U.S. Department of the Interior

U.S. Geological Survey

Professional Paper 1823

Long-Term Continuous Acoustical Suspended-Sediment 

Measurements in Rivers—Theory, Application, 

Bias, and Error
Cover: Array of three single-frequency side-looking acoustic-Doppler profilers at the 

Colorado River near Grand Canyon, Arizona, gaging station (study site CR87). Array has 

been raised out of the water for maintenance. From left to right: 2-MHz Nortek EasyQ, 

600-kHz Nortek Aquadopp, and 1-MHz Nortek EasyQ acoustic-Doppler profilers.
Long-Term Continuous Acoustical 

Suspended-Sediment Measurements in 

Rivers—Theory, Application, Bias, and 

Error

By David J. Topping and Scott A. Wright

Professional Paper 1823

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Director

U.S. Geological Survey, Reston, Virginia: 2016

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Suggested citation:

Topping, D.J., and Wright, S.A., 2016, Long-term continuous acoustical suspended-sediment measurements in 

rivers—Theory, application, bias, and error: U.S. Geological Survey Professional Paper 1823, 98 p., 

http://dx.doi.org/10.3133/pp1823.

ISSN 1044-9612 (print)

ISSN 2330-7102 (online)

ISBN 978-1-4113-4033-6
iii

Contents

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

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

Purpose and Scope ..............................................................................................................................4

Instruments, Study Sites, and Field Methods ...........................................................................................4

Acoustic-Doppler Profilers .................................................................................................................4

Study Sites .............................................................................................................................................7

Field Methods and Required Field Conditions .................................................................................8

Theoretical Framework.................................................................................................................................9

The Sonar Equation and Beam-Averaged Backscatter.................................................................9

Source Level...............................................................................................................................10

Two-Way Transmission Loss....................................................................................................10

Target Strength...........................................................................................................................12

Reverberation Level ..................................................................................................................13

Relative Backscatter.................................................................................................................14

Calculation of the Sediment Attenuation Coefficient and Beam-Averaged 

Backscatter...................................................................................................................15

Physical Basis for the Sediment Attenuation Coefficient............................................................15

Estimation of the Grain-Size Distribution and Wet Density of the Silt and Clay: 

Attenuation Constraint ........................................................................................................20

Evaluation of Target Strength from Pressure-Based Re-Derivation of the 

Sonar Equation ......................................................................................................................22

Unit Target Strength ..................................................................................................................26

Relative Unit Target Strength...................................................................................................29

Development of Base-Backscatter-Calibration Relations...........................................................31

Calculation of the Backscatter from Silt and Clay ........................................................................34

Estimation of the Grain-Size Distribution and Wet Density of the Silt and Clay: 

Backscatter Constraint .......................................................................................................38

Suspended-Sediment-Grain-Size-Distribution Effects on Relations between B , 

αS, log10(S), and log10(CXS-SAND) ..............................................................................................40

Procedure for Applying Method................................................................................................................43

Calibration Procedure for Each Single-Frequency ADP in an Array .........................................43

Multi-Frequency RUTS-Based Calculation of CXS-SAND and D50-XS-SAND Using the 

Single-Frequency Estimates of CXS-SAND..............................................................................43

Results ...........................................................................................................................................................43
iv

Introduction to the Analyses of Bias and Error ......................................................................................46

Error Analysis ...............................................................................................................................................46

Comparison of In-Sample and Out-of-Sample Relative Errors ...................................................47

Comparison of Relative Biases in Acoustical Measurements of Suspended-Sand 

Concentration Made Using Different Methods................................................................48

Dependence of Relative Errors in Acoustical Measurements of Concentration and 

Median Grain Size on Either Concentration or Grain Size .............................................50

Estimation of the 68-Percent- and 95-Percent-Confidence-Level Time-Varying Relative 

Errors in Individual Acoustical Measurements................................................................54

Importance of Errors from Slow Changes in the Spatial Structure of Suspended 

Sediment in the River Cross Section..................................................................................56

Conclusions...................................................................................................................................................62

Acknowledgments.......................................................................................................................................64

References Cited .........................................................................................................................................64

Appendix 1. Example Illustrating how Transmit-Pulse, Blanking, and Receive-Window 

Durations Together Determine the Locations and Sizes of the Measurement Cells ..........72

Appendix 2. Calibration Measurements, ADP-Array Configurations, and the Distance 

Between the Calibration Cross Section and the ADP Arrays at Each Study Site ...............73

Appendix 3. Effect of the Downing and Others (1995) Empirical Near-Field Correction.................74

Appendix 4. Comparison of the Effects of Water Temperature, Pressure, and Salinity on 

the Water Absorption Coefficient................................................................................................77

Appendix 5. Details of Grain-Size Distributions Used to Develop Theoretical B ́ Relations 

at the CR87 and RG-RGV Study Sites ..........................................................................................78

Appendix 6. A Method for the Detection and Correction of Discharge-Correlated Shifts in 

the BBC Relation at High Values of S..........................................................................................79

Appendix 7. Single-Frequency ADP Calibration Procedure................................................................84

Appendix 8. Multi-Frequency RUTS-Based Calculation of CXS-SAND and D50-XS-SAND ...........................95

Appendix 9. Plots of Predicted Versus Observed Values of CXS-SILT-CLAY, CXS-SAND, and 

D50-XS-SAND, with Data Segregated by Study Site ........................................................................97
v

Figures

1. Examples of different types of suspended-sediment hysteresis from three rivers...........2

2. Cartoon sketch of a river cross section showing the location of an ADP array................8

3. Examples of the steps in the conversion from the raw acoustical measurements 

to beam-averaged backscatter................................................................................................16

4. Effects of changes in sediment sorting and sediment wet density on αUNIT .....................18

5. Effects of acoustic frequency and grain-size-dependent differences in sorting 

on αUNIT ..........................................................................................................................................19

6. Comparison of empirical and theoretical relations between ADP-calculated αS and 

physically measured CXS-SILT-CLAY.................................................................................................20

7. Unit target strength (UTS), sediment component of the unit target strength

(UTSSED), and beam component of the unit target strength (UTSBEAM) plotted as a

function of D50 .............................................................................................................................28

8. Effects of changes in sediment sorting on the sediment component of the unit

target strength at 1 MHz............................................................................................................29

9. Relative unit target strength associated with the D50 of a suspended-sediment 

grain-size distribution at acoustic frequencies of 600 kHz, 1 MHz, and 2 MHz................30

10. Comparisons of measured and theoretical values of the relative unit target 

strength ........................................................................................................................................31

11. Base-backscatter-calibration relations..................................................................................32

12. A factor of 100 variation in the velocity-weighted suspended-sediment

concentration in the river cross section that is possible for any given value of B .......35

13. Theoretical behavior of B ́ under different silt and clay grain-size distributions,

sand grain-size distributions, and silt and clay wet densities ...........................................37

14. Comparison of empirical and theoretical values of B ́ on the Colorado River and 

Rio Grande ..................................................................................................................................38

15. Comparison of empirical and theoretical relations between B and the logarithm 

of CXS-SAND in the Colorado River and Rio Grande for different ranges of S.......................40

16. Behavior of theoretical relations between B and log10(CXS-SAND) for different

values of αS and behavior of theoretical relations between αS and log10(S) for

different values of B for different grain-size distributions of suspended 

silt and clay..................................................................................................................................41

17. Examples of theoretical and measured relations between αS and log10(S) ......................42

18. Predicted versus observed plots for the EDI or EWI and acoustical measurements 

of CXS-SILT-CLAY, CXS-SAND, and D50-XS-SAND .........................................................................................44

19. Relative errors in single-frequency and two-frequency acoustical measurements

of CXS-SAND plotted as a function of D50-XS-SAND ..........................................................................50

20. Relative error in acoustical measurements of CXS-SILT-CLAY at all six study sites

plotted as a function of EDI- or EWI-measured CXS-SILT-CLAY, and relative error in

acoustical measurements of CXS-SAND plotted as a function of EDI- or

EWI-measured CXS-SAND and D50-XS-SAND......................................................................................52

21. Relative error in acoustical measurements of D50-XS-SAND at all six study sites

plotted as a function of EDI- or EWI-measured D50-XS-SAND and CXS-SAND ..............................53
vi

22. Six plots showing 68-percent- and 95-percent-confidence-level relative errors in 

individual acoustical suspended-sediment measurements and in individual equaldischarge increment (EDI) or equal-width increment (EWI) measurements plotted 

as a function of the acoustical suspended-sediment measurements...............................56

23. Autocorrelation in the relative errors associated with acoustical measurements 

of CXS-SILT-CLAY ................................................................................................................................60

24. Autocorrelation in the relative errors associated with acoustical measurements 

of CXS-SAND .....................................................................................................................................61

3-1. Examples of measurements of the relative backscatter made with and without 

the Downing and others (1995) empirical near-field correction.........................................74

4-1. αW plotted as a function of water temperature in 1-degree increments over wide

ranges of pressure and salinity................................................................................................77

6-1. Examples of the presence and absence of significant relations between water

discharge and log10(S) at constant B and αS..........................................................................81

6-2. Examples of the presence and absence of significant correlations between water

discharge and CXS-SAND at constant B and αS..........................................................................82

6-3. Examples of the increased upward scatter in measured log10(S) relative to

theoretical log10(S) that arises from discharge-correlated BBC shifts .............................83

7-1. Examples of the effect on αS of limiting the number of cells...............................................87

7-2. Examples of the effect on B of limiting the number of cells ...............................................89

7-3. Summary of the effects of imposing lower maximum limits on the number of cells 

on the mean bias and variability in αS and B ........................................................................91

7-4. Example showing noise-floor offsets required to exclude acoustical measurements

in the curved part of the acoustic profile immediately above the instrument noise

floor...............................................................................................................................................91

7-5. Effect of the different noise-floor offsets depicted in figure 7-4 on calculations of 

αS and B .....................................................................................................................................92

7-6. Comparison between ADP measurements made at a bad deployment location in

a large lateral-recirculation eddy with ADP measurements made at a good

deployment location in uniform flow ......................................................................................93

7-7. Second comparison between ADP measurements made at the bad and good

deployment locations.................................................................................................................94

9-1. Predicted versus observed plots for the EDI or EWI and acoustical measurements 

of CXS-SILT-CLAY, CXS-SAND, and D50-XS-SAND, with data segregated by study site..........................97

Tables

1. Acoustic-Doppler profiler properties and settings.................................................................7

2. Effect of using different acoustical methods on the relative bias in acoustical

measurements of CXS-SAND ..........................................................................................................49

3. Levels of significance (pSIG) associated with F-tests conducted to determine whether

relative errors in the acoustical suspended-sediment measurements depend

significantly on concentration or sand median grain size (D50-XS-SAND)...............................51

4. Results from autocorrelation analyses conducted on relative errors in acoustical

measurements of suspended-sediment concentration (CXS-SILT-CLAY and

CXS-SAND) .........................................................................................................................................59
vii

Conversion Factors

Inch/Pound to International System of Units

Multiply By To obtain

Length

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Volume

ounce, fluid (fl. oz) 0.02957 liter (L) 

quart (qt) 0.9464 liter (L) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Mass

ounce, avoirdupois (oz) 28.35 gram (g) 

pound, avoirdupois (lb) 0.4536 kilogram (kg) 

Pressure

pound per square inch (lb/in2) 6.895 kilopascal (kPa) 

Density

pound per cubic foot (lb/ft3) 16.02 kilogram per cubic meter (kg/m3)

pound per cubic foot (lb/ft3) 0.01602 gram per cubic centimeter (g/cm3)

International System of Units to Inch/Pound

Multiply By To obtain

Length

millimeter (mm) 0.03937 inch (in.)

meter (m) 3.281 foot (ft) 

kilometer (km) 0.6214 mile (mi)

Volume

liter (L) 33.82 ounce, fluid (fl. oz)

liter (L) 1.057 quart (qt)

Flow rate

cubic meter per second (m3/s) 35.31 cubic foot per second (ft3/s)

Mass

gram (g) 0.03527 ounce, avoirdupois (oz)

kilogram (kg) 2.205 pound avoirdupois (lb)

Pressure

kilopascal (kPa) 0.1450 pound per square inch (lb/ft2) 

Density

kilogram per cubic meter (kg/m3) 0.06242 pound per cubic foot (lb/ft3) 

gram per cubic centimeter (g/cm3) 62.4220 pound per cubic foot (lb/ft3) 

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as 

°F=(1.8×°C)+32.

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as 

°C=(°F–32)/1.8.
viii

Abbreviations

ADP Acoustic-Doppler profiler

BBC Base backscatter calibration 

EDI Equal discharge increment

EWI Equal width increment

USGS U.S. Geological Survey

Notation

2TL two-way transmission loss (in units of dB)

a acoustical measurement of generic quantity in relative error equation (in units of

 mg/L or mm, depending on whether the error is in concentration or grain size)

aS radius of the median size of the suspended sediment (in units of m)

aT radius of the transducer (in units of m)

A acoustic signal strength, including noise (in units of counts)

AN instrument noise floor (in units of counts)

AS the part of the acoustic signal strength produced by the interaction of the sound

 emitted by the ADP with the suspended sediment in the moving water (in units of

 counts)

b EDI or EWI measurement of generic quantity in relative error equation (in

 units of mg/L or mm, depending on whether the error is in concentration or 

 grain size)

bSF scale factor used to convert counts to decibels (in units of dB/count)

b(θ,φ) transmitted beam pattern in spherical coordinates (θ,φ)

b’(θ,φ) received beam pattern in spherical coordinates (θ,φ)

B relative backscatter (in units of dB)

B beam-averaged backscatter (in units of dB)

Bbase

 beam-averaged base backscatter associated with CXS-SAND-REF (in units of dB)

BTHRESH threshold beam-averaged backscatter (in units of dB) above which a high-dB

 correction is required

B ́ excess backscatter, that is, the backscatter from silt and clay (in units of dB)

BF fluid-corrected backscatter (in units of dB)

B1-LOWf effective lower-frequency B associated with the initial lower-frequency estimate

 of CXS-SAND in the two-frequency RUTS-based calculation of CXS-SAND and D50-XS-SAND 

 (in units of dB)

B2-LOWf effective lower-frequency B associated with the initial higher-frequency estimate

 of CXS-SAND in the two-frequency RUTS-based calculation of CXS-SAND and D50-XS-SAND 

 (in units of dB)
ix

BHIGHf effective higher-frequency B associated with the initial higher-frequency estimate

 of CXS-SAND in the two-frequency RUTS-based calculation of CXS-SAND and D50-XS-SAND

 (in units of dB)

BHIGH-DEFECT value of RUTS along higher-frequency theoretical RUTS relation at the value of 

 D50-XS-SAND calculated in the two-frequency RUTS-based calculation of CXS-SAND and

 D50-XS-SAND (in units of dB)

BLOW-DEFECT B1-LOWf minus B2-LOWf (in units of dB)

c measured speed of sound in water (in units of m/s)

cN nominal speed of sound in water (in units of m/s)

C concentration of suspended sediment (in units of mg/L)

CSAND suspended-sand concentration (in units of mg/L)

CXS velocity-weighted suspended-sediment concentration in the river cross section

 (in units of mg/L)

CXS-SAND-REF velocity-weighted “reference” concentration of suspended sand in the river cross

 section (in units of mg/L)

CXS-SAND-HIGHf initial CXS-SAND estimated using the higher-frequency ADP in the two-frequency

 RUTS-based calculation of CXS-SAND and D50-XS-SAND (in units of mg/L)

CXS-SAND-LOWf initial CSAND estimated using the lower-frequency ADP in the two-frequency RUTS

 based calculation of CXS-SAND and D50-XS-SAND (in units of mg/L)

CXS-SAND velocity-weighted suspended-sand concentration in the river cross section (in

 units of mg/L)

CSILT-CLAY suspended-silt-and-clay concentration (in units of mg/L)

CXS-SILT-CLAY velocity-weighted suspended-silt-and-clay concentration in the river cross

 section (in unit of mg/L)

D sediment grain size, that is, diameter (in units of mm)

D50 median grain size of a suspended-sediment grain-size distribution (in units of mm)

D50-SAND suspended-sand median grain size (in units of mm) 

D50-XS-SAND velocity-weighted suspended-sand median grain size in the river cross section (in

 units of mm)

D50-XS-SAND-REF reference D50 for a velocity-weighted suspended-sand grain-size distribution in

 the river cross section (in units of mm)

D50-XS-SED velocity-weighted D50 of the suspended-sand, silt, and clay mixture in the river

 cross section (in units of mm)

D50-XS-SED-REF reference D50 for a velocity-weighted suspended-sediment grain-size distribution 

 in the river cross section (in units of mm)

DIT directivity index for the outgoing (transmitted) acoustic beam (in dB)

E68 relative form (in units of percent) of the 68-percent-confidence level true 

 proxy error

E68-ABS absolute form (in units of concentration [mg/L] or grain size [mm]) of the

 68-percent confidence level true proxy error
x

E95 relative form (in units of percent) of the 95-percent-confidence level true proxy

 error

E95-ABS absolute form (in units of concentration [mg/L] or grain size [mm]) of the

