution experiments both in the field and laboratory to evaluate the one-dimensional stochastic model 

given by equation 4. The bed configurations in these experiments were large dunes in the field and ripples in 

the laboratory flume. Using radioactive tracer particles, a series of longitudinal concentration-distribution 

curves were obtained at different times for a given flow 

condition. The longitudinal concentration-distribution 

function, <&(x;t), is defined to be the weight of tracer 

particles per unit volume of bed material as a function 

of longitudinal distance and time and is related to fix; f)

by

w

= f(x:t) (16)

in which WT is the total weight of the tracer particles 

placed in the channel, W is the channel width, h is
DEVELOPMENT OF THEORY

average depth of the zone of bed material movement, 

and f(x\t) is given by equation 4. Based on equation 16, 

the parameters k^ and k2 were estimated. With these 

estimates, Hubbell and Sayre reported that the 

theoretical and observed concentration-distribution 

functions agree reasonably well.

Yang (1968) carried out a set of experiments using 

radioactive tracer particles to verify the model given by 

equation 12. Experiments were performed with ripple 

and dune bed conditions in a laboratory flume O.H m 

wide by 18.3 m long. He reported that the shape of the 

experimental longitudinal dispersion curves are fairly 

well represented by equation 12. Yang also made 

preliminary runs with a single plastic particle in a 

small flume and found that the step lengths very closely 

follow a gamma distribution with the parameter r approximately equal to 2 and that the rest periods follow 

an exponential distribution very closely.

The first intensive experimental study on the movement of single particles was done by Grigg (1969). The 

experiments were conducted in a laboratory flume with 

two bed material sizes. The bed configurations were ripples and dunes. Using single radioactive tracer particles, he measured the step lengths and the rest periods 

directly and found the step lengths to be approximately 

gamma distributed and the rest periods to be approximately exponentially distributed as proposed by Yang.

Grigg found interesting correlations between: (1) 

Various properties of the step length distribution, the 

stream power (product of mean bed shear stress and 

mean flow velocity), and the distribution of bedform 

lengths; and (2) various properties of the rest period 

distribution and statistical properties derived from the 

variation of bed elevation with respect to time.

Based on an idea suggested by Hubbell and Sayre 

(1965), Grigg also made some progress toward experimentally testing the Sayre-Conover two-dimensional 

stochastic model. By analyzing a record of the bed 

elevation as a function of time, he showed that the conditional probability density function of the rest periods 

can be approximated by the exponential function,

?T\Y c\y) =

x D

and

(17)

(18)

in which a and /3 are constants and y measures bed 

elevation in terms of the standard deviation about 

mean bed elevation.

REMARKS

Based on the review given in the previous sections, 

the following remarks are offered.

1. The Sayre-Conover model given by equation 10 is 

the most general one-dimensional model. The rest of 

the one-dimensional models, which were previously discussed, can be obtained from this model by proper 

substitutions. Therefore, it may be rated as the best existing one-dimensional model.

2. The Sayre-Conover model given by equation 8 is 

the only existing two-dimensional stochastic model. 

The derivation of the Sayre-Conover model has been 

discussed by Lee (1973). To verify equation 8, a method 

of estimating the probability distribution of the elevation at which particles are deposited must be known. 

One of the purposes of this report is to present such a 

method.

3. In order for the stochastic model to serve a prediction purpose, the relation between flow conditions and 

the parameters describing the probability distributions 

must be known. Without such knowledge the stochastic 

models cannot contribute much to the prediction 

problem.

4. A great deal of effort is required to perform dispersion and single particle experiments. If another method 

can be developed to estimate the necessary probability 

distributions from more readily accessible data, considerable savings would result. The methods developed 

in this report require only records of bed elevation.

DEVELOPMENT OF THEORY

CHARACTERISTICS OF PARTICLE MOVEMENT 

OVER A DUNE BED

Dunes are one of the most common bed forms in 

alluvial channels. Field observations by Simons and 

Richardson (1966) indicated that dunes may form in 

any alluvial channel, irrespective of the size of bed 

material, if the stream power is sufficiently large to 

cause general transport of the bed material without exceeding a Froude number of unity. The longitudinal 

profile of a dune is approximately triangular in shape 

with a gentle upstream slope and steep downstream 

slope. The upstream slope depends somewhat on flow 

conditions, whereas the downstream slope is more dependent on the angle of repose of the bed material. The 

length of a dune ranges from about 0.61 m to several 

hundred meters, depending on the scale of the flow 

system. The Chezy discharge coefficient, Cl-Jg, ranges 

from 8 to 12, and the total bed-material discharge concentration ranges from 100 to 1,200 milligrams per liter 

for dune flow conditions. For further information 

readers may refer to Simons and Richardson (1966).

For dune flow conditions, a record of the bed elevation as a function of time at a particular location 

reveals an alternating sequence of periods during 

which either erosion or deposition is occurring. This
STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

type of record is commonly obtained from the output of 

a depth sounder which is at a fixed location and 

hereafter will be referred to as the yx(t) record, that is, 

the elevation of the bed, y, positive upward, as a function of time, t, at a fixed location, x. When deposition occurs, [dyx (t)/dt] >0, and when erosion occurs, 

[dyx (f)/df\ < 0, provided these derivatives exist. An instantaneous longitudinal bed profile may be characterized by an alternating series of erosion and deposition reaches. An instantaneous longitudinal profile can 

be obtained by mounting a depth sounder in a boat, provided that the speed of the boat is large relative to the 

speed of bed forms. These bed profiles will hereafter be 

referred to as the yt(x) records, that is, the elevation of 

the bed, y, positive upward, as a function of the 

longitudinal coordinates, x, at a given time, t. The 

longitudinal coordinate will be assumed to increase in 

the downstream direction; therefore, the reaches with 

positive slopes, [dyt(x)/dx]>0, represent the upstream 

or stoss sides of the dunes, and the reaches with negative slopes, [dyt(x)/dx]<0, represent the downstream or 

slip faces of the dunes. The dune crest is defined by a 

local maximum in the yt(x) record, and the dune trough 

is defined by a local minimum in the record.

Anyone who has an opportunity to observe closely the 

movement of sediment is aware that dunes move 

downstream owing to erosion from their, upstream face 

and deposition on their downstream face. That is, the 

bed forms migrate downstream because deposition occurs on the downstream face, where [dyt(x)/dx] <0, and 

erosion occurs on the upstream face, where 

[dyt(x)/dx] >0. It will be assumed throughout this report 

that no deposition occurs on the upstream sides of 

dunes and no erosion occurs on the downstream faces of 

dunes. This assumption is not strictly true physically 

but is necessary for the determination of the conditional step length distributions. If the assumption is 

true, each sediment particle on the stoss side of a dune 

must make a step in the downstream direction before it 

is deposited on the slip face of any dune. Once deposited 

it rests there until the dune has migrated downstream 

and it becomes reexposed on the stoss side. In other 

words, sediment particles are transported downstream 

in an alternating sequence of steps and rests of random 

length and duration. The frequencies and magnitudes 

of these steps and rests are of basic interest in understanding the nature of the movement of the sediments.

Because particles must be eroded from and deposited 

on the surface of the bed, the step length of a particular 

particle depends only on the elevation from which it is 

eroded, the elevation at which it is deposited, the number of dune crests which it passes before being 

deposited, and the scale and shape of the bed surface 

(yt(x) record) during the time of its movement. Likewise

the rest period of a particular particle depends on the 

scale and shape of the yx (t) record and on the elevation 

at which the particle is deposited. If the bed material 

size is not uniform, the elevation of deposition or erosion may also depend on the size of particles because of 

vertical sorting.

The intimate relationship between the bed-form 

shape, as measured by the yx(t) and yt(x) records, and 

the step lengths and rest periods of a bed-material particle allow the probability distributions of the step 

lengths and the rest periods to be estimated from the 

bed-form data. In the following three sections, a method 

of estimating the probability distributions of the rest 

periods, step lengths, and elevations at which a particle 

is deposited or eroded using the yx (t) and yt(x) records 

will be presented. In the last two sections the bedmaterial transport equations and a general two-dimensional bed-material dispersion equation will be derived 

as functions of these probability distributions. In the 

next chapter the transport equations will be tested 

using data from three flume runs and the results will be 

discussed.

ESTIMATION OF THE PROBABILITY DISTRIBUTIONS OF 

THE ELEVATIONS OF DEPOSITION AND EROSION

The probability that particles are deposited between 

the elevations 17and 17 may be written as

(

number of particles deposited 

within the interval (T^., i]. + ^ in time t ^ 

number of particles deposited 

over all intervals in time t

where

P [ ] = probability; 

YD = random variable describing the elevations at

which particles are deposited;

17,, 17,+ i = lower and upper class limits associated with 

the class mark of the elevation yJ9 respectively; and 

t = time during which the observations were

made.

The elevation at which particles are deposited will 

hereafter simply be referred to as the elevation of 

deposition, YD .

If the number of particles per unit volume of the bed, 

O, is constant, the flow is stationary (statistical sense), 

and both erosion and deposition cannot occur at the 

same point at the same time, the numerator and the 

denominator of equation 19 can be obtained from the 

yx(t) record. The total number of particles deposited per 

unit area within the class interval (17^, 17^+1] in time t, 

denoted by Nd(yj) is given by
DEVELOPMENT OF THEORY

m. 

]

= 1,2, .... n , (20)

k = 1

where

n —

m =

class mark for the realization of YD ;

number of class intervals for the realization of

YD ; 

vertical rise of the bed in the class interval

associated with yj for the kth deposition

period; and 

maximum number of bed forms contained in

the yx (t) record and which also contain some

deposition in the class interval associated

with .

Figure I illustrates the class marks, ^ , and the vertical rise of the bed, Ay*fe , within the class intervals, 

Aty, for a typical yx(f) record. It is clear that 

Aj^^l A;yy = T7y+1 -17,. The total number of particles 

per unit area deposited over all intervals, the 

denominator of equation 19, is designated by Nd and is 

obtained by summing equation 20 over all class 

marks:

Nd= _

j = 1 j = 1 k = I

Equation 19 now becomes

(21)

m.

lim

m. -» 

k = 1

. (22)

j = 1 k = 1

Similarly an analysis of the erosion periods 

to estimate the probability that particles 

between the elevations 7) ; and 7), +1,

can be used 

are eroded

m'.

k = 1

m i

, (23)

where

at

e

YE = random variable describing the elevations

which particles are eroded; 

yi = class mark for the realization of YE ; 

,ri i+ i = lower and upper class limits of yt , respectively;

y.(t)

Typical deposition 

period

FIGURE 1. Typical y (t) record illustrating the class marks for deposition and erosion.
STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

m- = maximum number of bed forms contained in 

the yx (t) record and which also contain 

some erosion in the class interval associated with y,;

Ne (yt) = total number of particles per unit area 

eroded from the interval (TJ,-, TJH-I] centered at Ji ; 

Ne — total number of particles per unit area

eroded over all intervals; 

n = number of class intervals for YE ; and 

&y~i~k = amount of erosion which occurred during the 

kth erosion period in the vertical class interval associated with yt (fig. 1).

In the limit as m^-and m- approach infinity for a stationary record, the distributions of P[f)j < YD ^ T)J+I] and 

P[rj ( < YE ^ T?H-I] must be identical. The elevation at 

which particles are eroded will hereafter simply be 

referred to as the elevation of erosion, YE .

To repeat, the following assumptions are necessary 

for equations 22 and 23 to be valid: (1) Flow is in 

equilibrium such that both deposition and erosion processes are stationary with respect to time t; (2) both erosion and deposition do not occur at the same point during the same time period; and (3) the number of particles per unit volume of the bed is constant.

If the measuring equipment were sensitive enough 

to detect the movement of single particles, the second 

assumption would not be necessary because it would 

be physically impossible for one particle to be eroded 

and another to be deposited at the same point and at 

the same time. For yx (t) records obtained from less 

sensitive equipment, of course, the assumption may 

not be strictly true. When the bed material is not 

uniform in size, equations 20 through 2-3 are not 

strictly true because the number of particles per unit 

volume of the bed, O, is a function of elevation due to 

a vertical sorting. However, equations 22 and 23 

should serve as first approximations to the true probabilities, P[-r)j<YD ^7)J+l } and P[rj, < YE ^ TJ ;+I], 

even for the nonuniform bed material.

If the number of particles per unit volume were 

known as a function of elevation, y, assumption (3) 

could be dropped. In this case, the counterpart of equation 22 is

m. 

] Ay+ 

y

lim

m. -* 

k = 1

m. (24)

J E **>,

j = 1 k = 1

in which fty is the number of particles per unit volume 

of the bed associated with 3^. The counterpart of equation 23 would be similar. The value of Oy could be obtained from core-sample segments taken from 

different elevations within the bed. In the next section, H will be assumed to be a constant in estimation 

of Pty < YD ^-r)j+1] and Pfo <YE ^ t) i+1]. Because 

the bed material was very uniform in size, however, 

the assumption should have been very good.

Equations 22 and 23 are estimated by use of the 

sample probability mass functions which are defined 

to be

Nd (y,)

- P[r\. < YD < r] ^ for a large m , (25)

and

P Y (yt )

E l

(y-) VJ

N

I P[TJ. < Y,, < T]. , ,] for a large m, , (26)

t Ci ~~" I "* J. t

in which PY (yj) is used to estimate equation 22 and 

PyE(yt} is used to estimate equation 23. Estimates of 

the mean and variance of the elevation of deposition 

are respectively

n ~

D J <*,.)

and 27)

D

in which E [ ] and Var [ ] denote estimates of the expected value and variance, respectively. Replacing the 

D's with E's and the j's"with i's in equation 27 gives the 

estimates for the mean and variance of the elevation of 

erosion.

The probability density functions for the elevations 

of deposition and erosion [fyD(y) and fy(y)] may be 

inferred from the probability mass functions [Pyjiyi) 

and PyE(yi)] by means of a statistical fitting procedure. This will be discussed later.

ESTIMATION OF THE PROBABILITY DISTRIBUTIONS 

OF THE REST PERIODS

In this study the rest period of a particle is defined 

as the time lapse between the burial and reexposure of
DEVELOPMENT OF THEORY

the particle. This definition is consistent with the 

assumption that erosion and deposition do not occur at 

the same point during the same time period, and it is 

also necessary in analyzing single particle measurements because the measurement techniques cannot 

detect momentary rests by the particle. Using the 

burial definition, the yx (t) record provides a means of 

estimating the probability density functions of particle 

rest periods conditioned on the elevations of deposition. The method is illustrated schematically in figure 

2, where the statistic {^ ; j = I, 2, . . ., n; k = 1, 2, . . ., 

rrijj Jmeasures the conditional rest period, the index k 

signifies a particular bed form. The term ra^ designates the maximum number of bed forms which are 

contained in the yx (t) record and which also contain 

both an up -crossing and a down -crossing at the elevation y-. This use of the yx (t) record was first suggested 

by Hubbell and Sayre (1965) and was partly evaluated 

by Grigg (1969).

A relative frequency analysis of the statistic {tj k } 

leads to a sample conditional probability mass function 

of the rest periods which is defined to be

D

j = 1 , 2 , . . . , n; a = 1 , 2 , . . . , r , (28)

T = random variable describing the rest 

periods;

where

ta , yj = class marks for Tand YD , respectively; 

r a , r a+l = lower and upper class limits of ta , respectively; and 

r = number of class intervals for T.

Equation 25 can be used to release the condition on 

equation 28 and to obtain the marginal sample probability mass function for the rest periods,

J = 1

a = l,2, ..., r . (29)

From equations 28 and 29 the corresponding probability density functions for the conditional rest 

periods, fT\ Y (t\y), and for the marginal rest periods, 

fT (t), may be approximated by means of a statistical fitting procedure.

The mean and variance of the conditional rest 

periods are estimated from the sample moments

m.. . J.J

and

= 1

Vai

1

m. .

J.J

•[T\YD = y.]

m. . J.J E V2

k = 1

m. .

j ' j

h E ;.*

>(30)

TIME OF RE-EXPOSURE 

AND MOVEMENT

FIGURE 2. Typical y (t) record illustrating the conditional rest periods of a particle.
10 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Estimates of the mean and variance of the marginal 

rest periods are respectively

E(T} =

and

Var[T] = E[T2 ] - (E[T]) 2 ,

(31)

in which

E[T

m. .

j * i — r (t- -, t i

k = I

D

The joint probability density function of T and YD , 

denoted by /^y (£,,y) is estimated from the sample joint 

probability mass function defined to be

PT y ^a' y ^ = P ^a < T -la + I' ^' < YD -^' + 1^' ^32^

where J = 1, 2,..., n and a = 1, 2, .. ., r. From equations 

25 and 28, P^y (£a , jy,) is completely determined such 

that

PT y ^a' -^ ~PT\y ^ \y-)Pv (y-) • (33)

Finally the correlation coefficient between Tand YD is 

estimated to be

'T,Y

E[TYD ] - E[T]E[yD ]

D

(34)

in which

* [TYD ]=

a = 1 j = I

E[YD ] and Var [ YD ] are given in equation 27, and 

and Var [T] are given in equation 31. The joint distribution expresses the relation between the rest period 

and the elevation of deposition and the correlation 

coefficient measures a degree of linear association between the rest period and the elevation of deposition.

If the shape of the yx (t) record is dependent on the 

flow conditions and bed-material properties, then the 

rest period statistics as determined by equations 28 

through 34 are also functions of flow conditions and 

bed-material properties.

In summary, the probability distribution for the 

marginal rest period of a sediment particle, the rest

period conditioned on the elevation of deposition, and 

the elevation of particle deposition and erosion can all 

be obtained from a continuous record of the bed elevation at a single point as a function of time. The only 

assumptions that are needed are: (1) Both erosion and 

deposition do not occur at the same point at the same 

time; (2) bed elevation is stationary (in the statistical 

sense); and (3) the number of particles per unit 

volume of the bed is constant. These assumptions are 

not severely restrictive, and the results are equally applicable to both field and laboratory analysis.

