Development and field application of a littoral processes monitoring system for examination of the relevant time scales of sediment suspension processes

UFL/COEL-TR/116

DEVELOPMENT AND FIELD APPLICATION OF A
LITTORAL PROCESSES MONITORING SYSTEM FOR
EXAMINATION OF THE RELEVANT TIME SCALES OF
SEDIMENT SUSPENSION PROCESSES

by

Eric D. Thosteson

Dissertation

1997

DEVELOPMENT AND FIELD APPLICATION OF A LITTORAL PROCESSES
MONITORING SYSTEM FOR EXAMINATION OF THE RELEVANT TIME
SCALES OF SEDIMENT SUSPENSION PROCESSES

By

ERIC D. THOSTESON

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1997

In loving memory of

H. Dale Henderson and Lois W. Heaster.

ACKNOWLEDGMENTS

To the Coastal Sciences Program of the US Office of Naval Research, I wish to

express my appreciation for providing the financial support for this research. For their

unique ability to teach the practical aspects of coastal engineering on a day to day basis

despite never-ending demands, constant criticism, and meager rewards, I wish to thank

Sidney Schofield, Chuck Broward, Vernon Sparkman, Victor Adams, and Jim Joiner.

Their expertise, dedication, and never ending assistance and good humor have been as

rewarding as any course offered during my academic career. I sincerely appreciate the

freedom, trust, and friendship granted to me while pursuing my research interests from

Dan Hanes, the chairman of my graduate committee. To the remainder of my graduate

committee, chosen for their love for the science of coastal engineering, which is apparent

in their teaching and unending assistance, I wish to express my gratitude.

For their constant support and encouragement, I am forever grateful to my mom,

Karen Thosteson, my brother, Pete Thosteson, and my sis', Melanie Bleigh. Additional

thanks go to Chris and Monica for their friendship and free food, the Dog-Boy for

providing the foreground music while I worked on this dissertation in the background,

and to all of the great friends I've made in this endeavor.

TABLE OF CONTENTS

page
ACKN OW LED GM ENTS..................................................................................................iii

LIST OF TABLES .............................................................................................................vi

LIST OF FIGURES ........................................................................................................... vii

KEY TO SYM BOLS .................................................................................................... x

ABSTRACT ..................................................................................................................... xiv

1 INTRODUCTION ................................................................................................. 1

2 SYSTEM DEVELOPMENT................................................................... 4

Instrum ents .............................................................................................................. 5
System Dem hands ................................................................................................... 8
System Architecture ................................................................................................... 11
Software and Algorithm s....................................................... ........................... 16
Sum m ary..................................................................................................................... 23

3 ACOUSTIC BACKSCATTER CONVERSION TECHNIQUE.............................. 25

Theory......................................................................................................................... 28
Develop ent.................. ......................................................................................... 31
Concentration inversion ..........................................................................................31
Size determ ination................................................................................................... 35
Verification of Technique................................................ ..... ................................ 39
Sum m ary..................................................................................................................... 45

4 SU SPENSION TIM E SCALES..............................................................................46

The SIS96 Project .............................................................................................. ...... 47
Size Determ nation ............................................................................................... 53
Dominant Frequency Band of Suspension Events ......................................... .... 59
Correlation with Velocities......................................................................................... 71
D discussion of Results ................................................................................................. 81

5 CON CLU SIONS....................................................................................................... 87

iv

A A COU STIC PARA M ETERS ............................................................. .................. 91

Log-norm al D istribution............................................................................................. 91
System Constant................................................................................................. 91
W after A ttenuation....................................................................................................... 91
Sedim ent A ttenuation............................................................................................... 92
Backscatter.................................................................................................................. 94
N earfield Correction ............................................................................................. 95

B LIST OF VEN D ORS ........................................................................................... 97

C MONLOG 1.0 PROGRAM LISTING.......................................... ..... ............99

REFEREN CES ................................................................................................................ 147

BIO GRA PH ICA L SKETCH ........................................................................................... 150

LIST OF TABLES

Table page
3-1 Values of sinh(B)/B for extreme concentrations ................................... ............ 30

4-1 Calibration constants for instruments used in present analysis ............................... 50

4-2 Conditions at the measurement site during experiments examined in this study...... 60

LIST OF FIGURES

Figure page
2-1 External components to acquisition system........................................... .............. 5

2-2 Real-time processing of collected back-scattered signal profiles............................ 10

2-3 Block diagram of internal system components...................................... ........... 12

2-4 TPU timing in synchronized pulse width modulation mode ................................. 14

2-5 Memory map of data memory on the model 7..................................................... 18

2-6 68000 series assembly necessary for binary search algorithm ................................23

3-1 Expected voltage for given concentrations, 5 MHz transducer............................... 27

3-2 Significance of sinh(B)/B term in acoustic backscatter equation............................ 30

3-3 Numerically generated profiles for (a) 1.0, (b) 2.25, and (c) 5.0 MHz transducers ..40

3-4 Resulting (a) concentration and (b) median grain sizes from inversion technique
using 2.25 and 5.0 M Hz profiles ...................................... ................................ 40

3-5 Resulting (a) concentrations and (b) median grain size from inversion technique
using 1.0 and 2.25 M Hz profiles ................................... ................................. 41

3-6 Resulting (a) concentrations and (b) median grain size from inversion technique
using 1.0, 2.25, and 5.0 M Hz profiles ................................... .................................41

3-7 Comparison of known and calculated concentrations (a) shown at distinct
concentrations and ranges, and (b) shown as the mean error of all concentrations
w ith range ............................................................................................................ 43

3-8 (a) Comparison of known and determined median grain size and (b) the resulting
error w ith range.................................................................................................... 43

4-1 Sensor Insertion System (SIS) ............................................................................. 48

4-2 Instrumentation used in project.................................................................................. 50

4-3 Calculated concentration profiles from (a) 2.25 MHz and (b) 5.00 MHz
calibration data using optimum system constant and DC offset versus known
concentration............................. ........................................................................... 52

4-4 Best fit normal cumulative distribution function to sieved grain size data; sample
taken at location of and prior to run 18................................................................. 52

4-5 (a) Profile taken from pier on north and south side with superimposed
experimental water depths and (b) corresponding sieved median grain size ...........53

4-6 Multiple roots in variance-size relation ................................................................ 54

4-7 Change in bottom location due to settling of instrument framework ...................... 56

4-8 Perceived median grain size profiles from run 19 and run 20 and corresponding
near bed concentration profiles......................................... ............................. 58

4-9 Surface elevation spectrum for run 18 with 80% confidence intervals ................... 61

4-10 Plot of the 100 mg/1 contour for whole time series of run 18................................ 62

4-11 Plot of the 1 g/l contour for whole time series of run 18 ....................................... 62

4-12 Time series of the squared bottom velocity magnitude and vertically integrated
concentration ...................................................................................................... 63

4-13 Concentration spectrum for run 18 ............................................................... 64

4-14 Bottom velocity spectrum for run 18............................................... ................ 66

4-15 Cumulative variance functions from concentration and surface elevation energy
spectra......................................................................................................................... 67

4-16 Cumulative variance functions from concentration and surface elevation energy
spectra..................................................................................................... .................. 69

4-17 T50 indicating less lower frequency (high period) relevance with increased
distance from bed ................................................................................................ 70

4-18 Surface displacement time series with envelope determined by Hilbert
transform ................................................... ........................................................ 71

4-19 Spectrum of envelope ......................................................................................... 74

4-20 Coherency function between square of bed velocity magnitude and near bed
concentration for run 18 ...................................................................................... 76

4-21 Concentration spectrum with areas of coherence > 60% indicated....................... 77

4-22 Phase of transfer function for run 18 ................................................. ............. 79

4-23 M agnitude of transfer function for run 18 .......................................... .......... ... 80

4-24 Results from run 23............................................................................................ 82

4-25 Results from run 25............................................................................................ 83

4-26 Results from run 27............................................................................................ 84

A-i Normalized total scattering cross section........................................... ............. 93

A-2 Form function...................................................................................................... 95

ix

KEY TO SYMBOLS

A Acoustic transducer system constant

a,a, Sediment particle radius (as determined by sieve analysis); mm

Radius of acoustic transducer's piezoelectric crystal; m
a,

B Total local attenuation of sound; Nepersm
m

c Sound speed in water; m/

C(z) Mass concentration; or kgm

o Concentration of suspended sediment at range z0; g or km

Co Empirical relation in form function

d Grain diameter (as determined by sieve analysis); mm

F(z) Backscatter function

Fo Backscatter function at range Zo

f Form function

f Frequency; Hz

fM Acoustic frequency; MHz

I Integrating factor used in finding explicit concentration solution

Iij Discrete form of integral

i Index (chapter 2 table element; chapter 3 transducer number)

j Index bin number

K, Constant in form function; Kf = 1.14 for non-cohesive sedimentary
material

Ka Constant in normalized total scattering cross-section; Ka = 0.18 for
quartz sediment

ko Thorne's (1993) acoustic transducer system constant;

m Number of acoustic transducers

N Number of elements included in RMS calculation

n Highest bin number (n +1 total bins)

n Power of inhomogeneous term in Bernoulli equation

P(z) Backscattered pressure; Pa

p(a) Probability distribution of sediment radii; Ym
mm Coefficient in Bernoulli equation (substitution)

p() Coefficient in Bernoulli equation (substitution); p

Q() Coefficient of inhomogeneous term in Bernoulli equation (substitution); p

Qz) Coefficient of inhomogeneous term in Bernoulli equation (substitution)
Coefficient of inhomogeneous term in Bernoulli equation (substitution);
-Q

Ri ith element in RMS lookup table

S Acoustic transducer system constant;

Sxx, S, Energy spectra; (time series units)2 s

T Temperature; C

t Substitution in Bernoulli equation solution; C-'

V(z) Output voltage from acoustic transducer; V

Vo Output voltage at range zo; V

x Dimensionless particle radius

xI, x2 Empirical constants in form function; x, = 1.4, x2 = 2.8

Zx Cumulative variance function

z Range from transducer; m

Zo Range to first point in analysis; m

z, Theoretical nearfield limit; m

a, (z) Sound attenuation due to sediment; Nepers
m

aw Sound attenuation in water; Nepersm

7 Constant of integration in explicit solution for concentration. Evaluated
using known values at zo

E Empirical constant in nearfield correction relation; E = 2

S(z) Local sediment attenuation proportionality; (NepersXm2)

771,772 Empirical constants in form function; 77, = 0.5,772 = 2.2

P c Mean concentration across transducers; or kgm

Uo Mean grain diameter in phi

V, ,v2 Empirical constants in form function; v, = 0.37,v2 = 0.28

P, Sediment density; kg/m

72 Variance
a7

c Standard deviation of concentration across transducers; or km

a7 Standard deviation of grain diameter in phi

T Acoustic pulse width; s

0Logarithmic grain diameter; 0 = log2 d

X Normalized total scattering cross section

Z (z) Nearfield correction parameter

0o Nearfield correction parameter at range Zo

Abstract of Dissertation Presented to the Graduate School of the University of Florida in
Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

DEVELOPMENT AND FIELD APPLICATION OF A LITTORAL PROCESSES
MONITORING SYSTEM FOR EXAMINATION OF THE RELEVANT TIME
SCALES OF SEDIMENT SUSPENSION PROCESSES

By

Eric D. Thosteson

December 1997

Chairman: Daniel M. Hanes
Major Department: Coastal and Oceanographic Engineering

A microcontroller-based system of oceanographic instrumentation providing a

comprehensive set of measurements relevant to sediment transport processes has been

developed. Analysis of the data provided by the system yields time series of vertical

profiles of mean sediment size and concentration, horizontal profiles of bedform

geometry, and single location measurements of flow velocity, pressure, turbidity, and

water temperature. Details of the system architecture, including capabilities provided by

both hardware and software contained within the system are given.

An improved method for the determination of suspended sediment size and

concentration from the system's acoustic backscatter intensity measurements is

presented. By retaining the size dependence throughout the derivation for an explicit

solution for concentration, a new explicit solution to the acoustic backscatter equation

results. This new concentration solution improves the technique for determining median

sediment size by incorporating sediment attenuation in the calculation. Because this new

technique relies on the minimization of the variance in concentration as determined by

different frequency transducers, the previous technique of pairing transducers of different

frequencies is replaced by a technique making use of any number of different frequency

transducers. The new size/concentration inversion technique is tested using both

simulated and laboratory data. Numerical precision is shown to be the only source of

error with the use of simulated data. Laboratory tests result in less than 20% error in the

determination of both concentration and size over a range of nearly one meter.

Finally, suspended sediment concentration data from the nearshore region

obtained from an experiment performed in Duck, North Carolina, are examined to find

the relevant time scales of sediment suspension. In this location, low frequency forcing

mechanisms are as significant in suspending sediment as the incident-band wave forces

typically used to model suspension. Like wave groups, this low frequency forcing results

from the linear superposition of velocity components in a narrow band of frequencies.

When these frequency interactions are considered, coherence greater than 60% is found

between the velocity squared and the near-bed concentration across most of the spectrum.

CHAPTER 1
INTRODUCTION

With continuing coastal development, the need for proper coastal planning and

management of its resources grows. Towards meeting this need, several models of beach

evolution are currently used as tools for predicting shoreline change and for design of

coastal structures. Improving these models requires a better understanding of the

underlying physical processes of sediment transport. The capability of a theory to

describe the basic physics of a process is dependent on the accuracy of measurements of

the process. Refinement of the theory of the basic mechanics of sediment transport thus

depends upon improved measurements of the processes.