 95-percent confidence level true proxy error

E68-FIT relative form (in units of percent) of the power-law fit to σ of the proxy relative

 error

E68-FIT-ABS absolute form (in units of concentration [mg/L] or grain size [mm]) of the power

 law fit to σ of the proxy relative error

E95-FIT-ABS absolute form (in units of concentration [mg/L] or grain size [mm]) of the power

 law fit to 1.96σ of the proxy relative error

E(EDI-EWI)68-ABS absolute form (in units of concentration [mg/L] or grain size [mm]) of the

 68-percent confidence-level combined field and laboratory-processing error in

 the concurrent EDI or EWI measurement

E(EDI-EWI)95-ABS absolute form (in units of concentration [mg/L] or grain size [mm]) of the

 95-percent-confidence-level combined field and laboratory-processing error in

 the concurrent EDI or EWI measurement

f nondimensional form function that describes the backscattering strength of

 sediment grains in water

f

REF value of f associated with the reference D50-XS-SAND, that is, D50-XS-SAND-REF

 (nondimensional)

f

SED value of f calculated for the grain-size distribution of a sand, silt, and clay mixture 

 (nondimensional)

I

0

 acoustic intensity at the reference distance 1 m from the transducer (in units

 of W/m2)

I

1 acoustic intensity at some greater distance from the transducer (in units of W/m2

)

I

INC incident acoustic intensity on the reverberating volume (in units of W/m2

)

I

R reflected acoustic intensity from the scatterers in the reverberating volume at a

 reference distance 1 m in front of the volume (in units of W/m2)

I

REF reference acoustic intensity in water at 1 μPa one meter from the transducer (in

 units of W/m2)

I

RL intensity of the measured reverberation, excluding noise (in units of W/m2

)

k wave number (in units of 1/m)

K constant in Thevenot and others’ (1992) constant-grain-size backscatter–

 concentration relation

K1 constant equal to –0.1(SL + UTSREF); this constant is also the regression

 determined y-intercept of the BBC relation

K1-HIGHf K1 for the higher-frequency ADP in the two-frequency RUTS-based calculation of

 CXS-SAND and D50-XS-SAND

K1-LOWf K1 for the lower-frequency ADP in the two-frequency RUTS-based calculation of 

 CXS-SAND and D50-XS-SAND

K2 regression-determined slope of the BBC relation
xi

K2-HIGHf K2 for the lower-frequency ADP in the two-frequency RUTS-based calculation of

 CXS-SAND and D50-XS-SAND

K2-LOWf K2 for the lower-frequency ADP in the two-frequency RUTS-based calculation of 

 CXS-SAND and D50-XS-SAND

K3 constant in the high-dB correction in appendix 7

LBS acoustic backscattering length (in units of m)

mCELL measurement cell number

M mass concentration of suspended sediment (in units of kg/m3)

n number of observations or measurements

N number concentration of suspended sediment (in units of particles/m3)

pNOISE instrument noise-floor pressure (in units of Pa)

pSIG statistical level of significance

pREF reference pressure in water (in units of Pa)

prms reverberation-level root-mean-square pressure measured at the transducer 

 (in units of Pa)

p0 source-level pressure at distance r

0

=1 m from the transducer (in units of Pa)

P power of the sound (in units of W)

Q water discharge (in units of m3/s)

QTHRESH threshold water discharge (in units of m3/s) in the discharge-weighting factor in

 appendix 7

r axial distance (range) along the beam from the transducer (in units of m)

r

0 reference distance of 1 m from the transducer (in units of m)

r

BLANK blanking distance (in m)

r

C

 critical (Fresnel) distance between the near-field and far-field regions (in units 

 of m)

r

D larger critical distance between the near-field and far-field regions used in the

 Downing and others (1995) empirical near-field correction (in units of m)

RL reverberation level (in units of dB)

RLOFFSET linear offset (in units of dB) required to shift decibels calculated using a floating

 scale within the ADP dynamic range to absolute decibels

RUTS relative unit target strength (in units of dB)

sV backscattering cross section of a unit reverberating volume (in units of 1/m3)

S ratio of CXS-SILT-CLAY to CXS-SAND (nondimensional)

SL source level (in units of dB)

t

P

 acoustic transmit-pulse (that is, ping) duration (in units of s)

TS target strength (in units of dB)

TSRI range-independent target strength (in units of dB)

TSSAND-REF range-independent target strength of CXS-SAND-REF (in units of dB)
xii

TSSED range independent target strength of a sand, silt, and clay mixture in suspension

 (in units of dB)

UTS unit target strength (in units of dB)

UTSBEAM beam component of the unit target strength (in units of dB)

UTSREF reference unit target strength associated with D50-XS-SED-REF or D50-XS-SAND-REF (in 

 units of dB)

UTSSED sediment component of the unit target strength (in units of dB)

V the reverberating volume (in units of m3)

α attenuation coefficient resulting from the sum of the water absorption and

 sediment attenuation coefficients (in units of dB/m)

αN attenuation coefficient resulting from the sum of the water absorption and

 sediment attenuation coefficients (in units of Nepers)

αS sediment attenuation coefficient (in units of dB/m)

αUNIT unit sediment attenuation coefficient (in units of [dB-L]/[m-mg])

αW coefficient of absorption for acoustic energy in water (in units of dB/m)

λ acoustic wavelength (in units of m)

σ sample standard deviation

σG geometric sample standard deviation

ρS density of suspended sediment (in units of kg/m3)

ρSAND density of suspended sand (set equal to quartz density, that is, 2,650 kg/m3)

ρSED density of a sand, silt, and clay mixture in suspension (in units of kg/m3)

ρW density of water (in units of kg/m3)

ψNF nondimensional near-field correction

φ	 	 unit of sediment grain size, defined as φ = −log2 D

Ψ solid angle of a two-way acoustic beam subtended at the face of the transducer

 (in units of steradians)

Ω solid angle (in units of steradians)
Long-Term Continuous Acoustical Suspended-Sediment 

Measurements in Rivers—Theory, Application, 

Bias, and Error

By David J. Topping and Scott A. Wright

Abstract

It is commonly recognized that suspended-sediment 

concentrations in rivers can change rapidly in time and 

independently of water discharge during important 

sediment-transporting events (for example, during 

floods); thus, suspended-sediment measurements at

closely spaced time intervals are necessary to characterize 

suspended-sediment loads. Because the manual collection 

of sufficient numbers of suspended-sediment samples

required to characterize this variability is often time and 

cost prohibitive, several “surrogate” techniques have been 

developed for in situ measurements of properties related to 

suspended-sediment characteristics (for example, turbidity, 

laser-diffraction, acoustics). Herein, we present a new 

physically based method for the simultaneous measurement 

of suspended-silt-and-clay concentration, suspended-sand 

concentration, and suspended-sand median grain size in rivers, 

using multi-frequency arrays of single-frequency side-looking 

acoustic-Doppler profilers. The method is strongly grounded

in the extensive scientific literature on the incoherent

scattering of sound by random suspensions of small particles. 

In particular, the method takes advantage of theory that relates 

acoustic frequency, acoustic attenuation, acoustic backscatter, 

suspended-sediment concentration, and suspended-sediment 

grain-size distribution. We develop the theory and methods, 

and demonstrate the application of the method at six study 

sites on the Colorado River and Rio Grande, where large 

numbers of suspended-sediment samples have been collected 

concurrently with acoustic attenuation and backscatter 

measurements over many years. The method produces

acoustical measurements of suspended-silt-and-clay and 

suspended-sand concentration (in units of mg/L), and 

acoustical measurements of suspended-sand median grain 

size (in units of mm) that are generally in good to excellent 

agreement with concurrent physical measurements of these 

quantities in the river cross sections at these sites. In addition, 

detailed, step-by-step procedures are presented for the general 

river application of the method. 

Quantification of errors in sediment-transport

measurements made using this acoustical method is essential 

if the measurements are to be used effectively, for example, to 

evaluate uncertainty in long-term sediment loads and budgets. 

Several types of error analyses are presented to evaluate 

(1) the stability of acoustical calibrations over time, (2) the 

effect of neglecting backscatter from silt and clay, (3) the bias 

arising from changes in sand grain size, (4) the time-varying 

error in the method, and (5) the influence of nonrandom

processes on error. Results indicate that (1) acoustical 

calibrations can be stable for long durations (multiple years), 

(2) neglecting backscatter from silt and clay can result in 

unacceptably high bias, (3) two frequencies are likely required 

to obtain sand-concentration measurements that are unbiased 

by changes in grain size, depending on site-specific conditions

and acoustic frequency, (4) relative errors in silt-and-clayand sand-concentration measurements decrease substantially 

as concentration increases, and (5) nonrandom errors may 

arise from slow changes in the spatial structure of suspended 

sediment that affect the relations between concentration 

in the acoustically ensonified part of the cross section and

concentration in the entire river cross section. Taken together,

the error analyses indicate that the two-frequency method 

produces unbiased measurements of suspended-silt-and-clay 

and sand concentration, with errors that are similar to, or larger 

than, those associated with conventional sampling methods.

Introduction

The instantaneous concentration of suspended sediment

and the instantaneous discharge of water are not well 

correlated in most rivers because of discharge-concentration 

hysteresis in one or more size classes of the suspended 

load. As observed by Gray and Simoes (2008), “a lack of 

synchronization between peaks of water discharge and 

sediment concentration over a flood hydrograph is more the

rule than the exception.” Discharge-concentration hysteresis 

is common in rivers because it arises from multiple processes. 
2 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

For example, it can arise from changes in the upstream 

sediment supply or lags between discharge and dune geometric 

adjustment during floods (Topping and others, 2000a, 2000b;

Kleinhans and others, 2007). Both of these causal mechanisms 

result in hysteresis in concentration that is systematically 

coupled to hysteresis in grain size, but in different ways. 

Depletion or enrichment of the upstream sediment 

supply leads to characteristic opposing styles of hysteresis 

in concentration and grain size during floods. Depletion of

the upstream sediment supply leads to clockwise discharge–

concentration hysteresis that is coupled to counter-clockwise 

discharge–grain-size hysteresis, herein referred to as Type 1

hysteresis (fig. 1A–B) (for example, Rubin and others, 1998;

Dinehart, 1998; Topping and others, 1999, 2000a, b, 2007a).

Conversely, enrichment of the upstream sediment supply 

leads to counter-clockwise discharge–concentration hysteresis 

coupled to clockwise discharge–grain-size hysteresis, herein 

referred to as Type 2 hysteresis (fig. 1C–D) (for example, 

Heidel, 1956; Dinehart, 1998; Kleinhans and others, 2007).

These two opposing styles of supply-regulation hysteresis

arise from the same physical process. Owing to Rouse-style 

mechanics (for example, McLean, 1992), finer size classes of

sediment disproportionately comprise the suspended load, as 

compared to their proportion on the bed. Therefore, changes

in the upstream supply of sediment are manifest to a greater 

degree in finer size classes of sediment (Topping and others,

2000b), leading to the above-described opposing styles 

of hysteresis. 

Changes in the upstream supply of sediment are not, 

however, required to produce discharge–concentration 

hysteresis. Rivers with dunes on the bed may exhibit another 

type of discharge–concentration hysteresis within size 

classes of sediment that have essentially unlimited upstream 

supplies. Because it takes longer for dunes to decay than to 

build in wavelength and amplitude during floods, dunes in

sand-bedded rivers tend to be relatively larger at an equivalent 

discharge of water during the recession of a flood than during

the rising limb (for example, Scott and Stephens, 1966;

Julien and Klaassen, 1995). This process of dune geometric

adjustment results in greater bed roughness arising from 

greater dune form drag during the recession than during the 

rising limb of a flood (Julien and others, 2002; Kleinhans and

others, 2007; Shimizu and others, 2009). Because of relatively

greater dune form drag (and therefore lesser skin-friction 

boundary shear stress) during the recession of floods, less

overall entrainment of sand and less upward diffusion (by 

Rouse mechanics) of the coarser size classes of sand occurs 

during the recession than during the rising limb (Kleinhans 

and others, 2007). This process generally leads to lower

suspended-sand concentrations and finer suspended-sand

median grain sizes at the same discharge during the recession 

than during the rising limb. Thus, the lag between discharge

and dune geometric adjustment during floods produces

clockwise discharge–concentration hysteresis coupled to 

clockwise discharge–grain-size hysteresis, herein referred to 

as Type 3 hysteresis (fig. 1E–F). 

Because the systematic variation between the discharge 

of water and suspended-sediment concentration exhibited by 

discharge–concentration hysteresis is common, calculation 

of accurate sediment loads in rivers requires that dischargeindependent measurements of suspended-sediment 

concentration be made at intervals shorter than the timescale 

over which this systematic variation occurs. Moreover, 

because this systematic variation commonly extends to 

sand-size sediment, calculation of accurate sand loads 

requires that these measurements include information on the 

grain‐size distribution of the suspended sediment. This type

of discharge-independent suspended-sediment measurement 

program using conventional sampling techniques is 

labor‐intensive (in the field and in the laboratory), logistically

difficult (floods may occur at night), and expensive.

Figure 1. (following page) Examples of different types of suspended-sediment hysteresis from three rivers; arrows indicate direction 

of hysteresis loop. A-B, Type 1 hysteresis evident in suspended-sediment data collected at the Sacramento River at Sacramento, 

California, U.S. Geological Survey (USGS) gaging station 11447500 (USGS, 2011a). Data collected between November 16, 1964, 

and April 6, 1965, during a winter flood that peaked at a discharge of 2,820 m3/s on December 25, 1964. A, Clockwise discharge–

concentration hysteresis in both suspended sand and suspended silt and clay. B, Counter-clockwise discharge–median-grain-size 

hysteresis in the suspended sand. C-D, Type 2 hysteresis evident in suspended-sediment data collected at the San Juan River near 

Bluff, Utah, USGS gaging station 09379500 (Iorns and others, 1964). Data collected between January 14 and 31, 1952, during a winter 

flood that peaked at a discharge of ~678 m3/s on January 19, 1952. On the basis of regional USGS stream-gaging data, most of the water 

and suspended sediment in this flood likely originated in then-ungaged Chinle Creek, which enters the San Juan River ~30 km upstream 

from this station. C, Counter-clockwise discharge–concentration hysteresis in both suspended sand and suspended silt and clay. D, 

Clockwise discharge–median-grain-size hysteresis in the suspended sand. E-F, Type 3 hysteresis evident in suspended-sand data and 

Type 1 hysteresis evident in suspended-silt-and-clay data collected at the Mississippi River at St. Louis, Missouri, USGS gaging station 

07010000 (Scott and Stephens, 1966; USGS, 2011b). Data were collected between February 13 and August 10, 1961, during the annual 

flood that peaked at a discharge of 16,700 m3/s on May 11, 1961. E, Clockwise discharge–concentration hysteresis in both suspended 

sand and suspended silt and clay. F, Clockwise discharge–median-grain-size hysteresis in the suspended sand. Error bars in A-F

indicate the 95-percent confidence interval combined field and laboratory processing errors estimated on the basis of the methods of 

Topping and others (2010, 2011).
Introduction 3

men16-3130_fig01

1

10

100

1,000

10,000

0 1,000 2,000 3,000

A

0.10

0.15

0.20

0.25

0.30

0 1,000 2,000 3,000

B

1,000

10,000

100,000

0 50 100 150 200 250

C

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0 50 100 150 200 250

D

10

100

1,000

10,000

0 2,000 4,000 6,000 8,000 10,000 12,000

E

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 2,000 4,000 6,000 8,000 10,000 12,000

F

Concentration, in milligrams per liter

Median grain size, in millimeters

Water discharge, in cubic meters per second

Suspended sand

Suspended silt and 

clay 

Figure 1.
4 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Thus, declining budgets contributed to a decline in this

type of intensive suspended-sediment monitoring by the 

U.S. Geological Survey (USGS) in the 1980s (Osterkamp 

and Parker, 1991). As daily sediment-sampling programs 

declined in the United States and Canada, it became more 

attractive to employ less expensive, statistically based 

monitoring programs that relied on the assumption of a 

strong correlation and random variability between water 

discharge and suspended-sediment concentration to reduce 

the number of required suspended-sediment measurements 

(Cohn, 1995). As stated in Cohn (1995), “A motivation 

for developing these methods is economics. Traditional

methods for estimating sediment loads (Porterfield, 1972)

call for near-daily sampling over many years. Such sampling 

programs appear to be increasingly vulnerable to government 

cost-cutting.” Unfortunately, the systematic variation 

exhibited by discharge–concentration hysteresis precludes 

the supposed cost-effective use of time-invariant sediment 

rating curves in the calculation of accurate sediment loads in 

many cases. Use of sediment rating curves in the calculation 

of accurate sediment or sand loads requires that the variation 

in suspended-sediment or suspended-sand concentration 

be random (for example, Cohn and others, 1989) about the 

best-fit relation between discharge and concentration, an

assumption that is clearly invalid in many cases. As a result of 

the nonrandom variability in concentration about these curves, 

Glysson and others (2001), in an analysis of sediment data 

from 10 USGS gaging stations located in different regions of 

the United States, showed that the absolute value of the error 

in the annual sediment loads predicted by these curves fit to

entire periods of record could equal 526 percent, and errors in 

the daily sediment loads predicted by sediment rating curves 

could be as high as 4,000 percent. 