ESTIMATION OF THE PROBABILITY DISTRIBUTIONS 

OF THE STEP LENGTHS

The yx (f) record contained the information necessary 

to estimate the probability distributions of the rest 

periods. Both the yx (t) and the yt(x) records are necessary to determine the step length statistics. Unfortunately, more assumptions are also necessary and 

these assumptions may be considerably more restrictive than the ones made up to this time.

As previously mentioned, it will be assumed that each 

sediment particle on the stoss side of a dune makes a 

step in the downstream direction before it is deposited 

on the slip face of any dune. Once deposited it rests 

there until it is reexposed on the stoss side. Let EitjtV be 

the event that a particle, eroded from elevation, yt , of 

the stoss side of a dune, passes v dune crests before it is 

deposited at elevation, 3^-. Then the statistic {Xij, v> k '•> 

ij = 1, 2, . . ., n; v = 1, 2, . . .; k = 1, 2, . . ., miJiV } (fig. 3) is 

the measure of the conditional step length of the event, 

Ei j V . The term mi ^ v represents the total number of 

possibilities of the event Eit ^ v contained in the yt(x) 

record and the index k specifies a particular possibility. 

In general, the term m^will be different for each combination of values i, j, and v.

A frequency analysis of the statistic (x^j^ k } gives a 

sample conditional probability mass function which is 

defined to be

+ E + + y

= 1, 2, . . . , s; i, j, = 1, 2, . . . , n; v = 1, 2, . . . , (35)

where

X= random variable describing the step

lengths;

Xp — class mark for the realizations of X; 

+i = lower and upper class limits of x^ , respectively; 

s = number of class intervals for the realizations of X; and
DEVELOPMENT OF THEORY 11

FIGURE 3. Statistic, [x^v^, i,j = l,2,...,n;v = l,2,...;k = l,2,..., m;j }, for the step length of a particle.

Ev = event that a particle passes v dune crests 

before it is deposited (fig. 3) .

The corresponding probability density function,

may be determined from

Px\YE ,YD ,Ev (xp \yi ,yjt v), 

and its mean and variance are estimated to be

m.

V yD = y," Ev 1 =

. . i,J,v

I/J ' V =

x. , , i,j,v,k

k = 1

and

m. . i,J,v

.,.

'(36)

in which

m. . i,J,v

If Y^, IQ , and E^are mutually independent (it seems 

to be reasonable that after a particle passes the crest of 

a dune it has probably lost track of where it came from), 

the density function [fx\YD ,Ev (x\y,v)] of the step 

lengths given that a particle is deposited at elevation y 

after passing v dune crests is estimated from the sample conditional mass function which is given by

(37)

i = 1

in which pyF(jj) is given by equation 26. The mean and
12 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

variance of the step lengths of a particle which is 

deposited at y^ after passing v dune crests are estimated 

to be

i=

and

Var[X\YD =

in which

(38)

Likewise, the following sample probability mass functions and corresponding means and variances are obtained;

v = 1

E[X\YD = y.] =

PX\YD ,Ev

v = 1

y = 1 

PX\E V v) =

K39)

(V

E[X\Ev ]= (y.) 

D

Var [X\E

(y. 

D

•(40)

and

X\E

v =

E[X] = E[X\E ]P[E

V V

Var [X] = E[X\E ]P[E

v = 1

- (E[xir.

(41)

The density functions, fx\ y^UAy), fx\Ev (x \ y)» and /*(*) 

are estimated from equation sets 39, 40, and 41, respectively.

The joint probability density function of X and YD , 

conditioned on the event Ev , [fx,YD \Ev (x> y\v)] can be 

estimated from a sample joint probability mass function,

/*r^'*-' ' D v

(xn ,y.\v)

X (42)

and

' ) ' (43)

Note that PYD \EV (yj\v) = PYD^ because YD and Ev 

were assumed to be independent. The correlation coefficient of Xand YD , conditioned on the event Ev , is then 

estimated to be

PX Y \E D v

in which

E[XYD \EJ

[X\E] >/Var

' and

$[YD], Var [YD], fi[X\£J, and Var [X\^J are given 

by equation sets 27 and 40. Similarly, the joint probability density function of Xand YD , [/^yD(JC, y)] and the
DEVELOPMENT OF THEORY 13

corresponding correlation coefficient, PX,YD > are 

mated to be

and

E[XYD ] - E[X]E[YD ]

(45)

(46)

in which

p = 1

The joint distributions (eqs. 43 and 45) express the relation between the step length and the elevation of 

deposition, and the correlation coefficients (eqs. 44 and 

46) measure the degree of linear association between 

the step length and the elevation of deposition.

The American Society of Civil Engineers (Task Committee on Preparation of Sedimentation Manual, 1962) 

defines bedload as that material moving on or near the 

bed. Accepting this general definition, it would appear 

consistent to count any sediment particle which was 

able to skip across a dune trough as suspended load, 

since it would be extremely unlikely that a particle 

would be able to pass the trough while moving on or 

near the bed. A very precise, and admittedly restrictive, 

definition of bedload is used for the purpose of this 

report. For the purposes of this report, bed load is 

defined as that part of bed material which is deposited 

on the downstream face of the dune from which it is 

eroded. Then the suspended load must be that material 

which is not deposited on the downstream face of the 

dune from which it is eroded, that is, all sediment particles which pass two or more dune crests before being 

deposited. The same particle could be counted as 

bedload during one step but as suspended load during 

the next step. By definition then, it follows that 

P [E^ = probability that a particle is transported as the 

bedload; and

probability that a particle is transported as the suspended load during any step. The probability distributions and moments for the step lengths of a bed-load 

particle may be obtained by putting v = I in the sets of 

equations 35 through 40 and 42 through 48.

For a bed material composed of coarse sand it seems 

reasonable to assume that all particles are transported 

as bed load, that is, all particles eroded from the stoss 

side of a dune will be deposited on the downstream side

of the same dune; and therefore, P [E^ = 1, and 

P [Ev] = 0 for v ^ 2. For this case, a frequency analysis 

of the statistic [xij^k \ iJ = 1, 2, . . ., n; & = 1, 2, . . ., 

mi j v \ v = 1} gives a sample conditional probability 

mass function which is defined to be

X

p = 1, 2, i, j = 1, 2,

The corresponding probability density function, 

fx\y <YD (x\y',y), may be approximated from 

Px\YE ,YD(xp\yi,yj). Denoting the statistic [xitjtVik ', 

ij = 1, 2, . . .; k = I, 2, . . ., miJfV • v = 1} simply as {xij<k ; 

ij = I, 2, . . ., n; k = 1, 2, . . ., m^} (fig. 3), the corresponding mean and variances are estimated to be

m.

k=

and

(48)

The term rni j represents the total number of bed forms 

in the sample for which the upstream side intersects 

the elevation ^ and the downstream side intersects the 

elevation yt . In general, the term mtj will be different 

for each combination of values of i and j (fig. 3).

Based on the statistic {Xij k } and assuming that YE 

and l^are mutually independent, the probability density functions, fx\YD(x \y)> fx^x^ an^ fx,Y (x>y) as we^ 

as the corresponding moments are estimated by setting 

P [EJ = 1 in equation sets 37, 38, 39, 40, 42, and 44.

Since the bed-form shape and rate of movement are 

dependent on the flow condition and bed material properties, it should be clear that equations 35 through 48 

are also functions of the flow condition and bed 

material properties. The statistic {x^} will be analyzed 

later to estimate the various probability distributions of 

the step lengths for a coarse sand for three different 

flow conditions.

Summarizing this section, the step length distributions can be estimated by combining the information 

contained in the yx (f) and yt(x) records. Additional 

assumptions are required however. These are: (1) No 

deposition occurs on the upstream sides of dunes, and 

no erosion occurs on the downstream faces of dunes; 

and (2) the elevation of particle erosion, YE , the eleva-
14 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

tion of particle deposition, YD , and the event that a particle passes v dune crests before it is deposited, Ev , are 

mutually independent. The first assumption may not be 

strictly true especially, due to flow separation, in the 

neighborhood of dune trough where both deposition and 

erosion may occur at the same point. For dune flow conditions, however, laboratory observation shows that 

such an area is small enough that the results should be 

applicable without an appreciable error. The second 

assumption seems to be reasonable because as a sediment particle passes a dune crest it likely loses the 

memory of where it came from. Estimation of P [Ev] 

would not appear to be a simple task. However, for a bed 

material composed of coarse sand, the assumption that 

all particles which are eroded from the stoss side of a 

dune will be deposited on the downstream side of the 

same dune seems to be reasonable.

BED-MATERIAL TRANSPORT EQUATIONS

The mean transport speed of a bed material particle, 

VT , is estimated to be

y _ Total distance traveled by a particle after m steps 

T Total time required for a particle to make m steps

mE[X] = E[X] 

mE[T] E[T]

for a large m (49)

.}p (y.)

in which VT denotes an estimate of the mean transport 

speed, VT,

n n

E E E

v = 1 j = 1 i = 1

m. . i,J,v

m. . i.j.v k = 1

xi,i,v,k

P Y (y.-)py (y i )P(E }

*E D ]

and

E[T] =

m.

m. . j,k

" J k= 1

In equation 49, the duration of particle movement is 

assumed to be negligible compared to the rest period. 

This assumption will be used throughout this section.

The mean transport speed could also be estimated to

be

VT= E[X\y =

(50)

where V? also estimates the mean transport speed of 

a bed material particle. In general it can be shown 

that VT ± V^ and the results of this study will indicate 

that VT^ VT • Now the question is: Which one will 

give the better estimate of the mean bed material 

transport rate? The difference between the two equations is the manner in which the events are averaged 

or weighted. So, in order to answer the question, one 

must depend upon physical arguments and reasoning. 

In equation 50, the average speed of a particle at each 

elevation is weighted by the number of particles with 

this speed. Equation 50 gives the best estimate of the 

arithmetic mean of individual particle speeds. On the 

other hand, equation 49 computes the estimate of the 

mean particle speed as the total distance traveled by a 

number of particles divided by the amount of time required to transport the same number of particles. In 

other words, the center of mass of a group of particles 

is translated through a distance,

in time,

Equation 49 will be used in this section because it is 

based on a mass flux concept and it gives an unbiased 

estimate of the mean sediment transport rate. The 

mean particle speed given by equation 50 will be useful 

in the study of bed material dispersion because it is 

based on individual particle speeds.

Defining the bed load and suspended load as given in 

the previous section, the mean transport speed of a bedload particle, VB , is estimated to be

E[X\£

VB = (51)

E[T] 

where VBis an estimate of VB and

EU^] . £ £

; = i = i

n

.j

Similarly, the mean transport speed of a suspended load 

particle, Vs, is estimated to be
DEVELOPMENT OF THEORY 15

vs =

I E E[X\E ]P[E] 

v v

(52)

E[T]

E[T]

v = 2

where

v = 2

/'N

VS = estimate of Vs;

Ev = union of event, Ev , for u 12= 2.

E [X\£?J is given in equation 40. Note that

TM* oo "" oo

P I I £ = 7""" \l v L-!

_v = 1 . 

and

v = 1

oo

* IK

v = 2 _

E^

v = 2

[X\Ev ]P[Ey ]

oo

T*"^

\ nr IT 1

v = 2 

Using equations 49, 51, 52, and 53,

VT =

(53)

E[T]

v = 2

E[T]

v = 2

) . (54)

If all bed material particles have identical transport 

characteristics, which is reasonable for uniformly sized 

bed material, the mean total bed material discharge is 

obtained by use of the continuity concept,

Q T = Y g (l - (55)

where

QT = estimate of the mean total bed material discharge in weight per unit time; 

y s = specific weight of the bed material; 

6 = porosity of the bed; 

W= width of the channel;

h = average depth of the zone in which bed material 

movement occurs; and V^is given in equation 49.

Similarly, estimates of the mean bed-load discharge 

and suspended load discharge are, respectively,

and

Q =y (l-G)WhV_P[EJ ,

= Ys (l -0)lVhVs (l -

(56)

(57)

s^, yx

where VB and Fs are given in equations 51 and 52, 

respectively. From equations 54 through 57,

QT - QB (58)

Although equations 55, 56, and 57 have the form of a 

continuity equation, the concept of continuity applies 

only in a statistical sense, because particles move only 

when they are exposed on the stoss side of a dune or 

when they are in suspension. Hubbell and Sayre (1964) 

proposed that the average depth of the zone of bed 

material movement, h, be estimated from the yt(x) 

record. For this method, the length of the reach for 

which h is to be determined is divided into sections. 

Starting from the upstream end, each section of length 

/j extends from the dune trough at which the section 

begins to the first trough downstream that is deeper 

relative to a line parallel to the plane of the mean bed 

surface. After sectioning, a mean depth of sand above 

the projected line for each section, /ijis determined, and 

the h for the total reach, Lx , is computed as the 

weighted average of the /i/s for each section. Expressed 

mathematically,

m

(59)

i = 1

The reasoning behind the procedure is based upon the 

assumption that although the individual dunes may 

change shape as they progress downstream, a statistical constancy of form exists over a long reach. Hence, 

quantitatively the particles subject to movement are 

those that would move if the entire profile were to 

progress downstream without changing form, and the 

depth of bed material movement is defined by lines that 

are parallel to the mean bed surface and extend 

downstream from the deepest trough.

If all bed material particles are assumed to be 

transported as the bedload,

(60)

where VB is determined from equation 51. For coarse 

sand P [EJ is expected to be very close to unity because 

the suspended load is negligible compared to the bed 

load. For a fine sand for which Pt^l =^1, equation 60 

would give only an approximation to the total load.
16 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Equations 55, 56, and 57 can be used with measured 

yx (i) and yt(x) records to compute the various transport 

rates. However, in order to apply the equations to the 

prediction of the bed material transport rate where the 

y^(t) and yt(x) records are not available, the relations, 

E[X\ YE = yt, YD = yj, Ev],

E[T\YD = yj\, P[EV], py ty), PyE(y^ and h, to pertinent hydraulic and sediment parameters must be 

established.

The mean transport speed of bed material particles 

deposited at elevation yj, which is denoted by V^j), 

(more precisely, deposited between the elevations TJ, 

and t)j+1, centered at yj ) may be estimated

(61)

where VT (f) estimates VT (j),

oo n

 E E

v = 1 i = 1 

and

m. . i.J.v

E x l 'J' v '

m. .

Based on equation 61, another transport equation can 

be developed as follows. Let £, denote the percentage of 

volume between elevations TJ, and TJJ+I occupied by

dunes over a given reach; then, 

from the yt(x) record (fig. 4),

~

can be estimated

(62)

where Lx is the total length of yt(x) record, and Xjft is 

defined in figure 4. Applying equations 61 and 62, the 

mean total bed material discharge can be expressed as

-9)W £

= y s (l-9)W

..

E[X\y =yJ

j j

(63)

where

QT = estimate of mean total bed material discharge; and

kyj = nonstandardized class width associated with 

elevation y/Aty = TJJ+I - 17,).

Equation 63 takes into account the local variation of 

the depth of the zone of bed material movement with 

respect to the elevation of deposition, and demonstrates 

to what extent each elevation contributes to the total 

transport rate. Similarly, the mean transport speed of a 

bed-load particle deposited at elevation yj} which is 

denoted by VB(j), can be estimated from equation 61 by 

considering P[£J = 1 and P [Ev] = 0 for v ^ 2. It can 

be shown, of course, that equations 54, 56, and 57 also 

apply at each elevation j as well as to depth-averaged 

values.

FLOW

X],2 Xj,3 . Xj,4 X),5 Xj,6

•*• x

FIGURE 4. Method for estimating the percentage of volume occupied by dunes between elevations ijy and TJJ+I ; f = ~ ^> ^
DEVELOPMENT OF THEORY 17

The third method to compute the mean total bed 

material discharge is based on the following reasoning:

F Number of particles"! [~ Weight "1 

QT = deposited per x per

L unit time and area J L particle J

("Mean distanced

x traveled by x

L a particle J

Width "1

of (64) 

..channel J

where QTis the mean total bed material discharge in 

weight per unit time. Restricting the attention to elevation yjt the terms in equation 64 are estimated as 

follows:

Number of particles deposited per 

unit time and area at y.

m .

J + 

Ay. , '

k = 1

[

Weight per particle

at yy

•]• Vs"' 6 ' 

"/

~1

. and

Mean distance traveled by a particle 

which is deposited at elevation yD

where Lt is the total length of yx(f) record, and all other 

symbols have been defined previously. Summing the 

product of the terms in equation 65 over all elevations,

YJ1 - 6)W

m. 

J

,(66)

where 

QT = estimate of mean total bed-material discharge;

and 

W = width of channel.

Equation 66 also illustrates the contribution of each 

elevation to the total transport, but its primary distinction is that the transport rate is computed from the 

sounding records with a minimum number of computations.

The relationship between the three transport equations, 55, 63, and 66, will now be demonstrated. First, 

the comparison of equations 55 and 66 is demonstrated. 

Combining equations 22, 25, and 66,

Ys d-e)w

j = 1

N

[X] , (67)

Multiplying and dividing by the marginal rest period 

and utilizing equations 20, 21, 25, 30, and 31,

'

m..

k E=

... ,(68)

As the value of Ay decreases the value of the last 

term can be approximated without appreciable error, as

m.

(69)

Strictly speaking m/A^ is equal to or slightly greater 

than the term

(fig. 1). Assuming a long record with small vertical 

class intervals such that equation 69 is valid and such

that m , =

Q" = Y (1-

? equation 68 reduces to

m.

E[T]

which would be equivalent to equation 55 provided that 

the average depth of the zone in which bed-material 

movement occurs, h' is defined by

h 1 = (71)

Equation 71, is similar to equation 59 except that it is 

based on the time record of depth, yx (t), while equation 

59 is based on the longitudinal profile, yt(x).