Developments in computers and instrumentation have not only improved the

capability of measurement of sediment transport phenomena in the laboratory, but have

made possible field measurements not possible a decade ago. While the control and

repeatability of laboratory experimentation is appealing, there are factors which influence

sediment transport in the field, which are neglected in laboratory studies. The evidence

of this is the disagreement often encountered between results from laboratory and field

experiments. Thus, it is desirable not only to verify laboratory results in the field, but

also to improve field experimentation apparatus and technique such that other elements

influencing the motion of sediment can be found.

In this dissertation, a new system of instrumentation developed for field

measurement of small-scale sediment transport and the associated hydrodynamics is

1

presented. Chapter 2 begins the presentation with details of the instrumentation and

architecture of the most recently developed system. This system was developed after

several refinements of earlier systems. Each refined system was developed in response to

newly discovered limitations of the prior system. Improvements from earlier versions

include new and better instrumentation, faster communications, and less restrictive data

storage requirements. In all, five systems were developed, beginning with the system

used in the Supertank project (Hanes et al., 1993; Thosteson, 1995). Subsequent systems

were used for the Vilano Beach project, the Duck94 project, and the SIS95 and SIS96

projects. During the writing of this text, the latest system is being used in the

SandyDuck97 project. In later chapters, the data presented are not that from the system

described in Chapter 2, but instead are from the next most recent version of the system.

In an effort to present the reader with the most recent and useful information, Chapter 2

presents the newer system. The overall architecture has changed little since the version

used in the Duck94 experiments, but the reader will be alerted to differences in the

systems when appropriate.

Perhaps the most powerful instrument and certainly the most demanding

computationally on the system is an acoustic backscatter system, used for obtaining high

resolution measurements, both spatially and temporally, of suspended sediment

concentration and size. Recent improvements to the technique of determining

concentration from the measured backscattered intensities are further extended in chapter

3 to include determination of the mean grain diameter of particles in suspension. The

new technique presented removes attenuation assumptions previously required in order to

determine the particle size and reduces the computational effort in determination of both

concentration and size. Also, this technique allows more than two unique frequency

transducers to be used in a single computation. This reduces the possibility of obtaining

multiple solutions for size, which can arise when only two unique frequencies are used,

and again reduces computational effort.

In chapter 4, the field experiments performed the fall of 1996 at the SIS96 project

are reviewed and the analysis of the resulting data is covered. Field application of the

acoustic analysis techniques presented in chapter 3 for determination of suspended

sediment size will be investigated. The remainder of the analysis is centered on

determining the frequency ranges in which sediment suspension is most prevalent. It is

seen that often the most dominant suspension events occur with a frequency much lower

than the frequencies of the incident waves, though very little energy is found in the

spectrum within the infragravity band. A coherency function analysis with the square of

the bottom velocity shows that the low frequency forcing responsible for the sediment

suspension results from the interaction between frequencies in the incident band of the

spectrum. Interaction between frequency components is reviewed, showing that the

resulting low frequency forcing is caused by wave groups. In chapter 5, conclusions

from this research and suggestions for future research will be presented.

CHAPTER 2
SYSTEM DEVELOPMENT

With more sophisticated oceanographic instrumentation comes the need for more

advanced data acquisition and instrument control systems. Evolving from acquisition

systems used in several earlier experiments, a new system has been developed for use in

the SandyDuck97 project, held at the Field Research Facility (FRF) in Duck, North

Carolina in late 1997 (Hanes et al., 1993). This new system delivers highly accurate

measurements of both the hydrodynamics and the resulting sediment processes with high

temporal and spatial resolution. The focus during the development of this system has

been on the ability to collect an unbroken record of data, consisting of highly

synchronous data from all instruments attached to the system, with duration of several

days. Such a record allows the investigators to utilize standard time series analysis

techniques to examine sediment processes on time scales ranging from seconds to days

without making assumptions typically required to account for gaps between data records.

In its current form, outlined in figure 2-1, the system utilizes a three frequency acoustic

backscatter system (ABS), two acoustic Doppler velocimeters (ADVs), three multiple-

transducer arrays (MTAs), an optical backscatterance sensor (OBS), a pressure sensor, a

compass, a tilt meter, an external temperature sensor, and an underwater video camera.

Internal monitoring includes leak detection, internal temperature, unregulated and

regulated shore supplied voltages, and the internal battery voltage. This chapter

Figure 2-1. External components to acquisition system.

describes the system architecture, essential algorithms responsible for data collection and

processing, and the system's capabilities.

2.1. Instruments

The choice of instrumentation used by the current system was made to most

accurately record as many parameters relevant to sediment suspension processes as

possible with minimal flow obstruction. Waves and currents are measured using a

Transmetrics pressure sensor and Sontek three-axis acoustic Doppler velocimeters

(ADVs). ADVs were chosen due to their ability to provide high-resolution velocity

measurements for long periods of time without the zero-drift difficulty associated with

electromagnetic devices an important consideration with the long records which will be

collected with this system. Tilt and compass transducers will ensure accurate positioning

of the ADVs and monitor any frame motion or failure. For a combined concentration and

turbidity measurement, a single D&A Instruments OBS is utilized. Three Seatek MTAs,

two operating at 2.00 MHz and the third at 5.00 MHz, are used for bottom bedform and

slope measurements. MTAs are linear arrays of acoustic transducers which provide high

resolution measurements of bedforms and the local slope of the seabed. The 5.00 MHz

unit contains 32 transducers spaced 1.5 cm apart, making it ideal for measurement of

small-scale bedforms (wavelengths of order 5 to 50 cm). Each 2.00 MHz unit contains

16 transducers spaced 6 cm apart, providing excellent measurement of larger scale

bedforms (wavelengths of order 20 to 200 cm). To this point in time, all three MTAs

have been used together to form a single 2.5 m array, giving bedform measurements

ranging from small to large scale (Jette and Hanes, 1997). Since orientation of bedforms

can not be determined when the MTAs are used in this manner, orientation is determined

from video observations with a DeepSea Micro-SeaCam 1050 when turbidity is low and

from an independent Simrad Mesotech model 900 rotating scanning sonar system, not

described in this text. Profiling for suspended sediment size and concentration is

accomplished using an acoustic backscatter system (ABS) made by the Centre for

Environment, Fisheries and Aquaculture Science (CEFAS, formerly known as The

Fisheries Laboratory), a description of which follows. Finally, to determine speed of

sound and acoustic attenuation parameters for use in evaluating data from the acoustic

instruments, water temperature is measured. Further information on the instrumentation

vendors is found in appendix B.

Simultaneous operation of multiple acoustic concentration profilers at different

frequencies has successfully provided measurements of profiles of both sediment

concentration and mean sediment size (Hay and Sheng, 1992). The present system

includes an ABS utilizing 1.0, 2.2 and 5.0 MHz transducers. Sync and trigger lines

provide precise control over selection and firing of the transducers. A single analog

signal line returns the envelopes of the backscattered analog voltages. Upon receipt of a

pulse on the sync line, the ABS begins a cycle such that the first pulse on the trigger line

fires the first transducer and switches the signal line to transmit data from the first

transducer to the acquisition system. The next pulse on the trigger line causes the second

transducer to be fired and switches the signal line to transmit the second transducer's

voltage envelope. The cycle continues with the third pulse selecting the third transducer.

Additional pulses on the trigger line restart the cycle, beginning again with the first

transducer.

Of the instruments listed above, the system is responsible for analog to digital

conversion and subsequent storage of the data from all but the MTAs and the video

camera. Responsibility of the system to the MTAs is limited to setting acquisition

parameters and collecting data through a digital serial connection shared between all

three MTAs and the main system. The video camera is switched on and off and provided

power by the system. The video signal simply enters the underwater package and exits

immediately through the shore cable. Of the remaining instruments, all but the ABS are

instruments that collect data from a single point in space only. For this reason, they will

henceforth collectively be referred to as the single point instruments (SPIs). The system

has been designed to provide data at a maximum rate of four final measurements per

second. Signals from each of the SPIs are passed through two-pole, linear-phase, anti-

aliasing filters with a cut-off near 10 Hz. This filter, combined with over-sampling of the

SPIs at 100 Hz and subsequent averaging to the desired final measurement rate,

eliminates the possibility of aliasing and minimizes signal degradation due to the

frequency response of the filter.

2.2 System Demands

Fundamental to the design of this system is the desire to minimize the travel

distance of analog signals from the instruments. The purpose of this is to minimize

filtering effects, both in amplitude and phase, and reduce the introduction of additional

signal degradation caused by the otherwise necessary modulation and demodulation of

the raw high-frequency signals. We therefore place the system offshore, which results in

new design demands related to power consumption, size, storage requirements, and

system monitoring functions. We also desire that the system be fairly self-contained,

requiring only minimal external equipment support and user-intervention. User-

intervention consists of configuring sampling schemes before data collection and

offloading of any data residing on the system after collection is complete. Finally, given

the specific modes of operation of the instruments to be used in the study as well as any

other demands imposed by them, synchronous measurements from all instruments is of

the utmost importance.

Ideally, each of the three ABS transducers should measure the identical profile at

the same instant in time, giving three coincident and collocated measurements of acoustic

backscatter. In practice, this is difficult for several reasons. First, physical limitations

due to the size of the transducers and the acoustic beams from the transducers prevent

achieving measurements from perfectly over-lapping profiles. Second, to statistically

reduce the random fluctuations in the backscattered signal due to coherence of the

returned signal, for each transducer, single profiles of the mean power are constructed

from several consecutive profiles. It is important to realize that the mean power must be

used rather than the mean voltage, as the concentration of suspended sediment is

proportional to the power of the backscattered signal. This is shown by the acoustic

backscatter equation written in terms of the mass concentration, C (Thorne et al., 1991).

C(z)= Z e4y (a,+a,) (2-1)

In this equation, P is the backscattered pressure (proportional to the voltage measured at

the crystal), z is the distance from the transducer, V is a function describing near-field

characteristics of the transducer, ko is the system constant, and aw and a, are water and

sediment attenuation parameters respectively. The system constant characterizes specific

physical and electrical properties of a particular transducer. These parameters and the use

of the acoustic backscatter equation are examined in more detail in chapter 3 and

appendix A. Since the sediment size and concentration profiles are in general functions

of time, additional fluctuations in the backscattered signal due to random reconfiguration

of the particles in the ensonified volume and due to changing sizes and concentrations

certainly result. To account for these limitations, several conditions shall be imposed on

the acquisition system. Profiles must be acquired at a high enough rate such that the

changes with time in size and concentration profiles are small. Within this period, a

sufficient number of profiles need to be collected such that a representative average will

statistically minimize the fluctuations. Finally, the ensemble of samples from any one

transducer should be well distributed over this entire period.

Acoustic concentration profilers very quickly produce large volumes of data. To

meet the three conditions imposed above, the current system collects 100 profiles per

second from each of the three transducers on the ABS. Each individual profile consists

of up to 120 12-bit samples from the analog signal, sampled at 100 kHz. For ease of

processing and storage, each 12-bit sample, hereafter referred to as a bin, is stored in a

zero-padded 16-bit word. As each profile is collected, the squared value of each bin is

calculated using a highly efficient table-lookup algorithm. This squared value is then

added to a running sum of squares, such that for each bin in the profile, the root mean

squared (RMS) value can be calculated after a specified number of profiles have been

collected. Again in the interest of minimizing computational time, RMS values are

calculated from the sums through the use of a binary search method through a previously

calculated table of sums. Figure 2-2 outlines the entire reduction procedure for the case

of computing RMS profiles of 120 points from 23 consecutive profiles a common

4 .. .
E, F:R return ds

120 points sampled at 100 kHz

i square d-:. '
profiles ...

23 profiles collected at 100 Hz
Figure 2-2. Real-time processing of collected backscattered signal profiles.

configuration of this system. Algorithms for computing the square and the RMS will be

detailed later in this text. The final outcome of this on-the-fly processing of the

backscattered signal profiles is a set of statistically meaningful profiles requiring

significantly less storage space and significantly less communications bandwidth to

transmit real-time digital data from the system.

2.3 System Architecture

To meet the control, communications, synchronization, processing, and data

collection requirements given, a low-power Onset Tattletale model 7 data logging engine

is employed. The model 7 provides 512 kilobytes of flash EEPROM storage and 256

kilobytes of static RAM for system and application software, as well as an additional 2

megabytes of pseudo-static RAM for data storage. In addition, it includes 28 digital I/O

lines, a 4 channel analog to digital converter, a parallel I/O port, RS-232

communications, a real-time clock, and 500 megabytes of hard drive storage. Central to

the model 7 is the Motorola 68332 microcontroller. The 68332 incorporates the CPU32

central processing unit executing a superset of MC68000 instructions. For use in the

present system, the microcontroller operates at 16 MHz. Also, the 68332 incorporates a

time processing unit (TPU), which is essentially a special purpose slave processor that

controls two timers and sixteen I/O lines. In the present system, this TPU is

indispensable for handling synchronization as well as hardware and software triggering.

Figure 2-3 outlines the additional interfacing made to the model 7 in the system.

Additional information and descriptions follow.