Fortunately, the increasing availability of optical and 

acoustical sensors and associated advances in optical and 

acoustical theory have opened the possibility that these 

sensors may be used in conjunction with smaller numbers of 

conventional suspended-sediment samples to allow accurate 

sediment loads to be calculated (Gray and Gartner, 2009). 

Under conditions where suspended-sediment concentrations 

are not extremely high (that is, not >>1,000 mg/L), 

and concentration-independent changes in grain size or 

sediment color do not occur (for example, Conner and 

DeVisser, 1992; Sutherland and others, 2000; Voichick and

Topping, 2014), measurements of optical-backscatterance

(Schoellhamer, 2001; Schoellhamer and Wright, 2003) and

turbidity (Rasmussen and others, 2009) have been shown 

to be highly correlated with conventional measurements of 

suspended-sediment concentration. Melis and others (2002, 

2003) and Topping and others (2004, 2006) showed that, if

maintenance were conducted at intervals frequent enough 

to maintain clean optics (every two days to two weeks, 

depending on location and season), in situ laser-diffraction 

and transmissometry-based measurements made using 

conventional-sample-calibrated LISST-100 instruments

(Agrawal and Pottsmith, 1994, 2000) could be used to measure 

the concentration and grain-size distribution of suspended 

sediment at reasonable accuracy. Topping and others (2004,

2006, 2007b), Wall and others (2006), and Wright and others 

(2010) showed that acoustical measurements made at a single 

frequency could be calibrated to yield suspended-sediment 

concentration measurements in rivers, although these simple 

single-frequency methods are only accurate under extremely 

limited ranges of sediment grain size (as shown herein).

Purpose and Scope

In this report, we develop and test a physically based 

multi-frequency acoustical method for making measurements 

of the concentration and aspects of the grain-size distribution 

of suspended sediment in rivers at 15-minute intervals over 

long-term (that is, decadal) time scales. The biases and

errors associated with this multi-frequency method and 

those associated with simpler single-frequency methods 

are evaluated. The multi-frequency acoustical method

described herein is developed to provide measurements 

that (1) are not limited to relatively low concentrations of 

suspended sediment (compared to optical-backscatterance, 

turbidity, and LISST‐100 measurements), (2) are relatively

unaffected by changes in sediment grain size (compared 

to optical-backscatterance, turbidity, and single-frequency 

acoustical measurements), (3) are unaffected by changes in 

sediment color (compared to optical-backscatterance and 

turbidity measurements), and (4) do not require the intense 

cleaning maintenance that LISST-100 measurements require.

Instruments, Study Sites, and 

Field Methods

Acoustic-Doppler Profilers

This report presents the results from a 12-year

study focused on the development and testing of a 

physically based method for the continuous measurement 

of suspended-sediment concentration and grain size at 

15-minute intervals in rivers using multi-frequency arrays of 

single‐frequency “side-looking” acoustic-Doppler profilers

(ADPs). These arrays typically consist of 1- and 2-MHz

single-frequency ADPs; at one study site, a 600-kHz ADP is

also included in the array. The term “side-looking” is used

to indicate an ADP with transducers that generate horizontal 

acoustic beams that are oriented roughly perpendicular to 

the main flow direction. The transducers in these ADPs

are ceramic circular piston transducers and are monostatic 

(meaning that they are used to both transmit and receive 

sound). Each transducer is mounted on backing disk such that 

no sound is transmitted behind the plane of the transducer 

face. The combined ceramic transducer and backing disk is

then enclosed within a transducer cup and covered by potting. 
Instruments, Study Sites, and Field Methods 5

The principal type of ADP used in this study is the

Nortek EasyQ (Nortek, 2002). Other ADPs used are the OTT

SLD (OTT, 2014) and the Nortek Aquadopp profiler (Nortek,

2008). During the course of this study in 2006, Nortek stopped 

making the EasyQ and transferred manufacturing of this 

instrument to OTT Hydromet. OTT changed the name of this

instrument to the SLD, and in 2008, changed the design of the 

instrument by removing one transducer, thereby eliminating 

one beam. These instruments were chosen for this study

because they all have internal power regulation (meaning the 

acoustic source level remains constant regardless of daily 

fluctuations in battery voltage), they allow measurements

to be made in a relatively large number of cells (≥50) along

each acoustic beam, and they make measurements along 

up to four acoustic beams. The EasyQs and SLDs have

transducers that generate two horizontal acoustic beams, 

oriented 50° apart, and one vertical beam; the Aquadopp has

transducers that generate only two horizontal beams oriented 

50° apart. The EasyQ came in two mounting configurations,

horizontal and vertical. The horizontal configuration was the

preferred configuration of this instrument because it had a

fourth transducer that generated an acoustic beam that was 

oriented downward at a 45° angle. This fourth beam, absent

in the SLD, aided in evaluating data quality, and sometimes 

allowed correction for sediment deposition on the transducer 

or organic blockage. If unobstructed by underwater bars or 

the opposing riverbank, the lengths of the beams along which 

measurements are made range from 10 to 25 m, depending on 

ADP frequency. 

All ADPs used in this study work by the same principles: 

	(1)	 A ping, that is, a short pulse of sound, at a fixed frequency

is transmitted through water along a beam. The duration

of this pulse is determined by the programmed cell size 

and speed of sound. 

	(2)	 The intensity of sound backscattered from particles in

the water following each ping is detected over specific

time intervals dependent on the programmed cell size and 

speed of sound. 

	(3)	 The beam-parallel velocity of water in each cell along

a beam is computed by measuring the Doppler shift 

in the frequency of this backscattered sound (Nortek, 

2002, 2008, 2013). The beam-parallel velocities in each

cell along the beam are then converted to velocities 

in each cell at the midpoint between the beams 

using trigonometry. 

If no particles (or air bubbles) were present in the water, 

no sound would be backscattered (described in the next 

section of this report). In the case of interest, the particles 

responsible for backscattered sound are mostly suspended 

sediment, with some additional organic particles. Because air 

bubbles also cause attenuation and backscatter of sound (see, 

for example, the discussion on p. 223–229 in Urick, 1975;

Medwin and Clay, 1997) and, under certain conditions, can 

result in greater attenuation and (or) backscatter of sound than 

suspended sediment, ADPs must be deployed at locations with 

minimal air bubbles.

EasyQs and SLDs can make acoustical measurements 

in two modes, a velocity-cell mode and a diagnostics mode 

(Nortek, 2002). This feature makes these instruments ideal

for purposes of making acoustical measurements for velocity 

and suspended sediment, especially when the conditions at a 

study site require different measurement-cell configurations

for these two purposes. In the velocity-cell mode, the 

EasyQ and SLD make acoustical measurements in 3–9 cells 

(depending on the model and year manufactured) along the 

two horizontal beams. The locations of these velocity cells

can be programmed by the user to be anywhere along the 

lengths of the acoustic beams. In the diagnostics mode, the 

1-MHz EasyQ and SLD make acoustical measurements in 

64 cells, and the 2-MHz EasyQ and SLD make acoustical 

measurements in 50 cells. The large number of closely spaced

measurement cells in the diagnostics mode is the feature that 

allows accurate measurements of acoustic attenuation and 

backscatter to be made by these ADPs, as will be documented 

below. Diagnostics-mode measurements are the acoustical 

measurements used in the methods developed in this report 

when the ADP is either an EasyQ or a SLD. 

For all of the ADPs used in this study, the horizontal 

measurement cells are located at the midpoint between the two 

horizontal acoustic beams. Because the two horizontal beams 

are oriented 50° apart, the beam-parallel blanking distances 

and cell sizes are thus ~10 percent larger (that is, larger by a 

factor of 1/cos25°) than those programmed by the user. For 

consistency, this 10-percent increase in blanking distance and 

cell size applies to all beams, including the vertical beams 

on the EasyQ and SLD and the 45° downward beam on 

the EasyQ. 

Each acoustical measurement made by an ADP transducer 

consists of a transmitted ping (that is, pulse), followed by a 

short period of no transducer activity (that is, blanking) to 

allow “settling” of the transducer before each receive cycle, 

followed by a series of receive windows of identical length 

during which the transducer is “listening” for backscattered 

sound (Nortek, 2008, 2013). The number of receive windows

is equal to the number of programmed cells along the beam 

generated by the transducer. To ensure that measurement cells

are located at consistent, fixed locations along an acoustic

beam, ADP transmit pulse, blanking, and receive windows 

are defined in terms of their lengths (in m). The durations of

the transmit pulse, blanking, and receive windows are then 

calculated on the basis of these defined lengths and the speed

of sound. As water temperature, pressure, and (or) salinity 

change, the durations of the transmit pulse, blanking, and 

receive windows are modified as a function of the speed of

sound so that the locations of the measurement cells remain 

constant along the beam (Nortek, 2013). 
6 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

 For the ADPs used in this study, the length of the 

transmit pulse and each subsequent associated receive window 

is identical, with both varying together as a function of ADP 

frequency1. Longer transmit pulses, and the therefore longer 

receive windows, are associated with lower frequencies. 

Blanking distance is generally constant among the ADPs 

used in this study. Cell size is determined by the convolution 

of the transmit pulse and receive window, both of which 

are rectangular functions. Because the length (in m) of 

the transmit pulse and each receive window is identical, 

convolution results in a triangular-weighting function 

(commonly referred to in the signal-processing literature as 

a triangular, hat, or tent function) in which the full extent 

of a measurement cell is equal to twice the length of the 

transmit pulse, but the majority of the backscattered acoustic 

signal is returned from particles located in only the central 

50 percent of a measurement cell (fig. 5-5 in Nortek, 2013).

Thus, although the full cell size is twice the transmit-pulse

length, because most of the backscatter is from the middle 

50 percent of the cell, the nominal cell size is essentially equal 

to the transmit-pulse length. In the ADPs used in this study, 

the nominal cell size is the cell size programmed by the user. 

Although the lengths of the transmit pulse and receive window 

are equal, the durations are not. Because, in a monostatic 

system, the distance sound travels from the transducer to each 

cell and then back to the transducer is twice the distance from 

the transducer to each cell, to maintain equal lengths between 

the transmit pulse and each receive window, the durations 

of the blanking and receive windows are doubled relative to 

the duration of the transmit pulse. An example is provided in 

appendix 1 to help explain how transmit-pulse, blanking, and 

receive-window durations together determine the locations and 

sizes of the measurement cells along an acoustic beam.

Convolution of the transmit pulse and receive window 

gives rise to the following series of general equations that 

describe the location and extent of a measurement cell along 

an acoustic beam for the case where the lengths of the transmit 

pulse and receive window are identical (Nortek, 2008; Nortek,

2013). First, the distance (that is, range) from the transducer to 

the midpoint of a measurement cell is:

r r = + BLANK P ct mCELL (1)

where 

r is distance from the transducer to the 

midpoint of a cell (in m),

rBLANK is the blanking distance (in m),

c is the speed of sound (in m/s),

t

P is the transmit-pulse duration (in s), and

mCELL is the cell number.

The quantity ctP in equation 1 is the transmit-pulse length 

and is equivalent to the nominal cell size (that is, thickness) 

programmed into the ADP by the user. The position of the near

boundary of a measurement cell, that is, the boundary of a cell 

nearest the transducer, is: 

r r = + BLANK P ct ( ) mCELL −1 (2)

where 

r is now the distance from the transducer to the 

near boundary of the full extent of a cell 

(in m);

whereas the position of the far boundary of a measurement 

cell, that is, the boundary of the cell farthest the transducer, is:

r r = + BLANK P ct ( ) mCELL +1 (3)

where 

r is now the distance from the transducer to 

the far boundary of the full extent of a cell 

(in m).

The position of the near boundary of a nominal cell,

where, by virtue of the triangular-weighting function most of 

the backscatter in a measurement cell originates, is:

r r = + ct m − 



 

 BLANK P CELL 

1

2

(4)

where

r is now the distance from the transducer to the 

near boundary of the nominal cell (in m);

whereas the position of the far boundary of a nominal cell is:

r r = + ct m + 



 

 BLANK P CELL 

1

2

(5)

where

r is now the distance from the transducer to the 

far boundary of the nominal cell (in m).

Each ADP was programmed to make time-averaged 

measurements for 4 minutes out of every 15-minute interval. 

The 4-minute measurement duration was chosen to be

short enough to not consume too much battery voltage 

and long enough to suitably time average over turbulent 

fluctuations in suspended-sediment concentration (Topping

and others, 2011). ADPs are thus capable of making 

acoustical measurements over a large horizontal slice of a 

river cross section for a duration sufficiently long to sample

1

Transmit pulses and receive windows are of equal length for all Nortek

Aquadopp acoustical measurements and diagnostics-mode Nortek EasyQ and 

OTT SLD acoustical measurements. EasyQ and SLD acoustical measurements

made in the velocity-cell mode may have transmit pulses and receive windows 

that differ in length depending on the programmed locations and sizes of the 

measurement cells along the acoustic beams.
Instruments, Study Sites, and Field Methods 7

the suspended‐sediment field for statistically stable estimates

of the suspended-sediment concentration and grain-size 

distribution in this ensonified slice. Because of these features,

after processing, the acoustical measurements made by 

the ADP arrays can yield suspended-sediment information 

with greater spatial and time averaging than optical or 

automatic-pump methods, both of which can only make point 

measurements. Table 1 provides a listing of the physical 

properties of the ADPs (and their settings) that are pertinent to 

the methods developed in this report. 

Study Sites

At each study site, ADPs were rigidly mounted to 

the riverbank, although in different array configurations

depending on the riverbank geometry at each site. The

locations along a river at a study site chosen for installation 

of ADP arrays had the most spatially uniform flow so that the

suspended‐sediment conditions ensonified by the acoustic

beams would be as uniform as possible. Ideally, ADPs are 

mounted adjacent to each other so that they ensonify the 

same part of a river cross section, and therefore “sample” 

the same suspended-sediment conditions. In addition to 

ADPs, automatic pump samplers were also deployed at these 

study sites, with Sequoia Scientific Laser In-Situ Scattering

and Transmissometry (LISST)-100 type C (Agrawal and

Pottsmith, 2000) and (or) LISST-25X type C (Sequoia

Scientific, 2010) laser-diffraction instruments also deployed

during the first eight years of this study (Melis and others,

2003; Topping and others, 2004, 2006). Description of

the calibration and error of the point suspended-sediment 

measurements made with the automatic pump samplers and 

LISST instruments are outside the scope of this report and are

the subject of a forthcoming paper. 

The study sites are located on the Colorado River

in Grand Canyon National Park, Arizona, and on the Rio 

Grande in Big Bend National Park, Texas. The locations

of the study sites were chosen on the basis of a need for 

suspended-sediment data to aid river managers and the 

location of existing USGS gaging stations. Three of the six

study sites were located at USGS gaging stations, two were 

located in reaches slightly upstream from gaging stations, 

and one site required installation of a new gaging station. 

The large range in sediment conditions across the six study

sites collectively made them ideal for the purposes of this 

study, which sought to develop a method that is generally 

applicable in rivers. The locations of the study sites, as well

as downloadable data, are available at http://www.gcmrc.

gov/discharge_qw_sediment/ or http://cida.usgs.gov/gcmrc/

discharge_qw_sediment/. Study sites on the Colorado River 

are at the following USGS gaging stations: “Colorado River 

near river mile 30, 09383050,” herein referred to as the CR30 

study site; “Colorado River above Little Colorado River

near Desert View, AZ, 09383100,” herein referred to as the 

CR61 study site; “Colorado River near Grand Canyon, AZ,

09402500,” herein referred to as the CR87 study site; and

“Colorado River above Diamond Creek near Peach Springs, 

AZ, 09404200,” herein referred to as the CR225 study site. 

Table 1. Acoustic-Doppler profiler properties and settings.

[For the Nortek EasyQ and OTT SLD, the settings described are those associated with the diagnostics mode]

ADP model

2 MHz 

Nortek

EasyQ

2 MHz

OTT SLD

1 MHz

Nortek 

EasyQ

1 MHz

OTT SLD

600 kHz

Nortek

Aquadopp

Number of transducers 4 3 4 3 2

Transducer diameter (m)1 0.0279 20.025 0.0279 20.045 0.045

Blanking distance (m)3 0.22 0.22 0.22 0.22 0.55

Nominal cell size (m)3 0.22 0.22 0.44 0.44 0.55

Number of cells along beam 50 50 64 64 50

Transmit-pulse length (m)3 0.22 0.22 0.44 0.44 0.55

Receive-window length (m)3 0.22 0.22 0.44 0.44 0.55

Ping rate (Hz) 12 12 4 4 4

Averaging window (s) 240 240 240 240 240 

Pings per averaging window 2,880 2,880 960 960 960

Measurement interval (s) 960 960 960 960 960

1

Diameters listed are the active diameters of the ceramic transducers that generate the horizontal beams.