To investigate the relation between equations 63 and 

66, we proceed as follows. The last term of equation 66 

can be approximated without an appreciable error, as 

rrij; Abusing equation 69. Replacing the last term by its 

approximate value, and multiplying equation 66 by the
18 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

right-hand side of equation 30 while dividing by the 

left-hand side,

. . . (72)

The number of bed forms contained in the yx (t) record 

and which also contains some deposition in the class interval, rrij, should be almost equal to the total number of 

bed forms with both an upcrossing and a downcrossing 

in the interval, mjj. Assuming nij = m^, equation 72 is 

identical to equation 63 except that the percentage of 

the volume in the class interval occupied by dunes is 

computed from the yx (t) record instead of from equation 

62 which is based on the yt(x) record.

In summary, three transport equations have been 

presented, equations 55, 63, and 66. Although the equations appear quite different in form, they are all based 

on similar assumptions. As the record length becomes 

long and the class intervals reduce to zero in the limit, 

the three equations would become identical provided 

that either the yx (t) record or the yt(x) record could be 

used to determine the active depth. (This should be true 

for equilibrium flow.) In the following section, the total 

load will be computed for three flow conditions using all 

three equations, and the results will be compared.

GENERAL TWO-DIMENSIONAL STOCHASTIC MODEL

FOR DISPERSION OF 

BED-MATERIAL SEDIMENT PARTICLES

Let us define the following stochastic processes:

N(t)

X (t) = ^ Xi = longitudinal position of a bed- 

; = o material sediment particle at 

time t in which X (0) = X^ = 0.

N(t) = counting process describing number of steps 

taken by a bed-material sediment particle 

in time t.

Xi = length of ith step of a bed-material sediment 

particle.

n

X(ri) = 2* %i = longitudinal position of a bed1 = 1 material sediment particle after 

n steps.

Y (t) = vertical position of a bed-material sediment

particle at time t.

YD(ri) = elevation at which a bed-material sediment 

particle is deposited after n steps.

The probability that the particle has, at time t, traveled 

a distance equal to or less than x and that it is located at 

an elevation equal to or less than y may now be expressed as the joint distribution function

F(x,y;t) = P(X(t) < x, Y (t)

> N(t) = n] • (73)

n = 0

Using the definition of conditional probability and 

assuming that the duration of the particle movement is 

negligible, equation 73 can be restated as

F(x,y;t) = x, YD (n) <y, N(t) =n]

x >

n = 0 

y

= f P[X(n) < x, N(t)

mm

(74)

where

jmin = lowest elevation of deposition; and 

fY (n) (y) = probability density function of YD(n).

The event, (N(t) = n }, can be expressed in terms of the 

rest period of a bed-material sediment particle

{N(t) = n} = {T(n) < Opj{T(n + 1) > t} (75)

where

{ } = events;

O = intersection of events;

<C 

T(n) = 2< Ti> and

i = l 

Tt = random variable describing the duration of

ith rest period of a bed-material sediment 

particle.

By virtue of equation 75, it follows that

P[N(t) = n] = P[T(n) < t, T (n + 1) > t]

= P[T(n) t - T(n)] . (76)

For further simplification of equation 74, the following assumptions are made: (1) X(n) and N(t) are 

mutually independent for every n. (2) Xt for i ^ 1 are 

independently and identically distributed according to 

the probability density function fx(x), where
DEVELOPMENT OF THEORY 19

0 ^ x < oo. Outside this range, fx(x) = 0. (3) Xjis independent of YD (f) for i ±j. (4) YD(i) for i ^ 1 are independently and identically distributed according to the probability density function fyD(y}, where ymin =^ jysS ymax. 

Outside this range, fyD(y} = 0. (5) T{ for i ^ 1 are independently and identically distributed according to the 

probability density function fT (t), where 0 ^ t < oo. 

Outside this range, fT (t) = 0. (6) Tt is independent of 

YD (j — 1) for i =£/'. In other words, assumption 1 states 

that the total distance X(ri) traveled by a sediment particle after n steps should not depend on which time interval within [Q,t] that these n steps occurred. The step 

lengths are always positive so that the particle always 

moves in downstream direction (part of assumption 2). 

Each step length depends on the elevation at which the 

particle is deposited at the end of that step (assumption 

3). The elevation at which the particle is deposited at 

the end of any step does not depend on the elevation at 

which it was deposited at the end of any previous step 

(assumption 4). Finally, the duration of each rest period 

depends on the elevation at which the particle was 

deposited at the end of the previous step (assumption 

5).

Utilizing assumption 1, equation 74 becomes 

y

F(x,y;t)=f <x\YD (n) =

y ^ n = 0

P[N(t) = n\Y(n) = y']/ D

(y')}dy'

P[X(0) <x\YD (0) =y<]

P(N(t) =0\YD (0) =y']/ y (0) (y')dy'

D

j 

f

=y']

-'mm V n = 1 

 P[N(t) =n\YD (n) = y']/y ( n) (y')}dy' .(77)

Under assumptions 2, 3, and 4, and using the concepts 

of joint and conditional probability,

P[X(n) <x\YD (n) =y'] 

= P[X(n-l) +X 

x

0

= y 1 ]

V D

x x' 

= f dx'/ f

0 0

X(n- l),Xn\YD (n) v ^"

and using assumptions 3 and 4,

P[X(n) < x\Yn (rO = y r

 L)

X X'

0 0

0 0

*X(n - 1) ( ** )fX\Y

D

(n -1)

(78) ~\* D

in which

n - 1

E

i = 0

X (n - 1) = £ X. , XQ =

(n -1)

and

(n - 2) 

fx (6)

(n)

- 6)d6 ; n = 3, 4, 5, . . .

(79)

In equations 78 and 79, fx(n -i),

denotes the joint probability density function of

(n-l)

X(n - 1) and Xn , conditioned on YD(n), fx (O is the 

(n -l)-fold convolution of the probability density function for the length of a single step, and it is equal to the 

probability density function for the distance traveled by 

the particle in (n -I) steps, and fx\YD(x \^) ig tne con" 

ditional probability density function for a single step 

length given that the particle is deposited at elevation y.

Turning now to the other part of equation 77 and 

using equation 76 and assumptions 4, 5, and 6,

P[N(t) = n\YD (n) =y']

= P[T(n) < t, Tn+l >t-T(n)\YD (n) = y1 ]

l

•//

0 t - t 1

t oo

r (n) r

= J fT U')dt' J

D

(t\y')dT (80)
20 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

in which

n

T(n) = £ Tj

i = 1

t 1 ^ 

(n) p (n-1)

f tt*\ f l-t 1 \ 1 f IC*\ f It* - (\\ Hfi 'T(n) IT") J J T (V> I T ( I tf)"tf -

0

n = 2. 3. 4, ... , >(81)

and

(n)

fT (t<) =fT (t) ; n= 1 .

In the above, Tre+1 is the random variable describing the

duration of the (rc+l)th rest period of a particle,

/T(w),Tn \r (rc)(*'»T \y') ig the joint probability density

function of T(n) and Tn+1, conditioned on YD(ri),

/T\YD (* \y) is the conditional probability density function for the duration of a rest period given that the par-

(n)

tide was deposited at elevation y, and f^t) is the ra-fold

convolution of the probability density function, f^t), for 

the duration of a single rest period and is equal to the

probability density function for the duration of n successive rest periods.

Similarly, the terms for n = 0 in equation 77 become:

P[X(0) <x\YD (0) =y'l =1 (82)

because X(0) = X0 = 0 and 0 ^ x < oo, and

P(N(t) = 0\YD (0) = y'J =P[T1 > t\YD (Q) = y1 ]

oo

= f /T, v (t'\y')dt' , (83) Jf T " YD

where TJ is the random variable describing the duration of the first rest period in time t. It is important to 

note that the initial condition, X(0) = XQ = 0, implies 

that the particle starts its first rest period at t = 0. 

Introducing equations 78, 80, 82, and 83 into equation 

77,

y

F(x,y;t) = f f (y')dy' f fT. v (t'\y')dt' 

J D J. N D

 ^min

\) OO / *Y* *Y* '

r r- f r r (n ~ 1} + f ry (y')dy' V / da:' / / (?)

J D ^"^ \ -^ -^

>Vn " =1 \° °

t °° \

^x^y,,^' - S\y')dC-/rT U')dt'/ fTXy (t\y')dt 1(84)

u 0 t - f u /

The first term of equation 84 represents the joint probability that the particle has not moved from its initial

position and that its initial elevation is equal to or less

than y, and it is not a function of x. Hence,

2 | 2 y

dxdy [F(x 'y;0]|n = 0 = a*ay J fy^^y1

^rnin

00 /wD (tlxyl)dt' EO 

The corresponding density function is therefore

f(x,y,t) = d F(x,y; t)

J 9oc9y ^

00 / oc

E

| r (i-D

" An X XV7D

n = 1 \ 0

t 00 \

. r (n) r \

J fT (t l )dV / /-TVy (t\y)dt .(85)

0 Y-f D J

If a large number of identical particles are initially at

rest at x = 0, y = y0, equation 85 expresses the

longitudinal and vertical distribution at time t of the

particles which have moved from their respective initial positions. It should be noted here that f(x, y; t) is

not a true probability density function because

00 ^max

J dx J f(x,y;t)dy = 1 - P(N(t) = 0] <1 .(86)

0 'mm y . 

That is to say, equation 85 applies only after the particle has moved from its initial position. The expression 

f(x, y; t) does not exist for x = 0. 

If we assume that X^is independent of YD(j) for all i 

and j and drop assumption 3, equation 85 reduces to 

equation 8,

f(x,y.t)

00 £ 00

(n) p (n) p

= fy (y)Y_,f\ (x) J /T U ' )dt'J fT\Y (t^)dt ' (8)

D n = l' 0 t - t' D 

which is the Sayre-Conover (1967) two-dimensional

stochastic model. The difference between equations 85

and 8 is that equation 85 takes some of the dependence

between X and YD into account whereas equation 8 is 

based on the independence of X and YD . Hence, the

Sayre-Conover model is a special case of equation 85.
DEVELOPMENT OF THEORY 21

The marginal case of equation 85 gives the 

longitudinal distribution at time t of the particles which 

have moved from their initial positions. Integrating 

equation 85 over y,

ymax <» x . .. t . . C » r (n - 1) (n)

f(x;t)= I f(x,y,t)dy= > f fv (£) d£ I fT (t') df J ** i J A ^ *

'min n = 1 0

t - V

/

yr

max

<t\y')f

mm 

By virtue of assumption 1,

.(87)

J

max

f

mm

r

max

min

= fx T (x- (88)

Substituting equation 88 into equation 87 and rearranging terms,

J

(n-1)

i 

J"/-T (t')dt' J^ fT (t)dt

Because

J

(n - 1) (n)

and from equation 76,

t °° 

P[N(t) =n] = C fT (t l )dt' J° (89)

0 tMeanwhile, equation 75 can be restated as

{N(0 = n} = (T(n) < t}f|{T(n + 1) > t}

= (T(n) < t} - {T(n 

= (T(n) < t} - {T(n

t}'

where (T(n+l) > t} c denotes the complement of the

event {T(n+D > t}. Because {T(n+l) ^ t} is a subevent of {T(n) ^ t}, it follows that

P[N(t) = n] =P[T(n) < t] - P[T(n + l) < t ]

1 (n) ' (n + 1) 

= J fT (t')dV - J fT (t>) dt< . (90)

0 0

From equations 89 and 90, we have the marginal probability density function,

f(x;t) = fx (x)P[N(t) = n]

(n)

n = 1

r 

J

(n)

dt 1 . (91)

Equation 91 is identical to the Sayre-Conover (1967) 

one-dimensional stochastic model which is given in 

equation 10. As with equation 85, here also f(x;t) is not 

a true probability density function because

OO

J f(x;t)dx = \ - P[N(t) = 0] <1 , 

0

where P [N(t) = 0] is the probability that the particle 

has not moved from its initial position.

In order to apply equations 84 or 85, the probability 

density functions /yD(y), fr\YD(t\y), and fx\yD(x\y) 

must be known. These density functions are estimated 

from equations 25, 28, and 39. The probability density 

functions fT (t) and fx (x) are determined by the relations

max

D

(y)dy (92)

min

and

/x u> - /

max

(y)cfy (93)

min

where ;ymin and ;ymax are estimated from the yx (t) record. 

Equations 92 and 93 are the continuous forms corresponding to equations 29 and 40 (or 41), respectively. 

With fT (t) and fx (x) determined, the corresponding

(n) (n-1)

convolutions, fT (t) and fx (x) are determined from equations 81 and 79, respectively.
22 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

ANALYSIS AND DISCUSSION OF RESULTS

EXPERIMENT AND BASIC DATA

Three dune runs were made in a tilting recirculating 

flume of rectangular cross section, 61 m long, 2.4 m 

wide, and 1.2 m deep. The flume has been described in 

detail by Williams (1971).

The bed material used in these experiments was a 

screened river sand (Cherry Creek sand), with a median sieve diameter, d5Q = 1.13 mm, and a geometric 

standard deviation, ag = 1.51. The size distribution, 

shown in figure 5, was obtained by a sieve analysis of 

3,000 grams of bed material.

After an equilibrium flow, as defined by Simons and 

Richardson (1966), was established, the yx (t) and yt(x) 

records, the total bed-material discharge, and the hydraulic properties of interest were measured. The 

methods and procedures of the measurements have 

been described in detail by Lee (1969). The summary of 

measured and derived data is given in table 1. The 

values of the water discharge, depth, energy slope, bed 

shape, and total bed-material concentration presented 

in table 1 are the average of several individual 

measurements. The sampled load was measured by a 

DH-48 sampler. The number of measurements was the 

same as the number of yt(x) charts.

99.0

= 1.13mm 

.72mm 

" 0.75mm

0.4 0.6 as 1.0 2.0 3.0 

SIEVE DIAMETER, IN MILLIMETERS

FIGURE 5. Size distribution curve of bed material.
ANALYSIS AND DISCUSSION OF RESULTS 23

TABLE 1. Basic data and computed parameters

Water discharge Flow depth Energy slope Water temperature 

Qw, m3 /s d, cm Se T, °C

Run Mean flow velocity 

M Standard M Standard U, cm/s M Standard M-.,,, Standard 

Mean deviation Mean deviation Mean deviation Mean deviation

4A 0.464 .003 31.1 0.6 61.3 0.00167 0.00009 20.0 0.3 

16 1.24 .006 90.8 0.9 55.8 0.00029 0.00021 22.8 0.2 

17 1.53 .008 89.3 0.6 70.4 0.00047 0.00005 22.0 0.2

Total Bed-material discharge

Sampled load 

Run Concentration Mean total concentration 

Cy, mg/L load mg/L

M Standard M Standard 

Mean deviation Mean deviation

Mean Chezy Mean bed Mean Streamjjpwer 

resistance shear stress shear velocity T bU> 

coefficient T f,, d/cm2 [/«, cm/s d/cm«s

C/Vg

4A 168.6 53.6 2.77 1.5 6.1 8.6 50.8 7.13 3110 

16 8.9 3.1 0.39 0.0 0.0 12.1 25.9 4.91 1450 

17 29.7 10.8 1.61 6.3 10.2 11.0 41.2 6.43 2900

yx(t) Record yjU) Record

Mean Time interval of Range of 

Run Froude number Length of record Lag interval Length of record Number of measurements lag interval 

Fr Lt , hours min Lx ,m charts hours cm

4A 0.35 312 2.4 1235 31 6 3.7 - 13.0 

16 0.19 80 1.2 1006 33 1 8.6 - 13.6 

17 0.24 109 0.6 983 30 1 7.2 - 11.8

The yt(x) charts were obtained by mounting a sonic 

depth sounder on the instrument carriage and traversing it along the centerline of the flume in the upstream 

direction. The sonic depth sounder has been described 

by Karaki, Gray, and Collins (1961). Although the 

duration of traverse was approximately 5 minutes, the 

yt(x) record was considered to be instantaneous. The 

yx (t) record was obtained by locating a sonic depth 

sounder at the centerline of the flume 42.1 m 

downstream of the headbox. Both the yt(x) and yx (t) 

records were digitized with an analog-to-digital converter at the lag intervals shown in table 1. The lag interval on the yt(x) charts were not constant because the 

speed of the carriage was somewhat different for each 

chart. The output of the converter was to computer 

cards so that all statistics could be processed on the 

digital computer.

PROBABILITY DISTRIBUTIONS OF THE ELEVATIONS 

OF DEPOSITION AND EROSION

The sample probability mass functions for the elevation of deposition and erosion were computed using 

equations 25 and 26 and the yx (f) records. The results of 

calculations for the three flume runs are presented in 

table 2.

The yx (t) record of each run was standardized so 

that the class mark, yt , measures the elevation of 

deposition or erosion in terms of the standard deviation about the mean bed elevation. The class width of 

0.4 was used for all class marks. The frequency 

histograms for the elevation of deposition and erosion 

are plotted in figure 6.

The truncated Gaussian probability density function, 

defined by

fy (y) = fy (y)

Y D E

i -\y2

c 1 ?

- ^?~ ~ i or7 1 e~2 y"

2.4 12 SZn

-±- I C e ~2 y dy* 

^ J-2A 

for -2.4 < y < 2.4

fv ^) = fv {y} = ° °therwise *

YD Y E )

appears to fit the data reasonably well. A symm 

triangular density function defined by

> 

fy (y) = fy (y) =-^y + ^4 forO< y < 2.4

YD YE 2.4 *' q

fy (y) -fy (y) = -Jry + ri for ~ 2A <y < °

YD E 2 A L '^

fv (y) = fv (y) = 0 otherwise , 

Y D E

also appears to fit the data reasonably well. Equa 

94 and 95 are both plotted in figure 6. In equatioi 

and 95, /yfl(y) and fyE(y) are tne probability de 

functions of the elevation of deposition and ere 

respectively, and y is the standardized elevation.