Further extending the features of the model 7 is a Daedulus Research MUX32

board. This board further multiplexes each of the four analog to digital channels by

eight, allowing the model 7 to sample up to 32 analog signal lines. These 32 channels

will be referenced by a port, numbered 0 to 3, corresponding to the original channel

number on the analog to digital converter, and a channel number numbered 0 to 7, now

referring to one of the newly multiplexed lines. Besides increasing the number of analog

lines, the MUX32 also buffers the 16 TPU digital I/O lines and replaces the model 7's

analog to digital reference with a voltage reference of higher precision.

Figure 2-3. Block diagram of internal system components.

Interfaced to the model 7 is a filter board, which provides the anti-aliasing for the

analog signals, as previously mentioned, a power board, and several communications

components. On the shore end, the DC voltage supplied to the package is such that 30 V

is input to the system package offshore with all instruments turned on. For example, if

the resistance in 500 meters of cable is such that 15 V is lost due to this resistance, 45 V

would be supplied from shore. Given 30 V input, the power card regulates power for all

components of the system in addition to providing power switching for each of the

instruments independently. Also, the power card keeps 24 V of NiCd batteries charged,

for use in supplying short bursts of high current to power the model 7's hard disk drive.

These batteries also can power the system for a short time in the event the power

connection from shore is compromised. Communications from the package to shore is at

57.6 kilobaud through two 50 ohm coaxial cables using RS-422 transceivers on both ends

of the connection. Such a connection has been shown in previous experiments to be

reliable through 500 meters of coaxial cable. For reasons discussed shortly, RS-232

communications from the model 7 go through an Onset Tattletale model 8. Between the

model 7 and model 8, serial communications are at 38.4 kBd. The serial communications

path then goes from the model 8's 57.6 kBd RS-232 connection to the RS-422 transceiver

on the communications card.

By default, the main clock, TCR1, on the TPU of the 68332 runs at one fourth of

the CPUs clock frequency. Instead, with a CPU clock frequency of 16 MHz, TCR1 is set

during system initialization to operate at 1 MHz due to the range of clock frequencies that

must be generated by the TPU. Without making this change, the risk of losing

synchronization between channels because of counter overruns within the TPU increases.

MonsterLog 1.0 (MONLOG1O.C) TPU timing diagram

TCR 1 -
1 MHz
Base frequency/16

TPU channel 14 -
100 kHz
A to D converter trigger

TPU channel 9 -
100 Hz
F3 interrupt

TPU channel 10 -
100 Hz
F2 interrupt

TPU channel 11 -
100 Hz
F1 interrupt

TPU channel 13 -
400 Hz
Hardware trigger

TPU channel 12 -
100 Hz
Hardware sync pulse

.................... --------,I---- -I- -I
S II I I ,,I l

I i l I I I K
7 US

R^ ...F llL...1LL nnnnnn... nnnnnnV
F... .. ...
5 us 5 us

S 30 us

5 us

-, I*-
S30 usus

*5 us
305 us 5 us 5 us

5us
E -

C Ci C i 4 C ; q "-t C C C 6 i
C C C C d S 5

Figure 2-4. TPU timing in synchronized pulse width modulation mode.

TPU channels are used in the present system for hardware synching and triggering of the

ABS, hardware triggering of the analog to digital converter, and for generation of the

interrupts necessary for the triggering of software exceptions used for data acquisition.

By using a mode of operation on the TPU known as synchronized pulse width

modulation, several TPU lines are set to run continuously as perfectly synchronized

clocks. Figure 2-4 illustrates the clock rates, pulse widths, and timing between channels.

Once these clocks have been started, they will run continuously and perform their

associated tasks without any intervention from the CPU. Analog signals are continuously

sampled, so the program simply decides which samples to keep. Data acquisition is

performed entirely within exceptions, described in more detail later, so it can be started

by simply enabling the interrupts generated by TPU channels 9 through 11.

Since both ACP profiles and SPI data are collected at 100 Hz and subsequently

processed down to the desired final data acquisition rate, at some instant immediately

after processing, a large burst of data is both saved to the data memory within the model

7 and output from the model 7. This burst of data, up to approximately 1000 bytes of

data, must be transmitted from the model 7 within roughly 10 ms to prevent interfering

with the timing of the data acquisition. Rather than complicating the communications

requirements by use of a MBd serial line, these data are simply output to the existing

parallel port on the model 7 into a FIFO buffer and then into the model 8 mentioned

above. By continuously monitoring the parallel and serial connections to the model 7, the

model 8 buffers all communications and allows the use of a single, 57.6 kBd, serial

connection to the package. This 57.6 kBd line is sufficient to allow transmission four

times per second of 120 point profiles from each of the three ACPs and the additional 14

channels of interest.

2.4 Software and Algorithms

A program called Monlog 1.0 controls the system. Monlog is written mainly in

C, with additional assembly coding of the actual data acquisition and data processing

routines for speed. Most of the code handles user-interface functions, such as creating

menus, handling user I/O, setting the time, setting record numbers, defining the sampling

scheme, and offloading stored data records. The algorithms necessary to handle these

functions are trivial and will not be covered in this text, although a full listing of the

program is given in appendix C.

Typical operation of the system begins at power-up. The system software handles

initialization of the core components of the Model 7. To customize specific portions of

the initialization, such as the necessary modification of the TPU configuration register to

establish the TPU's 1 MHz clock rate, the contents of the Model 7's serial EEPROM are

modified. Immediately following the system initialization, control is given to Monlog.

Monlog starts with system specific initialization, enabling charging of the system

batteries and ensuring that all interfaced instruments are started in a known state.

Communications baud rates are established and system variables initialized. The TPU

channels are set up and started as was shown in figure 2-4, with the interrupts they

generate initially disabled, and appropriate default values are set to govern data

acquisition. Finally, the internal monitoring channels are sampled, and the results, along

with the main system menu are displayed to the user.

Once the main menu is displayed, several options are available to the user. A

standard input routine is used throughout the program, such that a timeout feature can be

utilized at each opportunity for input. This guarantees that any involuntary selections,

such as could be caused by line noise, will not leave the program in an indeterminate

state. Upon any timeout, the program returns control to the main menu routine,

maintaining original values for any values that may have inadvertently been changed.

Included in the main menu are options that allow the user to power each of the

instruments interfaced to the system individually and to subsequently test them in an

interactive test mode, displaying results to the user in real time. Additional options are

used to offload data resident on the systems hard drive to shore using the Xmodem data

transfer protocol and to define the sampling scheme and system modes of operation.

An emergency storm-mode can be enabled from the main-menu. When enabled,

Monlog will detect loss of communications while in the main menu routine, and should

such a condition be detected, the program will begin acquiring data using a conservative

pre-defined acquisition scheme. Since loss of communication most likely will coincide

with loss of shore-supplied power, this scheme is defined for collection of short data files

at regularly spaced intervals in an attempt to conserve battery power for a time period

roughly equal to that of a storm.

Preparation for acquisition begins with the user setting the time and date and

defining the sampling scheme. Parameters such as the number of points per ABS profile,

number of profiles to include in an ensemble RMS profile, final sampling rates for the

ABS, SPIs, and MTAs, and sampling durations are all defined by the user. Based on this

information, Monlog allocates all available data memory and sets up pointers to

18

appropriate memory locations to create data headers, store data, create look-up tables for

use by the squaring and RMS algorithms, and temporarily store sums during data

acquisition. Figure 2-5 shows the structure of these pointers in memory based on values

determined from the user configuration. Note that the ranges labeled TMU and PAD

together form a Mathwork's Matlab data file in the memory of the model 7. Such

Monster Package memory map & setup

SOBASE
x' lookup table
RTBASE 4096x4
RMS lookup table
F1SUM 4096x4
F1 summation memory
F1SQR NPP'4
F1 XF summation memory
F2SUM NPP'4
F2 xsummation memory For use in data processing not
F2S0R NPP'4
6 -1 -- --- Nincluded in MAT file.
F3SUM 2 NPPs mo 8*NOSP+24'NPP+32768
F3 summation memory
F3SQR NPP'4
F3 X summation memory
SPSUM NPPO4
SPI x summation memory
SPSQR NOSP'4
SPI x' summation memory
F1HED NOSP'4
F1 Matlab data header
FIRMS 26
F1 RMS prolilie tim nes
F2HED NPP'NFP'2
F2 Mallab data header
F2RMS 26
F2 RMS profile lime series
F3HED NPP.NFP'2 TMA128 variable
F3 Mallab data header Stored as single data file on TT
F3RMS 26
F3 RMS profile time series
SPHED NPP'NFP'2
SPI Mallab header
SPMEN 26
SPI time series
MT1HED NOSP'NFP'2 TMU variable
S MTA # Matlab header NFP*(NPP'6+NOSP*2)+
MT1BAS 26 MTA NUMSCANS*152+
MTA #1 profile time series 26*9+24
MT2HED MTA NUMSCANS72
MTA #2 Matlab header
MT28AS 26
MTA #2 profile time series
MT3HED MTA_NUMSCANS'40
MTA #3 Matlab header
MT3BAS 26
MTA #3 profile time series
TMHED MTA NUMSCANS'40
Matlab Time header
TMBAS 26
Data ile start/end times
DMHED 24
Matlab Dummy header
DMBAS 26

OFFEND+1 Dummy variable space PAD variable

Figure 2-5. Memory map of data memory on the model 7.

compact structuring beforehand allows one to offload the entire range of memory

sequentially to the shore-based computer and immediately load the data record into the

analysis software for data inspection and processing. Note also that although the MTA

data are not collected directly by the model 7, space is allocated within the data file for

each of the MTAs. MTA data are offloaded from each of the MTAs after the rest of the

data file has been filled. This data format is convenient since the data from all of the

instruments are stored in the same file along with both the starting and ending times and

dates of the collection. In prior systems, all data collected by the system resided in these

data files. With the added ability to collect unbroken data files of length greater than

what could possibly be stored in this memory segment, only the first portion of the

collected data, whatever portion will fit within the 2-megabyte limit, is stored in this

format.

After the dynamic memory configuration, all that remains for Monlog to do is

enable the interrupts generated by the TPU channels and monitor the acquisition. Four

exception routines handle all of the data acquisition. Outside of these four exceptions,

the main process (that process running prior to the CPU's receipt of any interrupt)

monitors the progress of the exception routines and at the appropriate time, performs the

data analysis and storage. Following the timing diagram shown in figure 2-4, the order of

events following the memory initialization is as follows:

1) Interrupts generated by TPU channels 9 to 11 are set to execute the same

exception routine, entitled NOTHING.

2) Interrupts are enabled.

3) Upon identifying an interrupt from TPU channel 9, that interrupt associated with

the acquisition of transducer F3, the NOTHING routine reassigns each interrupt

its own exception routine. This insures that a profile from transducer Fl will be

the first to be sampled.

4) A hardware sync pulse is sent from TPU channel 12 to the ABS, effectively

resetting the ABS to sample transducer Fl on the next received trigger pulse.

5) A hardware trigger is sent from TPU channel 13 to the ABS, triggering transducer

Fl.

6) Shortly after sending the trigger to the ABS, TPU channel 11 generates an

interrupt, which starts the exception routine responsible for sampling transducer

Fl.

7) As data are continuously being sampled from the analog to digital converter in

relation to the triggering from TPU channel 14, each sample is acquired from the

converter, squared by use of a previously calculated lookup table, and added to a

running sum appropriate for the location in the profile.

8) Transducer Fl's exception routine completes and acknowledges the interrupt,

returning control back to the main process, which to this point, continues to

monitor the progress of the exceptions.

9) The above process repeats with TPU channel 13 triggering transducer F2, and

TPU channel 10 generating the interrupt that executes transducer F2's sampling

routine.

10) Channel 10's interrupt is acknowledged, again returning control to the main

process.

11) Again, the process repeats, with TPU channel 13 triggering transducer F3, and

TPU channel 9 generating the interrupt that executes transducer F3's sampling

routine.

12) This sampling routine additionally selects and samples each SPI attached to the

system, adds the result to an appropriate running sum for the particular SPI, and

decrements the counters being monitored by the main process.

13) Once the main process has detected that the appropriate number of profile

acquisitions have occurred, it again reassigns the interrupts to execute the

NOTHING exception routine. This routine has the task of keeping a tally of the

number of times TPU channel 9 interrupts occur. After a designated number of

occurences, this routine reassigns the interrupts to again sample analog data.

14) Immediately following the reassignment of the interrupts to the NOTHING

exception, the main process performs data processing and storage. Note that the

NOTHING exception will be called a sufficient number of times to allow the data

processing and storage tasks to complete before data sampling resumes.

This process continues until the time limit specified in the sampling scheme definition is

surpassed.

To this point, the details of the data analysis have been neglected. The first

analysis algorithm in need of description is the lookup table used to square the incoming

samples from the ABS. Normal multiplication is far too costly in processing time, so a

table look-up algorithm is employed. This table is simply a 4096 element (corresponding

to each possible value from the 12-bit analog to digital converter) list of squared values.

Given a value to be squared, this value is used as an index into the list, or table. The

value located at the given index is the square of the index. Next, the second algorithm is

responsible for providing the RMS value from a given running sum of squared values.