2

On OTT SLDs, the diameter of the transducer that generates the vertical (stage) beam is smaller than the diameters of the transducers that

generate the horizontal beams. 

3

Distances and lengths are beam-parallel and are not the values programmed by the user.
8 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

The numbers in these abbreviations refer to the USGS river

mile downstream from Lees Ferry, Arizona, the standard 

geographic convention used for locations along the Colorado 

River in Grand Canyon National Park. The study sites on the

Rio Grande are at the following USGS stations: “Rio Grande 

above Castolon, TX, 08374535,” herein referred to as the

RG-CAS study site; and, “Rio Grande above Rio Grande

Village, TX, 08375295,” herein referred to as the RG-RGV

study site. These study sites are respectively located in reaches

upstream from the “Rio Grande near Castolon, TX, 08374550”

and “Rio Grande at Rio Grande Village, TX, 08375300”

USGS gaging stations. In addition to the standard 1- and 

2-MHz ADPs, a 600-kHz ADP is also deployed at the CR87 

study site. ADPs were deployed at the Colorado River study 

sites in 2002, and deployed at the Rio Grande study sites in 

2010. Detailed descriptions and maps of the Colorado River 

study sites are provided in Griffiths and others (2012).

Field Methods and Required Field Conditions

Calculation of sediment loads requires integrated 

measurements of suspended-sediment concentration over the 

entire river cross section, that is, of the velocity-weighted 

suspended-silt-and-clay concentration, suspended-sand 

concentration, and suspended-sand median grain size. By 

convention, integrated measures of suspended-sediment 

concentration over an entire river cross section are made 

using isokinetic depth-integrating samplers that collect a 

physical sample of the water–suspended-sediment mixture as 

the samplers move through the water column (Edwards and 

Glysson, 1999; Topping and others, 2011). When submerged,

these samplers continuously collect the water–suspendedsediment mixture at the local flow velocity, hence the term

“isokinetic,” until the bottle or bag inside the sampler is full 

(Edwards and Glysson, 1999; Topping and others, 2011; Sabol

and Topping, 2013). Because the suspended sediment is being

collected at the local flow velocity, the suspended‐sediment

concentrations measured by these samplers are 

“velocity-weighted.” For brevity, “velocity-weighted 

suspended-sediment concentration in the river cross section” is 

hereafter referred to as CXS; “velocity-weighted suspended-siltand-clay concentration in the river cross section” is hereafter 

referred to as CXS-SILT-CLAY; “velocity-weighted suspended-sand

concentration in the river cross section” is hereafter referred 

to as CXS-SAND; and “velocity-weighted suspended-sand median

grain size in the river cross section” is hereafter referred to as 

D50-XS-SAND. The “XS” in the subscripts of these terms indicates

that these concentration and median-grain-size values are 

associated with the entire river cross section; “XS” does not

appear in the subscripts for concentrations or median grain 

sizes that are more localized in nature, for example, in only the 

part of the cross section ensonified by the acoustic beams.

The method developed herein calibrates acoustical

measurements made along horizontal acoustic beams that 

“slice” through part of a river cross section in order to 

calculate the suspended-sediment conditions in an entire 

river cross section (fig. 2). For this method to work well and 

produce accurate suspended-sediment measurements, the 

following conditions must be met:

(1) suspended-silt-and-clay concentration must be 

approximately uniform along the horizontal acoustic 

beams and, ideally, relatively uniform across the entire 

river cross section;

(2) the concentration of organic particles and air bubbles must 

be minimal, and ideally constant, along the horizontal 

acoustic beams;

(3) although variation in suspended-sand concentration is 

permissible between cells along the horizontal acoustic 

beams, no substantial net change in suspended-sand 

concentration should occur along the acoustic beams 

(meaning suspended-sand concentration should not 

generally increase or decrease along the beams in a net 

sense); and,

Figure 2. Cartoon sketch of a river cross section showing the location of an acoustic-Doppler profiler array

(green). For the method described herein to produce the most accurate results, suspended-sediment conditions 

(that is, concentration and grain-size distribution) within the part of the river cross section ensonified by the 

horizontal acoustic beams (red) should be well-correlated with suspended-sediment conditions in the entire 

river cross section (blue), as described in the text.
Theoretical Framework 9

(4) the suspended-silt-and-clay concentration, 

suspended-sand concentration, and suspended-sand 

grain-size distribution in the part of the cross section 

ensonified by the acoustic beams should be respectively

well correlated with these parameters in the entire cross 

section, with no strong dependence on an external 

nonsediment parameter. This requirement means, for

example, that suspended-sand concentration should vary 

over time in the ensonified part of the cross section in

approximately the same way that it varies on average in 

the entire cross section, without this variation changing 

substantially as a function of water discharge.

Because the goal is to calibrate the acoustical 

measurements made in part of a river cross section to the 

suspended-sediment conditions in an entire river cross section, 

we skip the intermediate step of calibrating the acoustical 

measurements to the suspended-sediment conditions in the 

ensonified part of the cross section “sampled” by the acoustic

beams. The ramification of this calibration approach is that

it is not possible to segregate variation in the calibration, and 

therefore error in the calibration, that arises from only the 

acoustical measurements from variation (and error) in the 

calibration that arises from time-varying differences between 

suspended-sediment conditions within the ensonified part of

the cross section and the entire cross section. Thus, the error

we report in the ADP calibrations arises from both processes 

and cannot be separated. 

Equal-discharge-increment (EDI) or equal-widthincrement (EWI) suspended-sediment measurements 

(Edwards and Glysson, 1999), made using depth-integrating 

samplers, form the basis of the cross-section calibrations of 

the ADP measurements. The river cross section in which these

measurements were made is herein termed the “calibration 

cross section.” Because EDI or EWI measurements cannot 

typically be made over the entire range of suspended-sediment 

conditions at each site (most notably the conditions of highest 

CXS-SILT-CLAY), cross-section-calibrated data collected by 

automatic pump samplers are used to aid in the development 

of ADP calibrations, with greater preference being given to 

the more accurate EDI or EWI measurements. The suspendedsediment measurements used to calibrate ADPs were made in 

cross sections at or near the locations of the ADP arrays. In the 

ideal case, ADPs were located within and therefore ensonify 

part of the calibration cross section, thus likely resulting 

in the least possible error in the ADP calibrations. As the 

longitudinal distance between the calibration cross section and 

the location of the ADP arrays increases, the error in the ADP 

calibrations is expected to increase. As this distance becomes 

extremely large (for example, >500 m), there is a chance that 

the suspended-sediment conditions are only poorly correlated 

between the calibration cross section and the part of the 

different cross section ensonified by the ADPs. Descriptions

of the calibration measurements, the ADP-array configuration,

and the distance between the calibration cross section and the 

ADP arrays at each study site are provided in appendix 2. 

At each study site, the period of data collection was 

separated into two parts, an initial calibration period where 

suspended-sediment measurements were used to develop 

ADP calibrations, and a later verification period where

suspended-sediment measurements were used to verify ADP 

calibrations. In terms of the analyses presented herein, the 

calibration period was used to calculate in-sample errors 

and the verification period was used to calculate out-ofsample errors. Depending on the range of measured sediment 

conditions, the duration of a calibration period at a study site 

may be one or more years.

Theoretical Framework

The Sonar Equation and Beam-Averaged 

Backscatter

The initial theoretical development that ultimately

led to the use of acoustics to measure suspended-sediment 

concentrations and grain-size distributions occurred during 

the early and middle part of the 20th century, and much of 

this work occurred in the 1940s during World War II (Urick, 

1975). Among the most important contributions of this early 

research were the derivation and formalization of the sonar 

equations (National Defense Research Committee [NDRC],

1946; Urick, 1962). In our study, the following form of the

active-sonar equation from Urick (1975) for a monostatic 

system is used: 

SL − + 2TL TS = RL (6)

where 

SL is the source level, 

2TL is the two-way transmission loss, 

TS is the target strength, and 

RL is the reverberation level in each cell 

measured by the ADP. 

This equation is hereafter referred to simply as the sonar

equation. By standard convention, each of the terms in the 

sonar equation is generally expressed as 10 times a base-10 

logarithmic ratio of acoustic intensity, in units of decibels. 

These intensities are time-averaged quantities, expressed in

units of watts per square meter (W/m2). 

Because the objective of this study was to use 

measurements of acoustic attenuation and backscatter to 

calculate suspended-sediment concentration and grain size, 

all terms in equation 6 were derived for the case where all 

scattering and all particle-related attenuation arise from only 

suspended sediment. Organic particles and air bubbles can 

dominate scattering and attenuation under certain conditions 

(for example, Urick, 1975; Medwin and Clay, 1997).
10 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Backscatter from air bubbles can exceed the backscatter 

from suspended sediment in the Rayleigh scattering regime 

(fig. 8.1.2 in Medwin and Clay, 1997). In addition, when the

resonant frequency of the air bubbles is near the frequency 

of the ADP, the backscatter from air bubbles will most likely 

mask the backscatter resulting from high concentrations 

of suspended sediment or any other particle (for example, 

fig. 8.10 in Urick, 1975; fig. 8.1.2 in Medwin and Clay, 1997).

With respect to attenuation, bubbles in the ocean have been 

measured to result in acoustic attenuation exceeding 60 dB/m 

(p. 289 in Medwin and Clay, 1997), a value slightly higher 

than the highest value of the sediment attenuation coefficient

measured in this study. Therefore, the methods developed on

the basis of the derivations of the terms in the sonar equation 

in this section of the report are limited to cases where only 

minimal organic particles and air bubbles are present. 

Knowing the exact values of all the terms in equation 6 

is extremely difficult and not required to calibrate an ADP to

measure suspended-sediment concentration. In many studies, 

only the values of 2TL, RL, and the range-dependent part 

of the TS are used to develop approximate calibrations (for 

example, Thevenot and others, 1992; Gartner, 2004; Wall and

others, 2006; Gartner and Wright, 2010; Wright and others,

2010; Wood and Teasdale, 2013). These calibrations are

referred to as “approximate” because they do not take into 

account how changes in the suspended-sediment grain-size 

distribution affect TS. Depending on ADP frequency and the 

grain-size distributions in suspension, neglecting effects of 

changing grain size on TS can lead to substantial biases in 

ADP measurements of suspended-sediment concentration. 

In our study, RL, 2TL, and all components of the TS are 

used to develop calibrations that incorporate the effects of 

changes in both the concentration and grain-size distribution 

of the suspended sediment. Because SL is constant for a 

given ADP, we use this term only in the development of the 

theoretical framework; in the calibration of each ADP, we

combine SL, the constant part of the RL, and the constant 

nonsediment parts of the TS, and replace this combined term 

with an empirically determined constant. On the basis of 

NDRC (1946) and Urick (1962, 1975), detailed definitions of

the terms in equation 6 are provided in the subsections below. 

SL is a function of the ADP and is described first, 2TL is a 

function of the medium the sound is passing through (water 

plus suspended sediment), TS is a function of the properties 

of the suspended sediment and the acoustic beam, and RL is 

measured by the ADP.

Source Level

Source level is described by

SL I

I = DI 



 



10log10  + 0

REF

T (7)

where

I0 is the acoustic intensity at the reference 

distance 1 m from the transducer, 

DIT is the transmitting directivity index for the 

outgoing (transmitted) acoustic beam, and

IREF, the reference acoustic intensity in water at 

1 μPa, is:

I p

c REF

REF

2

W N

= ρ

(8)

where 

pREF = 1×10-6 Pa is the reference pressure in 

 water, 

	 ρW = 1,000 kg/m3 is the density of water, and 

cN = 1,500 m/s is the nominal speed of sound in

water (reported to two significant digits).

For the circular piston transducers in the ADPs used 

in this study, the transmitting directivity index ranges from 

35 to 40 dB on the basis of table 3.2 in Urick (1975) and 

table 1 in this report. The ADPs used in this study transmit

approximately 20 W (Nortek, 2002, 2008). Because intensity 

is power divided by area, and the area of a sphere is equal 

to 4πr2, where r is distance from the acoustic source (that 

is, transducer) in m, this transmit power corresponds to an 

I0 = 1.59 W/m2. Substitution of these values into equation 7 

yields a source level of ~219 to 224 dB. 

Two-Way Transmission Loss

Transmission loss is described by

TL I

I = 



 



10 10  log 0

1

(9)

where 

I0 is, again, the acoustic intensity at the 

reference distance 1 m from the transducer 

(in W/m2), and

I1 is the acoustic intensity at some greater 

distance from the transducer (in W/m2). 

Transmission losses occur because of geometric

spreading of the sound with distance from the source and 

because of losses of sound in the medium through which the 

sound propagates. The geometric part of the TL varies as a 

function of the logarithm of the distance from the source, 

whereas the “medium loss” part of the TL varies linearly 

with distance (assuming the properties of the medium are 

constant along the beam). When the spreading of sound is 

unconstrained, the geometric part of the transmission loss is 

best characterized as a spherical spreading loss. Because no 

medium loss occurs in this first part of the TL, the power of the 
Theoretical Framework 11

sound passing through concentric spheres around the acoustic 

source must be equal. Therefore, because power is equal to

intensity multiplied by area, and the area of a sphere is equal 

to 4πr2, equation 9 can be rewritten as: 

 

TL

P r

P r

r

r

r = ( )

( )

















= 



 



10 10  = 20 10

0

2

2 10

0

2

10 log log log

π

π r0





 





 

 (10) 

where 

P is the power of the sound (in W),

r is distance from the transducer (in m), and

r0 is the reference distance of 1 m.

In the region very near the transducer, referred to as 

the near field, the spreading loss is not spherical owing to

a complicated oscillating pressure field in this region in

which the pressure amplitude increases to a maximum at a 

critical distance from the transducer (Lockwood and Willette, 

1973). Beyond this critical distance, in the far field, the

pressure decreases as a function of distance, as described 

by equation 10. By convention in the acoustical literature 

(Kino, 1987), the critical (that is, Fresnel) distance, rC, is 

defined as the square of the transducer radius divided by the

acoustic wavelength: 

r a C

T =

2

λ

(11)

where

aT is the transducer radius (in m), and

	 λ 	 the acoustic wavelength (in m).

As indicated by Medwin and Clay (1997), this definition of the

critical distance is only approximate, leading some researchers 

(for example, Downing and others, 1995) to define a larger

critical distance, rD, where:

r r a D C

T = = π

π

λ

2

. (12)

To correct for the near-field departure from spherical

spreading, Downing and others (1995) derived an empirical 

correction that is commonly used among those making 

acoustical suspended-sediment measurements (for example, 

Wall and others, 2006; Wood and Teasdale, 2013; Latosinski

and others, 2014). This correction and its effects on

measurements made by ADPs in this study are provided in 

appendix 3. Upon inclusion of a generic near-field correction,

the geometric part of the transmission loss in equation 10 

becomes

TL r

r = 



 



20 10  +10 ( ) 0

10 log log ψ NF (13)

where 

ѱNF is the nondimensional near-field correction.

Because (1) the ADPs used in this study do not have 

measurement cells located at distances less than the critical 

distance rC, (2) use of the Downing and others (1995) 

correction generally degrades the ADP measurements of 

relative backscatter (appendix 3), and (3) use of no correction 

provides better results at our study sites (appendix 3), we use a 

near-field correction of ѱNF=1 (that is, no correction). 

Medium transmission losses arise from the absorption 

of the sound by water and from the attenuation of the 

sound from suspended sediment, organic particles, and 

(or) air bubbles in the water. Medium transmission losses 

vary linearly with distance, except for cases where the 

concentration of suspended sediment is exceedingly high 

(Hay, 1991). In this study, we assume that the relation 

between suspended-sediment concentration and attenuation 

remains approximately linear over the entire range of 

suspended-sediment concentrations over which ADP 

acoustical measurements are possible, an approximation that is 

supported by the results from this study. 

Because, in a monostatic system, transmission losses 

occur between the transducer and each cell twice, that is, once 

when sound travels from the transducer to the cell and once 

when sound travels from the cell back to the transducer, the 

geometric and medium components of the TL are multiplied 

by 2, summed, and then referred to as two-way transmission 

losses, 2TL (Urick, 1962, 1975): 

 2 40 20 2 2 10

0

TL 10

r

r = r r 



 



log l  + og ( ) ψ α NF + + W S α (14)

where 

40 10

0

log r

r





 



 is the two-way spherical spreading loss term 

(in dB), 

20log10(ѱNF)	 is the two-way near-field correction term

(in dB),

	 αW 	 is the coefficient of absorption for acoustic

energy in water (in dB/m), and 

	 αS 	 is the sediment attenuation coefficient

(in dB/m).