Both distributions assume nonzero values on] 

-2.4 ^y^ 2.4 and the two models postulate 

YD and YE are identically distributed. The trunc 

limits on these distributions are rather arbitrary

K94)

tetric

>(95)

tions 

ns 94 

nsity 

>sion,

y for

that 

ation
24 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Run 4A

in 

c>

S

ro 

C>

CM

C>

1 1 I 1

T \

Run 16

in 

c>

£u to 

o 6

S CM 

> o 

cr

</> 

m 

o

o 

c>

Run 17

Triangular

———— Truncated Gaussian

-30-2.0 -1.0 0.0 1.0 2£> 3.0 -3.0-2.0 -1.0 0.0 1.0 2.0 3.0

STANDARDIZED ELEVATION, y

FIGURE 6. Frequency histograms, triangular density function, and truncated Gaussian density function for the elevation

of deposition and erosion.
ANALYSIS AND DISCUSSION OF RESULTS 25

The mean and variance for the truncated Gaussian 

density are

E[YD ] =E[Y£] =0 ,

and

2.4 

: 1.017 f y 2g(y)dy = 0.891

-2.4

(96)

where g (y) = e . For the triangular density

and

E[YD ] =E(YE ] =0

D ] = Var[y£ ] = E[Y (97)

The variances of these distributions are quite sensitive 

to the assumed truncation limits.

A goodness of fit test using the chi-square statistic indicated that neither model would be rejected for runs 16 

and 17 at a significance level of 0.05. For run 4A, 

however, both models were rejected at the same level of 

significance. As seen in figure 6, the truncated Gaussian density appears to give a slightly better approximation to fyD(y) and /V£(;y); but, the triangular density 

is much easier to handle in analytical treatments. The 

variance of the triangular distribution is even more 

sensitive to the assumed limits than is the variance of 

the truncated Gaussian distribution. Therefore, the 

triangular distribution probably should not be used in 

predicting the variance.

For stationary processes, continuity requires that the 

probability of erosion equal the probability of deposition 

for all elevations. Therefore, the density functions for 

the elevations of deposition and erosion must be identical. The mean and variance of sample histograms as 

well as the total number of points available for analysis, 

^mi , are shown in table 2. Little data were available for 

run 16, only 134 crossings compared to 2,167 for run 4A 

and 708 for run 17. Although run 16 was continued for 

33 hours, the very low transport rate (table 1) and slow 

movement of the bed forms limited the number of crossings available for analysis. It should also be pointed out 

that equilibrium flow was never attained for this flow 

which was barely above the initiation of motion stage.

REST PERIOD DISTRIBUTIONS

The sample conditional probability mass function of 

the rest periods were computed by determining the

difference betweeen the time of reexposure and movement and the time of burial of the center of each class 

mark for each crossing event, ra^; , that occurred in the 

yx (t) record (fig. 2). The results of the measurements 

are presented in tables 3 through 5, and examples of 

the mass functions are presented in figures 7, 8, and 9. 

The standardized yx (t) record was used and the class 

width of the elevation was taken to be the same as that 

used in determining the probability distribution for the 

elevation of deposition, 0.4.

The mean and variance of the conditional rest 

periods were computed using equation 30, and the 

results are presented in table 6. These results are also 

plotted as a function of bed elevation in figure 10. As 

can be seen from figure 10, both the conditional mean 

and variance of the rest periods decrease with increasing elevation of deposition. Inspection of figure 2 indicates that the conditional mean should decrease with 

increasing elevation of deposition. However, the 

decrease of the variance is not so obvious. Because the 

mean value is decreasing with increasing elevation, the 

decrease in the variance is not too meaningful. The 

coefficient of variation (standard deviation/mean) is 

probably a better measure of the variability of the rest 

periods. Restricting our attention to runs 4A and 17, 

for reasons to be discussed later, the coefficient of 

variation remained roughly constant in the range of 

0.6 0.75 for elevations above the mean bed elevation, 

and it increased with decreasing elevation to a value 

of about 1.5 at 2.4 standard deviations below the mean 

bed elevation. Thus the variability of the rest period, 

as measured relative to its mean, also decreases with 

increasing elevation at least up to the mean bed elevation.

As seen from figure 10, both the mean and variance

of the conditional rest periods may be approximated by 

an expression of the form,

E[T\TD =

and

=Ae

= Ce

~By

~Dy

(98)

The constants A, B, C, and Din equation 98 were determined by a regression analysis of the data plotted in 

figure 10, and the resulting values are presented in the 

figure. The values A and C represent measures of the 

mean and variance of the rest period, respectively, for 

the mean bed elevation. The values of B and D are 

measures of the rate of change of the mean and 

variance of the rest period with bed elevation, respectively.

The distributions of the conditional rest periods were 

approximated by the two-parameter gamma probability
26 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

O

0.03

0.02

0.01

50 100 

T, IN MINUTES

z

UJ 

O

O 

UJ

>

tr

UJ c/>

CD 

O

0.02

0.01 -

0.04-

0.02

200 400 80

0.012

0.006

YD=-..6

=0.006 

=0.742

0.08

0.04 -

0 200 400 0

T, IN MINUTES

FIGURE 7. Sample probability mass functions of the conditional rest periods with fitted two-parameter gamma

functions (run 4A).
ANALYSIS AND DISCUSSION OF RESULTS 27

0

0 400 800 

T, IN MINUTES

YD =-0.8 

K 2 = 0.006 

r? = 1-353

0.02

800

0.01 -

YD =0- 8

K 2 =0.023 

r 2 = 1.407 

m; ;=I8

100 200

0.008 -

0.004 -

YD =-1.6

K 2 = 0.003 

r 2 = 2.494 

rrv, ;= 6

A AA

0 1000 2000

0.02-

0.01 -

0

T, IN MINUTES

FIGURE 8. Sample probability mass functions of the conditional rest periods with fitted two-parameter gamma functions

(run 16).
28 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

0.03

0.02-

0.01

0 50 100 

T, IN MINUTES

0.02 H

co 0.01 -

UJ 

Q

O 

UJ

UJ 

CO 

CD 

O

200 400

YD = 0.8 

K 2 S 0.131 

r 2 = 2448

20 40 60

0.015-

0.010 -

0.005-

500 1000

T, IN MINUTES

K 2 S 0.186 

1-747 

32

40

FIGURE 9. Sample probability mass functions of the conditional rest periods with fitted two-parameter gamma

functions (run 17).
ANALYSIS AND DISCUSSION OF RESULTS 29

I0«

IO

CO

UJ

z

UJ

IO

10*

A B

O RUN 4A 40.4 0.91

A RUN 16 134.2 1.08

DRUN 17 41.6 1.10

a 

o

a 

o

8

@ e

-2.0 -1.0

L ' ' ' ' C ' D

P O RUN 4A 1210 2.18

In6 L A RUN 16 IOI60 2.93 - IO r n

DRUN 17 1110 2.79

! A A

CO 

UJ

ID

i io4

UJ

cc

1

CO

VARIANCE, IN

6 

ro

10°

2 a

- V8 

a A

8 o A 

o

0 0

a

A

-

a

1.0 2.0 -2.0 -1.0 

STANDARDIZED ELEVATION

1.0 2.0

FIGURE 10. Variation of the conditional mean and variance of rest periods with bed elevation.

density function which has the form,

v'1 " (k2 v) t >y >y

where

F(«) = gamma function; and 

^2, y rz, y = scale and shape parameters, respectively.

The scale and shape parameters were estimated by 

using the method of moments,

L 2,y

E[T\yp =yl 

Var[T\YD = y]

and

2,y Var[T\YD =

(100)

and the data contained in table 6. The variation of k2 y 

and r2 .yWith bed elevation are presented in table 7 

along with the results of a chi-square goodness of fit 

test. The ability of the two-parameter gamma distribution to fit the measured mass functions is illustrated in 

figures 7, 8, and 9.

From table 7, as well as from figures 7, 8, and 9, both 

the scale and shape parameters increase with increasing bed elevation, with a few exceptions for the shape 

parameter. The shape of the conditional density of the 

rest periods (figs. 7, 8, 9) approaches a J -shape and 

becomes more peaked as bed elevation decreases. 

Therefore, the exponential density might fit better 

than the two-parameter gamma density below the 

mean bed elevation (y < 0). The exponential density 

function is a special case of the gamma density with 

r2y = 1. The better fit of the exponential density seems 

to be consistent with the fact that all rejections of the 

chi-square test (6 rejections out of 22 at a significant 

level of 0.05) occurred below the mean bed elevation 

(table 7). It would appear that the exponential form for 

the conditional rest period as proposed by Grigg (1969) 

is only valid for elevations below the mean bed elevation.

A major factor in determining the degree of fit between the measured density functions and the fitted 

curves in figures 7, 8, and 9 appears to be the number of 

points available from which the distribution was constructed. In general, if more than 100 points were 

available, mti , the fit is pretty good. The weakness of 

the data for run 16 is very apparent. Even at the mean 

bed elevation, only 18 crossing events were observed.
30 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Combining equations 98 and 100, the scale and shape 

parameters can be estimated using only the constants 

A, B, C, and D.

AePy

and

2,y

M101)

The sample joint probability mass functions of the 

rest period and the elevation of deposition were computed from equation 33 using the results presented in 

tables 2 through 5. The results of these computations 

are presented in tables 8, 9, and 10. The correlation 

coefficients were computed by using equation 34, along 

with the data contained in tables 2, 6, 8, 9, and 10. The 

values of the correlation coefficients were -0.27, -0.53, 

and -0.26 for runs 4A, 16, and 17, respectively. The 

rest period and the elevation of deposition are 

negatively correlated, but the degree of their linear 

association is not strong.

The sample marginal probability mass functions, 

Pr(O> were computed by use of equation 29 and the 

data contained in tables 2, 3, 4, and 5. The results of 

these computations are also presented in tables 8, 9, 

and 10. The sample frequency histograms for the 

marginal rest periods are plotted in figure 11. The 

mean and variance of the marginal rest periods were 

computed by use of equation 31. These results are also 

presented in figure 11. The variance values appear to be 

extremely large. For example, the standard deviation 

for run 4A is almost four times the mean value. The 

computed variance values are extremely dependent on 

the long rest periods, the extreme events generally occur at low bed elevations. For example, by ignoring rest 

periods of greater than 2,000 minutes, which have a 

probability of occurrence of only 0.0015, the variance is 

reduced from 42,000 to 12,000.

Also shown in figure 11 are exponential density functions with a mean equal to the computed marginal 

mean. The exponential density function fits the data 

reasonably well; however, there would appear to be 

room for improvement. A gamma density fitted by the 

method of moments would be an extremely poor fit of 

the data. A gamma distribution, estimated by the maximum likelihood method may fit the data reasonably 

well.

The marginal distribution of the rest periods could 

also be estimated by

2.4

= \ (102)

J

f

-2A D

(t\y)f (y)dy 

D

where fT \ Y & \3^ is tne two-parameter gamma density 

(eq. 101) with parameters given by equation 101, and 

fY ( y) is given by equation 94, or it could be obtained by 

fitting the frequency histograms contained in figure 11 

with some assumed distribution.

STEP LENGTH DISTRIBUTIONS

The yt(x) record was standardized after removing a 

straight line trend. The trend determined by the 

method of least squares accounted for the possibility 

that the sand bed in the flume was not parallel to the 

instrument carriage rails supporting the sonic sounder. 

In standardizing the yt(x) record, the standard deviation obtained from the yx (f) record was used. With these 

standardized data, the statistic {Xijt k\ was analyzed 

(fig. 3) to estimate various probability distributions of 

the step lengths.

The sample probability mass functions given the 

elevations of deposition and erosion were computed by 

using equation 47, and the results are presented in 

tables 11 through 56. Examples of these mass functions 

are shown in figures 12, 13, and 14. The corresponding 

means and variances were estimated by equation 48 

and summarized in tables 57, 58, and 59.

It can be seen from tables 57, 58, and 59, as well as in 

figures 12, 13, and 14, that the conditional mean of the 

step length decreases with an increase in either the 

elevation of deposition or of erosion. This result could be 

expected simply from the typical shape of the dunes. 

Likewise the conditional variance of the step length 

tends to decrease with an increase in the elevation of 

either deposition or erosion. The above statements essentially imply that longer step lengths are associated 

with lower elevations at which a sediment particle is 

eroded and deposited, and vice versa.

The distribution of conditional step lengths were approximated by the two parameter gamma probability 

density functions,

(x\y,y')

k,

IMrj 

where

Uy<y" (103)

y and y' = arguments of YE and YQ, respectively; and

&: vv ,andr1:yv > = scale and shape parameters, 

respectively.

The parameters k^ y v . and r1> y v . were estimated by the
ANALYSIS AND DISCUSSION OF RESULTS 31

0.03

0.02

0.01 •

RUN 4A

E [TJ = 53.8 min 

Vtr[T]= 42,110 min 2

0.03

z 0.02

UJ

o

o

UJ

UJ (/>

00

o

0.01 •

RUN 16

E [T] = 180.5 min 

viY[T]=93,077min 2

0.03

0,02 • _f—|

0.01

RUN 17

E [T] =60.1 min 

v£r[T]= 60,469 min 2

100 200 

REST PERIOD, IN MINUTES

FIGURE 11. Frequency histograms for the marginal rest period and exponential fits.

300
32 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

0 1 2 

STEP LENGTH, IN METERS

0 1 2 

STEP LENGTH, IN METERS

UJ 

O

O 

UJ

CE 

UJ 

(/> 

CD 

O

1-

YE =0.0

YD =0.8

m: -=468

M I

0 1 2

STEP LENGTH, IN METERS

FIGURE 12. Sample probability mass functions of the conditional step lengths given the elevation of erosion is 0.0 with Gamma fits (run 4A).
ANALYSIS AND DISCUSSION OF RESULTS 33

0 1 2 

STEP LENGTH, IN METERS

0 1 2 

STEP LENGTH, IN METERS

0 1 2 

STEP LENGTH, IN METERS

FIGURE 13. Sample probability mass functions of the conditional step lengths given the elevation of erosion is 0.0 with Gamma fits (run 16).
34 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Z 

UJ

o 

o

LJ

cc

LJ

to

00

o

1 -

0 1 2 

STEP LENGTH, IN METERS

0 1 2 

STEP LENGTH, IN METERS

<O

Z 

UJ

o

s 1

> 

or

UJ 

CO

03

o

m; ; = 469 '»J

0 1 2 

STEP LENGTH, IN METERS

FIGURE 14. Sample probability mass functions of the conditional step lengths given the elevation of erosion is 0.0 with Gamma

fits (run 17).
ANALYSIS AND DISCUSSION OF RESULTS 35

method of moments, using data contained in tables 57, 

58, and 59 and the expressions

and

Var

Var [

= E[X\y_. = y, y_ = y']fc1 £ J ' D l,y,y j

The variation of klt v y • and rx y y , with the elevations of 

erosion and deposition are shown in tables 60, 61, and 

62. These approximations are also shown in figures 12, 

13, and 14.

The chi-square test for goodness of fit was used to 

test these gamma approximations. The results of these 

tests are summarized in tables 63, 64, and 65. None of 

the 81 distributions tested could be rejected at the 0.05 

level of significance. In other words, there is no good 

statistical reason to reject the hypothesis that the probability density functions for the step lengths, given the 

elevation of deposition and erosion, are distributed according to the two-parameter gamma distribution. The

1.0

0.8 •<

in 

cc

LU

0.6 -

S 0.4

0.2 -

o

k D 8

o « o D w O

D D

O RUN 4A 

A RUN 16 

D RUN 17

-2.0

fitted gamma distributions are also plotted and the 

example mass functions presented in figures 12, 13, 

and 14. These figures also help to illustrate the ability of the two-parameter gamma distributions to fit the 

measured conditional step length distributions.

The sample conditional mass functions, given the 

elevation of deposition, were computed based on equation 37 and the data contained in tables 2 and 11 56. 

These mass functions are presented in tables 66, 67, 

and 68. The corresponding conditional means and 

variances were computed using equation 38 and are 

presented in table 69 as well as being plotted in figure 

15. Again, the general decrease in the expected value of 

the step length with an increase in the elevation of 

deposition is apparent.

The sample joint probability mass function of the 

step length and the elevation of deposition was computed by equation 43, and the results are shown in 

tables 70, 71, and 72. The correlation coefficients were 

computed by equation 44, and their values were 0.15, 

-0.15,and-0.20 for runs 4A,16, and!7, respectively, indicating that the step length and the elevation of deposition are negatively correlated, but the degree of their 

linear associations is not strong. The sample marginal 

probability mass functions, Px(xp)> computed using 

equation 40, are also shown in tables 70, 71, and 72. The

0.3

<r

0.2 

LU 

O

O RUN 4A 

A RUN 16 

D RUN 17

A A

D O D

O O 

D D

2.0 -2.0 

STANDARDIZED ELEVATION

2.0

FIGURE 15. Variation of the conditional mean and variance of step lengths with bed elevation; E[X\ YD = y].
36 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

sample frequency histograms for the marginal rest 

periods are plotted in figure 16. The mean and variance 

of the marginal rest periods were computed by use of 

equation 41. These results are also presented in figure 

16. The range of the means is fairly small, only 0.610 to 

0.799 m. The mean dune lengths, as measured by the 

distance between trough points, for the three runs were 

1.19, 1.66, and 1.23 m respectively for runs 4A, 16, and 

17. The mean step lengths were, therefore, 54, 48, and 

49 percent of the mean dune lengths. Grigg (1969) 

found the mean step lengths of single tagged particles 

to be about 60 percent of the mean dune length. Of 

course, Grigg was working with a much finer sand, .33 

to .45 mm, as compared to 1.15 mm for this study. Also 

shown in figure 16 are gamma density functions for 

which the parameters k and r were determined from 

the mean and variance shown in the figure. The gamma 

functions appear to fit the data very well for all three 

runs. The value of the parameter r ranged from 4.05 for 

run 4A to 4.59 for run 17. This is slightly more than 

twice the value estimated by Yang (1968) from the step 

length distribution of a single plastic particle.