Again, standard algorithms involving multiplication and division demand too much

processor time, so an alternative approach is used. Since the number of elements used in

the calculation of the RMS value is known in advance in the given situation, a table of

sum of square values can be calculated prior to data acquisition. If N elements are to be

used, then the value in the table corresponding to the ith index is given by the following:

R, =N(i+0.5)2 (2-2)

By finding the lowest number in the table greater than a given running sum of squares

value, the resulting index to that element of the table is a very close approximation to the

RMS value for the given sum. For example, if an RMS value is to be determined from 23

elements, a table of borderline sums of 23 squared values can be generated. The first

element in the table, corresponding to index 0, would be as follows:

R0 = 23x(0.5)2 = 5.75 (2-3)

If a given sum of squares, after 23 samples have been added to it, is less than this value,

than the RMS value is closely approximated by the index into the table. Finally, since the

elements of the table are ordered by value, rather than searching each of the 4096

elements of the table for the first value greater than that given, a binary search algorithm

is utilized. The search begins at the center of the table, and the values are compared.

Based on the results of this comparison, the next comparison will be with the value

centered in either the upper or lower portion of the table. With each successive

comparison, the number of table elements remaining to compare is cut in half. Since all

divisions are by 2, binary shift operations are used to perform the division. For a 4096

element table, only 12 comparisons need to be made. The assembly code used for this

comparison is shown in figure 2-6. Note that the variables used in the beginning of the

routine as memory pointers were initialized prior to this segment of code.

routine for F1

nidxres move.l _flsqr,a0
move.l _tflbas,al
move.1 _rtbase,a2
;RMS routine Fl
avloopfl move.l (a0),d0
move.l #0,(a0)+
move.l #$0800,dl
move.l #$0400,d4
move.l #0,d3
move.l #0,d2
sredofl lea ($0000,a2,dl.w*4)
TABLE[TEST]
cmp.l (a5),d0
bge felsefl:
Y>=TABLE[TEST]
move.w dl,d2
bra dunfl:
felsefl move.w dl,d3
dunfl move.w d4,d5
or.w d3,d5
move.w d5,dl
Isr.w #l,d4
bne sredofl:
bcs sredofl:
move.w d2,(al)+

;Set base of Fl's sum(x^2)
;Set base of Fl's RMS storage
;Set base of rms table

;get byte from sum(x^2)
;initialize sum(x^2) storage
;initialize TEST
;initialize HALF
;initialize BASE
;initialize ANSWER
,a5 ;location of

;branch to else portion if

;ANSWER=TEST
;skip else portion
;BASE=TEST -> else portion
;WORK=HALF
;WORK=HALF or BASE
;TEST=HALF or BASE
;shift HALF right 1 bit
;branch back 10 times
;branch back 1 final time
;store rms value

Figure 2-6. 68000 series assembly necessary for binary search algorithm.

2.5 Summary

In this chapter, a system of oceanographic instrumentation capable of providing a

broad set of measurements relevant to sediment transport processes is presented. All

;Initialize RMS

24

instruments are in close proximity to the data system, maximizing data integrity.

Extremely efficient analysis algorithms written in C and assembly language provide data

processing and reduction during collection. Use of multiple processors allows

measurements to be both stored internally on the systems internal storage and exported to

shore during collection through serial communications. Data stored on the system are

offloaded using an error correction protocol to a compact, binary data file that can be

immediately viewed and interpreted by analysis software.

CHAPTER 3
ACOUSTIC BACKSCATTER CONVERSION TECHNIQUE

Use of a high frequency underwater acoustic transducer to profile the vertical

distribution of sediment concentration has been demonstrated under both laboratory and

field conditions by several investigators (Hanes et al., 1988; Hanes et al., 1993; Hay and

Sheng, 1992; Thorne et al., 1991; Green and Vincent, 1991). Basically, profiles of the

intensity of backscattered sound from suspended sediment are collected, and this intensity

is related to the suspended sediment concentration at each point in the profile. A typical

technique for determination of concentration from acoustic backscatter data requires

inversion of the acoustic backscatter equation for concentration, which yields an implicit

equation needing an iterative technique for solution. In order to obtain profiles of

concentration from a single transducer operating at a fixed frequency, knowledge of the

sediment grain size distribution is required prior to applying the inversion procedure. In

addition, the assumption that this size distribution either remains constant with range or

with a predetermined form at all ranges is required, as the scattering and absorption

properties of the sediment are dependent upon grain size (Thorne, 1993). Recently,

several investigators have demonstrated effective measurement of both the profiles of

sediment concentration and of the median grain size of the distribution (Hay and Sheng,

1992; Crawford and Hay, 1993). The technique requires the use of multi-frequency

acoustic transducers. Since the absorbing and scattering properties of sediment depend

on both the grain size and upon the frequency of the incident sound, each unique

25

frequency transducer provides an independent measurement of the backscattered intensity

profile. Hence, each point in the profile can be described by a number of independent

equations equal to the number of coincident and collocated measurements of unique

frequency. Typically, three transducers of unique frequency are used to collect

coincident intensity profiles. Although in theory, use of three frequencies suggests that at

each measured point, concentration and two parameters of the grain-size distribution can

be determined, typically, only concentration and one parameter of the distribution are

obtained. The sensitivity of the equations to small variations in intensity and also the

nonlinearity of the sediment size response functions are responsible for this limitation.

Lee and Hanes (1995) presented an explicit solution for concentration, to be

referred to in this paper as LH95, from the acoustic backscatter equation, significantly

reducing the computational effort by removing the need for iteration. An added benefit

derived from use of the explicit solution is removal of the ambiguity in concentration

solutions obtained by the iterative solution to the implicit equation. Figure 3-1 shows the

relation between concentration and transducer voltage (proportional to the square root of

intensity) as calculated by the acoustic backscatter equation (3-2) assuming a constant

concentration profile. From this figure, it is apparent that a single voltage value from the

transducer may result from two different concentrations. Physically, this can be

described with the following argument. At low concentrations, the sound attenuation due

to sediment in the sound path is low, resulting in an increase in the intensity of the

backscattered sound with increasing concentration. As this attenuation becomes more

dominant, the intensity of the backscattered sound begins to decrease with increased

concentration. So, from the two concentration solutions obtained from the implicit

Theoretical voltage from concentration; 5 MHz

0 1 1 ----
0 1 2 3 4 5 6 7 8
Concentration (g/l)
Figure 3-1. Expected voltage for given concentrations, 5 MHz transducer.

9 10

equation, the choice of the correct solution depends upon the magnitude of the

attenuation. The explicit solution for concentration yields only one concentration, as this

dependence on attenuation is accounted for by integration of the intensity profile.

In spite of the benefits presented by use of the LH95 explicit solution, it is

restricted in its use to concentration only. Its use requires that the grain size distribution

be known and constant with range. Holdaway and Thorne (1997) extended the

functionality of the solution by allowing the size distribution to vary but to retain a

predetermined form with range from the transducer. In the following derivation, a

similar explicit solution for concentration retaining the dependence on the grain size

distribution is found by following the LH95 development. Based on this explicit form,

the methods used for evaluating parameters of the grain size distribution are re-examined,

and an improved method, at least in computational effort, is introduced. In addition, the

existing method for evaluating the grain size parameters by pairing ACPs of different

frequencies is extended to utilize an arbitrary number of unique frequency transducers.

3.1 Theory

The equation that relates the intensity of the backscattered acoustic signal to the

concentration and size distribution of the scatterers in suspension is referred to as the

acoustic backscatter equation. This equation has been presented in several forms, each

nearly equivalent. Presented here is a general form of the equation, based jointly on the

form presented by Hay (1991) and Thorne (1993).

2 2 ( 2B (3-1)

Zo
aS Z )C(2)d2 (3-2)

The variables in equations (3-1 to 3-2) are defined as follows:

V = voltage read from transducer

z = distance from transducer

S = system constant

c = speed of sound, assumed constant with distance

T = acoustic pulse width (s)

fy = nearfield correction term (see appendix)

F = backscatter parameter (see appendix)

C = mass concentration, (sediment density in F)

a, = water attenuation parameter (see appendix)

a, = sediment attenuation parameter

B =(a, +C(z)C(z))

zo = range from transducer at which first concentration and size is evaluated.

( = local sediment attenuation proportionality constant (see appendix)

The nearfield correction term, y is included here for completeness, and it can easily be

included in the following derivation by temporarily absorbing it in the backscatter

parameter. Since it is simply another function of z, it does not complicate the derivation.

It has not been included in the following derivation, because it was not used in the

subsequent numerical simulations or laboratory tests. The complete solution, including

this term, is presented in the appendix.

In equation (3-1), the final term on the right side of the equation, (sinh B)/B,

presents difficulty when trying to obtain an explicit solution for concentration. This term

accounts for the difference in the magnitude of the sediment attenuation from the portion

of the ensonified volume closest to the transducer to the portion furthest from the

transducer. Figure 3-2 shows the magnitude of this term versus the term B, and table 3-1

shows the magnitude of this term for several cases using a distribution of quartz sand

with median grain diameter, u, = 2.65 and standard deviation, oa = 0.25. Note these

parameters are given in units of phi, defined as 0 = -log2 d where d is the grain

diameter in mm. The transducer frequencies listed in table 3-1 are the highest

frequencies used in this study. A 30 g/l mass concentration corresponds to roughly a 1%

1.07

m
c
r
"' 1.03

0 0.1 0.2 0.3 0.4 0.5 0.6
B value

Figure 3-2. Significance of sinh(B)/B term in acoustic backscatter equation.

Table 3-1. Values of sinh(B)/B for extreme concentrations.

Transducer Concentration (g/l) sinh B
B
B B
frequency (MHz)

5 30 0.5591 1.053

5 5 0.0994 1.002

2.25 30 0.0956 1.002

2.25 5 0.0172 1.000

concentration by volume, the approximate upper limit of concentration before multiple

scattering must be considered, and the 5 g/1 concentrations are the highest tested in this

study. From these results, it can be seen that for the highest concentrations for which the

present theory applies, this term can be significant when the highest frequency

transducers are used. When operating transducers of lower frequency with suspensions

of lower concentration, this term is very close to unity. Additionally, since the sediment

attenuation term within the exponential function is an empirically determined parameter,

determination of its value without including the (sinh B)/B term may compensate for its

absence. So, for the remainder of this text, the acoustic backscatter equation will be

approximated by the following expression:

v2z2 = AF(z)C(z)ez-4,a+a()
(3-3)

In equation (3-3), the system sensitivity constant, S, the speed of sound, c, and the pulse

width, r, have been combined into a single system constant, A. The system constant can

be later separated back into these constituents in order to correct for sound speed

variations.

3.2 Development

3.2.1 Concentration inversion

Beginning with the general form of the acoustic backscatter equation (3-4), the

concentration dependence is removed from the sediment attenuation term, a,, giving a

form in which the multiple term dependence on concentration is more obvious (3-5).

AF(z)C(z)= V2 (z)z2 exp(4z(a, +a,(z)))
(3-4)

AF(z)C(z)=V2(z)z2 exp 4(aw + (2)C(2))d
ZOJ

Following the LH95 derivation, first, the natural logarithm of the equation is found

In A+ln F +lnC= 2ln(Vz) + J (4a, + 4C)d

(3-6)

and then the derivative, denoted by (').

F' C V+V
-+--=2 + 4o, + 4C
F C Vz

(3-7)

Upon arranging the terms of equation (3-7), a nonlinear differential equation of the

Bernoulli type results.

C'[+ F_2V+V 4a, C= 4C2

(3-8)

Rewriting in standard form, equation (3-8) becomes the following:

C'+ p(z)C = Q(z)c

P(z=[ F' 2(Vz+V 4,,]
p(z) LF 2 Vz-

Q(z)= 44
n=2.

(3-9)

(3-10)
(3-11)
(3-12)

Following the standard method of solution for a Bernoulli equation, the following

substitutions can be made:

t=C-" = C-1; C =t-1

with

(3-5)

(3-13)

dC dC dt
dz dt -2 d
dz dt dz

(3-14)

These substitutions result in a readily solved first order linear inhomogeneous differential

equation.

- t-2t + pt-1 = Qt-2
t' + pt =Qt
t+ fit =

(3-15)

(3-16)

Where, in equation (3-16), the following apply:

p=-p
Q=-Q

(3-17)

Solution to equation (3-16) is found by first determining the integrating factor.

I=exp(q dz)

SPdz=-~ F- 2( Vz+V 4ad z=-InF +4az +21n(Vz)

I =exp(21n(Vz)+4azz-lnF)= exp(4atz)
F

(3-18)

(3-19)

(3-20)

After multiplying the equation by the integrating factor, an exact differential results,

which can then be integrated for solution.

d (Vz)2 exp(4a = -4 exp(4taz)
dz F ) F

()2 exp(4,zz)t = 4( exp(4a,w)di~ + y
F F
Zo

(3-21)

(3-22)

Equation (3-22) is then solved for t, and then finally for the concentration, C.

y-z 4( W exp(4a,2)di
t = zo (3-23)
(Vz) exp(4a,z)
F
(Vz)2 p(4
1 F exp(4alz)
C t z-(V,, (3-24)
t y- 4 exp(4a,)d2
Zo

Next, the boundary conditions (3-25) are applied for solution of the integration constant.

C=Co
V = Vo at z = zo
V- a (3-25)
F=Fo

Y= F Cexp(4ao) (3-26)

The concentration Co at the nearfield limit z0, which is the closest point to the transducer

not in the nearfield, may be estimated in many applications by assuming the

concentration and size are constant within the nearfield. With this assumption, the

implicit form of the acoustic backscatter equation, equation (3-5), simplifies somewhat,

giving equation (3-27).