In general, αW depends on water temperature, salinity, and 

pressure. Under the relatively shallow-water and low-salinity 

conditions in rivers, however, αW meaningfully depends only 

on water temperature (appendix 4). In our study, we use the 

αW of Schulkin and Marsh (1962), which is calculated on the 

basis of varying water temperature while pressure and salinity 

are held constant at appropriate low values. αS depends on 

the properties of the suspended sediment and is discussed in 

detail in the “Physical Basis for the Sediment Attenuation 

Coefficient” section below. 
12 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Target Strength

In this study, target strength, TS, is defined as the echo

returned from a given suspension of particles in a volume of 

water. In this definition, TS is composed of two components: a 

part related to the properties of the suspended sediment, and a 

part related to the geometry of the reverberating volume in the 

acoustic beam. In this usage,

TS I

I = s V 



 



10log l 10  =10 og10 ( ) +10log10 ( ) R

INC

V

(15)

where 

IR is the reflected acoustic intensity from the

scatterers in the reverberating volume at 

a reference distance 1 m in front of the 

reverberating volume (in W/m2),

I

INC is the incident acoustic intensity (that is, 

the intensity of sound in the transmitted 

acoustic beam) upon the reverberating 

volume (in W/m2),

sv is the backscattering cross section of a unit 

reverberating volume, and 

V is the reverberating volume, that is, 

measurement cell (in m3).

The backscattering cross section of a unit reverberating

volume has units of 1/m3 and depends only on the properties of 

the scatterers in suspension. For the purposes of this study, the 

scatterers are assumed to be exclusively sediment grains with 

a concentration, grain-size distribution, and wet density. The

backscattering cross section of a unit reverberating volume 

is derived below in the “Evaluation of Target Strength from

Pressure-Based Re-Derivation of the Sonar Equation” section 

of this report. 

The reverberating volume depends on the properties of

the acoustic beam and, at an instant in time is

P 2

2 = ct V r Ψ (16)

where 

c is the speed of sound (in m/s), 

t

P is the acoustic transmit-pulse duration (in 

seconds), and

Ψ is the solid angle of a two-way acoustic beam 

subtended at the face of the transducer 

(in steradians), that is, the two-way 

beam width.

The reverberating volume is the volume of water containing

the scatterers that gives rise to sV. At an instant in time, this 

volume is centered distance r from the transducer and has 

range-gated thickness ctP 2 (fig. 8.4 in Urick, 1975). The

acoustical measurements made by ADPs are not at an instant 

in time, however, but rather are made in measurement cells of 

finite thickness 2ctP, with this larger thickness arising from the 

convolution of transmit pulses and receive windows of equal 

length. Thus, the acoustic signal strength reported by an ADP

in a given measurement cell does not reflect the intensity of

the backscattered sound at an instant in time, but instead is 

the average of the intensity of the sound backscattered over 

the entire duration associated with the measurement cell, with 

most of the backscattered intensity in the average arising from 

the middle 50 percent of the cell (fig. 5-5 in Nortek, 2013).

The classic derivation of the reverberating volume in Urick

(1975) is for the different condition where the length of the 

receive window is much less than the length of the transmitted 

acoustic pulse. Therefore, the quantity ctP 2 in equation 16 

should be replaced by ctP when using measurements made 

by the types of ADPs used in this study. For convenience in 

the derivations of the relative backscatter and fluid-corrected

backscatter below, replacement of the quantity ctP 2 in 

equation 16 with ctP, substitution of equation 16 into 15, and 

rearrangement yields

 ( ) ( ) ( ) 10 V 10 P 10 TS s =+ + 10log 10log 20log ct Ψ r . (17)

By definition, the cross-sectional area of the reverberating

volume formed by the intersection of the two-way acoustic 

beam at distance r with a sphere centered on the face of the 

transducer is thus 2 Ψr , where

( ) ( ) 4

0 = , , ′ ∫

π

Ψ Ω bb d θφ θφ (18)

where 

b(θ,φ) is the transmitted beam pattern in spherical 

coordinates (θ,φ), 

b′(θ,φ) is the received beam pattern in spherical 

coordinates (θ,φ), and

Ω denotes a solid angle (in steradians).

As a result of the beam shaping that occurs between the 

transmitted and received pulses, that is, how the beam width 

evolves as a function of the interaction between b(θ,φ) and 

b′(θ,φ) (for example, fig. 8.3 in Urick, 1975), the two-way

beam width is likely ~20 to 30 percent smaller than the width 

reported by vendors for the transmitted beam (for example, 

EdgeTech, 2015).

The integral in equation 18 is difficult to solve and has

been evaluated using several different methods. One of the 

more common methods in the literature is to estimate the 

two-way beam width over only the part of the main lobe 

of the beam where the acoustic power exceeds –3 dB (that 

is, exceeds half) of peak power (for example, Hay, 1991;

Hay and Sheng, 1992; Thorne and others, 1993). However,

because suspended sediment or other particles outside this 

part of the main lobe still contribute small amounts to the TS, 

it is desirable to solve equation 18 over a larger part of the 
Theoretical Framework 13

two‐way beam pattern. The first way this has been done is

through application of the concept of the equivalent two-way 

beam width (Urick, 1975; Simmonds and MacLennan, 2005).

As defined in Urick (1975), the equivalent two-way beam

width is the two-way width of an idealized acoustic beam 

where the response is unity within this width and zero outside 

this width. Mathematically, this method results in b(θ,φ) and 

b ́(θ,φ) both being replaced by one and equation 18 becoming:

( ) ( ) ( )( ) 4

0 0 , , 11 Ψ

= ′ = ∫ ∫ π

Ψ ΩΩ bb d d θφ θφ . (19)

A different, but mathematically equivalent, version of the 

integral in equation 18 was solved by Thorne and Hardcastle

(1997) over the entire main lobe of the acoustic beam for 

transducers typical of the type used in this study; their solution

to this integral is used in this study and is described in the 

context of equation 15 in the “Evaluation of Target Strength

from Pressure-Based Re-Derivation of the Sonar Equation” 

section of this report below.

Reverberation Level

The reverberation level is defined as

RL I

I = b A A RL 



 



10log10  = − ( ) + RL

REF

SF N OFFSET , (20)

where

IRL is the intensity of the reverberation, excluding 

noise, measured by the ADP in a cell on a 

beam,

I

REF is the reference intensity in water at 1 μPa, as 

defined in equation 8,

A is the ADP-recorded strength of the returned 

acoustic signal, including noise, in a cell 

on a beam (in units of counts), 

AN is the instrument noise floor (in units of

counts),

bSF is a scale factor (in units of dB/count) used to 

convert counts to decibels using a floating

scale within the dynamic range of the ADP, 

and

RLOFFSET is a linear offset (in units of dB) required to 

shift decibels calculated using a floating

scale within the ADP dynamic range to 

absolute decibels. 

A is a time-averaged quantity measured in each cell along 

each beam and is commonly referred to as the received signal 

strength indication, that is, RSSI (Deines, 1999; Lohrmann,

2001); the locations and dimensions of each cell are

determined by c and t

P

, as described in the previous section. 

Upon reception at the transducer, the returned acoustic signal 

is amplified by the electronics in the ADP (Nortek, 2008;

Nortek, 2013). Measured in units of counts, the RSSI is an 

inverse measure of the amount of amplification (that is, gain)

that must be applied to a given returned acoustic signal, where 

the logarithm of the gain setting is inversely proportional 

to the number of counts (Nortek, 2008). Weaker acoustic 

returns require greater gain, and are therefore recorded as a 

lower number of counts. For the ADPs used in this study, the 

dynamic range in the quantity bSFA is ~90 dB (Nortek, 2002, 

2008). The relation between the logarithm of the gain setting

and A is linear and has an accuracy of 1 to 2 dB over a 70-dB 

range (Lohrmann, 2001). Outside this range, that is, at values 

of the quantity bSFA much greater than ~90 dB, the relation 

between the logarithm of the gain setting and A is nonlinear 

(Lohrmann, 2001)2. A is recorded in the ADP datalogger to the 

nearest integer. Therefore, the precision of the RL is limited 

by the value of bSF. For the ADPs used in this study, 1 count is 

equal to 0.40 to 0.45 dB within the linear range between the 

logarithm of the gain setting and A (Nortek, 2008). For these 

ADPs, the precision in measured A is therefore 0.45 dB within 

this linear range (Nortek, 2002, 2008). 

A is the sum of two components, both measured in 

counts, such that

A A = + N S A (21)

where

AS is the part of the RSSI produced by the 

interaction of the sound emitted by the 

ADP with the suspended sediment in the 

moving water. 

The instrument noise floor, AN, typically ranges from 25 to 

30 counts among the ADPs used in this study, and is a 

combination of noise produced by the ADP electronics and 

environmental noise. AN is directly measured by the ADP 

every 15 minutes (that is, the same interval as measurements 

of A) by recording the intensity of the backscattered sound 

in receive windows that are associated with a zero-transmit 

pulse. Manufacturers do not typically calibrate ADPs so that 

a given value of A corresponds to a fixed value of decibels.

Thus, although the relation between counts and decibels is

precise to within 0.45 dB and is accurate to within 1–2 dB in 

a relative sense within the linear range between the logarithm 

of the gain setting and A, the relation between counts and 

decibels is somewhat unconstrained in an absolute sense. In 

other words, a change in A of +1 count equates to a change 

in the RL of +0.40 to +0.45 dB, but this change could be, for 

2

Acoustical measurements made in this nonlinear range between the 

logarithm of the gain setting and A should be avoided. If measurements are 

made in this range, they may need adjustment as described in appendix 7.
14 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

example, from 50 dB to 50.40–50.45 dB, or from 60 dB to 

60.40–60.45 dB. This “floating-scale” bias is accounted for

by the RLOFFSET term in equation 20. In addition to removing 

the floating-scale bias in RL, the value of RLOFFSET also 

compensates for any bias in SL. On the basis of full solution of 

the sonar equation using the version of equation 6 re-derived 

in the “Evaluation of Target Strength from Pressure‐Based

Re-Derivation of the Sonar Equation” section of this report 

below, the value of RLOFFSET was found to typically be in 

the 75- to 85-dB range among the ADPs used in this study, 

although in several cases it was much larger. RLOFFSET is 

required only to close the sonar equation and is not needed 

or used to develop ADP suspended-sediment calibrations, 

because the calibrations are instrument specific such that the

value of RLOFFSET is embedded in the calibrations. 

Because the particles associated with AS are moving 

with the water, a Doppler shift is produced, and the frequency 

associated with AS is slightly different than the transmitted 

acoustic frequency. For ADP deployments in rivers, this 

frequency shift is small and has negligible effect on the 

accuracy of the methods described in this report. For example, 

under typical values of c in rivers, when the velocity of water 

through the acoustic beams is ~3 m/s, the frequency associated 

with AS could differ from the transmitted acoustic frequency 

by as much as 0.2 percent, depending on the angle between the 

acoustic beams and the velocity vector. 

The value of bSF varies substantially between different 

ADPs (Deines, 1999) and may vary substantially between 

different beams on the same ADP (Wall and others, 2006). For 

example, Deines (1999) reported a range of 0.35 to 0.55 in 

bSF among different model acoustic-Doppler current profilers,

and Wall and others (2006) reported a range of 0.41 to 0.46 

among different beams on the same acoustic-Doppler profiler.

This small variation in bSF is not important, however, because 

knowing the exact value of bSF is not required to develop 

accurate ADP calibrations using the methods described in 

this report, so long as bSF is kept constant for a given ADP. 

Therefore, in this study, bSF is set equal to 0.43 on the basis of 

Lohrmann (2001). 

Relative Backscatter

The two quantities calculated on the basis of ADP

measurements that are used in the methods described in 

this report to measure suspended-sediment concentration 

and grain size are the sediment attenuation coefficient, αS, 

and the beam-averaged backscatter, B. To calculate these

quantities from the raw measurements of A in each cell on 

each ADP beam, it is useful to first develop a relation between

the relative backscatter, B, and r. Relative backscatter, B, 

represents the backscatter recorded by the ADP adjusted 

for losses that occur along the beams; it is the sum of the

reverberation level, RL, instrument noise floor (converted to

dB using bSF), and the two-way transmission loss, 2TL, minus 

the range-dependent part of the TS (that is, the 20log10r term) 

in equation 17, and the reverberation level offset, RLOFFSET. 

Following the convention of Thevenot and others (1992), the

relative backscatter in each cell on each beam is, thus,

B R = + ( ) L bSFA R N O − L T FFSET + − 2 2 L r 0 ( ) 10 log . (22)

Note that, in this form, the first three terms on the right side

are simply equal to bSFA by equation 20. Equation 22 is written 

in this form because RLOFFSET is an unknown positive constant 

and AN is not recorded in the datalogger of every model 

ADP, whereas A is. Substitution of equations 14 and 20 into 

equation 22 then yields

B b = + A r ( ) + +r r − ( ) r SF NF W S 20 2 2 40 10 10 0 log l ψ α α og . (23)

Because r0=1 m, the 40log10(r0) term on the right side 

of equation 23 is zero, and is therefore dropped from 

equations 24 and 25 below. As everything needed to calculate 

the first three terms on the right side of equation 23 is stored

in the ADP datalogger, for simplification it is convenient to

define another term, the fluid-corrected backscatter (Wright

and others, 2010), as

B b A r r F S = + F N 20log( ) ψ α F W + 2 , (24)

which allows equation 23 to be written in the following 

simplified form:

B B = + r F S 2α , (25)

where 

B is the relative backscatter (in dB),

BF is the fluid-corrected backscatter (in dB),

	 αS is the sediment attenuation coefficient (in

dB/m), and

r is the distance from the transducer to the 

midpoint of each measurement cell (in m).

When two horizontal beams are used on an ADP, BF in 

each cell is calculated on the basis of A averaged among 

equivalent cells on each beam. This “beam averaging” among

equivalent cells allows greater accuracy in the calculation of 

αS (described below) because the ensonified volume used to

calculate αS is doubled in comparison to that associated with 

only one beam.
Theoretical Framework 15

distribution, and sediment density (Urick, 1948; Flammer,

1962; Flammer and others, 1969; Moore and others, 2013).

Flammer (1962) and Flammer and others (1969) divided the 

scattering-loss region into three parts: a scattering-loss region 

where losses were produced primarily by redirection of sound 

away from the direction of the detector, a diffraction-loss 

region where larger sediment grains “shadowed” other grains 

in addition to redirecting the sound away from the detector, 

and a transition-loss region between the scattering and 

diffraction loss regions. Most recent investigators have treated 

Flammer’s (1962) scattering, transition, and diffraction-loss 

regions as one scattering-loss region (Sheng and Hay, 1988;

Hay, 1991, Thorne and others, 1993; Schaafsma and Hay,

1997; Thorne and Buckingham, 2004; Thorne and Meral,

2008; Moore and others, 2013). In this study, we follow this

more-recent convention and distinguish only between viscousand scattering-loss regions.

In this study, we calculate the sediment attenuation 

coefficient (αS) for each ADP measurement using the 

least-squares linear regression between BF and r, described 

in the previous section of the report, and then compare this 

empirical measurement of αS with theoretically predicted 

values of αS at multiple acoustic frequencies to estimate the 

grain-size distribution of the suspended sediment that gives 

rise to the acoustic attenuation. Our study uses the conventions 

of Urick (1948), Flammer (1962), Hay (1983), and Moore 

and others (2013) to calculate this theoretical value of αS that 

includes the effects of both viscous and scattering losses. In 

addition, we use the approach of Thorne and Meral (2008)

and Moore and others (2013) to account for the effect on αS

of multiple grain sizes of sediment. Finally, we follow Moore 

and others (2013) and correct for the fact that probability 

density functions describing grain-size distributions measured 

in the laboratory are not equivalent to those derived in the 

acoustical literature. 

Acoustical theory typically treats sediment grain-size 

distributions as “number size distributions,” where particles 

are treated identically regardless of their size. Sediment 

grain-size distributions are, however, always measured in the 

laboratory in terms of either the mass or volume of sediment 

in each size class. Thus, a sediment grain-size distribution

measured in a laboratory must to be converted to a “number 

size distribution” before either the form function or scattering 

cross section (Thorne and Meral, 2008) associated with this

grain-size distribution can be calculated. This mathematical

operation is conducted by simply dividing the measured 

amount (mass or volume) of sediment in each size class by the 

volume of a sediment grain in that size class. So long as the 

sediment density is constant across the grain-size distribution, 

dividing by volume has the equivalent effect of dividing by 

mass in making this correction, regardless of whether the 

grain-size distribution was analyzed by volume or mass in 

the laboratory.

Calculation of the Sediment Attenuation 

Coefficient and Beam-Averaged Backscatter

For the condition where the concentration, density, 

and grain-size distribution of suspended sediment are all 

approximately constant along the horizontal acoustic beams, 

there is no net change in B along the beams. Under this 

assumed condition, all variation in B should average out 

along the beams (or at least over most of the beams), and B

can therefore be treated as constant along the beams for any 

single ADP measurement. When B is held constant along the 

beams, the sediment attenuation coefficient αS can then be 

calculated using a least-squares linear regression between 

BF and r (after Topping and others, 2006, 2007b) while

iteratively solving for the effective noise floor. The effective

noise floor is related to curvature in the measurements of A

along the acoustic beams as the noise floor is approached, and

is greater than the instrument noise floor, AN. Determining 

the effective noise floor is an iterative process described

completely in appendix 7. In this regression, αS is equal to 

–1/2 times the slope of the relation between r and BF. Once αS

is known, B is then calculated in each cell where A exceeds 

the effective noise floor (using equation 23) and then averaged

among the equivalent cells in each beam used. Once this first

beam-averaging step is completed, the values of B in each cell 

are then averaged among all cells along the beams where A

exceeds the effective noise floor. This final averaged quantity

is termed the beam-averaged backscatter, B , used extensively 

below. Figure 3 illustrates the steps required to compute the 

fluid-corrected backscatter, sediment attenuation coefficient,

relative backscatter, and beam-averaged backscatter for 

several measurements from the CR87 study site. 