BED-MATERIAL TRANSPORT

The following assumptions and conditions were used 

to estimate the mean total bed-material transport rate 

by equations 55, 63, and 66: (1) Because the bed 

material was coarse sand (fig. 5), all sediment parades 

are assumed to be transported as bed load. Expressed 

mathematically, P[E^\ = 1. (2) y s(l - 6) = 1602 kg/m3 . 

(3) A^= QASy everywhere. By virtue of item 1, it follows 

that VT = VB, VT(J) = %(/), and QT = QB. In item 3, sy 

is the standard deviation of the bed elevation computed 

from the yx (t) record.

All parameters and statistics which are required by 

equations 55, 63, and 66 are summarized in tables 73 

and 74. The average depth of the zone of bed material 

movement, h, was determined by equation 59. It was 

found that one chart of the yt(x) record (about 34 m) is 

sufficient to obtain a reliable value of h, although over 

30 charts of the yt(x) record were used in this study. 

Each chart contained about ten dunes. Using equation 

62, £,, the percentage of volume between elevations 17/ and Tly+i occupied by dunes (hereafter 

will be referred to as the effective volume ratio) was obtained from the yt(x) record. The results 

are presented in table 74 and plotted in figure 17. As 

shown in figure 17 ^ is nearly independent of flow condition. As long as the bed forms are dunes, ^ does not 

change appreciably. It is also shown in figure 17 that £j 

is nearly unity and zero at jy, = 2.4 and jj = + 2.4, 

respectively. This is partial justification for the upper 

and lower limits of the elevations of erosion and deposition used in equations 94 and 95.

Another effective volume ratio can be obtained from 

the yx (0 record. Denoting this ratio by £,-,

m. 

J =Jr E'/. (105)

fc= 1

where

Lt = total length ofyx (t) record;

nij = maximum number of bed forms contained in

the yx (t) record which also contains some

deposition at elevation ^; and 

tj k = measurement of the conditional rest period.

There is no significant difference between £, and 

t,j (table 74) except for depths greater or less than 2.0 

standard deviations from the mean. The longitudinal 

profiles (yt(x) records) appear to contain a larger number of extreme events than the time record at a given 

point (the yx (i)). The explanation for this is probably 

that the flow was fairly stationary but that it was not 

longitudinally uniform.

A comparison of measured and computed total bedmaterial transport rates is shown in table 75. It is seen 

that:

1. For run 4A, all three equations provide excellent 

estimates to the observed mean total bed-material discharges.

2. Equation 55 provided an excellent estimate to the 

mean total bed-material discharge for run 17. However, 

the other two equations overestimated the discharge by 

more than 25 percent. The reason for the differences in 

the equations is not understood.

3. None of the equations gave good estimates of the 

mean total bed-material discharge for run 16. The consistently overestimated discharge ranged from 64 percent for equation 63 to 80 percent for equation 55. It 

should be remembered, however, that the mean total 

bed-material discharge was less than 9 mg/L during 

this run, that the flow was not in equilibrium as illustrated by the large variation of energy slope (table 1), 

and that very few rest period statistics were available 

for analysis (fig. 8).

Taken as a whole, the results are very encouraging. 

Although equation 55 gave the most accurate results 

for run 17, it should be noted that equations 63 and 66 

gave very consistent results for all runs when they are 

compared one with the other. The discharge predicted 

by equation 66 was 8.3, 8.4, and 8.5 percent larger than 

that predicted by equation 63 for runs 4A, 16, and 17, 

respectively. Although equation 66 is probably simpler 

to evaluate than equation 63, it appears that some accuracy has been sacrificed. The main difference between equations 63 and 66 is the way in which the 

effective depth or effective volume ratio (eq. 62, 71) is
ANALYSIS AND DISCUSSION OF RESULTS 37

RUN 16

E^X] = 0.799m 

Vtr [X]= 0.140m 2

I 2 

STEP LENGTH, IN METERS

FIGURE 16. — Frequency histograms for the marginal step length with Gamma fits.
38 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

TABLE 73. — Variation of various statistics with stream power

Run

4A

16

17

7 V

(d/cm-s)

3,110

1,450

2,900

E[X]

(m)

0.649

0.799

0.610

VarfX]

(m 2 )

0.104

0.140

0.081

E"[T]

(mm)

53.8

180.5

60.1

Var[T]

(min )

42,110

93,077

60,469

,", _ E[X]

T A

1 E[T]

(cm/s)

0.0201

0.0073

0.0169

V<

(cm/s)

0.0378

0.0166

0.0372

h

(cm)

9.66

6.89

7.13

s

y

(cm)

4.26

3.01

3.61

TABLE 74. — Comparison of the effective volume ratios at elevation yj; 

t,j, fromyt(x) record and£j, fromyx (t) record

'f,

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

Run 4A

V

0.968

.963

.949

.925

.877

.791

.668

.512

.344

.189

.078

.015

.001

.000

_____

— _--

_____

0.903

.883

.865

.787

.675

.529

.379

.222

.112

.044

.009

Run 16

V V

1.000

.995

.975

.935

.865

.778

.670

.535

.385

.230

.103

.036

.012

.004

.001

———

—— —

0.890

.840

.719

.656

.567

.388

.230

.097

.037

.008

.001

Run 17

V

1.000

.998

.989

.964

.900

.801

.669

.522

.364

.202

.079

.020

.002

.000

.000

-----

———

———

0.943

.867

.756

.645

.522

.380

.234

.120

.046

.013

.001

-----

computed, and these functions were similar (table 74); 

therefore, the consistency of their final result was expected. Equation 55 had the lowest average absolute error for all three runs; however, equation 63 gave the 

most accurate result on two out of three runs. Because

of the similarity of equations 55 and 63, it would be 

difficult to say one was more accurate than the other. 

Their relative accuracy probably depends on chance occurrence of extreme events in one or the other records 

of bed elevation.

In table 75, $T is the mean total bed-material discharge in weight per width and time, and it was obtained by dividing equations 55, 63, and 66 by the width 

of the channel, W.

If we define q'B (j) as the mean bed-load discharge 

associated with elevation yj, then based on equation 63,

(n?.Ay. (106)

where $B(j) estimates q'B(j) and VB(j) is an estimate of 

the mean transport speed of a bed-load particle at 

elevation yj. The mean transport speed, VB(j), is given 

by equation 61 provided that the suspended load is 

negligible. With equation 106, the variation of bed-load 

discharge with bed elevation may be investigated. This 

variation is shown in figure 18 for all three runs. It is 

seen that the maximum bed-load discharge is associated with the mean bed elevation and that an insignificant portion of the bed-load movement appears to occur 

for 3^ -2.4 and ;y, 25 +2.4.

TABLE 75. — Comparison of measured and computed total bed-material transport rates

4A

16

17

4A

16

17

Measured total bed-material discharge (t/day-m)

Number of Maximum Minimum

measurements

54 5.30 0.91

32 0.72 0.18

32 3.04 0.59

Computed mean, q^, (t/daym)

Eq. 55 Eq. 63 Eq. 66

2.68 2.72 2.94

0.70 0.64 0.67

1.67 2.05 2.18

Computed

Eq. 55

0.970

1.801

1.035

Standard

deviation

0.88

0.14

0.59

mean /measured

Eq. 63

0.983

1.641

1.269

Mean

2.77

0.40

1.61

mean

Eq. 66

1.066

1.725

1.354
ANALYSIS AND DISCUSSION OF RESULTS 39

< o

cr

5 d 

o

UJ O

o

UJ O

Explanation 

O Run 4A 

A Run 16 

D Run 17

-3.2 -2.4 -1.6 -Q8 0.0 0.8 1.6 2.4 3.2 

STANDARDIZED ELEVATION, y.

FIGURE 17. — Effective volume ratio as a function of bed elevation, y,;

(0tr

UJ 

0. ^

I 3

UJ

z — Q o

UJ

i o 

o

<2 CVJ 

Q °

3 Q

Q °

UJ

00

Explanation 

O Run 4A 

& Run 16 

D Run 17

-3.2 -2.4 -1.6 -0.8 0.0 0.8 1.6 2.4 

STANDARDIZED ELEVATION, y.

3.2

FIGURE 18. — Variation of bed-load discharge with bed elevation.

VARIATION OF VARIOUS STATISTICS

WITH FLOW CONDITIONS AND A RELATION

BETWEEN THE STEP LENGTH

AND THE REST PERIODS

Although three flume runs are not sufficient to 

establish a reliable relation between the various 

statistics and flow conditions, some qualitative trends

can be determined from table 73. The stream power 

(product of mean bed shear stress and mean flow 

velocity) was used as a measure of the flow conditions. 

From table 73, it is seen that:

1. The mean transport speed of a bed-material particle ( VT, V'T), the average depth of the zone in which bed 

material movement occurs (h), and the standard deviation of the bed (sy), appear to increase with increasing

stream power (r~b U).

2. The marginal mean of the step lengths (E [X\), the 

marginal variance of the step lengths^CVar [X\), the 

marginal mean of the rest periods (E[T]), and the 

marginal variance of the rest periods (Var[T]), appear 

to decrease with increasing stream power within the 

range of stream power investigated here.

The variation of the ratios of the conditional mean 

step length to the conditional mean rest period

( VB (y) = (£(X\ YD = y]) I (£[T\ YD = y})) 

and of the conditional variance of the step length to the 

conditional variance of the rest period

(Var [X\ YD = y]/Var[T\ YD = y}} 

with bed elevation, y, is shown in figures 19 and 20. 

From these figures it is seen that both ratios increase 

with increasing bed elevation.

TWO-DIMENSIONAL STOCHASTIC MODEL

FOR DISPERSION OF 

BED-MATERIAL SEDIMENT PARTICLES

A two-dimensional stochastic model for dispersion of 

bed-material sediment particles was derived earlier and 

was given by equation 85,

(85)

The one-dimensional model as a marginal case of equation 85 was

f(x;t) -

(n) / T

J /

dt<

Note that y is the standardized elevation. In order to apply equations 85 and 91, the probability density functions,

fx(x), andfxlx) must be specified.

Although probability density functions for all these 

distributions have not been determined in this report, 

the measured probability mass functions have been 

presented in tables 2, 3-5, 8-10, and 66-68, respec-
40 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Explanation 

O Run 4A 

Run 16 

D Run 17

<>"Q

-3.2 -2.4 -1.6 -0.8 0.0 0.8 1.6 2.4 

STANDARDIZED ELEVATION, y

FIGURE 19. — Mean transport speed of a bed-load particle as a function of bed elevation, y.

o

Explanation 

O Run 4A 

A Run 16 

D Run 17

-3.2 -2.4

-0-6

-1.6 -0.8 0.0 Q8 1.6 2.4 

STANDARDIZED ELEVATION, y

FIGURE 20. — Ratio of the conditional variance of step lengths to the 

conditional variance of rest periods as a function of bed elevation.

lively. Equations for determining the /i-fold convolutions of pT(t) andpx(x) can be obtained from equations 

82 and 79 with proper substitutions (Parzen, 1967). 

Further progress in the solution of either equation 85 or 

91 could proceed along either of two lines. First, all 

probability density functions could be replaced with the 

corresponding sample probability mass functions, the 

integrals approximated by summations, and the solutions obtained numerically. Alternately, the mass functions could be fitted by density functions of some 

assumed form and an analytical solution attempted. 

Lee (1973) used various fitting procedures to obtain all 

the probability density functions required to solve equation 85, but the integration of the equation appears 

quite formidable.

SUMMARY AND CONCLUSIONS

Stochastic models were developed which can be used 

to predict the transport and dispersion of bed-material 

sediment particles in an alluvial channel. These models 

are based on the proposition that the movement of bedmaterial sediment particles consists of a series of steps 

separated by rest periods and, therefore, their application requires a knowledge of the probability distributions of the step lengths, the rest periods, and the elevation of particle deposition and erosion.

The probability distribution of the rest periods, conditioned on the elevation of particle deposition and the 

probability distributions of the elevation of particle erosion and deposition, were obtained from a record of the 

bed elevation at a fixed point as a continuous function 

of time \yx(t) record]. The necessary assumptions were: 

(1) Equilibrium flow; (2) both erosion and deposition do 

not occur at the same point during the same time 

period; and (3) the number of particles per unit volume 

of the bed is constant.

The probability distribution of the step lengths, conditioned on the elevation of particle erosion and the 

elevation of particle deposition, was obtained from a 

series of instantaneous longitudinal bed profiles \yt(x) 

record]. The required assumptions were: (1) All bedmaterial sediment particles which are eroded from the 

upstream face of a dune will be deposited on the 

downstream side of the same dune; and (2) no deposition occurs on the upstream sides of dunes, and no erosion occurs on the downstream faces of dunes. These 

assumptions appeared to be reasonable at least for a 

dune-covered bed composed of a coarse sand.

Introducing an additional assumption that the elevation of particle erosion and the elevation of particle 

deposition are mutually independent, various related 

probability distributions were obtained. These distributions included: (1) The marginal distributions of the 

rest periods and the.step lengths; (2) the joint distribution of the rest periods and the elevation of particle 

deposition; and (3) the joint distribution of the step 

lengths and the elevation of particle deposition.

A two-dimensional stochastic model for dispersion of 

bed-sediment particles was then derived (eq. 85). In 

order to apply the model, the probability distributions of 

(1) the step lengths given the elevation of particle 

deposition; (2) the rest periods given the elevation of 

particle deposition; and (3) the elevation of particle 

deposition, must be known. The mass functions of these 

distributions were estimated; however, the integrations 

required by the model remained unsolved.

Applying the concept of continuity, three bedmaterial transport models were presented. Application 

of these models requires the estimation of: (1) The con-
REFERENCES CITED 41

ditional means of the rest periods and the step lengths; 

(2) the probability distribution of the elevation of 

deposition; (3) the average depth of the zone of bedmaterial movement; and (4) the effective volume ratio. 

These were all obtained from the yx (t) and yt(x) records. 

In the derivation of the models, the bed load was 

defined as that part of bed material which is deposited 

on the downstream face of the dune from which it is 

eroded, and the suspended load was defined as that part 

of bed material which passes two or more dune crests 

before being deposited. These definitions are very precise compared to the definitions prepared by the Task 

Committee on Preparation of Sedimentation Manual 

(1962).

Based on flume experiments with a coarse sand, the 

following conclusions were drawn:

1. The elevation of particle erosion and the elevation 

of particle deposition can be considered to be identically 

distributed, and their distribution can be approximated 

by either a truncated Gaussian density function or a 

symmetric triangular density function. In general, the 

truncated Gaussian density provides slightly better 

results; although the triangular density is much easier 

to handle analytically.

2. The conditional probability distribution of the rest 

periods, given the elevation of deposition, can be well 

described by the two-parameter gamma density function. The shape of the conditional density approaches a 

J -shape and becomes more peaked as bed elevation 

decreases.

A. Both the conditional mean and variance of the 

rest periods increase with decreasing bed elevation. These relations can be expressed by exponential functions.

B. Both the scale and shape parameters for the 

conditional distribution of the rest periods increase 

with increasing bed elevation, and they can be 

described by exponential functions of bed elevation.

C. The correlation coefficient between the rest 

periods and the elevation of deposition indicated 

that the rest periods and the elevation of deposition 

are negatively correlated, but the degree of their 

linear association is not strong.

3. The conditional probability distribution of the step 

lengths, given the elevation of deposition and the elevation of erosion, can be approximated by the twoparameter gamma distribution. The shape of the conditional density is strongly dependent on the elevation of 

deposition and erosion.

A. For a fixed elevation of deposition, both the 

double conditional mean and variance of the step 

lengths increase with decreasing elevation of erosion. In other words, longer step lengths are associated with lower elevation at which a sediment particle is eroded or deposited and vice versa.

B. The correlation coefficient between the step 

lengths and the elevation of deposition indicates 

that they are negatively correlated, but the degree 

of their linear association is not strong.

4. All three bed-material transport models are found 

to be quite satisfactory except for run 16.

A. The effective volume ratio can be obtained 

from either the yt(x) record or the yx (t) record, and 

it appears to be nearly independent of flow condition.

B. The maximum bed-load movement is associated with mean bed elevation, and little movement 

occurs for y ^ —2.4 and y 5= +2.4.

5. The mean transport speed of a bed-material particle, the average depth of the zone of bed material movement, and the standard deviation of bed elevation increased with increasing stream power, whereas the 

marginal means and variances of the rest periods and 

the step lengths decreased with increasing stream 

power.

Figures 10 and 15 suggest that the step lengths and 

the rest periods are positively correlated in an average 

sense, but the degree of linear association was not 

strong.

REFERENCES CITED

Crickmore, M. J., and Lean, G. H., 1962, The measurement of sand

transport by means of radioactive tracers: Proc. Royal Soc. of

London, ser. A, v. 266, p. 402-421. 

Einstein, H. A., 1937, Der Geschiebetrieb als Wahrscheinlichkeitsproblem [The bed-load movement as a probability problem]: Mitteilung der Verschsanstalt fur Wasserbau, an der Eidgenossische

Technische Hochschule in Zurich, Verlag Rascher and Co., 110 p. 

___ 1950, The bed load function for sediment transportation in

open channel flows: U.S. Dept. Agriculture Tech. Bull., no. 1026,

70 p. 

Grigg, N. S., 1969, Motion of single particles in sand channels: Ph. D.

dissert., Colorado State Univ., Fort Collins, 162 p. 

Hubbell, D. W., and Sayre, W. W., 1964, Sand transport studies with

radioactive tracers: Am. Soc. Civil Engineers Proc., v. 90, no.

HY3, p. 39-68.