Co- (Vz0)2 e4Z+) (3-27)
AF

This can be solved iteratively for concentration for a given grain size, using a zero

sediment attenuation form of the equation for an initial estimate.

C-(Vzo)2 e4o.
Co e (3-28)
AF

3.2.2 Size determination

The technique originally introduced by Hay and Sheng (1992) for determining the

median size of particles in suspension involved first approximating the acoustic

backscatter equation (3-5) by neglecting the sediment attenuation. In this way ratios

could be constructed from the approximate equations for any pair of unique frequency

transducers. Since the concentration dependence of the attenuation is removed by

neglecting the sediment attenuation, the remaining concentration terms in the equation

cancel in the formation of the ratio. Hence, the only remaining unknowns in the ratios

are functions of the size distribution of the suspended particles. By assuming the particle

size can be described by a two-parameter distribution, the log-normal distribution, and by

further assuming one parameter is constant, the ratios can be evaluated over a range of

the other parameter. The standard deviation is the parameter assumed constant and the

ratios are determined over a range of median particle sizes. Median particle size is then

found by minimizing the difference between the ratios and its known value with respect

to the median particle size. Crawford and Hay (1993) improved the technique by solving

the approximate equations first for those terms that are not functions of the size

distribution or transducer frequency. These terms are equal in all of the equations,

regardless of transducer frequency, so equating the remaining terms in the other

equations eliminates the concentration dependence. Again, the minimization technique is

applied as before to determine the median particle size.

First by solving each transducer's equation only for concentration, and then by

minimizing the variance in the concentrations predicted by any number of transducers

with respect to median sediment size, Crawford and Hay's technique is here slightly

modified. By using the explicit solution for concentration, (3-24), there is no longer the

need to neglect the sediment attenuation. In addition, once the median particle size is

found by the minimization technique, the mean concentration is readily computed from

the existing concentration solutions. The exact procedure is as follows.

The explicit solution for concentration derived above is discretized to represent

each frequency of m transducers with n+1 bins outside the nearfield region of each

transducer.

(V exp(4a,iz) (3-29)
SF.j i = 1,2,..., m
i--ligj j = 1,2,..., n

In the denominator of this expression, I is the discrete form of the integral, given by the

following:

Iij = 2 i,k exp4ak ik exp(4aiZk,1 k-1) (3-30)
k=1 Fi .k F'k-

The integration constant is defined again at point zo.

i = -(V,0 exp(4a,zo) (3-31)

It should be mentioned that zero concentration in suspension, which results in zero

voltage read at the transducer, will result in an indeterminate value of the integration

constant. The location of z0 should thus be set at the first range with non-zero

concentration, determined by the following iterative technique.

The initial concentration in the profile, located at the first point outside all of the

transducer nearfields, is found for each transducer by the former iterative technique.

Co =(Vz)2 e4zo(a+co) (3-32)
AF

Two considerations should be taken when determining this initial concentration.

First, because of attenuation, the magnitude of the voltage read from the transducer is

limited in magnitude from the above expressions, as is apparent in examination of

figure 3-1. Due to the statistical fluctuations in the backscattered signal and also to

instrument noise, it is possible that the actual signal is higher in magnitude than this

theoretical limit. In such cases, the iterative technique will not converge to a solution. A

simple divergence test in the iteration algorithm will reveal this condition. Minimization

of the difference between the initial concentration guess and that returned by equation

(3-32) will produce a good concentration approximation in such cases. Second, it is

important to realize that equation (3-32) will produce two concentration solutions for the

reasons discussed previously. For the lowest frequency transducers typically used, the

higher magnitude solution is regularly above the expected range of applicability of the

present theory, and can safely be ignored. For the higher frequency transducers, the

decision of which concentration to use must be based on physical arguments or by

comparison with the results from lower frequency transducers. In field measurement of

suspended sediment, the transducer is usually a sufficient distance from the bed, such that

the higher magnitude solution is again outside the expected range and can be safely

ignored. For example, a typical 5.0 MHz transducer with a 16 cm nearfield will give two

solutions at zo = 16 cm. As seen in figure 3.1, the lower magnitude solution will fall

between 0 g/1 and 1.6 g/l, and the higher above 1.6 g/l. Should the transducer be located

a half meter from the bed, concentrations above 1.6 g/1 will not likely be found at zo = 16

cm (34 cm above the bed), and the lower magnitude solution is the most probable

concentration.

The complete technique for solution is as follows. Equation (3-32) is solved

iteratively for the initial concentration for each transducer over a range of median grain

diameters. Note that both F and are functions of the grain size distribution, typically

assumed to be log-normal. Calculation of these parameters first requires one to

determine the distribution based on the given median grain diameter. The form of the

log-normal distribution is given in the appendix. The standard deviation of the grain size

distribution is assumed constant and is determined by other physical arguments. In the

case of field measurement of sediment suspension, the standard deviation is generally

assumed equal to that of the distribution of sediment in the seabed below the transducer.

For each median grain diameter, the mean concentration and the variance in the

concentration between transducers is calculated.

1 2
cm. (3-33)

2 1 = i (3-34)

The median grain diameter is recognized as that with the minimum concentration

variance, and the concentration is given by the corresponding mean concentration. If

only two transducers are used, more than one solution for the median grain diameter is

possible. In this case, determination of size is still possible if the range of grain sizes is

restricted and appropriate transducer frequencies are selected in advance.

After the initial concentration is found, the integration constant, y, can be found

for each transducer from equation (3-31). The solution for the remainder of the bins in

the profile proceeds by solving equation (3-29) for each transducer for a range of median

grain diameters, and then by selecting the correct grain diameter by minimization of the

concentration variance between transducers, given by equation (3-34). Again, the

concentration is given by the corresponding mean value from all transducers.

3.3 Verification of Technique

As an initial test of the calibration technique, ideal voltage profiles for 1.0, 2.25,

and 5.0 MHz transducers were simulated using the acoustic backscatter equation (3-4)

(a) (b)

0.7 6, 1.42
0.5

504 0o8

> 0.3 >0.6

0.2 0.4

0. 02

0 01 02 0.3 04 0 06 07 08 0.9 1 0.1 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9

Distance (m)
2.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Distane (m)

Figure 3-3. Numerically generated profiles for (a) 1.0, (b) 2.25, and (c) 5.0 MHz
transducers.

with a predetermined log-normal distribution. For simplicity, the generated profiles

contained no nearfield, or in other words, the acoustic backscatter equation is assumed to

be valid at the face of the transducer and beyond. Figure 3-3 shows these numerically

generated profiles. The concentrations used were 0.01, 0.02, 0.04, 0.08, 0.16, 0.32,

0.63,1.25, 2.5, and 5.0 g/l, and the grain size distribution was assumed to have a median

grain diameter /. = 2.66 and a standard deviation o = 0.25. Since in the absence of

attenuation, the voltage read from the transducer increases with increasing suspended

particle concentration, the concentrations are easily distinguishable at zero distance.

Higher attenuation with higher operational frequency and with higher concentration is

apparent from the figures. Particularly, the profiles from the highest concentrations in the

5 MHz simulation are attenuated so heavily that away from the transducer little, if any,

signal remains.

a) b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0. 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Distance (m) Distance (m)

Figure 3-4. Resulting (a) concentration and (b) median grain size from inversion
technique using 2.25 and 5.0 MHz profiles. Asterisks indicate known values.

Shown in figure 3-4 are the concentration and size profiles resulting from

applying the new technique using double precision calculations (64 bit) with only the

10,

10 -
o!10

10o'

10-4 H-

~_ I

2.25 and 5.0 MHz voltage profiles from figure 3-3. Since only two frequencies were

used in this test, the range of median grain sizes was restricted within +/- 2 standard

deviations of the initial known distribution. In figure 3-4b, the size profiles overlap for

all cases except the cases involving the highest two concentrations. Not surprisingly, the

technique accurately produced the initial concentration, as in Holdaway and Thorne's

(1997) simulations, and size in the majority of the cases. This test does illustrate the

(a) (b)
102 250

10 200

10- 150

0 00

10-2 50-

0 0.1 02 0-3 0.4 0.5 0.6 0.7 08 0o9 0 0.1 0.2 0.3 014 0.5 0.6 0.7 0.8 0.9
Distance (m) Distance (m)

Figure 3-5. Resulting (a) concentrations and (b) median grain size from inversion
technique using 1.0 and 2.25 MHz profiles. Asterisks represent known values.

(a) (b)
102 250 .

10o 20D

O .01o o -- *3 150
10 150

10 100

10"2 I 501

0 0.1 0o2 0!3 0.4 0.5 06 07 0.8 09 1 0 0.1 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9 1
Distance (m) Distance (m)

Figure 3-6. Resulting (a) concentrations and (b) median grain size from inversion
technique using 1.0, 2.25, and 5.0 MHz profiles. Asterisks represent known values.

difficulty though in using the highest frequency transducer through significant ranges of

high concentrations. As noted before in figure 3-3, at the highest concentrations, the high

sediment attenuation results in large signal loss away from the transducer. In figure 3-4,

the error induced in evaluating the concentration and size from this small signal is

apparent. Figure 3-5 shows the results from use of the 1.0 and 2.25 MHz signals. The

results in this test showed excellent agreement with both the original size and

concentrations, even through a one-meter range of relatively high concentration. The

final inversion, shown in figure 3-6, uses all three of the simulated signals. Again,

because of the signal loss in the high concentration 5 MHz data, the results exhibit similar

behavior to those in figure 3-4. For this reason, it is important to be aware of signal loss

when working with the highest frequency transducers, such that this can be considered in

the inversion algorithm.

Laboratory tests were performed in a recirculating calibration chamber which

produces a suspension of sediment of approximately constant concentration and constant

grain size distribution (Hanes et al., 1988). Transducer frequencies of 1.0, 2.2, and 5.0

MHz were used for the measurements. Backscattered intensity signals were collected at

100 Hz with each transducer, and the ensemble RMS was determined from 1 minute's

data for each concentration. As in the numerical simulations, the distribution parameters

for the sediment were a median grain diameter y. = 2.66 and a standard deviation cr =

0.25. The concentrations used were 0, 0.1, 0.2, 0.3, 0.4, and 0.5 g/l. Figure 3-7 shows

the agreement between the known concentration and that determined with the new

inversion technique. Several factors explain the form of the error curve with range. First,

concentrations in the chamber were determined by adding calculated dry masses of

43

(a) (b)
0.6 20
Is
+ 18
0.5 o
16
10

03.4 104
o 12

0.3 0.1 0 03 0.4 0. 05 06 0.7 08 0
0.2
x 6-
4
0.1
2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.4 0.5 0.6 0.7 0.8 0.9
Known concenlraion (01) Distace (m)

Figure 3-7. Comparison of known and calculated concentrations (a) shown at distinct
concentrations and ranges of (*) 40 cm, (o) 50 cm, (x) 60 cm, and (+) 70 cm; and (b)
shown as the mean error of all concentrations with range.

(a) (b)
200 20

190 18

1801
16
170

S160 -------------------------------- -
& I12
150
10
140

130 8

.4 0.5 0.6 0.7 0.8 0.9 .4 0.5 0.6 0.7 08 0.9
Distance (m) Distance (m)

Figure 3-8. (a) Comparison of known (broken line) and determined (solid line) median
grain size and (b) the resulting error with range.

sediment to the known volume of water. Due to hindered settling within the funnel at the

base of the chamber, actual concentrations in the tube may be slightly lower than those

calculated. Next, the initial concentration measurement is located 40 cm from the

transducer face due to both nearfield effects from the transducers and from complications

introduced by amplifier saturation at shorter ranges. Determination of the initial

concentration must therefore be done with a signal that has already experienced 40 cm of

water and sediment attenuation through significant concentrations. The sensitivity of the

concentration measurement to attenuation increases the likelihood of error, particularly

for high concentrations, long attenuation paths, and high operational frequencies.

Finally, for calibration of the acoustic transducers, the error between known and

calculated concentrations was minimized in the range from 40 cm to 90 cm. In this

minimization technique, approximately half of the calculated concentration profile

typically falls below the known value and half above. This effect is apparent in the error

profile of the present concentration evaluation, Figure 3-7b, which shows the best

agreement in the center portion of the profile. If the transducer calibration were

performed at just a single range, as is often described in the literature, and the

measurements presented in this text evaluated using the single point calibration

information, the error in the determination of concentration is less than 5%.

Figure 3-8 compares the known median grain size with that measured in the

circulation chamber. Again, if the calibration is performed at a single range and the size

determined from the single point calibration, the error in evaluation of median grain size

is less than 10%. Even in this case, the evaluated median grain size is slightly higher

than the known value. Errors result in this evaluation from use of somewhat low

concentrations for determination and from differences in various sediments not accounted

for explicitly in the empirical form function. Use of low concentrations was made

necessary by operation of the 5.0 MHz transducer with an initial concentration evaluation

point located a significant distance from the transducer. Presently, one form function is

said to describe noncohesive quartz sediment (see appendix), but it is expected that grain

properties of a given sediment sample will modify the form function slightly. Empirical

evaluation of the form function for a given sediment type would likely improve the error

in determination of the median grain size.