Physical Basis for the Sediment Attenuation 

Coefficient

Acoustic attenuation caused by the presence of suspended 

sediment arises from two distinctly different physical 

processes that vary in importance largely as a function of 

ADP frequency, sediment grain size, and sediment density 

(Flammer, 1962; Flammer and others, 1969). These include

losses arising from viscous drag between the water and 

sediment grains (Urick, 1948), and losses arising from the 

scattering of sound by sediment grains in directions other 

than “back” toward the detector (Lamb, 1945; Urick, 1948;

Morse, 1948; Sheng and Hay, 1988; Hay, 1991; Schaafsma

and Hay, 1997; Thorne and Buckingham, 2004; Thorne and

Meral, 2008). For typical ADP frequencies, viscous losses 

dominate when the suspended sediment is relatively fine,

whereas scattering losses dominate when the suspended 

sediment is relatively coarse. The transition between these

two loss regions depends largely on acoustic frequency, 

the characteristics of the suspended-sediment grain-size 
16 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

men16-3130_fig03

αS = 0.0283 dB/m

αS

 = 0.0549 dB/m

αS = 1.31 dB/m αS

= 11.6 dB/m

B = 77.4 dB

B = 87.2 dB

B = 97.4 dB

B = 89.2 dB

0

50

100

150

200

250

Effective noise floor = 41 counts

Instrument noise floor = 25 counts

30

40

50

60

70

80

90

100

70

75

80

85

90

95

100

0 5 10 15 20 25 30

Distance from transducer along horizontal beams, in meters

B, in decibels

BF , in decibels

A, in counts

A

B

C

1 37.9 0.116 ±0.011

2 0.116 ±0.011

3 0.115 ±0.011

4

76.4

1,940

17,500

±2.7 23.2

±5.4 335

±140 4,330

±1,000 281 0.090

±3.1

±41

±530

±29 ±0.009

Measurement

number

CXS-SILT-CLAY CXS-SAND D50-XS-SAND

Figure 3. Examples of the steps 

in the conversion from the raw 

acoustical measurements (in counts) to 

beam-averaged backscatter (in dB) for 

four very different suspended-sediment 

conditions at the CR87 study site. 

Acoustical measurements are from 

the 1-MHz acoustic-Doppler profiler 

(ADP). Measurement 1 was made at 

17:34 on March 2, 2007; measurement 2 

at 17:47 on January 29, 2008; 

measurement 3 at 15:17 on March 6, 

2008; and measurement 4 at 6:35 on 

August 25, 2012. Indicated values of 

CXS-SILT-CLAY (in mg/L), CXS-SAND (in mg/L), 

and D50-XS-SAND (in mm) with 95-percentconfidence-level errors are from 

concurrent equal-discharge increment 

measurements. A, Acoustic signal 

strength, A, averaged between the 

corresponding cells (1–64) in the two 

horizontal acoustic beams plotted as a 

function of r. Shown are the instrument 

noise floor of 25 counts (dashed 

horizontal line) and the effective noise 

floor of 41 counts (solid horizontal 

line), calculated on the basis of an 

iteratively determined best noise-floor 

offset of 16 counts. B, Fluid-corrected 

backscatter, BF, plotted as a function 

of r. Shown for each measurement are 

the values of αS determined by leastsquares linear regressions fit to the 

values of BF. C, Relative backscatter, 

B, plotted as a function of r. Shown for 

each measurement are the values of 

the beam-averaged backscatter, B .
Theoretical Framework 17

Following Urick (1948), αS is the product of the unit 

sediment attenuation coefficient, αUNIT, and the concentration 

of suspended sediment, C:

α α S = UNITC . (26)

In our study, αUNIT is defined as αS at a suspended-sediment 

concentration of 1 mg/L, where αS is expressed in units 

of dB/m and C expressed in units of mg/L. αUNIT is the 

combined ensemble-averaged viscous- and scattering-loss 

attenuation coefficient and is derived in this study using

equations 4–10 in Moore and others (2013), and making 

the appropriate conversions, such that αUNIT is expressed in 

units of decibel‐liter per meter-milligram or [dB-L]/[m-mg].

Importantly, the following four conditions are assumed 

for equation 26 to apply: the suspended-sediment grains 

are randomly distributed within the ensonified volume of

water, the scattering is incoherent, the effects of multiple 

scattering are unimportant, and the effects of extremely high 

concentrations on attenuation are also unimportant.

Changes in the sorting of the suspended-sediment 

grain-size distribution have a major effect on αUNIT. As 

illustrated by Moore and others (2013) and suggested by 

Flammer (1962), a grain-size distribution of sediment with 

median grain size D50 has different attenuation characteristics 

than does sediment of uniform size; the grain-size distribution

tends to have lower peaks and higher troughs in the D50–αUNIT

relation (fig. 4A). As the sorting of the suspended-sediment 

grain-size distribution decreases (meaning the geometric 

sample standard deviation, σG, of the distribution increases), 

the magnitudes of the viscous-loss maximum and 

scattering-loss maximum in αUNIT both decrease while the 

magnitude of the minimum in αUNIT between the viscous- and 

scattering-loss regions increases. As the magnitudes of the 

two maxima decrease, the positions of these maxima shift 

to the right in figure 4A, meaning that the D50 associated 

with the viscous- and scattering-loss maxima in αUNIT both 

increase slightly. In contrast, as the magnitude of the minimum 

increases, the position of the minimum between the viscousand scattering-loss regions shifts to the left and the D50

associated with this minimum in αUNIT decreases. Ultimately, 

as the sorting becomes extremely poor and σG increases 

to ~4φ, the minimum in αUNIT between the viscous- and 

scattering-loss regions disappears. In this report, all grain-size 

distributions are approximated as log-normal distributions, 

and sorting of sediment is measured using φ units, defined as:

φ = −log2 D, where D is the diameter of the sediment grain in 

mm (Krumbein, 1936). 

Changes in the density of suspended sediment (ρS) also 

result in large changes in the magnitudes of the maxima in 

αUNIT and a slight shift in the position of the minimum in αUNIT

between the viscous- and scattering-loss regions (fig. 4B). In 

cases where the suspended sediment is dominated by clay, 

and depending on whether the clay in suspension is composed 

dominantly of smectite-, kaolin-, illite-, or chlorite-group 

minerals, the density of the suspended sediment can vary 

from ~1,800 kg/m3 to ~3,300 kg/m3 (DeWit and Arens, 1950;

Deer and others, 1966; Gartner and Carder, 1979). The wet

density of montmorillonite determines the lower limit of 

this range and the (wet or dry) density of chlorite determines 

the upper limit of this range. Not only does the density vary 

between different clays, but the density of kaolin- illite-, and 

smectite-group clays generally decreases when wet as water 

gets absorbed into and adsorbed onto the clay structure (DeWit 

and Arens, 1950; Deer and others, 1966; Gartner and Carder,

1979). Among the clays that decrease in density when wet, 

smectite-group clays have the largest decrease in density 

and kaolin-group clays have the least decrease in density. 

Voichick and Topping (2014) showed empirically that, absent

any major change in silt and clay grain-size distribution, 

changes in the amount of smectite comprising the suspended 

clay results in large changes in αS. Their analysis showed

that, in cases where the suspended clay was composed of 

less smectite than normal, αS was approximately twice that 

under more typically smectite-rich silt-and-clay conditions. 

To illustrate why this observed difference occurred, an

example of a theoretical D50–αUNIT relation for three sediment 

densities is shown in figure 4B. The three densities used in

this example are: (1) the standard 2,650 kg/m3 quartz density 

that best approximates the typical densities of sand and silt, 

(2) the 3,300 kg/m3 density that is the upper limit of chlorite 

density, and (3) the density of 1,800 kg/m3 (DeWit and Arens, 

1950; Gartner and Carder, 1979) that best approximates the

fully saturated wet density of montmorillonite, a dominant 

smectite-group clay mineral found in the suspended sediment 

of the tributaries of the Colorado River in the study area 

(Beverage and Culbertson, 1964). For simplicity, the values 

of αUNIT in this example were calculated for constant sediment 

density across the entire grain-size distribution. 

The example in figure 4B shows that changes in 

sediment density result in major changes in the magnitudes 

of the viscous-loss (left peak) and scattering-loss maxima 

(right peak). In addition, changes in sediment density 

also result in slight changes in the D50 associated with 

these maxima (especially in the case of the viscous-loss 

maximum). As sediment density increases from 2,650 to 

3,300 kg/m3, the magnitude of the viscous-loss maximum 

in αUNIT is predicted to increase by ~31 percent and the 

magnitude of the scattering-loss maximum in αUNIT is 

predicted to decrease by ~17 percent. Conversely, as 

sediment density decreases from 2,650 to 1,800 kg/m3, the 

magnitude of the viscous-loss maximum in αUNIT is predicted 

to decrease by roughly a factor of two and the magnitude of 

the scattering loss maximum in αUNIT is predicted to increase 

by roughly a factor of 1.5. Although this increase in the 

scattering-loss maximum may seem unrealistic, because the 

density of sand-size sediment typically remains constant at 

~2,650 kg/m3 in rivers, flocs of low-density clay can approach
18 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

1 mm in diameter under the right conditions, especially in 

estuaries and marine environments (Dyer and Manning, 1999;

Hill and others, 2001; Curran and others, 2007), and the

formation of flocs can substantially alter the acoustic

backscattering and attenuation characteristics of suspended 

clay (MacDonald and others, 2013; Rouhnia and others, 2014;

Thorne and others, 2014).

In addition to the controls of sediment sorting and 

density, acoustic frequency also exerts a strong control on 

αUNIT (fig. 5A). For a given sorting, as acoustic frequency 

increases, the magnitude of αUNIT increases and the positions 

of the maxima and minimum in the D50–αUNIT relation shift to 

the left. Because of this leftward shift, the D50 associated with 

the transition from viscous (left peak) to scattering (right peak) 

losses decreases as ADP frequency increases. For example, 

for grain-size distributions with σG=1φ, the 600-kHz transition 

from viscous to scattering losses dominating αUNIT occurs 

at a D50 of ~0.1 mm, whereas the 3-MHz transition from 

viscous to scattering losses dominating αUNIT occurs at a D50 of 

~0.025 mm (fig. 5A). 

From the example in figure 5A, it is evident that there 

exists an optimal range of acoustic frequency, suspendedsand D50, and suspended-silt-and-clay D50 where the sand 

contributes very little to αUNIT and the sediment attenuation 

coefficient can be approximated as

α α S ≈ UNIT S C ILT-CLAY . (27)

In many rivers, the D50 of the suspended sand ranges from 

0.0625 to ~0.25 mm (very fine to fine sand) and the D50 of 

the suspended silt and clay ranges from ~0.0005 to ~0.01 mm 

(fine clay to fine silt). For the sorting of the grain‐size

distributions portrayed in figure 5A, these D50 result in a fair 

degree of separation between the value of αUNIT associated 

with the sand and the value of αUNIT associated with the silt 

and clay, especially at the 600-kHz and 1-MHz acoustic 

frequencies. For these D50 and ADP frequencies, CXS-SILT-CLAY

can be reasonably accurately predicted by αS so long as the 

following three conditions are met: (1) the suspended-silt-andclay concentration (CSILT-CLAY) is approximately uniform within 

the ensonified volume along the acoustic beams, (2) the spatial

distribution of suspended silt and clay in the river cross section 

is approximately uniform such that CSILT-CLAY in the ensonified

volume is approximately equal to CXS-SILT-CLAY, and (3) the 

suspended-sand concentration (CSAND) does not greatly exceed 

the suspended-silt-and-clay concentration (CSILT-CLAY) along the 

men16-3130_fig04

0.0001 0.001 0.01 0.1 1

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0001 0.001 0.01 0.1 1

αUNIT, in decibel-liter per meter-milligram

D50, in millimeters

A B ρS

 = 2,650 kg/m3

ρS = 3,300 kg/m3

ρS = 1,800 kg/m3

σG = 0.1φ

σG = 0.5φ

σG = 1φ

σG = 2φ

σG = 3φ

σG = 4φ

Figure 4. Effects of changes in (A) sediment sorting and (B) sediment wet density (ρS) on αUNIT, at an acoustic 

frequency of 1 MHz. A, Predicted values of αUNIT at the 1-MHz frequency for the median grain size (D50) of log-normal 

suspended-sediment grain-size distributions with geometric standard deviations (σG) of 0.1φ, 0.5φ, 1φ, 2φ, 3φ, and 

4φ. The wet density of the suspended sediment is held constant at 2,650 kg/m3 (quartz density) in this example. 

B, Predicted values of αUNIT at the 1-MHz frequency for the D50 of a log-normal suspended-sediment grain-size 

distribution with σG=1φ and sediment densities of 2,650 kg/m3 (quartz density), 3,300 kg/m3 (chlorite density), and 

1,800 kg/m3 (fully-saturated montmorillonite wet density). Although αUNIT is predicted to decrease as wet density 

decreases for silt-and-clay-sized sediment, αUNIT for sand-size flocs of fully saturated montmorillonite is predicted to 

exceed the values of αUNIT for quartz-density sand.
Theoretical Framework 19

acoustic beams. This result allowed Topping and others (2006,

2007b) and Wright and others (2010) to develop their first‐cut

single-frequency approximate method of using acoustic 

attenuation to measure CXS-SILT-CLAY and acoustic backscatter to 

measure CXS-SAND, as reviewed by Hanes (2012, 2013). 

Sorting differences between the grain-size distributions 

of the suspended sand and the suspended silt and clay will 

typically increase the separation between the value of αUNIT

associated with the sand and the value of αUNIT associated 

with the silt and clay, making the key assumption in the 

method of Topping and others (2006, 2007b) and Wright and

others (2010) expressed by equation 27 all the more valid, 

especially at the 600-kHz frequency. In the Colorado River 

and Rio Grande basins, the sorting of the suspended sand is 

much better than the sorting of the suspended silt and clay, a 

condition that is likely in most rivers. For example, in each 

of the 22 rivers selected by Wright and others (2010) to be 

representative for rivers in all regions of the United States, σG

for the suspended silt and clay was much larger than σG for the 

men16-3130_fig05

D50, in millimeters

0.0001 0.001 0.01 0.1 1 Silt-sand break 0.0008 mm 0.125 mm

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.0001 0.001 0.01 0.1 1

A B

αUNIT, in decibel-liter per meter-milligram

600 kHz

1 MHz

2 MHz

σG = 1φ at 600 kHz

σG = 1φ at 1 MHz

σG = 1φ at 2 MHz

σG = 1φ at 3 MHz

Figure 5. Effects of acoustic frequency and grain-size-dependent differences in sorting on αUNIT . A, 

Predicted values of αUNIT at acoustic frequencies of 600 kHz, 1 MHz, 2 MHz, and 3 MHz for the D50 of log-normal 

suspended-sediment grain size distributions with σG=1φ. B, Predicted values of αUNIT at acoustic frequencies of 

600 kHz, 1 MHz, and 2 MHz for the likely typical case where the suspended silt and clay is much more poorly sorted 

than the suspended sand and has slightly lower density than the suspended sand. σG=3φ and ρS=2,500 kg/m3 for all 

silt-and-clay-sized sediment and σG=0.63φ and ρS=2,650 kg/m3 for all sand-size sediment in this plot. The unrealistic 

downward step changes in the values of αUNIT at the silt-sand break are artifacts mostly of the abrupt decrease in 

σG at the silt-sand break. In reality, the decrease in σG across this break is likely more gradual, resulting in more 

gradual decreases in the values of αUNIT across the silt-sand break. Indicated are the typical D50 values measured for 

the suspended sand (0.125 mm) and calculated for the suspended silt and clay (0.0008 mm) at the CR87 study site. 

Among 1,770 equal discharge increment (EDI) measurements made at this site between 1999 and 2013, the mean 

D50 of the suspended sand was 0.125 mm (to the nearest 1⁄4-φ increment) and the mean σG of the suspended sand 

was 0.63φ.

suspended sand (on the φ scale). At the Colorado River and 

Rio Grande study sites, σG of the suspended sand measured 

among many EDI and (or) EWI measurements was ~0.63 to 

0.65φ, whereas σG of the suspended silt and clay was estimated 

to range from ~2.5 to 3.0φ. An example of how these sorting 

differences increase the separation between the sand αUNIT and 

the silt and clay αUNIT at the CR87 study site is provided in 

figure 5B. 