___ 1965, Closure to: Sand transport studies with radioactive tracers: Am. Soc. Civil Engineers Proc., v. 91, no. HY5, p. 139-149. 

Karaki, S. S., Gray, E. E., and Collins, J., 1961, Dual channel stream

monitor: Am. Soc. Civil Engineers Proc., v. 87, no. HY6, p. 1-16. 

Lee, B. K., 1969, Laboratory study of an alluvial stream at one-foot

depth: M.S. thesis, Colorado State Univ., Fort Collins, Civil Eng.

Dept., 57 p. 

___1973, Stochastic analysis of particle movement over a dune

bed; Ph. D. dissert., Colorado State Univ., Fort Collins, 220 p. 

Parzen, E., 1960, Modern probability theory and its applications: New

York, John Wiley and Sons, Inc., 464 p. 

___1967, Stochastic processes: San Francisco, Holden-Day, Inc.,

324 p.
42 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Sayre, W. W., and Conover, W. J., 1967, General two-dimensional 

stochastic model for the transport and dispersion of bedmaterial sediment particles: Internat. Assoc. Hydraulic 

Research, 12th Cong., Fort Collins, Colo., Proc., v. 2, p. 88-95.

Shen, H. W., and Todorovic, P. N., 1971, A general stochastic model 

for the transport of sediment bed material: 1st Internat. Symposium on Stochastic Hydraulics, ed. Chao-Lin Chiu, Proc., Pittsburgh, Penn., 1. 426-448.

Simons, D. B., and Richardson, E. V., 1966, Resistance to flow in

alluvial channels: U.S. Geol. Survey Prof. Paper 422-J, 61 p.

Task Committee on Preparation of Sedimentation Manual, Committee on Sedimentation, 1962, Sediment transportation 

mechanics: Introduction and properties of sediment: Am. Soc. 

Civil Engineers Proc., v. 88, no. HY6, pt. 1, p. 78.

Williams, G. P., 1971, Aids in designing laboratory flumes: U.S. Geol. 

Survey open-file report, 294 p.

Yang, T., 1968, Sand dispersion in a laboratory flume: Ph.D. dissert., 

Colorado State Univ., Fort Collins, 162 p.
SUPPLEMENTAL DATA 

TABLES 2-72
44 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Table 2. Sample probability mass functions of elevations of deposition and erosion

Elevation 

"*

-3.6

-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

7 m. or 7 m'. L i L ti i

E[rD ] or E[jy

V^orvtrpy

Run 4A

*1D W \^

0.000

.000

.000

.006

.025

.060

.091

.125

.152

.160

.156

.120

.070

.025

.009

.001

.000

2,167

- .130

.813

0.000

.000

.000

.006

.032

.060

.088

.129

.140

.166

.154

.116

.068

.032

.009

.001

.000

2,167

- .133

.850

Run 16

Py Cj/^) Py (y^J)

0.000

.006

.007

.011

.019

.038

.062

.104

.122

.133

.197

.137

.097

.046

.016

.005

.000

134

.055

.999

0.000

.006

.007

.013

.025

.044

.065

.109

.116

.134

.189

.130

.097

.044

.016

.005

.000

134

.006

1.046

Run 17

*1D W \^

0.000

.000

.001

.005

.017

.044

.089

.129

.153

.154

.154

.117

.073

.041

.019

.004

.000

708

- .039

.863

0.000

.000

.002

.006

.017

.047

.089

.129

.148

.157

.153

.118

.072

.041

.017

.004

.000

708

- .043

.870

Triangular 

Density

———

0.004

.028

.Q56

.083

.111

.139

.158

.139

.111

.083

.056

.028

.004

———

Truncated 

Gaussian

-----

-----

———

0.006

.022

.046

.079

.118

.148

.162

.148

.118

.079

.046

.022

.006

———

m. is the total number of bed forms contained in the yx (£) record and which 

1 also contain some deposition in the class interval associated with the 

elevation y^.

m'. is the total number of bed forms contained in the yx (f) record and which 

1 also contain some erosion in the class interval associated with the 

elevation yj.
standardized elevation, y .

,,,,,,,

rc^o^rooo^o^oo^^o^oo

o o o o

O I— M 4*.

oooooooowt/iviwooo

o o o o 

o t— o ro

OOOOOOOOW<OVii-'OOO

00000

o o o t-> H*

OOOOOOOOW-JViJ^J^OO

§

0 0 0 

^- 0 *»• 

OOOOOOOOO-JO--|4^OO

000 

OOOOOOOOOWVi*--OOO

o o o o o to

O >— W NJ '-J ^ 

OOOOOOOOW<OOt—.t*.tOO

o o o o o o

O O 4*. M VI tO 

OOOOOOOOW-jO^l-'tOOO

000 

»-• 4*. »-• 

OOOOOOOOOOViW-frOO

000

o to to

oooooooooovi»-'<ooo

o o o o o

b b 

ooooooooooo^-*».oo

bob

OOOOOOOOOOO4^«3C3O

o o 

oooooooooooo*.oo

'vj

OOOOOOOOOOOOONJO

-^

o-BasSgSSSS^So

OOP

a 1 s

Wi O O

-J OO ^

Vi O O

Vi O O

a s g

hj ^ ro 

tn o o

ro vi o

Vi O O

-J O Vi 

Vi O O

4*. tn 4*. 

Vi O O 

0 O 0

Vi O O 

0 O 0

t~. 

OO O O>

o o o

0 O 0

\n O O 

o o o

0 O 0

o o o 

o o o 

o o o

^3

oo^oo^rooo^

-HSIa

°sii§l

oot—t— to

OOrOWO^*O4^

OOOt—

|sJ*fc-gO

ooo

OOOOO^Oi-'

-groo

OO

OOOOOJi.**.<

0 

t-- 

OOOOOO'-jO

o 

OOOOOOOV

OOOOOOO

OOOOOOOO

oooooooo

OOOOOOO
46 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Table 4. Sample conditional probability mass function of rest periods, p^iy (.t |j/0 (Run II

1

1

t

*

C

n

8

u

•o

M

•o

T3 

§

4-1

T

T

t

3?

C

a

a+1

a

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

a+1

a

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0

20

10

0

0

0

0

.2500

.1333

.1765

.1111

.4138

.2778

.5385

.2000

.5000

1.0000

0

220

240

230

0

0

0

0

0

.0666

0

0

.0345

0

0

0

0

0

0

20

40

30

0

0

0

0

0

.0667

.0588

.0556

.1034

.1667

0

.4000

.5000

0

0

240

260

250

0

0

0

0

0

6

.0588

.1111

0

0

0

0

0

0

0

40

60

50

0

0

0

0

0

.0667

0

.1111

.0690

.1111

.2308

.4000

0

0

0

260

300

280

0

0

0

0

.1250

.0666

.1176

.1111

0

0

0

0

0

0

0

60

80

70

0

0

0

0

0

0

.0588

.0556

.0345

.1111

0

0

0

0

0

300

400

350

0

0

0

.1667

0

.1333

.0588

0

0

0

0

0

0

0

0

80

100

90

0

0

0

0

0

0

.0588

.0556

.0345

.0556

.2308

0

0

0

0

400

500

450

0

0

0

0

.1250

.0667

.0588

0

0

0

0

0

0

0

0

100

120

110

0

0

0

0

0

.0667

.0588

.0556

.2069

.0556

0

0

0

0

0

500

600

550

0

0

0

0

0

.0667

0

.0555

0

0

0

0

0

0

0

120

140

130

0

0

0

0

0

.0667

.0588

.1111

.0345

.1666

0

0

0

0

0

600

700

650

0

0

0

.1667

0

.0667

.0588

0

0

0

0

0

0

0

0

140

160

150

0

0

0

.1667

.1250

.0667

.0588

.1111

0

0

0

0

0

0

0

700

800

750

0

0

0

.1667

.1250

0

0

0

0

0

0

0

0

0

0

160

180

170

0

0

0

0

0

0

.0588

.0555

.0345

.0556

0

0

0

0

0

800

1,300

1,050

0

0

0

.1666

.1250

0

0

0

0

0

0

0

0

0

0

180 200

200 220

190 210

0 0

0 0

0 0

0 0

0 0

0 .0666

0 .0588

0 0

.0345 0

0 0

0 0

0 0

0 0

0 0

0 0

1,300

1,800 m. .

-Z.,1

1,550

0 0

0 0

1.0000 2

.1666 6

.1250 8

0 15

0 17

0 18

0 29

0 18

0 13

0 5

0 2

0 1

0 0

, T ,. and t are in minutes. ' a+1 a

. is the total number of bed forms contained in the yx (t) record and which 

>l also contain both an up-crossing and a down-crossing at the elevation y,.
SUPPLEMENTAL DATA TABLES

Table 5. Sample conditional probability mass function of pest periods, p*,\ v C* lj/0 (Run j|i_ a t

47

T

Ta+l

t 

a

-2.8

-2.4

-2.0

iri -1.6

c - 1 ' 2

•H -0.8

> -0.4

« 0.0

1 0-4

•g 0.8

•3 1.2

w 1.6

2.0

2.4

2.8

T a

Ta+l

*a

-2.8

-2.4

-2.0

^ -1.6

1 ^

> -0.4

« 0.0

-o 

« 0.4

•3 o.s

| 1.2

£ l - 6

2.0

2.4

2.8

0

10

5

0

0

0

.0322

.0328

.0652

.0385

.0789

.1892

.2625

.4286

.6250

.7273

1.0000

0

130

140

135

0

0

.1000

.0322

.0164

.0109

0

0

0

0

0

0

0

0

0

10

20

15

0

0

0

.0322

.0164

.0652

.1442

.2281

.3063

.3250

.2857

.2500

.2727

0

0

140

150

145

0

.3333

0

.0322

0

.0109

0

0

0

0

0

0

0

0

0

20

30

25

0

0

0

.0322

.1311

.1630

.2404

.2544

.2703

.2500

.1786

.1250

0

0

0

150

200

175

0

0

0

.0322

.0328

.0217

0

0

0

0

0

0

0

0

0

30

40

35

0

0

0

.1290

.0984

.1522

.2115

.2105

.1261

.0875

.0893

0

0

0

0

200

250

225

0

.3334

0

.1613

.0328

0

0

0

0

0

0

0

0

0

0

40

50

45

0

0

.1000

.0968

.1639

.1739

.1250

.1228

.0631

.0500

.0179

0

0

0

0

250

300

275

0

0

.1000

.0322

0

.0109

.0096

0

0

0

0

0

0

0

0

50

60

55

0

0

.1000

.0322

.0492

.1304

.1058

.0351

.0270

.0250

0

0

0

0

0

300

400

350

0

0

0

.0322

.0328

.0109

0

0

0

0

0

0

0

0

0

60

70

65

0

0

0

0

.1639

.0978

.0288

.0526

.0180

0

0

0

0

0

0

400

500

450

0

0

.1000

.0645

.0164

0

0

0

0

0

0

0

0

0

0

70

80

75

0

0

0

.0322

.0656

.0326

.0385

0

0

0

0

0

0

0

0

500

1,000

750

0

0

.1000

.0645

.0164

0

0

0

0

0

0

0

0

0

0

80

90

85

0

0

0

0

.0328

.0217

.0096

0

0

0

0

0

0

0

0

1,000

2,000

1,500

0

0

.1000

0

0

0

0

0

0

0

0

0

0

0

0

90

100

95

0

0

0

.0323

.0328

0

.0192

.0088

0

0

0

0

0

0

0

2,000

3,000

2,500

0

0

.1000

0

0

0

0

0

0

0

0

0

0

0

0

100 110 120

110 120 130

105 115 125

000

000

0 .1000 .1000

.0323 .0968 0

.0328 .0164 .0164

.0109 .0217 0

.0096 0 0

.0088 0 0

000

000

000

000

000

000

000

3,000

4,000 m. .

3,500

0 0

.3333 3

0 10

0 31

0 61

0 92

0 104

0 114

0 111

0 80

0 56

0 32

0 11

0 3

0 0

T ., and t are in minutes.

is the total number of bed forms contained in the yx (.t) record and which also contain 

both an up-crossing and a down-crossing at the elevation y..
48 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Table 6. Variation of conditional mean and variance of 

rest periods vith elevation of deposition;

and

Standardized 

Elevation

1

-2.8 

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2

O

Er

Err

Run 4A

iv»<i.

Run 16

min

Run 17

1,400.8

252.2

120.5

83.4

58.2

40.2

27.4

21.0

16.7

13.9

11.4

8.5

—————

1,550.3

714.8

505.6

230.9

185.8

151.7

64.5

61.5

36.0

35.6

19.8

1,443.7

610.6

198.0

92.5

53.5

40.4

29.8

22.3

19.0

14.0

9.4

7.5

CT 0

2,167 134 708

Var[r|W, min2

Run 4A Run 16 Run 17

4,179,378

109,148

19,552

8,696

3,529

1,288

409

193

127

82

49

30

—————

84,679

204,840

246,658

39,416

28,686

16,715

4,030

2,691

1,039

351

2

4,671,616

762,079

74,793

14,193

2,548

1,129

316

188

145

107

50

18

2,167 134 708

Table 7. Estimates of parameters and the results of goodness of fit test fcr the 

conditional pest periods (two-parameter gamma)

! ,,..'•• b-r-l

Elevation

-0.4 

0.0 

0.4 

0.8

min" '

.031 

.067 

.109 

.131

Run

->,/'

583

1.255 

1.835 

2.276 

2.193

4A

m. . ^,^

314 

361 

338 

248

Goodness

° f 3/ 

Fit Test^7

x2 > x2

o

a

a 

x2 > x2

X2 <X2 

X XQ

a

X2 < X2

o

0

Run 16

k Goodness 

'^ r m. . of 

min" 1 "^ tr ' t" Fit Test

.006 1.204 17 x2 < x2 

.009 1.376 18 x 2 < x2 

.016 1.031 29 x2 < x2 

.023 1.407 18 x2 < x2

min- 1

.036 

.094 

.118 

.131

Run

"*.„

1.444 

2.811 

2.638 

2.488 

1 827

17

m . .

104 

114 

111 

80

Goodness 

of 

Fit Test

x r2 < xxo2

x2 > x2

o

T2 > X2 x xo

X2 >X2 

X2 < X2

o

x T2 < Xx2o

X2 < X2 

T- 2 < X2 X X^

0

x2 = critical chi-square value at 

0 a significant level of 0.05
•a

a_^

0

o 

-~i

§

00

o 

o

o 

o

§

o 

o

o

o

s

0

o 

*fc 

4k

§

C^

O

1

O

0

*fc

0

o 

-J

§

-J

§

Standardized elevation, y.

o • •••>><

e»*fcOa^r\)Oo-tkO*»c»NJa^oj^oo

o o o o

0000

ooooooooorororoooo

o o o o 

§000

o o o o o 

o o o o o

OOOOQOOOOH-'OOOOO

in o i/n <o ^

o o o o 

o o o o

OOOOOOOOOH-'OOH-'OO

§

0 0 

a o

OOOOOOOOOOH-l— OOO

o o o o o o 

§o o o o o 

ro N> »-> H- »-• o

l/l l/l 00 W <O U"l

o o o o o o 

o o o o o o 

ooooooaooH-*ki-'>-oo

§

b o 

0 0

oooooooooo>— roooo 

*k a^ **

o o o

000

ooooooooooo^-ooo

§

0000 

o o o o

OOOOOOOOOOH-HJOOO

in -fe oo oo vi

o o 

o o

OOOOOOOOOOOt—i— OO

o o o 

o o o 

oooooooooooor^oo

§

0 

0 

OOOOOOOOOOOOH-OO

§ OOOOOOOOOOOOOHJO

rf H H

moo

-J oo -J

moo

Ul O O 

Ul O O

l/l O O

-J O l/l

moo 

ro m O

Ol O O 

Ul O O

l/l O O 

000

O1 O O 

000

o o o

000

l/l O O

o o o

000

o o o 

o o o

O 0 0

T3

R_^

CT^

00

1

0

It—

o 

*-

-^1

o 

b

0

b 

o

o

0

s

o 

o 

-J 

-J

o 

o

3

00

0

o 

-J

0

§ 

2

Standardized 

o 

oo.uo<*ro».