It should be noted that evaluating concentration and size with a constant

concentration profile is actually a more demanding application of the technique and

system than is typically experienced in field measurements, due to the propagation of

error through the profile. In measurements of sediment suspension above the seabed, the

transducer is typically far enough from the seabed such that the concentrations near the

transducer are low.

3.4 Summary

A new technique of determining both concentration and the median grain

diameter of suspended particles has been presented. The significant advantage of the

technique is that by using an explicit solution for concentration, the median grain

diameter can be found without having to neglect sediment attenuation. In addition,

because incorporating the correct median grain diameter in the explicit solution will

produce an identical concentration regardless of the operational frequency of the

transducer, the concentration variance between any number of transducers can be

minimized to find the median grain diameter. Numerical simulations show the technique

produces both the expected concentrations and grain diameters. In addition, laboratory

results from a recirculating calibration chamber verify that the technique applies well in

determining sediment size and concentration from measurements of backscattered

acoustic intensity.

CHAPTER 4
SUSPENSION TIME SCALES

In the previous chapters, a new system of instrumentation capable of accurate

depiction of sediment suspension processes with high spatial and temporal resolution has

been described. In addition, a new, robust process of data conversion from

multifrequency acoustic backscatter data to concentration and median sediment size has

been introduced. In the fall of 1996, a system similar to that described in the first chapter

was deployed from the Sensor Insertion System (SIS) at the Army Corps of Engineers

Field Research Facility (FRF) in Duck, North Carolina. In this chapter, the time series of

concentration profiles obtained from the acoustic backscatter measurements collected at

this project are examined in relation to the instantaneous hydrodynamic measurements.

It is common in the study of sediment suspension in a wave environment to

decompose the concentration into steady and fluctuating components (Nielsen, 1992).

The significance of each component in a sediment transport calculation depends on the

relative importance of the two transport mechanisms: transport by currents or transport by

waves. In the longshore direction, the sediment flux computed from the product of the

mean concentration and steady current velocity has been used successfully for

determining the mean longshore rate of transport (Hanes and Huntley, 1986). In general,

determination of the mean vertical suspended sediment concentration profile typically

involves computation of the near bed concentration by use of a reference concentration

model and computation of concentrations above by a vertical distribution model. Of the

46

many models of reference concentration, a simple linear relation between the bed shear

stress and reference concentration is shown to work best (Smith and McLean, 1977;

Thosteson, 1995). The mean vertical concentration distribution is best described by a

model incorporating both turbulent diffusion and vertical convection due to vortex ripples

(Nielsen, 1992; Lee, 1994). In examination of cross-shore transport, the fluctuating

component of concentration becomes more important (Huntley and Hanes, 1987).

There is significant evidence of the importance of low frequency water wave

motion in the process of cross-shore sediment transport (Huntley and Hanes, 1987; Beach

and Sternberg, 1991; Osborne and Greenwood, 1992). This has been attributed to the

fixed phase difference between the components of velocity and concentration, where the

low frequency velocities result from free and group-bound infragravity waves. Low

frequency variation in concentration has been shown to be associated with wave groups

(Hanes, 1991). The aim of the present investigation is to examine the significance of the

concentration variation at various frequencies and to further examine suspension by wave

groups. It is hoped that this will aid in development of future models that predict the

fluctuating components of concentration.

4.1 The SIS96 Project

Shown in figure 4-1 is the SIS on the FRF's pier. All instrumentation is deployed

from the SIS, which consists of a crane mounted on tracks that extend along the length of

the pier. Instruments attached to an arm (called the "bah") at the end of the crane's boom

can be positioned with reasonable precision at locations near the seabed. Four bayonets

located at the end of arm closest to the pier are forced into the seabed by the weight of the

crane to stabilize the bah. Cross-shore movement of the SIS allows data collection to be

.4.- -- .-" -C*C- c ~ -

Figure 4-1. Sensor Insertion System (SIS)

performed in areas with varying sediment composition and wave conditions. The

bulkiness of this arrangement immediately suggests that the framework will interfere

with the processes to be measured. While there is certainly an effect from the presence of

the structure, precautionary measures are taken to minimize this impact. The arm itself

consists of pipe of smaller diameter than the main structure, and is just massive enough

not to flex by wave forcing. Next, the arm is distanced as far from the bed as possible

considering the range limitations of the acoustic instruments. Orientation of the arm is

longshore, such that it will have the minimum influence in the vertical and cross-shore

directions. In the presence of long-shore currents, measurements are taken on the

appropriate side of the pier to be upstream of the pier and the main structure of the crane.

Finally, the instruments themselves are located at the end of the arm farthest from the

main structure and pier. Though these measures minimize the influence of the supporting

structure and the pier itself, it is expected that some influences endure. This is accepted

as a trade-off for the ease of repositioning the test site and recohfiguring instrumentation.

It should be mentioned that even after the longest deployments of the instrumentation

(approximately 1 hour), no scour hole could be seen with the MTA measurements,

suggesting that the influence of the arm itself was minimal.

As mentioned previously, the system of instrumentation used in this project was

slightly different from that described in chapter one. In fact, this project inspired many of

the new features of the newer system. The fundamental limitation of the system

deployed in this project was the constraint on the duration of data collection. As will be

recognized later, the analysis of the data from these experiments indicates that long

records of suspension must be examined to capture low frequency events that can

dominate the record. The other difference in the systems is in the instruments utilized.

Figure 4-2 shows the instruments as they were positioned on the arm of the SIS during

the project. Of the instruments shown in figure 4-2, the following instruments were a

part of the system and used in this investigation: 2 Simrad Mesotech model 810 ACPs, 1

Sontek ADV, 1 TransMetrics P21LA-25 PSIS pressure sensor, 3 Seatek MTAs, 1

DeepSea MicroSeaCam 1050, and 1 D&A OBS 3. The remaining instruments shown in

the figure were owned and operated by the FRF. Note that two individual Simrad

Mesotech ACPs with frequencies of 2.25 and 5.0 MHz were used instead of the ABS

system described in the previous chapters. As will be seen later, this introduced

complication into trying to determine size from the backscatter data.

50

SIS96 Instrument Arrangement

View from South
when on pier

h- 35 i

CJ

Figure 4-2. Instrumentation used in project.

Table 4-1. Calibration constants for instruments used in present analysis.

Instrument Gain Offset System constant

Pressure sensor 8.35x10-3 mount -6.77 m

(salt water) (salt water)

Current X 1.187 x10-3 ms count 2.457 m/

Current Y 1.203x10-3 count -2.492 m/
s-count /s

OBS 1.409 x 10-3 ,count -0.164/ -

2.25 MHz ACP 11 mV 0.464

5.00 MHz ACP 2 mV 0.929

Calibration of the instruments which measure from only a single point in space,

termed single point instruments (SPIs), is straight forward. Hydrostatic measurements

are used for calibration of the pressure sensor, and the ADVs are calibrated using a

moving cart in a tank of still water. The OBS sensor is calibrated in a sediment-

recirculating calibration tank with the ACPs. Table 4-1 shows the calibration constants

for the various instruments. As described in the previous chapter, the only undetermined

parameter for the ACPs is the system constant. Due to a slight DC offset in the output

signals from the ACPs, table 4-1 also lists the optimum offset for each transducer. An

optimization process is utilized to determine both the system constant and DC offset

which produces the concentration profiles closest to the known concentrations in the

sediment-recirculating tank. The predicted concentration profiles versus the known

values of concentration for the two ACPs are shown in figure 4-3. Across all locations

and concentrations, the mean errors for the 2.25 MHz and 5.0 MHz transducers are 8.8%

and 17.9% respectively.

(a) (b)
2.25 MHz S=0.464 DC=O.011 volts 5.0 MHz S=0.929 DC=0.002 volts
10o 100

o o

10-, -----

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Range (m) Range (m)

Figure 4-3. Calculated (solid lines) concentration profiles from (a) 2.25 MHz and (b)
5.00 MHz calibration data using optimum system constant and DC offset versus known
concentration (dashed lines).

For four days, from October 29 until November 1, 1996, experiments were

performed from various locations on the pier, ranging from water depths of 1.4 to 7.0

meters. Measurements were attained with wave conditions ranging from H,,= 0.35 to

1.0 m, due to the capability of moving the SIS. Waves were a mixture of locally

generated seas with peak periods near 6 seconds and an underlying swell component near

11 seconds. The local component was most prevalent at the start of the week, with the

swell component becoming more dominant in the latter days. Grab samples of sediment

were collected at the start of each data collection run, and sieve analysis indicated the

median grain sizes ranged from 120 to 200 microns (3.06 and 2.32 respectively on the <

scale). Sieve analysis also indicated that the grain size distributions of the samples from

the runs investigated in the present study (identified in section 4.3) were described well

by the log-normal distribution. Shown in figure 4-4 is the best fit normal cumulative

distribution function to the sieve data from run number 18. Figure 4-5 shows the

Mean phi= 2.242 Std. dev. = 0.4482

0.9 -
0.8
0.7

i 0.6
S0.5

0.4

0.3
0.2

0.1
0
0 0.5 1 1.5 2 2.5 3 3.5
Phi

Figure 4-4. Best fit normal cumulative distribution function to sieved grain size data;
sample taken at location of and prior to run 18.

53

(a) (b)
I 210 x
x
6 200.
4 190 xc

2 1180

-2 I160
._.8
-4. 150
-6 140
-8 ~ 130
-10c '1- O 120 v
0 100 200 300 400 50S 600 0 100 200 300 400 500 600
Pier location (m) Pier location (m)

Figure 4-5. (a) Profile taken from pier on north (solid) and south (dashed) side with
superimposed experimental water depths (*) and (b) corresponding sieved median grain
size.

variation of the median sediment grain size with pier location and the beach profile as

taken over the edge of the pier with a plumb bob. Elevation in this figure is relative to

the mean water level at the time of the survey. Jette presents additional information

regarding the SIS96 project (1997).

4.2 Size Determination

As described in the previous chapter, the median grain diameter of the particles in

suspension can be determined by computing the concentrations from each frequency

transducer over a range of sizes. Again, the size is determined to be that which

minimizes the variance in the computed concentrations across transducers.

Determination of size from field data is complicated by several factors that sometimes act

in conjunction. These complicating factors include the following: multiple zeros in the

variance versus size relation, poor measurement resolution, statistical fluctuation in the

backscattered signal returns, and spatially separated acoustic beams.

Because of the nonlinearity of the form function, it is possible that the variance is

zero at more than one grain size. Increasing the number of unique sound frequencies

decreases the likelihood of the variance having multiple roots. In this project, only two

transducer frequencies were used, making the determination of the correct root difficult at

times. It is not uncommon to obtain a variance versus size relation as that shown in

figure 4-6. In such a case, a physical argument based on the sizes that a bed sample

contains is used to choose the more likely sediment size. If the backscattered signal is

mainly from a washload component in the suspended sediment (small suspended particles

x 10-8

6-

4 \
._
I-
CO

02

02-

1

0
80 100 120 140 160 180 200 220 240 260 280
Median grain diameter (microns)

Figure 4-6. Multiple roots in variance-size relation.

not found locally, but instead advected from another region), than certainly this choice

will be in error.

Next, resolution of the transducers and sampling resolution must be considered.

For instance, in the present experiments, the transducers are sampled with a 12-bit analog

to digital converter. At small amplitudes, a one-count change in the measured

backscattered signal can result in a change of order in determined concentration. Since

ultimately it is the difference in concentration measurements across transducers which

determines the median sediment size, this large change in concentration drastically

changes the resulting size evaluation. For this reason, a minimum value of the

backscattered signal strength is required before the size evaluation can be trusted. In

practice, only sizes obtained from concentrations greater than 50 mg/l are used.

Because the measurement of the returned signal from suspended sediment is a

random process, many instances or profiles must be collected to obtain statistically

meaningful results. As described in chapter 2, the root mean square (RMS) of a

predetermined number of profiles is generated by the acquisition system. The number of

profiles included in this RMS profile is chosen such that the error in the concentration

measurements, proportional to the reciprocal of the square root of the number of profiles,

is minimal. Again, because of the sensitivity of the size evaluation to small concentration

differences, a small error in concentration may result in a relatively large error in the

evaluation of size. For this reason, an RMS computed from a significant number of

measured signals (averaging of the measured intensities) must be used in order to obtain a

reliable size evaluation. It is essential that the sea bottom location in the profile be

determined prior to the RMS process, since movement of the bottom with time will make

it difficult to determine its location from an RMS profile. For this same reason, with

regard to bed location, the closest reliable size estimate is that just above the highest bed

location over the averaging period. In the present experiment, there was significant

movement of the bed location in the profile, as indicated in figure 4-7. The change in bed

location was due not to accretion, but instead to settling of the framework into the seabed.

This was verified by examination of the MTA bottom profile, which showed a uniform

movement of the profile with little change in the ripple field.

Bottom location relative to that at start
Rr

0 I I II I
0 200 400 600 800 1000 1200 1400 1600 1800 2(
Elapsed time (s)
Figure 4-7. Change in bottom location due to settling of instrument framework.