Figure 5B illustrates that the separation between the 

sand αUNIT and the silt and clay αUNIT is much greater at 

lower acoustic frequencies. At 2 MHz, the silt and clay 

αUNIT is 1.4 times the value of the sand αUNIT; at 1 MHz, the

silt and clay αUNIT is 4.5 times the value of the sand αUNIT;

and at 600 kHz, the silt and clay aUNIT is 7.3 times the value 

of the sand αUNIT. Thus, because of the greater validity of

the approximation in equation 15 at lower frequencies, 

we generally use the values of αS measured at the lowestfrequency present at a study site to calculate CXS-SILT-CLAY. 
20 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Estimation of the Grain-Size Distribution and 

Wet Density of the Silt and Clay: Attenuation 

Constraint 

Two constraints are used to estimate the grain-size

distribution and the wet density of the suspended silt and 

clay. The first of these constraints is to use multiple acoustic

frequencies to solve for the values of the D50, σG, and ρS for 

the silt and clay grain-size distribution that best agree with the 

values of αUNIT empirically determined using ADP-calculated 

values of αS and measured values of CXS-SILT-CLAY (fig. 6). This

first constraint is a slightly modified version of the method

devised by Moore and others (2013). Whereas Moore and 

others (2013) held the density of the silt and clay constant at 

quartz density, we allow it to vary between the reasonable 

bounds set by the density of chlorite (3,300 kg/m3) 

and the wet density of montmorillonite (1,800 kg/m3). 

The second constraint is to use multiple acoustic frequencies

to solve for the values of the D50, σG, and ρS for the silt and 

clay that, in combination with the known values of the D50, 

σG, and ρSAND for the suspended sand (appendix 5), best predict 

the observed values of the excess backscatter, B ́, associated 

with higher relative concentrations of suspended silt and clay 

(described below in the section called “Estimation of the 

Grain-Size Distribution and Wet Density of the Silt and Clay: 

Backscatter Constraint”). Both of these constraints must be 

met to achieve a reasonably accurate estimation of the silt and 

clay grain-size distribution and density. 

For example, on the basis of using three acoustic 

frequencies to satisfy both of these constraints at the CR87 

study site, the estimated typical median grain size of the silt 

and clay is 0.0008 mm, the estimated typical σG of the silt 

and clay is 3φ, and the estimated typical wet density of the 

silt and clay is 2,500 kg/m3. These estimations of the D50

and σG for the suspended silt and clay are consistent with 

the measurements of the suspended silt-and-clay grain-size 

distribution reported in the USGS National Water Information 

System database for this study site and for the tributary river 

(that is, the Little Colorado River) that supplies most of the silt 

and clay transported in suspension at this study site, and are 

consistent with 210,228 measurements of the suspended-siltand-clay grain-size distribution made at this study site using 

a LISST-100 at 15-minute intervals between 2002 and 2010.

These measurements indicate that most of the silt and clay in

suspension at this study site is composed of clay-size particles. 

In addition, the estimated wet density of 2,500 kg/m3 that is 

slightly lower than the density of quartz is consistent with 

lower-density clays comprising part of the suspended silt 

and clay at this study site. The effects of this silt and clay

grain-size distribution and density on backscatter at the 

600-kHz, 1-MHz and 2-MHz frequencies at the CR87 study 

site are depicted in figure 14 below in the section of this report 

called “Estimation of the Grain-Size Distribution and Wet 

Density of the Silt and Clay: Backscatter Constraint.” 

Figure 6. (following page) Comparison of empirical and theoretical relations between αS calculated by the acoustic-Doppler profiler 

(ADP) and physically measured CXS-SILT-CLAY at the CR87 study site at the (A) 600-kHz, (B) 1-MHz, and (C) 2-MHz acoustic frequencies. 

Empirical relations are least-squares linear regressions forced through the origin; theoretical relations are derived for a log-normal 

grain-size distribution of silt and clay with D50= 0.0008 mm, σG=3φ, and ρS=2,500 kg/m3. Differences in the goodness-of-fit, R2, values 

between the empirical and theoretical relations are insignificant when defined over such a large range in αS and CXS-SILT-CLAY. The high 

R2 values indicate that, on the basis of Moore and others (2013), the density and grain-size distribution of the suspended silt and clay 

at this study site is fairly constant. αS error bars are 95-percent-confidence-level error bars that include a 2 percent estimated error in 

the ADP-calculated values of αS and the 95-percent-confidence-level error in the mean value of the αS time-averaged over the 1-hour 

interval centered on the temporal midpoint of the time of each EDI or calibrated-pump measurement. Silt-and-clay-concentration 

error bars are 95-percent-confidence-level error bars that include both field and laboratory-processing errors. As expected, the 

approximation that α α S U ≈ NITCXS-SILT-CLAY is better at the 600-kHz and 1-MHz frequencies than it is at the 2-MHz frequency, as indicated by 

the greater increase in the variance in CXS-SILT-CLAY in C that is absent in A and B during high-sand-concentration conditions. An expanded 

view of the lower-left part of C is provided in D to better show the effect of higher suspended-sand concentrations on the relation 

between αS and CXS-SILT-CLAY at the 2-MHz frequency. The 992 measurements made between September 29, 2006, and November 25, 2012, 

are plotted in A, of which 135 are equal-discharge-increment (EDI) measurements and 857 are calibrated-pump measurements; 752 

measurements made between August 25, 2005, and March 12, 2008, are plotted in B, of which 96 are EDI measurements and 656 are 

calibrated-pump measurements; 564 measurements made between February 10, 2006, and March 12, 2008, are plotted in C, of which 

76 are EDI measurements and 488 are calibrated-pump measurements.
Theoretical Framework 21

men16-3130_fig06

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

0 2 4 6 8 10 12 14 16

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

0 5 10 15 20 25

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

0 10 20 30 40 50 0

1,000

2,000

3,000

4,000

5,000

0 2 4 6 8 10

αS, in decibels per meter

CXS-SILT-CLAY, in milligrams per liter

A B

C D

Empirical relation R2 = 0.987

Theoretical relation R2

 = 0.987

Empirical relation R2

 = 0.987

Theoretical relation R2

 = 0.987

Empirical relation R2

 = 0.988

Theoretical relation R2

 = 0.984

High αS

 values in this region arise from 

high-sand-concentration conditions

High αS

 values in this 

region arise from sand 

concentrations ranging from 

1,000 to 5,000 mg/L

Figure 6.
22 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Evaluation of Target Strength from Pressure-Based Re-Derivation of the 

Sonar Equation

Target strength, TS, is determined by the amount and nature of the particles in suspension 

and by the dimensions of the ensonified volume (NDRC, 1946; Urick, 1975; Reichel and

Nachtnebel, 1994). Much research has been completed on the acoustic scattering effects of 

individual and, later, concentrations of particles in suspension (Rayleigh, 1896; NDRC, 1946;

Anderson, 1950; Hickling, 1962; Urick, 1975; Sheng and Hay, 1988; Hay, 1991; Hay and

Sheng, 1992; Thorne and Campbell, 1992; He and Hay, 1993; Thorne and others, 1993, 1995,

1997; Thorne and Hanes, 2002; Thorne and Buckingham, 2004; Thorne and Meral, 2008;

Moore and Hay, 2009). Although the initial work in this field was conducted on regularly

shaped particles (for example, spheres, cubes), sufficient work using natural sand grains (in

both single and mixtures of grain sizes) has allowed a sufficiently robust theory on how sound

is backscattered by suspensions of sediment in water. To allow a greater understanding of the

physics underlying the sonar equation (equation 6), we re-derive equation 6 on the basis of 

the equation relating the acoustic pressure backscattered from sediment in suspension to the 

incident acoustic pressure. As a result of this re-derivation, the heretofore-undefined part of the

TS in equation 15, associated with only the amount and nature of the sediment in suspension, 

that is, sV (the backscattering cross section of a unit reverberating volume), can be defined.

The starting point for this re-derivation is the following version of the equation relating

the root-mean-square acoustic pressure backscattered from a volume of water containing 

sediment in suspension to the acoustic pressure produced by a transducer; this equation is

formulated on the basis of all of the literature cited in the previous paragraph. It is assumed that 

the suspended-sediment grains are randomly distributed within the ensonified volume of water,

the scattering is incoherent, the effects of multiple scattering are unimportant, and the effects of 

extremely high concentrations on attenuation are also unimportant. In this equation, 

 p p r L N r b b d dr

r

r t c

r t c

rms

0 BS

NF

e N

P

P

= ( ) ′( ) −

−

+

∫ 0

4

2 0

4

4

4

ψ

θ φ θ φ

α π

, , /

∫ Ω 











+ pNOISE

, (28)

where 

prms is the reverberation-level root-mean-square backscattered pressure measured at 

the transducer (in Pascals),

p0 is the source-level incident pressure (in Pascals) at distance r0=1 m from the 

transducer, 

pNOISE 	 is the instrument noise-floor pressure (in Pascals),

LBS is the acoustic backscattering length (in m) defined below,

ѱNF 	 is the nondimensional near-field correction that accounts for nonspherical

spreading very near the transducer,

N is the number concentration (in particles/m3) of suspended sediment,

r is the range (that is, distance) along the beam from the transducer (in m), 

t

P is the acoustic pulse (ping) duration (in seconds), 

c is the measured speed of sound (in m/s), 

	 αN is the attenuation coefficient (in nepers/m) resulting from the sum of the water

absorption and sediment attenuation coefficients,

b(θ,φ) is the transmitted beam pattern in spherical coordinates (θ,φ), 

	 b′(θ,φ) is the received beam pattern in spherical coordinates (θ,φ), and

Ω denotes a solid angle (in steradians).

Some of the variables in equation 28 have already been defined in previous sections of this

report and are defined again here simply for convenience. Also, although different in form, the

double integral in equation 28 is mathematically equivalent to the triple integral in equation 5 in 

Thorne and Hanes (2002), given the condition that the region behind the plane of the transducer

face contributes zero to the solid angle of the two-way acoustic beam subtended at the face of 
Theoretical Framework 23

the transducer. This condition is commonly referred to as the

“infinite planar baffle” approximation and originated with

Rayleigh (1896). Finally, the integration limits of r-t

p

c/4 and 

r+t

p

c/4 arise from the previously described t

p

c/2 range-gated 

thickness of the reverberating volume at an instant in time.

As described in Medwin and Clay (1997), the acoustic 

backscattering length, LBS, originated with the work of 

Anderson (1950), and is defined as

L a f BS

S = 2 , (29) 

where 

aS is the radius of the median grain size of the 

suspended sediment (in m),

f is the nondimensional form function that 

describes the backscattering strength of 

the grain-size distribution of suspended 

sediment as a function of kaS, and

	 k = 2π/λ	 is the wave number (in 1/m), and

	 λ 	 the acoustic wavelength (in m).

As defined by Anderson (1950), the aS/2 scaling in equation 29 

is the geometric scattering length, that is, the length scale 

that best describes the scattering of sound by a sphere with 

radius aS in the kaS>1 geometric scattering regime. Anderson 

(1950) did not use the letter f, nor did he use the term “form 

function” to describe the f in equation 29, but rather referred 

to this parameter as the “reflectivity factor.” Hickling (1962)

later used f∞ for this nondimensional function that he termed 

the “reflection function,” which described echo form, and

only later did it become convention to refer to this function 

as the form function (Medwin and Clay, 1997). In essence, 

Anderson’s (1950) reflectivity factor (later the form function)

is the nondimensional parameter that allows equations like 

equation 28 to be solved continuously over both the Rayleigh 

(kaS<1) and geometric scattering regimes. 

The form function for grain-size distributions of

natural, nonspherical suspended sediment is very different 

than that for a single size class of spheres. Because it was 

derived for grain-size distributions of suspended-sediment 

with naturally shaped grains, the form function, f, used in 

our study is that derived using equation 3 in Thorne and

Meral (2008). The input to this equation is a number size

distribution; in our study, laboratory-measured grain-size

distributions are approximated as log-normal distributions 

and then converted to equivalent number size distributions 

using the approach described above in the Physical Basis for 

the Sediment Attenuation Coefficient section. The Thorne

and Meral (2008) form function takes into account both the 

effect of the nonspherical shape of natural sediment grains and 

the effect of multiple grain sizes. Relative to form functions 

evaluated for single-size spheres, these two effects combine to 

result in a substantial increase in f in the Rayleigh scattering 

regime (kaS<1), and a smaller decrease in the geometric 

scattering regime (Thorne and Buckingham, 2004; Thorne and

Meral, 2008). 

The Thorne and Meral (2008) form functions used

in this study are derived for either unimodal or bimodal 

grain-size distributions of suspended sediment, depending 

on the situation. When the form function is derived for 

sand-size sediment only, the grain-size distribution of the 

suspended sand is assumed to be unimodal. This assumption is

justified because the vast majority of the laboratory grain‐size

analyses of the suspended-sand samples used in this study 

are approximately unimodal. It is important to note, however, 

that this unimodal approximation may limit the ability 

of the method developed herein to accurately predict the 

concentration and median grain size of the suspended sand in 

cases where the suspended sand in a river is highly bimodal. 

When the form function is derived for grain-size distributions 

of suspended sediment consisting of a sand mode and a silt 

and clay mode, the form function derived is that for a bimodal 

distribution composed of a unimodal grain-size distribution 

of suspended sand combined with a unimodal grain-size 

distribution of suspended silt and clay. 

Over the next several paragraphs, the steps involved in 

re-deriving the sonar equation (equation 6) from equation 28 

are shown in detail. Herein we use the solution of Thorne and

Hardcastle (1997) for the range-dependent integral modified

for the larger reverberating volume (that is, measurement cell) 

associated with the types of ADPs used in this study,3

e

e N

P

P P N

−

−

+ − = ∫ 4

2 2 2

2 4

α

α

r

r t c

r t c r

r dr t cr

/ . (30)

Substitution of this solution, along with equations 18 and 

29, into equation 28, and squaring (because I p = c 2 ρ W and 

pressure must be ultimately converted to intensity) yields:

N

rms

22 2

2 2 00 S P 4 2

2 2 NOISE

NF

e

4

α

ψ

  − =   +  

pr a r t c p fN p r Ψ . (31)

Using the equation for the volume of a sphere, the number 

concentration in equation 31 can be converted into a mass 

concentration by

N M

a = 34 3 π ρS S

, (32)

where 

M is the mass concentration (in kg/m3) of 

suspended sediment, and

	 ρS is the density of the suspended sediment (in 

kg/m3).

3

Please note that the original form of this range-dependent integral in 

Thorne and Hardcastle (1997) and within equation 28 is for acoustical

measurements made in the smaller reverberating volumes associated with 

an instant in time, and is: e

e N

P

P N p

−

−

+ − = ∫ 4

2 2 4

4 4

2

α

α

r

r t c

r t c r

r dr

t c

r

/
24 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Employing equations 3 through 7 in Thorne and Hardcastle (1997) for the integral in

equation 18 yields, as found by Mouraenko (2004),

 

2

T

0.96 2π

  =     ka

Ψ . (33)

Equation 33 calculates two-way beam widths that agree within 2 percent of those estimated 

for the equivalent two-way beam in table 8.1 in Urick (1975) on the basis of Reverberation 

Group, University of California Division of War Research (1943). The two-way beam widths

calculated by this equation are, however, ~63 percent of the vendor-estimated one-way -3 dB 

beam widths for the ADPs used in this study and 54 to 72 percent of the one-way -3 dB beam 

widths measured in the laboratory by Hay (1991). These results are in general agreement with

the statement by EdgeTech (2015) that beam shaping results in two-way beam widths that are

~72 percent of the one-way beam widths. Making further substitutions of equations 32 and 33 

into equation 31, with slight rearrangement, yields the following equation: 

 p p p r f M

a

t c

r ka rms NOISE

NF S S

P

T

e 2 2 0

2

0

2

2

2

2

2

3

16

2 0 96 − = 



 



 −

ψ π ρ

π . 4α Nr 















. (34)

The next steps in the re-derivation of the sonar equation involve conversion of equation 34

to the logarithmic form of equation 6, where each term is expressed as 10 times a base-10 

logarithmic ratio of acoustic intensity. The first step in this process is to multiply equation 34

by unity in the form of the quantity (r/r0)2/(r/r0)2. This operation allows for the re-derived terms

in equation 6 match the definitions of those terms provided above. Following this operation,

equation 34 becomes

 p p p r f M

a

t c

r ka rms NOISE

NF S S

P

T

e 2 2 0

2

0

2

2

2

2

2

3

16

2 0 96 − = 



 



 −

ψ π ρ

π . 4 0

2

0

2

α Nr r r

r r

















( )

( )

















. (35)

Following some rearrangement, equation 35 can then be rewritten in the following 10 times 

base-10 logarithmic, decibel form: 

 10 10 20 40 10

2 2

10 0

2

10 10 log l p p og p r log log ( ) rms − NOISE = ( ) − ( ) ψ NF − ( ) + ( )

+





 



 +

20

10 3

16

10 2 0 96

10

10

2

10

log

log log .

r

f M

a t c

π ρ ka

π

S S

P

T





 





















− ( )

+ ( ) − ( )

2

10

10 0 10 0

40

40 20

α Nr e

r r

log

log log .