§

00000

OhJrOUlW

oooooo

moowoow-

ooooo

§

Qh-t\)WW

tOWO'-'O

oooo

OOO»-*I

bbb

OOOOON)t-«

m^t-*

oo

OO

§

OOOOOON)

o 

o 

OOOOOOOt— 

OOOOOOO*—

OOOOOOOO

OOOOOOOO

oooooooo*

oooooooH-

OOOOOOOO

oooooooo
50 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Table 9. Sample joint probability mass function of rest periods and elevation of deposition,

pT.rJ*«>yf) (Run 16)

Ta

Ta+l

t 

a

-2.8

-2.4

-2.0

»" -1-6

g' -1.2

3 -0.8

£ -0.4

" 0.0

•a

3 0.4

"2 0.8

| 1.2

« 1.6

2.0

2.4

2.8

MV

Ta

Vi

*a

-2.8

-2.4

-2.0

^ -1.6

e -1.2

w -0.8

£ -0.4

" 0.0

S 0.4

"2 0.8

| 1.2

« 1.6

2.0

2.4

2.8

Pr cy

0

20

110

0

0

0

0

.0151

.0138

.0216

.0148

.0817

.0381

.0524

.0093

.0082

.0048

0

.2598

220

240

230

0

0

0

0

0

.0069

0

0

.0068

0

0

0

0

0

0

.0137

20

40

30

0

0

0

0

0

.0069

.0072

.0074

.0204

.0229

0

.0185

.0082

0

0

.0915

240

260

250

0

0

•0

0

0

0

.0072

.0148

0

0

0

0

0

0

0

.0220

40

60

50

0

0

0

0

0

.0069

0

.0148

.0136

.0153

.0225

.0185

0

0

0

.0916

260

300

280

0

0

0

0

.0075

.0069

.0144

.0148

0

0

0

0

0

0

0

.0436

60

80

70

0

0

0

0

0

0

.0072

.0074

.0068

.0153

0

0

0

0

0

.0367

300

400

350

0

0

0

.0063

0

.0138

.0072

0

0

0

0

0

0

0

0

.0273

80

100

90

0

0

0

0

0

0

.0072

.0074

.0068

.0076

.0225

0

0

0

0

.0515

400

500

450

0

0

0

0

.0075

.0069

.0072

0

0

0

0

0

0

0

0

.0216

100

120

110

0

0

0

0

0

.0069

.0072

.0074

.0409

.0076

0

0

0

0

0

.0700

500

600

550

0

0

0

0

0

.0069

0

.0074

0

0

0

0

0

0

0

.0143

120

140

130

0

0

0

0

0

.0069

.0072

.0148

.0068

.0229

0

0

0

0

0

.0586

600

700

650

0

0

0

.0063

0

.0069

.0072

0

0

0

0

0

0

0

0

.0204

140

160

150

0

0

0

.0063

.0075

.0069

.0072

.0148

0

0

0

0

0

0

0

.0427

700

800

750

0

0

0

.0063

.0075

0

0

0

0

0

0

0

0

0

0

.0138

160

180

170

0

0

0

0

0

0

.0072

.0074

.0068

.0076

0

0

0

0

0

.0290

800

1,300

1,050

0

0

0

.0063

.0075

0

0

0

0

0

0

0

0

0

0

.0138

180

200

190

0

0

0

0

0

0

0

0

.0068

0

0

0

0

0

0

.0068

1,300

1,800

1,550

0

0

.0191

.0063

.0075

0

0

0

0

0

0

0

0

0

0

.0329

200

220

210

0

0

0

0

0

.0069

.0072

0

0

0

0

0

0

0

0

.0141

, and £ are in minutes.
SUPPLEMENTAL DATA TABLES

Table 10. Siwiple joint probability noss function of rest periods and elevation of deposition,

PT.Y C^y) CRun 17)

51

T

a

Ta+ l

t 

a

-2.8

-2.4

-2.0

^ -1.6

«=• - 1 ' 2

•2 -0.8

> -0.4

* 0.0

"8 0.4

•3 0.8

| 1.2

M 1 '^

2.0

2.4

2.8

Pr cy

T

a

Vi

*o

-2.8

-2.4

-2.0

s^ -1 - 6

c - 1 - 2

0

•£ -0.8

> -0.4

* 0.0

1 °- 4

"S °- s

| 1.2

2 1.6

2.0

2.4

2.8

p r (ta)

0

10

5

0

0

0

.0014

.0029

.0085

.0059

.0121

.0294

.0309

.0312

.0256

.0137

.0035

0

.1651

120

130

125

0

0

.0017

0

.0015

0

.0015

0

0

0

0

0

0

0

0

.0047

10

20

15

0

0

0

.0014

.0015

.0085

.0220

.0351

.0476

.0382

.0208

.0102

.0051

0

0

.1904

130

140

135

0

0

.0017

.0014

.0015

.0014

0

'0

0

0

0

0

0

0

0

.0060

20

30

25

0

0

0

.0014

.0017

.0211

.0367

.0392

.0420

.0294

.0130

.0051

0

0

0

.1996

140

150

145

0

.0018

0

.0014

0

.0014

0

0

0

0

0

0

0

0

0

.0046

30

40

35

0

0

0

.0056

.0088

.0197

.0323

.0324

.0196

.0103

.0065

0

0

0

0

.1352

150

200

175

0

0

0

.0014

.0029

.0028

0

0

0

0

0

0

0

0

0

.0071

40

50

45

0

0

.0017

.0042

.0146

.0226

.0191

.0189

.0098

.0059

.0013

0

0

0

0

.0981

200

250

225

0

.0018

0

.0070

.0029

0

0

0

0

0

0

0

0

0

0

.0117

50

60

55

0

0

.0017

.0014

.0044

.0169

.0162

.0054

.0042

.0029

0

0

0

0

0

.0531

250

300

275

0

0

.0017

.0014

0

.0014

.0015

0

0

0

0

0

0

0

0

.0060

60

70

65

0

0

0

0

.0146

.0127

.0044

.0081

.0028

0

0

0

0

0

0

.0426

300

400

350

0

0

0

.0014

.0029

.0014

0

0

0

0

0

0

0

0

0

.0057

70 80

80 90

75 85

0 0

0 0

0 0

.0014 0

.0058 .0029

.0042 .0028

.0059 .0015

0 . 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

.0173 .0072

400 500

500 1,000

450 750

0 0

0 0

.0017 .0017

.0028 .0028

.0015 .0015

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

.0060 .0060

90

100

95

0

0

0

.0014

.0029

0

.0029

.0013

0

0

0

0

0

0

0

.0085

1,000

2,000

1,500

0

0

.0017

0

0

0

0

0

0

0

0

0

0

0

0

.0017

100

110

105

0

0

0

.0014

.0029

.0014

0

.0013

0

0

0

0

0

0

0

.0070

2,000

3,000

2,500

0

0

.0017

0

0

0

0

0

0

0

0

0

0

0

0

.0017

110

120

115

0

0

.0017

.0042

.0015

.0028

.0029

0

0

0

0

0

0

0

0

.0131

3,000

4,000

3,500

0

.0018

0

0

0

0

0

0

0

0

0

0

0

0

0

.0018

, and t are in minutes.
52 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

E

£

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u

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r-I^HOOOOO^H^

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rt rt o O

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SUPPLEMENTAL DATA TABLES 53

O O \D

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in

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in

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in

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1 0

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54 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

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SUPPLEMENTAL DATA TABLES 55

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56 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

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58 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

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SUPPLEMENTAL DATA TABLES 59

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60 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

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SUPPLEMENTAL DATA TABLES 61

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Sample conditional probability mass function of step lengths, p Y \ Y Y (xK \y.,y .")

'ED 

*D-Vj - 0-4

(Run 17) Table 52. Sample conditional probability mase function of 

V^

-0.8

P*|yfl f«sl»j0

0.5

0.0022

.196

1.5

.2405

.390

Class Mark,

2.5

.5367

.292

*B . ft

3.5 4.5 5.5

.2004 .0178 0.0022

.107 .015 .000

mi,o

449

647

—

£= I

2 8

0

0

P,lVW

0.5

.201

1.5

.382

Class Mark,

2.5

.304

XB' 

3.5 

.10

Note: The class widths for step lengths are equal to 1 foot for all 6. Note: The class widths for step lengths are eq

Table 53. Sample conditional probability mass function of step lengths, p v \ v „ (x<,\y • ,y - A i JE' D B 3

(Run 17)

v»<

-1.6 

-1.2 

-0.8 

-0.4

^iV^'v

0.5

0.0050

.187

1.5

0.0444 

.1414 

.2258 

.3682

.384

Class Mark,

2.5

5882

.4889 

.5455 

.6258 

.5274

.327

Xg, ft

3.5 4.5 5.5

.4222 .0444 ———— 

.2828 .0303 ———— 

.1419 .0065 ———— 

.0945 .0050 ————

.092 .010 ————

T-,3

45 

99 

155 

201

———

Note: The class widths for step lengths are equal to 1 foot for all B.
SUPPLEMENTAL DATA TABLES 63

!

O O O 

O Ul O 

» O M O

i m \o ^o

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i O O O O O *-t i

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o o o o o ooooo
64 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Tables?. Conditional means and variances of step lengths; £{X\XE -yi, ^D m Vj] and Var[Ar|jrg «y^, Xp'ti 4A)

^\;

-3.6

-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

EWv^j

vSrDr| V,.]

-3.6 -3.2 -2.8

—— - —— - .....

_ __ _ .._ _ . __ _

——— ——— ———

——— ——— 3.636

——— ——— 3.298 

——— ——— .60S

——— ——— 2.848 

——— ——— .507

——— 2.861 2.704 

——— ——— .766

——— 2.611 2.478 

——— .054 .392

—— - 2.082 2.434 

——— .098 .836

——— 1.892 2.557 

——— ——— .609

——— ——— 2.312 

——— ——— .943

——— ——— 1.692 

——— ——— .010

—— - -—— —— -

——— ——— .; ...

----- : ---- -----

——— 1.194 2.172

——— 1.396 1.232

-2.4

.....

.....

———

3.215

3.843 

.978

2.944 

1.114

2.848 

1.009

2.588 

.620

2.506 

.796

2.300 

.584

2.289 

.401

2.133 

.892

1.731 

.020

1.430

"~-

2.397

.884

-2.0

.....

.....

5.792

4.730

4.214 

1.409

3.417 

1.665

3.018 

1.125

2.747 

.942

2.589 

1.251

2.400 

1.011

2.365 

.985

2.257 

1.076

2.279 

.755

1.518 

.241

-----

2.577

1.376

-1.6

.....

3.974

4.172 

1.570

4.024

3.771 

.986

3.239 

1.175

3.136 

1.439

2.785 

1.137

2.519 

1.229

2.274 

1.069

2.159 

.986

2.108 

1.091

2.188 

1.127

1.779 

1.693

:::::

2.486

1.415

-1.2

—— „

3.038 

1.246

3.823 

2.148

3.605

3.407 

1.133

3.173 

1.176

3.063 

1.390

2.653 

1.145

2.455 

1.197

2.221 

1.084

2.054 

.910

1.933 

.837

1.903 

.889

1.687 

.942

1.670

2.384

1.277

-0.8

2.842 

1.417

3.620 

1.918

3.476

3.261 

1.074

3.076 

.991

2.976 

1.210

2.583 

1.063

2.336 

1.001

2.083 

.950

1.903

.771

1.775 

.690

1.756 

.626

1.558 

.539

1.377 

.000

2.257

1.141

-0.4

4.189

3.150 

1.235

3.385 

1.529

3.409

3.133 

1.069

2.989 

.974

2.854 

1.046

2.505 

.887

2.180 

.925

1.929 

.854

1.728 

.723

1.604 

.634

1.581 

.608

1.503 

.646

1.375 

.084

2.119

1.084

0.0

4.057

3.647 

.101

3.239 

1.021

3.459

3.102 

.872

2.956

.779

2.808 

.872

2.498 

.759

2.170 

.772

1.797 

.813

1.595 

.652

1.439 

.550

1.373 

.501

1.288 

.424

1.086 

.120

2.025

1.035

0.4

4.000

3.549 

.141

3.207 

.839

3.357

3.036 

.783

2.898 

.714

2.785 

.806

2.507 

.683

2.197 

.656

1.843 

.666

1.437 

.637

1.306 

.485

1.203 

.429

1.142 

.397

.954 

.125

1.972

1.008

0.8

3.934

3.440 

.191

3.195 

.941

3.309

2.935 

.665

2.804 

.599

2.717 

.726

2.466 

.623

2.205 

.602

1.916 

.581

1.521 

.495

1.127 

.416

1.036 

.313

.932

.273

.869 

.128

1.938

.935

1.2

3.857

3.283 

.302

2.965 

.360

3.184

2.909

.557

2.773 

.555

2.710 

.682

2.503 

.603

2.275 

.563

2.006 

.542

1.661 

.435

1.244 

.367

.870 

.301

.812 

.275

.785 

.132

1.983

.892

1.6

—— -

:::::

2.768 

.395

3.199

3.000 

.707

2.820 

.559

2.733 

.669

2.562 

.669

2.360 

.653

2.081 

.540

1.774 

.429

1.394

.357

.968 

.297

.595

.270

.687 

.147

2.056

.915

2.0

.... -

:::::

--.-.

5.346

4.308 

1.503

3.485 

.987

3.175 

1.055

2.879 

1.155

2.604 

1.225

2.350 

1.296

2.087 

1.192

1.829 

1.106

1.508 

1.035

1.081 

.829

.460 

.203

2.458

1.671

2.4

—— . .

:::::

-- —

.....

:::::

-—

- —

-----

— „

-----

— —

— —

-----

:::::

„—

.....

Note: Upper values are the means, in feet, and lower values are the variances, in feet squared.
SUPPLEMENTAL DATA TABLES

Table 58. Conditional means and variances of step lengths; E[J?|y-,»«., *„•«.] and V*r[X\!-, =w.. *„=«.] (Run 16)

& 1* D 3 E tr D j

65

^?\:

-3.6

-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

talV«^

D 3

-3.6 -3.2 -2.8

----- -----

.---. —— .- .. ——

——— ——— 4.600

——— ——— 4.334 

——— ——— .032

——— ——— 4.185 

——— ——— .067

——— ——— 4.649 

——— ——— 1.942

- —— ——— 4.390 

——— ——— 1.397

3.410 3.256 3.919 

——— ——— 1.062

3.197 4.391 3.823 

——— 3.638 1.307

2.917 3.802 3.384 

——— 2.159 .958

2.534 3.273 2.893 

——— 1.598 .860

2.266 2.844 2.413 

——— 1.072 .739

1.728 1.574 1.431 

——— ——— .213

—— - - ——

----- ——— - ——

2.073 2.512 3.008

1.463 3.476 2.290

-2.4

-----

4.516

4.415 

.467

4.477 

.545

4.646 

1.396

4.513 

1.076

4.020 

.837

3.633 

1.236

3.076 

1.092

2.727 

.696

2.317 

.576

2.067 

.601

———

2.924

1.635

-2.0

3.655

3.442

3.852 

.678

4.548 

1.059

4.623 

.809

4.438 

1.074

4.295 

1.038

3.961 

.982

3.608 

1.108

3.132 

1.020

2.769 

.737

2.285 

.599

2.127 

.428

3.097

1.530

-1.6

3.592

3.379

4.551 

1.068

4.612 

.979

4.626 

.760

4.524 

1.032

4.269 

1.079

3.878 

1.077

3.443 

1.069

2.969 

1.042

2.692 

.745

2.300 

.591

2.058 

.481

.160

.020

3.049

1.512

-1.2

3.531

3.318

4.447 

1.179

4.621 

1.077

4.514 

.699

4.369 

.931

4.113 

.960

3.726 

1.131

3.294 

1.121

2.851 

1.028

2.559 

.745

2.207 

.629

1.966

.527

.297

.029

2.927

1.499

-0.8

3.469

3.496

4.293 

.930

4.480 

1.053

4.437 

.752

4.327 

.950

4.039 

.918

3.632 

1.100

3.188 

1.126

2.731 

1.021

2.437 

.760

2.104 

.617

1.844 

.510

.229

.072

2.825

1.509

-0.4

3.407

3.594

4.142 

.832

4.363 

1.001

4.399 

.904

4.261 

1.067

3.942 

.978

3.553 

1.082

3.079 

1.131

2.617 

1.050

2.316 

.756

1.987 

.582

1.713 

.468

.242

.067

' 875

2.716

1.547

0.0

3.338

3.427

4.027 

.868

4.217 

.869

4.223 

.746

4.075 

.953

3.783 

.880

3.399 

1.012

2.936 

1.059

2.446 

1.032

2.155 

.733

1.846 

.558

1.598 

.434

.232

.080

2.567

1.477

0.4

3.263

3.102

3.833 

1.047

4.063 

.892

4.083 

.728

3.981 

.829

3.694 

.748

3.356 

.833

2.927 

.869

2.436 

.851

1.963 

.766

1.718 

.551

1.498

.427

.232

.084

2.473

1.432

0.8

3.186

3.042

3.696 

.983

3.905 

.913

3.931 

.716

3.834 

.790

3.562 

.705

3.287 

.783

2.917 

.762

2.457 

.714

1.960 

.675

1.519 

.612

1.388 

.410

.234

.083

2.402

1.381

1.2

3.111

2.897

3.830 

.573

3.909 

.843

4.012 

.656

3.987 

.733

3.685 

.678

3.474 

.747

3.072 

.656

2.638 

.585

2.174 

.516

1.684 

.465

1.175 

.416

.232

.083

2.517

1.392

1.6

.....

———

.....

3.098

3.591 

.199

3.851 

.620

3.689 

.665

3.481 

.609

3.116 

.524

2.674 

.462

2.262 

.411

1.824 

.367

1.358 

.349

TQT

.237

.083

2.500

1.234

2.0

.....

———

.....

3.051

3.126

.223

3.816 

1.163

3.568 

1.074

3.519 

.840

3.108 

.603

2.582 

.584

2.296 

.498

1.938 

.459

1.544 

.433

.276

.084

2.516

1.252

2.4

.....

- ——

———

.....

4.430 

.858

3.952 

.842

3.580 

.507

3.129 

.483

2.657 

.502

2.421 

.424

2.116 

.358

1.788 

.312

.183

.096

2.550

1.409

2.8

.....

—— -

———

———

4.379

3.243 

.904

2.933 

.824

2.698 

.733

2.459 

.611

2.219 

.433

1.999

.337

1.769 

.308

.139

.017

2.308

1.164

3.2

.....

-----

.....

.....

———

2.391

2.112

1.913

1.726

1.574

1.409

1.197 

" 998

1.495

.368

Note: Upper values are the means, in feet, and lower values are the variances, in feet squared.
66 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Table 59. Conditional means and variances of step lengths; l[X\XE *y., J-'y.] and VarfATJ I^-y^, YJ) =y,] (Run 17)

^SQ

-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

iWv^]

varWv ,.]

-3.2 -2.8 -2.4

—— . ——— —— --

..... ..... .....

——— ——— ———

——— ——— 3.655 

——— ——— .456

——— ——— 3.066 

——— ——— 1.202

——— ——— 2.656 

——— ——— 1.053

——— ——— 2.516 

——— -- —— .833

——— 3.396 2.878 

——— ——— .223

——— 2.110 2.131 

——— .965 .344

——— 1.108 1.636 

——— ——— .439

-- —— ——— 1.817

——— ——— 1.226

-—— --—— ---

..... ..... .....