000

uhirfill H I

A

Finally, it is often difficult in the field to obtain acoustic backscatter

measurements from transducers with collocated beams. Physical restrictions due to the

size of the transducers require that the beams be spatially separated. Since a significant

amount of temporal averaging is performed before the analysis, this introduces no

difficulty should the seabed be flat. If ripples exist, as they did in nearly every

experiment in this project, then it is likely that one transducer will receive backscatter

from a concentration profile which is higher or lower spatially. This spatial separation

corresponds to the transducers being positioned above different areas of the ripple

wavelength. In such a case, the differences in concentrations between the two

transducers results not only because of sediment size variations, but also because the

measurements are located at different elevations from the bed. Away from the bed, the

resulting error in size is probably small, as advection and mixing will remove horizontal

gradients in concentration. However, near the bed, the comparison of concentrations

measured from different elevations from the bed will result in notable error in the size

evaluation. The backscatter profiles can be aligned based on bed elevation as opposed to

alignment by transducer face locations, but this will result in the neglect of any

differences in the true vertical concentration and size profiles over a ripple wavelength.

Despite careful consideration of each of the above complexities, a consistent

estimate of the median sediment size could not be obtained from the data collected in this

project. In figure 4-8, the resulting size profiles from two separate data runs are shown

together with completely different results. The data for both runs were collected at the

same location, with one hour between the start times of the runs. Both size profiles result

from RMS profiles of the entire 16 minute runs. Inspection of the time series of the bed

58

(a) (b)

5 5

2 1.4 221.6 221.8 222 222.2 222.4 222.6 222.8 120 130 140 IS0 160 170
Sao (micons) Size (micrn)

RUm#19 Rmn#20
K K
S4 4

11
K K

0 0 ------- --- -----
221.4 221.6 221.8 222 222.2 222.4 222.6 222.8 120 130 140 150 160 170 11
Size (microns) Size (microns)

Run #19 Run 820

1 10 10 105 1 1
Concentraion (g/) Concetration (g4)

Figure 4-8. Perceived median grain size profiles from (a) run 19 and (b) run 20 and
corresponding near bed concentration profiles.

elevations from each transducer indicates the bottom location was the same for each

transducer. A sediment sample taken from the bed indicates a median grain diameter of

200 microns was present at this location. Although neither of the results seems

unreasonable, examination of the concentration variance across transducers indicates

agreement in concentration is never achieved. Over the entire range of the trial grain

sizes, selected to be from plus or minus three standard deviations of the local median

grain size, no applied size will result in equal concentration readings across transducers.

As to which transducer yields the higher concentration, there is no consistency. Either

transducer is just as likely to respond with a higher concentration measurement.

Recall that the beams of the two transducers used in this project were close

together (within 4 inches of one another) but not collocated. For this reason, it is

possible, and the results seem to indicate that a horizontal gradient in concentration exists

between the transducers. Although this result seems unlikely with time averaged data,

the likelihood of occurrence increases if the time series of concentration is dominated by

a few infrequent suspension events a hypothesis which is verified by the following

analysis. Furthermore, this result indicates the importance of having truly collocated

beams if an evaluation of median grain size is to be performed.

4.3 Dominant Frequency Band of Suspension Events

In order to study the most dominant time scales of sediment suspension, four runs

of the 30 runs collected will be examined. The investigation is limited to these four runs,

because these were the only runs of sufficient length to provide confidence in the low

frequency portions of the spectral analysis to be presented. Recall that in the system used

for this project the duration of data collection is limited by the available memory in the

data logger, unlike the newer system described in chapter 1. Table 4-2 shows the

conditions under which the experiments were performed. In this table, the wave height is

determined by correcting the pressure time series for depth attenuation using linear wave

theory and is then verified using that obtained by correcting the velocity time series,

again using linear wave theory (Dean and Dalrymple, 1984). In this calculation,

contributions from wave periods less than 3 seconds and greater than 20 seconds are

removed, albeit examination of the spectra prior to removal shows little energy in these

Table 4-2. Conditions at the measurement site during experiments examined in this
study.

Run Date Time Duration Pier side Location d50 Depth
(EDT) (MM:SS) (m) (4m) (m)

18 10/31/97 12:39:36 32:14 South 207 211 2.41

23 11/01/97 09:01:08 43:44 South 226 194 3.09

25 11/01/97 10:57:54 37:56 North 226 187 3.20

27 11/01/97 12:35:24 37:56 North 238 194 3.60

Run Hmo Tpeak Opeak I U direction 11 ripple ripple
(m) (s) (0) (cm/s) (0) (cm) (cm)

18 0.53 10.0 97 16 350 0.5 13

23 0.57 10.9 88 10 11 0.5 15

25 0.39 11.7 88 9 3 1.5 14

27 0.51 10.1 88 5 5 1.2 11

portions. The angles given are in a reference frame so 0 is directed longshore to the left

of an onshore observer looking out at sea, and they increase in the clockwise direction.

Wave direction is found using the maximum entropy method with the pressure and

horizontal velocity data, and is given as the direction from which the waves propagate.

In all cases listed, the instruments were placed offshore of the breakpoint such that the

instruments remained submerged for the duration of each run. In addition, collection

away from breaking waves avoids contamination of the concentration measurements by

bubbles. Due to the difficulties mentioned above in evaluating the grain size profiles, the

assumption of constant grain size is utilized in the conversion from the backscattered

signal to concentration. The grain size used in the analysis is that obtained from a grab

61

3.5,

3 -

2.5

0. !

2-

03

1

0.5 i

0 0.05 0.1 0.15 0.2 0.25
Frequency (Hz)

Figure 4-9. Surface elevation spectrum for run 18 with 80% confidence intervals.

sample taken in the vicinity of the instruments prior to data collection. Figure 4-9 shows

the surface elevation spectrum from run number 18. In this case and in each case that

follows, the spectrum is found using the entire record of surface elevation. A Bartlett

spectral window is then used to smooth the spectrum, resulting in a spectral estimate with

approximately 6 degrees of freedom. Note that the most significant portion of the energy

is found in the incident band of the surface elevation spectrum. Also, note the lack of

energy in the lowest frequency band that at the extreme left in the figure. Logged

observations indicate that the waves consisted of swell from an offshore low-pressure

II ~II~ i I

J~ ~

III I Ii 1b I I II I
0 200 400 600 800 1000 1200 1400 1600 1800
Elapsed time (s)

Figure 4-10. Plot of the 100 mg/1 contour for whole time series of run 18.
I I I I I I

Figure 4-11. Plot o

200 400 600 800 1000 1200 1400 160(
Elapsed time (s)

f the 1 g/l contour for whole time series of run 18.

0 1800

-1 1 1 1 11 1~ III1 __,_A

1I11 I1I

u I

mal ill halu l

m

" '-'Y-'UI"

system with little or no locally generated wind waves, in agreement with the measured

spectrum.

A qualitative observation of the concentration profile time series indicates that

suspension events in general tend to be intermittent with only infrequent occurrences of

high-concentration events, as described by Hanes (1988). This can be seen in figure 4-

10, where the 100 mg/l contour as determined from the 2.25 MHz transducer is plotted.

For comparison, the same time series is shown in figure 4-11, but the one gram per liter

contour is instead plotted. Inspection of figure 4-11 shows that the high concentration

events are indeed less frequent than the lower concentration events seen in figure 4-10.

- 0.8 -

2 0.6

gc0.4

. 0.2

01
1200 1300 1400 1500 1600 1700 1800

E 15
C.
E
C
. 10

aU
o
o 5

0
U)
U)

I I II

S1200 1300 1400 1500 1600 1700 1800
Elapsed seconds (s)

Figure 4-12. Time series of the squared bottom velocity magnitude and vertically
integrated concentration.

In addition, these infrequent high concentration events stay confined to the region very

close to the bed. Although these high concentration events are few in number, they

generate concentrations which greatly outweigh the typical concentrations found between

events. The infrequency of the high concentration events is again apparent in the bottom

time series of figure 4-12. Shown in the upper time series for comparison is the square of

the bottom velocity magnitude. This comparison will be examined in more detail in

section 4-4.

Next, the concentration time series is brought into the frequency domain by use of

the Fast Fourier Transform (FFT). Transformation of the concentration time series to the

^5
6

-2

04-
0)4

Ca
C,
.2 2

r--
0
o

0 0.05 0.1 0.15 0.2 0.25
Frequency (hz)
Figure 4-13. Concentration spectrum for run 18.

frequency domain results in what will be termed the concentration spectrum. This

concentration spectrum reveals the relative importance of each frequencies contribution

to the total variation in concentration. Based on the observations just made, the lowest

frequencies should show the highest contribution to the total variation in concentration.

In figure 4-13, the concentration spectrum is plotted from the time series of vertically

integrated concentration profiles of run 18, so the relative magnitudes of the total

concentration contributions at each frequency can be examined. As expected, the largest

portion of the suspended sediment concentration fluctuation, or variance, is accounted for

by variation at the lowest frequencies.

In order to examine this further and to look at the significance of this outcome at

different elevations from the bed, the cumulative variance function (CVF) is introduced.

The CVF, denoted by Za, indicates at a particular frequency the portion of the total

variance accounted for by lower frequencies. In analytical form, Zx is given by the

following expression:

I
jS.(f')df f
z, (f )= =1 j (f')df' (4-1)
fIS,(f')df' o
0

In discrete form, the lower limits of integration are replaced by the lowest resolvable

frequency, 1/T where T is the record duration. Likewise, the upper limit in the

expression in the denominator is replaced by the Nyquist frequency. For both the

concentration spectrum and the bottom velocity spectrum, the CVF is determined, and in

each case, the percentage contribution to the total fluctuation overeach band is easily

determined. Since the concentration can be expected to be more closely related to the

C\1 10

E
V i
8-
(D

6-

S4 i

2 -

0
0 0.05 0.1 0.15 0.2 0.25
Frequency (Hz)

Figure 4-14. Bottom velocity spectrum for run 18.

near bed velocity than to the surface elevation, the bottom velocity spectrum, shown in

figure 4-14, is used instead of a surface spectrum. It is found by attenuating the

measured velocity spectrum to the bed by use of linear wave theory (Dean and

Dalrymple, 1984). A comparison of the CVFs generated for the bottom velocity in figure

4-14 and the concentration spectrum from figure 4-13 is shown in the first plot of figure

4-15. In addition, the second plot makes the same comparison, but with the CVF

generated from the concentration time series measured 1 cm from the bed. Because the

highest concentrations are found near the bed, the integrated concentration is dominated

by the contribution from the near bed concentrations. So, it comes as no surprise that the

0 0.05 0.1 0.15 0.2 0.25
Frequency (hz)

1

0.9

:0.8- '

S0.7 -

5 0.6 -

-0.5

S0.4 Concentration (1 cm)
c- ---- Surface elevation
" 0. -

0.1

0
0 0.05 0.1 0.15 0.2 0.25
Frequency (hz)
Figure 4-15. Cumulative variance functions from concentration and bottom velocity
spectra for run 18.

two plots indicate nearly the same behavior for the integrated and near-bed

concentrations. In both cases, a significant portion (nearly one third) of the total variation

in concentration is accounted for in the low frequency band. In comparison, very little of

the energy of the surface spectrum, figure 4-9, or the bottom velocity spectrum, figure

4-14, is found at low frequencies. In order to clearly show the significance of lower and

incident frequencies, the plots are cut off at 0.25 Hz. Within the frequency range from

0.25 to 1.00 Hz, the variation in velocity and in concentration is uniformly distributed.

These results indicate the near-bed suspended sediment concentration time series

has a very significant low-frequency component. Furthermore, the forcing mechanism is

not apparent at the low frequencies, since this region is poorly represented in the surface

elevation spectrum. Although figure 4-15 indicates that the depth-integrated suspended

sediment concentration has similar behavior to, and is likely dominated by the near-bed

concentration, it is still instructive to examine the behavior further from the bed. Figure

4-16 makes the same comparison as in figure 4-15, but instead uses concentrations

measured 5 and 10 cm from the bed. Note that the low frequency contribution diminishes

and the variation in concentration is more uniformly distributed across the spectrum.

Also, with increasing height from the bed, the contribution to the total concentration

variation by frequencies greater than 0.25 Hz becomes more relevant. This and the

diminishing low frequency contribution both indicate that the variation becomes more

uniformly distributed with frequency as the distance from the bed increases. Above 10

cm, the concentration variation becomes still more uniform with frequency, but the

concentrations become so small that signal to noise ratio of the concentration

measurement becomes too low.

69

0.9- -

>0.8- 8 '
F-

S 0.7

05 0.6

0.5- I ---"

. 0.3

u- 0.2 -

0.1
0

0 0.05 0.1 0.15 0.2 0.25
Frequency (hz)

1

0.9 --

-0.8 -

0.7 -
/
65 0.6 /

0.5

0.4 /
O 0.3

Lu 0.2 Concentration (10 cm)
/ ,---- Surface elevation
0.1

0-
0 0.05 0.1 0.15 0.2 0.25
Frequency (hz)
Figure 4-16. Cumulative variance functions from concentration and bottom velocity
spectra for run 18.

0 2 4 6 8 10 12 14 16 18 20
T50 (s)
Figure 4-17. Tso indicating less lower frequency (high period) relevance with increased
distance from bed for run 18.

To further examine this dependence on the distance from the bed, the frequency at

which the CVF of the concentration spectrum is equal to 0.5 is determined at each

measurement elevation above the bed. The corresponding period is that at which 50% of

the variation in concentration occurs above and below, and is designated by Ts0. A high

value of To therefore indicates that the low frequency variation prevails over the higher

frequencies in the suspension time series. A plot of Ts0 shown in figure 4-17 reinforces

previous observations showing the diminishing contribution to the total concentration

variation by lower frequencies with distance from the bed.

4.4 Correlation with Velocities

Long waves have been shown to drive sediment transport in the nearshore

environment (Beach and Sternberg, 1991). In the present experiments, the bottom

velocity spectrum and the surface elevation spectrum show very little energy at the low,

or long wave, frequencies where significant concentration variation exists. In this case,

the mechanism resulting in the low frequency concentration variation is not apparent.

Simple examination of the wave energy spectrum from a wave record can reveal the

frequency components contributing to the sea state, but will not provide information on

0.5 i i

0.4

0.3

0 .2 :: : .

0.1

C-o

-0.2

-0.3 -

500 550 600 650 700 750 800 850 900 950 1000
Elapsed time (s)

Figure 4-18. Surface displacement time series from run 18 with envelope determined by
Hilbert transform.

amplitude modulation resulting from the interaction of waves at various frequencies. The

interaction between components at the surface results in low frequency amplitude

modulation of the wave record, termed 'groupiness' due to the 'groups' of waves formed.

By use of the Hilbert transform, the envelope of the wave record can be found (Haller

and Dalrymple, 1995). Such an envelope is shown superimposed on a portion of a

surface elevation time series in figure 4-18. Wave groups are generally believed to

contribute to the forcing of long waves (Longuet-Higgins and Stewart, 1964) and the

suspension of sediment in the nearshore region (Hanes, 1994). Because visual

observations at the time of the experiments indicated that wave groups were present, it is

speculated that the same interactions that result in wave groups result also in the low-

frequency suspension events seen in the previous section. In the following, the source of

these interactions is examined.

Begin by considering the sum of only two components of slightly different

frequency.

77 = a cos(k,x aot) + a cos(k2x a2t) (4-2)

The wave numbers and angular frequencies alternatively can be represented by the

following:

k, =k --
2

2 Ak (4-3)
k =k+ +
2

and

SAC
,=a 2

SA (4-4)
2

Substituting these expressions and simplifying gives

Ak AC _(4-5)
r= 2aco A-x---t cos(kx- t)45)

Squaring this expression and simplifying shows each interaction term.

2 1 + cos(Akx ACr)+ cos(2(kx -t))+ cos((2 +Ak)x -(2 +A+)t) (4-6)
[+ cos((2k- Ak)x (21- Ao)t)

Examining the terms within brackets one at a time, the first term, being independent of

frequency, is simply a 'DC' offset. The second term, of greatest interest in this study, can

be rewritten as

cos((k2 k,)x--(2 -a)t) (4-7)

which clearly shows this term results from the the difference of the original frequency

components. Similarly, the second and third term can be rewritten as

cos((k2 + k,)x (2 + )t) (4-8)

which again clearly shows this term results from the sum of the original frequency

components. The final two terms are harmonics of the two original frequency

components, respectively, as can be seen by rewriting the fourth term as follows:

-cos(2k2x 22t) (4-9)

Of the four components mentioned above, the frequency difference between components

results in wave groups. In order to examine the frequencies of the wave groups, the

envelope presented in figure 4-18 is transformed into the frequency domain,

0.18

0.16

0.14

e 0.12

>, 0.1

0.08

u 0.06

0.04-

0.02

0 I I
0 0.05 0.1 0.15 0.2 0.25
Frequency (hz)
Figure 4-19. Spectrum of envelope from run 18.

resulting in the envelope spectrum shown in figure 4-19. From figure 4-19, it is apparent

that the frequencies best represented in the envelope spectrum are the same as those

represented in the lowest portion of the concentration spectrum, figure 4-13. Re-

examination of figure 4-13 also reveals an active range of concentration variation at

frequencies higher than the incident wave frequencies. The components in this range

likely result from the harmonics and the frequency sum terms discussed above. This

further supports the conjecture that interaction terms contribute significantly to sediment

suspension.

Considering that all of the interactions are seen in the concentration variation, it is

worthwhile to examine the relation between the square of the bed velocity magnitude and

the near bed concentration. As before, the bed velocity magnitude is determined by

combining both horizontal velocity components, which are found by attenuating the

velocity measurements using linear wave theory. Concentration measurements made 1

cm above the bed are utilized in the following analysis. To this point, the results shown

in the figures have been restricted to those from run 18 for clarity. It should be noted that

the results from the other 3 runs used in this investigation show still greater significance

in the lowest frequencies of the concentration spectra. Application of the following

methods of analysis will verify this, and this analysis' results from data runs 23, 25, and

27 will be shown in figures 4-24 through 4-26 respectively.

In order to examine the relation, the coherency function is utilized. The

coherency function indicates whether one signal can be expressed as a linear function of

another signal. It is calculated by performing auto and cross-spectral analysis over

sections of the signals, and then determining the linear relation between sections.

{c,o (04)} + {Q", (0)} (4-10)
(oM)S,, (O)

In this expression, Sx and Syy are the autospectral densities of the respective signals, and

Cy and Qy are the cospectrum and quadrature spectrum the real and imaginary

components of the cross-spectral density function. Ochi (1990) provides thorough

explanations concerning the development and application of each of these functions as

well as the additional spectral analysis techniques presented in this dissertation. If the

same linear relation holds between various sections, the coherency function will return a

value of one. If the sections are related by nearly linear relations, the coherency function

will still return a value close to one. Should the sections have completely different linear

relations or should no linear relation exist for certain sections, the coherency function will

be zero. Ideally, the signal should be broken into as many sections as possible,

providing many degrees of freedom in the analysis. In addition, it is desirable to obtain

good frequency resolution at the lower frequencies in the spectral analysis, requiring that

the sections be sufficiently long. In the present case the length of the data files was

limited by the available memory in the data logger. Consequently, the number of degrees

of freedom in the analysis is limited to approximately 10, which allows a maximum

period of 3 minutes to be resolved. Coherency function values of better than 60% were

found in most of the comparisons between the square of the velocity and the near bed

a 0.5
(D
O 0.4
0

0.3

0.2

0.1

0.05 0.1 0.15 0.2

0.25

Frequency (Hz)

Figure 4-20. Coherency function between square of bed velocity magnitude and near bed
concentration for run 18.

concentrations, particularly at the lowest frequencies in the spectrum, as seen in figure

4-20 and plot (c) of figures 4-24 through 4-26. There is significant variation in the

magnitude of the coherency function across frequencies, due, in part, to the limited

degrees of freedom in the analysis. In addition, it is difficult to determine a linear

relation in the regions of the spectra in which there is little energy, due to limited

instrument resolution. Therefore, the value of the coherency function in these regions is

not as meaningful. For this reason, the results of the coherency analysis are presented in

another form. In figure 4-21, the concentration spectrum for the nearbed concentration is

shown. The curve is shown as solid in those regions in which the coherency function,

Near-bed concentration vs velocity magnitude squared

0.8
"'
t-"

C
0.6

C"

0
o
.m 0.4

o
o
o

0 0.05 0.1 0.15 0.2 0.25
Frequency (Hz)
Figure 4-21. Concentration spectrum (dotted) with areas of coherency function > 60%
indicated (solid) for run 18.

determined from the near bed concentration and square of the bed velocity magnitude,

returned a value greater than 60%. These marked regions are the most dominant regions

of the concentration spectrum, covering a total of 76% of the total variation of the

spectrum. This indicates that a linear relation between the velocity squared and near bed

concentration is likely. In each of figures 4-24 through 4-26, plot (b) verifies this result,

showing high coherence in the portions of the spectrum with the highest magnitudes of

variance. Though only run 18 showed significant variation in concentration at incident

wave band frequencies, it should be noted that significantly higher coherence was found

in this region as the number of degrees of freedom in the analysis was increased. Since

increasing the number of degrees of freedom simply means increasing the number of

instances used in the evaluation, this could suggest that the statistical means of the

processes are linearly related at these frequencies. More runs with significant variation in

this frequency band need to be analyzed to confirm such a hypothesis.

Since the coherency analysis indicates a linear relation exists between the square

of the bed velocity magnitude and the near bed concentration, it is interesting to examine

the transfer function.

Y(w)= X((o)H(o) (4-11)

The transfer function, H(o), is simply a function which relates a linear system's input,

X(o), the square of the bed velocity magnitude in this case, to the system's output, Y(o),

the near bed concentration (Ochi, 1990). In the frequency domain, the input signal

multiplied at each frequency by the transfer function produces the output signal.

Considering that the input and output signals can vary in both magnitude and phase, the

transfer function evaluated at a particular frequency in general produces a complex

number. The phase variation is indicative of the time lag between the processes. Figure

4-22 shows the phase of the transfer function calculated for run 18. Examination of

figure 4-22 and plot (d) of figures 4-24 through 4-26 show a relatively small, and

typically negative, transfer function phase. There is a slight trend to decrease in phase as

frequency increases in the low-frequency band of the relation. Such a trend indicates that

the time lag between the signals is nearly constant. It is expected that this time lag is

related to the upward sediment flux from mixing and to the fall velocity of the sediment.

Future investigations which include measurements of sediment size will aid in verifying

such a relation.

20 .

0-

-20 :

S-40
0) I"

-60

-80

-100 i
0 0.05 0.1 0.15 0.2 0.25
Frequency (Hz)

Figure 4-22. Phase of transfer function for run 18. Negative phase indicates
concentration lags square of velocity magnitude.

0 0.05 0.1 0.15 0.2 0.25
Frequency (Hz)

Figure 4-23. Magnitude of transfer function for run 18.

Finally, in figure 4-23 and in plot (e) of figures 4-24 through 4-26, the magnitude

of the transfer function is shown. For each particular run, the magnitude of the transfer

function at the frequencies in which the coherence was high varies little with frequency.

The relative difference in the transfer function magnitudes for different runs is attributed

to the difficulty in quantifying the bed location in the concentration profiles. In other

words, because the spatial resolution of the concentration measurement was limited to

0.75 centimeters by the sampling rate of the system, so was the resolution in determining

the bed location. With an exponential height variation in concentration, a small error in

determining the bed location can result in a significant change in concentration. For

example, referring back to figure 4-8, the concentration changes from 1.5 g/1 at the

closest measurement to the bed to approximately 0.5 g/l at the next closest measurement

0.75 cm higher. Regardless, it is promising to see that the value is nearly constant across

frequency for a particular run, and that this value is of similar magnitude between runs. It

is expected that this parameter too is related to sediment parameters and local

hydrodynamic conditions, such as bed roughness or bedforms. In relation with the

calculation of the time mean reference concentration, future research could attempt to

relate the magnitude of the sediment resuspension coefficient, since a linear relation

seems likely (Smith and McLean, 1977; Thosteson, 1995).

4.5 Discussion of Results

Although nearly every possible complexity preventing accurate evaluation of the

median grain diameter of suspended sediment was considered there were insurmountable

difficulties in determining grain size from the data collected in this experiment.

Horizontal gradients in concentration between measurements are believed to be

responsible for the difficulties in determining grain size. Vortex ripples were measured

in every data run, and may be responsible for this horizontal variation in concentration.

Because of the difficulties encountered, the constant grain size assumption was made in

the remainder of the analysis. Concentration measurements from the 2.25 MHz

transducer were used as opposed to those from the 5.0 MHz transducer, since for the

grain sizes encountered in the region, the 2.25 MHz unit shows less sensitivity to grain

size. It should be noted that errors resulting from this assumption are likely small in the

present analysis, because the active portion of the profile remained small. Namely, the

concentration profile stayed confined to the area very close to the bed, so analysis errors

82

(a) (b)

3.5
41.89% high cohefnce
3-

2.5

13

2

05

0 0.05 01 0.15 0.2 025 0 0.05 0.1 0.15 02 0.25
Frequency (Hz) Frequency (Hz)

(c) (d)
S* I 200
0.9 :

0.850
100
0.7

0.4 / v -501 .P 1-. ": .. : :

-100
02

0.1 -150

0 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25
Frequency (Hz) Frequency (Hz)
(e)

2.5

2

0.5

0 0.05 0.1 0.15 0.2 0.25
Frequency (Hz)

Figure 4-24. Results from run 23: (a) bottom velocity spectrum; (b) concentration
spectrum; (c) coherence; (d) transfer function phase; (e) transfer function magnitude.

83

(a) (b)
3i 25

2.5
i20

2- .66.6% high cohrenc.
~1.515
1 5

10

C --'^ f- -^ ----- -------- -..---'- '- ^ -- I --------- i ---- ----------*** -----**.-* -- ----
0 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 02 0.25
Frequency (Hz) Frequency (Hz)

(c) (d)
1 200

0 9 1 0
008
100
07

0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.( 0.15 0.2 0.25
0.6

04

0.3

Frequency (Hz) Frequency (Hz)

(e)

10

02

20

0 0.05 0.1 0.15 02 0.25
Frequency (Hz)

Figure 4-25. Results from run 25: (a) bottom velocity spectrum; (b) concentration

spectrum; (c) coherence; (d) transfer function phase; (e) transfer function magnitude.

0.05 0.1 0.15
Frequency (Hz)

0 005 0.1 0.15
Frequency (Hz)

02 0

100

50

-100

-150
/
0 0.05 0.1 0.15 0.2 0.2
Frequency (Hz)

0.2 0.25

Figure 4-26. Results from run 27: (a) bottom velocity spectrum; (b) concentration
spectrum; (c) coherence; (d) transfer function phase; (e) transfer function magnitude.

K.

PA

lil ~ ....- 1_ -

~C~C

0

i

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