——— .989 2.311

——— 1.815 1.104

-2.0

— ——

3.984

4.118 

.395

3.693 

.320

3.369 

.700

3.013 

.935

2.772 

.664

2.523 

.563

2.123 

.457

1.888 

.238

1.584 

.090

1.285 

.106

—— -

.....

2.473

1.069

-1.6

— ——

3.909

3.699 

.335

3.686 

.824

3.408 

.632

3.017 

.733

2.709 

.613

2.379 

.522

2.091 

.370

1.846

.237

1.621 

.220

1.520 

.248

.....

2.440

1.010

-1.2

———

4.026 

1.057

3.441 

.725

3.471 

.751

3.117 

.611

2.790 

.638

2.461 

.583

2.153 

.496

1.896 

.360

1.696 

.245

1.526 

.253

1.540 

.2§4

1.557 

.060

.....

2.271

.845

-0.8

4.657

3.773 

.427

3.319 

.575

3.320 

.623

3.008 

.629

2.662 

.681

2.354 

.616

2.041 

.516

1.763 

.398

1.573 

.289

1.400 

.245

1.289 

.155

1.366 

.053

.....

2.153

.861

-0.4

3.553

3.354 

.407

3.285 

.596

3.210 

.565

2.922 

.589

2.574 

.610

2.232 

.608

1.934 

.522

1.652 

.405

1.464 

.279

1.280 

.248

1.190 

.183

1.088 

.108

- ——

2.042

.849

0.0

3.157

3.188 

.414

3.104 

.546

3.076 

.546

2.818 

.552

2.499 

.530

2.141 

.529

1.792 

.502

1.513 

.382

1.332 

.257

1.146 

.214

1.038 

.174

.948 

.150

.941

1.925

.811

0.4

2.792

3.191 

.504

3.051 

.531

3.056 

.475

2.816

.477

2.476 

' .452

2.129 

.445

1.765 

.435

1.377 

.403

1.219 

.256

1.048 

.210

.945 

.168

.866

.148

.796

1.866

.818

0.8

2.663

3.048 

.608

3.000 

.542

2.983 

.421

2.769 

.453

2.465 

.408

2.151 

.383

1.814 

.351

1.421 

.318

1.069 

.272

.941 

.203

.841 

.161

.793 

.147

.682

1.841

.797

1.2

3.206 

.491

3.030 

.456

2.932 

.327

2.704 

.411

2.441 

.342

2.187 

.354

1.877 

.312

1.521 

.260

1.135 

.241

.786 

.212

.725 

.157

.714 

.139

.577

1.847

.762

1.6

-----

3.580 

.365

3.223 

.360

2.979 

.208

2.707 

.416

2.411 

.330

2.219 

.407

1.942 

.358

1.625 

.288

1.291

.247

.891 

.202

.545 

.161

.629 

.123

.494

1.899

.765

2.0

———

3.308 

.340

2.868 

.245

2.633

.247

2.515 

.706

2.259 

.391

1.998

.287

1.798 

.220

1.527 

.181

1.396 

.182

.984 

.147

.639

.133

.494 

.102

.411

1.777

.606

2.4 2.8

——— ———

::::: :::::

—— ——

—— ——

— .- —---

-— .....

-. — — ..

— - .....

1.624 — ——

1.234 —— —

.967 —— —

.753 —— —

.546 —— —

.328 -----

.506 —— —

.424 —— -

Note: Upper values are the means, in feet, and lower values are the variances, in feet squared.
SUPPLEMENTAL DATA TABLES

Table 60. Estimates of parameters describing too-parameter garma distribution for conditional step lengths (Run 4A)

67

\, rD-»j

V"i\

-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

-3.2 -2.8 -2.4

————— ———— 2.391

-2.0

2.098

-1.6

1.932

-1.2

2.309

-0.8

2.572

-0.4

7 787

2.530

0.0

2.616

0.4

2.693

0.8

2.709

1.2

3.390

1.6

3.905

2.0 2.4 2.8

1.654 —— —— —— ---

Note: Upper values are k , and lower values are r
68 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Table 61. Estimates of parameters describing two-parameter gonna distribution for conditional step lengths (Run 16)

^\y^- 

vv\

-3.6

-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

-3.6 -3.2 -2.8

::::: ::::: :::::::

——— ——— 135.437 

——— ——— 586.986

——— ——— 62.463 

——— ——— 216.406

——— ——— 2.394 

——— ——— 11.129

——— ——— 3.142 

——— ——— 13.795

—— - ——— 3.690 

——— ——— 14.462

——— 1.207 2.925 

——— 5.300 11.182

——— 1.761 3.532 

——— 6.695 11.954

——— 2.048 3.364 

—— -- 6.704 9,732

——— 1.671 3.265 

——— 4.752 7.879

——— ——— 6.718 

——— ——— 9.614

—— -- - —— - —— —— -

——— ——— —— ——

----- ----- -------

-2.4

::::::

9.454 

41.739

8.215 

36.777

3.328 

15,462

4.194 

18.928

4.803 

19.308

2.939 

10.678

2.817 

8.665

3.918 

10.685

4.022 

9.320

3.439 

7.109

------

------

-2.0

5.681 

21.885

4.295 

19.532

5.714 

26.418

4.132 

18.339

4.138 

17.772

4.034 

15.977

3.256 

11.749

3.070 

9.617

3.757 

10.403

3.815 

8.716

4.970 

10.570

13.908 

25.145

————

-1.6

4.261 

19.393

4.711 

21.727

6.087 

28.158

4.384 

19.832

3.956 

16.890

3.601 

13.964

3.221 

11.089

2.849 

8.460

3.613 

9.727

3.892 

8.951

4.279 

8.805

5.723 

10.188

11.319 

20.498

------

-1.2

3.772 

16.773

4.291 

19.827

6.458 

29.150

4.693 

20.503

4.284 

17.622

3.294 

12.275

2.938 

9.679

2.773 

7.907

3.435 

8.790

3.509

7.744

3.730 

7.334

4.523 

7.630

5.784 

9.938

------

-0.8

4.616 

19.817

4.254 

19.060

5.900 

26.179

4.555 

19.708

4.400 

17.771

3.302 

11.992

2.831 

9.026

2.675 

7.305

3.206 

7.814

3.410

7.175

3.616 

6.667

4.388 

7.144

7.319 

12.266

19.639 

27.769

-0.4

4.978 

20.620

4.359 

19.017

4.866 

21.406

3.993 

17.016

4.031 

15.889

3.284 

11.667

2.722 

8.382

2.492 

6.523

3.063 

7.095

3.414 

6.784

3.660 

6.270

4.330 

6.413

6.227 

9.384

------

0.0

4.639 

18.683

4.853 

20.464

5.661 

23.906

4.276 

17.424

4.299 

16.263

3.359 

11.416

2.772 

8.140

2.370 

5.797

2.940 

6.336

3.308 

6.107

3.682 

5.884

4.134 

5.726

6.116 

8.679

14.600 

17.053

0.4

3.661 

14.032

4.555 

18.507

5.608 

22.900

4.802 

19.117

4.938 

18.243

4.029 

13.521

3.368 

9.859

2.862 

6.973

2.563 

5.030

3.118 

5.357

3.508 

5.255

3.837 

4.977

5.776 

7.740

12.857 

13.886

0.8

3.760 

13.897

4.227 

16.702

5.490 

21.582

4.853 

18.607

5.052 

17.997

4.198 

13.799

3.828 

11.166

3.441 

8.455

2.904 

5.691

2.482 

3.770

3.385 

4.699

3.574 

4.342

5.423 

6.882

12.217 

12.387

1.2

6.684 

25.600

4.637 

18.126

6.116

24.537

5.439 

21.686

5.435 

20.028

4.651 

16.156

4.683 

14.386

4.509 

1 1 . 896

4.213 

9.159

3.622 

6.099

2.824 

3.319

3.276 

3.650

5.164 

6.186

11.434 

10.851

1.6

-----

18.045 

64.800

6.211 

23.920

5.547 

20.464

5.716 

19.897

5.946 

18.529

5.788 

15.477

5.504 

12.449

4.970 

9.065

3.891 

5.284

2.470 

2.336

4.679 

5.189

10.590 

9.309

2.0

::::::

-----

14.018 

43.820

3.281 

12.521

3.322 

11.853

4.189 

14.742

5.154 

16.019

4.421 

11.416

4.610 

10.586

4.222 

8.183

3.566 

5.506

3.064 

3.662

3.478 

3.339

9.548 

7.657

2.4

::::::

———

5.163 

22.873

4.694 

18.549

7.061 

25.279

6.478 

20.270

5.293 

14.063

5.710 

13.824

5.911 

12.507

5.731 

10.247

5.706 

8.206

5.760 

6.070

7.000 

4.704

2.8

::::::

-----

———

3.587 

11.634

3.559 

10.440

3.681 

9.931

4.024 

9.896

5.125 

11.372

5.932 

11.858

5.744 

10.160

6.780 

10.252

9.043 

11.367

"---

3.2

:::::

— -

——

——

-—-

-----

——

— -

-----

Note: Upper values are fc , and lower values are z^ , .
SUPPLEMENTAL DATA TABLES

Table 62. Estimates of parameters describing toe-parameter gamma distribution for conditional step lengths (Run 17)

69

\V"; v»i\

-3.2

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

-3.2 -2.8 -2.4 -2.0

...... ......

——— ——— —— —— 10.425 

——— ——— —— —— 42.931

——— ——— 8.015 11.541

——— ——— 2.551 4.813 

——— ——— 7.820 16.214

——— ----- 2.522 3.222 

——— ——— 6.699 9.709

——— —— -- 3.020 4.175 

——— ——— 7.599 11.572

——— ----- 12.906 4.481

——— 2.186 6.195 4.646 

——— 4.614 13.201 9.862

——— ——— 3.727 7.933 

——— ——— 6.097 14.977

——— ——— ———— 17.600 

----- -. —— ------ 27.878

——— ——— ———— 12.123 

- —— ——— ———— 15.578

——— .- —— ------ —— ——

-1.6

::::::

11.042 

40.844

4.473

5.392 

18.377

4.116 

12.418

4.419 

11.972

4. 557

5.651 

11.817

7.789 

14.379

7.368 

11.944

6.129 

9.316

————

-1.2

3.809 

15.335

4.746 

16.332

4.622

5.101 

15.901

4.373 

12.201

4.221 

10.388

4.341

5.267 

9.986

6.922 

11.740

6.032 

9.204

6.063 

9.337

------

-0.8

8.836 

33.338

5.772 

19.158

5.329

4.782 

14.385

3.909 

10.406

3.821 

8.996

3.955

4.430 

7.809

5.443 

8.562

5.714 

8.000

8.316 

10.719

-0.4

8.241 

27.640

5.512 

18.106

5.681

4.961 

14.496

4.220 

10.861

3.671 

8.194

3.705

4.079 

6.739

5.247 

7.682

5.161 

6.606

6.503 

7.738

10.074 

10.960

0.0

7.700 

24.594

5.685 

17.646

5.634

5.105 

14.386

4.715 

11.783

4.047 

8.665

3.570

3.961 

5.993

5.183 

6.904

5.355 

6.137

5.966 

6.192

6.320 

5.991

0.4

6.331 

20.203

5.746 

17.530

6.434

5.904 

16.624

5.478 

13.563

4.784 

10.186

4.057

3.417 

4.705

4.762 

5.805

4.990 

5.230

5.625 

5.316

5.851 

5.067

0.8

5.013 

15.280

5.535 

16.605

7.086

6.113 

16.926

6.042 

14.893

5.616 

12.080

5.168

4.468 

6.350

3.930 

4.201

4.635 

4.362

5.224 

4.393

5.394 

4.278

1.2

6.530 

20.934

6.645 

20.134

8.966

6.579 

17.790

7.137 

17.422

6.178 

13.511

6.016

5.850 

8.898

4.710 

5.345

3.708 

2.914

4.618 

3.348

5.137 

3.668

1.6

9.808 

35.113

8.953 

28.855

14.322

6.507 

17.615

7.306 

17.615

5.452 

12.098

5.425

5.642 

9.169

5.227 

6.748

4.411 

4.327

3.385 

1.845

5.114 

3.216

2.0 2.4

9.729 ——— 

32.185 ———

11.706 —— -- 

33.573 ———

10.660 ———

3.562 ----- 

8.959 ———

5.777 ——— 

13.051 ———

6.962 ——— 

13.909 ———

8.173 ———

8.436 ——— 

12.882 ———

7.121 ——— 

9.229 ———

6.694 ——— 

6.587 -----

4.804 ——— 

3.070 —— --

4.843 —— -- 

2.392 ———

2.8

:::::

——

-—

:::::

.....

.....

.....

-----

- — -

——

-----

Note: Upper values are k , and lower values are r
70 STOCHASTIC ANALYSIS OF PARTICLE MOVEMENT OVER A DUNE BED

Table 63. Results of goodness of fit test for conditional step lengths (Run 4A)

YD^J

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

fcl y v ,y

ft- 1

2.449

2.317

2.430

2.824

3.291

3.671

3.958

4.151

3.830

V»* =

ri,y,y'~

6.822

5.147

6.276

7-074

8.221

9.202

9.761

10.390

9.811

-0.8

m . .

128

217

336

423

457

420

307

181

48

Goodness

° f 3/

x2 < x2 

e

x 2 < ^

X2 < X2

c

X2 <X2

X2 < X2

a

X2 < X2

a

X2 < X2

a

x 2 < x2

x2 < x2 

c

k

ft- 1

2.127

2.049

2.193

2.259

2.210

2.767

3.298

3.701

3.854

V»t =

r ,

4.837

4.551

4.567

4.357

3.972

5.100

6.319

7.424

8.020

0.0

"V ,-

146

263

440

603

725

666

468

253

56

Goodness

of

x2 < x2 

o

X2 < X2

a

X2 < X2

X2 <X2

X2 < X2

a

x2 < x2

x2 < x2

x2 < x2 

a

x r2 < < xr2c

k

ft- 1

1.932

2.309

2.257

2.530

2.616

2.693

2.709

3.390

3.905

r , m. .

4.073 60

4.464 124

4.566 213

4.048 306

3.765 403

3.517 455

3.053 470

4.217 284

5.443 67

Goodness

of

X 2 < X2

X 2 < X2

a

T2 < r2 x xo

x2 < X2 

a

x2 < x2

X2 < X 2

c

X2 < X2

a

X 2 <X2

X2 <X2

=y, J=y'] Var[*|y =y, I

x2 = critical chi-square value 

at a significant level of 

0.05

Table 64. Results of goodness of fit test for conditional step lengths (Run 16)

1 •»•!/' 

ft' 1

-1.6 3.601

-1.2 3.294

-0.8 3.302

-0.4 3.284

0.0 3.359

0.4 4.029

0.8 4.198

1.2 4.651

1.6 5.716

'M,*-'

13.964

12.275

11.992

11.667

11.416

13.521

13.799

16.156

19.897

"i,J

155

239

293

341

365

337

272

146

52

Goodness 

Fit Test-7

X2 <X2

X2 < X2

c

X2 < X2

X2 < X2

a

x2 < x2

Q

X2 < X2

o

X2 < X2

X2 < X2

a

x2 < x2 

a

k , Goodness k , Goodness 'y*y f , m, . of *y y y •? , m. . of ft -l l.y.y T-.3 pit Test ft- 1 ^y-y l ' i> Fit Test

2.849 8.460 184 x2 < x2 3.892 8.951 124 x2 < x2

2.773 7.907 294 x2 < x2 3.509 7.744 200 x2 < X2

2.675 7.305 376 x2 < x2 3.410 7.175 254 x2 < x2

2.492 6.523 442 x2 < x2 3.414 6.784 297 x2 < X2

2.370 5.797 482 x2 < x2 3.308 6.107 317 x2 < x2

2.363 6.973 444 x2 < x2 3.118 5.357 325 x2 < x2 

a o

3.441 8.455 350 x2 < x2 2.482 3.770 331 x2 < X2

4.509 11.896 197 x2 < x2 3.622 6.099 207 x2 < X2

5.788 15.477 77 x2 < x2 4.970 9.065 83 x2 < x2

(t[x\YE-y, yfl

Y=y']

x2 = critical chi-square value 

0 at a significant level of 

0.05

Table 65. Results of goodness of fit test for conditional step lengths (Run 17)

V*,

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

v»t • -°- 8

l.y>y' 

ff 1

4.116

4.373

3.909

4.220

4.715

5.478

6.042

7.137

7.306

^y.y12.418

12.201

10.406

10.861

11.783

13.563

14.893

17.422

17.615

mi,o

109

259

406

487

502

449

324

155

53

Goodness

° f 3/ Fit Test-'

x2 < x2

x2 > x2 

a

x2 > x2 

a

x2 < x2

X2 < X2

c

~2 < ~2 X X^Q

X2 < X2

a

X2 < X2

o

X2 < X2

Yfy. - 0.0

fc l ' 

ff 1

4.557

4.341

3.955

3.705

3.570

4.057

5.168

6.016

5.425

r , m . .

10.842 129

9.346 301

8.073 493

7.165 632

6.397 704

7.161 647

9.375 469

11.292 235

10.535 80

Goodness 

of 

Fit Test

x2 > x2

x2 < x2 

o

X2 < X2

c

X2 < X2

c

X 2 > X2

o

X2 > X2 

0

X2 < X2

o

X2 < X2

c

X 2 < X2

a

k \ ' 

ft- 1

7.789

6.922

5.443

5.247

5.183

4.762

3.930

4.710

5.227

'sTi •

r ,

14.379

11.740

8.562

7.682

6.904

5.805

4.201

5.345

6.748

0.8

mi,3

68

159

267

357

411

425

427

247

85

Goodness 

of 

Fit Test

x2 < x2

Q

X2 < X2

c

x2 < x2

X2 <X2

X2 < X2

x2 < x2

X 2 <X2

X2 < X2

c

X2 < X2

c

r2 = critical chi-square value 

c at a significant level of 

0.05
SUPPLEMENTAL DATA TABLES 71

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