 (36)

Even though more simplification is possible at this stage, the r and r0 terms are kept in four 

separate logarithms in equation 36 so that they can be properly assigned to the 2TL and TS

terms in the re-derived sonar equation. Because 1 neper = 20log10(e) = 8.686 dB, following

conversion of αN to α in units of dB/m, separation of α into αW and αS, and more rearrangement, 

equation 36 becomes:

 10 10 20 40 10

2 2

10 0

2

10 10 log l p p og p log log r

r ( ) rms − NOISE = ( ) − ( ) ψ NF −

0

10

2

10 10 3

16

10 2 0 96





 





+





 



 +





 



log log . f M

a t c

π ρ ka

π

S S

P

T



















− −

+





 





2

10

0

2 2

20

α α W S r r

r

r log .

 (37)
Theoretical Framework 25

Conversion of p p rms

2 2 − NOISE to IRL, the intensity of the reverberation measured by the ADP in

each cell, and conversion of p0

2

 to ISL, the intensity of the transmitted acoustic signal 1 m from

the transducer, requires that both of these terms be divided by ρWc. Furthermore, because

RL =10 ( ) I I 10 log RL REF , SL =10 ( ) I I 10 log SL REF , and IREF (defined in equation 8) is the same

quantity in both RL and SL, it is possible to simply replace 10 10

2 2 log p p ( ) rms − NOISE in equation

37 with RL and replace 10 10

2 log p( ) 0 in equation 37 with SL, such that equation 37 becomes

 RL SL r

r = − r r 



 



 + ( ) + +

















40 20 2 2 +10 10

0

10 log log ψ α NF W S α log

log .

10

2

10

2

3

16

10 2 0 96

f M

a

t c

ka

π ρ

π

S S

P

T





 





+





 





















+





 

























20 10

0

log . r

r

 (38)

Substitution of equation 14 into equation 38 and minor rearrangement finally allows for

 RL SL TL f M

a r

t c

ka = − +





 



 +

 2 10 3

16

10 2 0 96 10

2

0

2 10 log log .

π ρ

π

S S

P

 T

 





















+ ( )





















2

10 20log . r (39)

Comparison of equation 39 with equation 6 reveals that the target strength (when sediment 

comprises the only particle in suspension and zero air bubbles are present),

 TS f M

a r

t c

ka = 



 



 +





 







10 3

16

10 2 0 96 10

2

0

2 10

2

log log .

π ρ

π

S S

P

T 













+ ( )





















20 10 log , r (40)

and can be solved only when M >0. On the basis of the early work by NDRC (1946) and 

Urick (1975), TS is a negative number of order 10 to 100 for sand- and finer-sized particles,

and calculation of TS using equation 39 for the 600-kHZ, 1-MHz, and 2-MHz ADPs used 

in this study is consistent with this prediction. Because TS =10log l 10 ( ) s V +10 og10 ( ) V, by 

equation 15, the backscattering cross section of a unit reverberating volume is

 s f M

a r V

S S

= 



 



 2

0

2

3

16π ρ , (41)

in units of 1/m3, as required on the basis of equation 15. The reverberating volume is

 V t c

ka = r 



 



2  0 96 2

2

P

T

π . , (42)

in units of m3, as required also on the basis of equation 15. Because, by convention, sediment 

grains are generally measured in terms of their diameter in mm and not their radius in m, and, 

in this study, the central tendency used to best describe a log-normal grain-size distribution is 

the median grain size, the following relation is used, where M is now the mass concentration of 

all sediment in the grain-size distribution with median grain size D50 (in mm):

 s f M

D r V

S

= 



 



 2

50 0

2

3000

8π ρ . (43)
26 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

To be consistent with the use of D50 to characterize the central tendency of the grain-size 

distribution in suspension, f in equation 43 is the form function associated with a log-normal 

suspended-sediment grain-size distribution with median grain-size D50 and geometric standard 

deviation σG, derived on the basis of Thorne and Meral (2008), as described previously.

The fact that f and D50 are associated with a grain-size distribution and not just a single 

grain size has major implications with respect to the derivation and physical interpretation 

of TS. The TS derived in equation 40 is therefore that for the entire grain-size distribution 

of sediment in suspension. In cases where the amount of silt and clay greatly exceeds the 

amount of sand in suspension, the TS will approach that for the suspended-silt-and-clay 

grain-size distribution and will be much different from the TS for the suspended-sand grainsize distribution. 

The typical sortings of the suspended-sand grain‐size distributions at the study sites are

used to derive the Thorne and Meral (2008) form functions used in this study for the sandsize sediment. On the basis of the EDI and EWI measurements, the value of σG that best 

characterizes the suspended-sand grain-size distributions in the Colorado River at the CR30, 

CR61, CR87, and CR225 study sites is 0.63φ, and the value of σG that best characterizes the 

suspended-sand grain-size distributions in the Rio Grande at the RG-CAS and RG-RGV study 

sites is only slightly larger at 0.65φ. Because grain-size analyses published in the USGS NWIS 

database and in Voichick and Topping (2014) indicate that the suspended silt and clay is much

more poorly sorted than the suspended sand, versions of the Thorne and Meral (2008) form

function used in this study for silt-and-clay-sized sediment are derived for broader grain-size 

distributions with values of σG ranging from ~2φ to ~3φ. As shown below, the effect of this 

broader grain-size distribution is to increase the target strength of the suspended silt and clay 

relative to what it would be if the grain-size distribution of the silt and clay were as narrow as 

that of the suspended sand. 

To finish the derivation of TS such that it is compatible with the backscatter–sedimentconcentration relations derived from the sonar equation for constant grain size by Thevenot

and others (1992) and Gartner (2004), we convert M from SI units into the more conventional 

sediment-concentration units of mg/L and replace M with C. All other parameters are retained 

in SI units. This conversion and the previously described conversion from aS to D50 results in 

equation 40 becoming 

 TS f C

D r

t c

ka = 



 



 +





 







10 3

8

10 2 0 96 10

2

50 0

2 10

2

log log .

π ρ

π

S

P

T 













+ ( )





















20 10 log . r (44)

Unit Target Strength

Because the range-dependent part of the TS is included in the relative backscatter, B, and 

therefore is not used to determine the relative effects of changes in concentration or grain size 

on the TS, it is useful to define another term, the range-independent target strength,

 TS f C

D r

t c

ka RI

S

P

T

= 



 



 +





 



10  3

8

10 2 0 96 10

2

50 0

2 10 log log .

π ρ

π

2 













 (45)

and then separate this new quantity into two components to separate the effects of changes in 

suspended-sediment concentration on TSRI from all other controls on TSRI:

 TS C UTS RI =10log10 ( ) + , (46)

where UTS, the unit target strength, is

 UTS f D r

t c

ka = 



 



 +





 







10 3

8

10 2 0 96 10

2

50 0

2 10

2

log log .

π ρ

π

S

P

T 













. (47) 
Theoretical Framework 27

Thus, UTS = TSRI when C = 1 mg/L. Based on equations 45

and 46 (and under constant t

P

, c, k, and aT), if the grain-size 

distribution and density of the suspended sediment remain 

constant, a tenfold change in C will result in a 10-dB change 

in the target strength. Similarly, if the grain-size distribution 

and concentration of the suspended sediment remain constant, 

a factor of 2 change in ρS will result in a 3-dB change in the 

target strength. Changes in the grain-size distribution affect 

both f and D50 in equation 45, resulting in a more complicated 

influence on target strength than do either changes in C or ρS. 

Finally, for maximum clarity in the examples plotted 

below, it is useful to further subdivide the UTS into two 

components: UTSSED, associated with the control on the UTS

of the suspended-sediment properties (grain-size distribution 

and density); and UTSBEAM, associated with the control on 

the UTS provided by the range-independent aspects of the 

geometry of the acoustic beam.

UTS U= + TS UTS SED BEAM , (48)

where

 UTS f D r SED

S

= 



 



10  3

8 10

2

50 0

2 log

π ρ , (49)

and

 UTS t c

ka BEAM P

T

= 



 



















 10 2 0 96 10

2

log . π . (50)

Evaluation of the relative importance of UTSSED and UTSBEAM

in calculations of the UTS is provided in figure 7 for the 

different model ADPs used in this study. 

The more complicated influence on TS of changes in 

the grain-size distribution under constant C is best illustrated 

by UTSSED. Based on the behavior of UTSSED (fig. 7B), 

backscatter measurements made with higher-frequency ADPs 

are generally less sensitive to changes in suspended-sand D50

(that is, D50-SAND) than are backscatter measurements made 

with lower-frequency ADPs, at least over the 600-kHz to 

2-MHz range in frequency of the ADPs tested in this study. 

As shown in figure 7B, UTSSED increases rapidly as a function 

of increasing D50 over most of the Rayleigh scattering 

regime, and only begins to plateau around kaS ~0.5. The

main implication of this result is that, when kaS<0.5, use of 

single-frequency acoustics to measure CSAND will be highly 

biased as a result of concentration-independent variation in 

D50-SAND, as illustrated by the following examples. D50-XS-SAND

measured in this study typically ranged from about 0.09 mm 

to 0.25 mm. Over this range in D50-SAND, UTSSED associated 

with 2-MHz ADPs increases by ~5.2 dB, with most of this 

increase occurring between 0.09 and 0.125 mm. Given that 

a 10-dB increase in TS is associated with a factor of 10 

increase in sediment concentration at constant D50, neglecting 

the influence of changing grain size on backscatter over

the 0.09- to 0.25-mm range in D50-SAND will lead to factor of 

4 biases in CSAND calculated on the basis of backscatter at only 

the 2-MHz frequency. As frequency decreases, the increase in 

UTSSED becomes much larger over the 0.09- to 0.25-mm range 

in D50-SAND, leading to much larger biases in CSAND calculated 

on the basis of backscatter. Over this range in D50-SAND, the 

UTSSED associated with 1-MHz ADPs increased by ~9.2 dB 

and the UTSSED associated with 600-kHz ADPs increased by 

~11.5 dB. These increases in dB lead to factor of ~8 and factor

of ~14 respective potential biases in backscatter-calculated 

CSAND if grain-size effects on backscatter are neglected. 

In addition to the above-described influence of changes in

D50 on UTSSED at different frequencies, changes in the sorting 

of a grain-size distribution in suspension also have profound 

effects on UTSSED. As a grain-size distribution of suspended 

sediment broadens, the difference in UTSSED between silt-andclay-sized and sand-size sediment lessens as a result of the 

effect of decreased sorting on the form function (fig. 8). In 

cases where σG of a grain-size distribution is less than ~1.5φ, 

the UTSSED associated with silt grain-size distributions will 

be much less than the UTSSED associated with sand grain-size 

distributions. In these cases, backscatter will be dominantly 

from sand-size sediment. Conversely, in cases where σG of a 

grain-size distribution exceeds ~1.5φ, the UTSSED associated 

with silt grain-size distributions becomes a larger fraction 

of the UTS associated with sand grain-size distributions, 

and measurable backscatter will be from silt-size sediment 

in addition to the backscatter from sand-size sediment. 

Ultimately, in cases where σG of a grain-size distribution 

exceeds ~3φ, the UTSSED associated with grain-size 

distributions of silt and clay will be nearly equal to the UTSSED

associated with grain-size distributions of sand. Under these 

conditions, backscatter from the suspended silt and clay is 

nearly as much as that from the suspended sand. Because 

the sorting of suspended silt and clay (σG= 2 to 3φ) is much 

broader than the sorting of suspended sand (σG= 0.63 to 0.65φ) 

at the study sites, again a condition likely in most rivers as 

described in the previous section, the effect of silt and clay on 

backscatter must be accounted for under conditions when the 

concentration of the suspended silt and clay greatly exceeds 

that of the sand.
28 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

men16-3130_fig07

0.01 0.1 1

D50, in millimeters

D50, in millimeters

UTS, in decibels UTSBEAM, in decibels

–140

–130

–120

–110

–100

–90

–80

–70

0.01 0.1 1

–70

–60

–50

–40

–30

–20

–10

0

0.01 0.1 1

–110

–100

–90

–80

–70

–60

–50

–40

UTSSED, in decibels

A B

C

Silt Sand

2 MHz

kaS = 1

Silt Sand

Silt Sand

1 MHz

kaS = 1

600 kHz

kaS = 1

600-kHz ADPs

1-MHz ADPs

2-MHz ADPs

600-kHz Nortek

Aquadopp

1-MHz Nortek EasyQ

1-MHz OTT SLD

2-MHz Nortek EasyQ

2-MHz OTT SLD

600-kHz Nortek

Aquadopp

1-MHz Nortek EasyQ

1-MHz OTT SLD

2-MHz Nortek EasyQ

2-MHz OTT SLD

Figure 7. A, Unit target strength, UTS, for the various model and frequency acoustic-Doppler profilers (ADPs) used in this 

study plotted as a function of D50 of a grain-size distribution of suspended sediment with σG=0.63φ. The break between silt and 

sand is indicated. B, Sediment component of the unit target strength, UTSSED, plotted as a function of the D50 of a grain-size 

distribution of suspended sediment with σG=0.63φ. The break between silt and sand is indicated, as are the values of D50 at the 

three acoustic frequencies where kaS=1, the boundary between Rayleigh and geometric scattering. C, Beam component of the 

unit target strength, UTSBEAM, plotted as a function of the D50 of a grain-size distribution of suspended sediment with σG=0.63φ. 

At a given frequency, UTSBEAM depends only on transducer radius, transmit-pulse length, and receive-window length, and 

therefore does not depend on D50. 
Theoretical Framework 29

men16-3130_fig08

Silt Sand

–100

–90

–80

–70

–60

–50

–40

–30

–20

0.01 0.1 1

D50, in millimeters

UTSSED, in decibels

kaS = 1

σG = 0.630φ

σG = 0.315φ

σG = 1.260φ

σG = 1.890φ

σG = 2.520φ

σG = 3.150φ

Figure 8. Effects of changes in sediment sorting of the 

sediment portion of the unit target strength, UTSSED, at 1 MHz. 

Shown are calculated values of UTSSED for D50s of suspended 

sediment with σG=0.63φ, 0.315φ, 1.26φ, 1.89φ, 2.52φ, and 3.15φ. 

These values of σG were chosen for this example because 

they are all multiples of 0.63φ, that is, the measured typical σG

for the suspended sand at the CR30, CR61, CR87, and CR225 

study sites. As the sorting of the suspended sediment gets 

poor and the σG approaches 3φ, the difference between the 

UTSSED of silt- and sand-size sediment is greatly reduced, 

and suspended silt can be associated with almost as much 

backscatter as suspended sand.

Relative Unit Target Strength

The behavior of UTSSED as a function of frequency 

and D50 is the physical process that allows accurate, that 

is, relatively unbiased, backscatter-based measurements of 

CXS-SAND and D50-XS-SAND to be possible when multiple acoustic 

frequencies are used. To more easily use the UTSSED to this 

end, we define a new term, the relative unit target strength

(RUTS), that is, the UTS relative to the UTS associated 

with a reference D50 in the river cross section, denoted as 

D50-XS-SED-REF or D50-XS-SAND-REF. D50-XS-SED-REF is the median 

grain size at a given location that best characterizes the D50

of the velocity-weighted suspended-sediment grain-size 

distribution in the river cross section over the widest possible 

range in concentration. Likewise, for sand-size sediment only, 

D50-XS-SAND-REF is the median grain size at a given location that 

best characterizes D50-XS-SAND over the widest possible range in 

concentration. In the calibration procedure described below, 

D50-XS-SAND-REF is used to determine the base-backscattercalibration relation for a given frequency ADP that is used 

with the backscatter measured at other frequencies to solve for 

both CXS-SAND and D50-XS-SAND in the multi-frequency calibration 

approach. Many measurements of the suspended-sand 

grain-size distribution over a wide range in concentration are 

required to determine D50-XS-SAND-REF, as a great deal of variation 

in D50 is possible. For convenience, the UTS associated with 

either D50-SAND-REF or D50-XS-SAND-REF is abbreviated as UTSREF. 

RUTS is calculated by simply subtracting UTSREF from the 

UTS for all values of D50:

RUTS = − UTS UTSREF , (51)

and is depicted in figure 9. Comparison of the theoretical 

RUTS with measurements can then be conducted to evaluate 

whether the derivation of TS in this section of the report 

is realistic. 

Of all our study sites, the site with the largest 

concentration-independent range in D50-XS-SAND is the 

CR30 study site. Thus, data from CR30 were used to test

the ability of the theory developed herein to predict the 

observed values of RUTS. Comparisons between measured 

and theoretically determined values of RUTS at CR30 

are good (fig. 10). As justified in the next section of this

report, to avoid cases where RUTS could be substantially 

affected by relatively large CXS-SILT-CLAY, only those cases 

where the EWI-measured CXS-SILT-CLAY was less than twice 

the EWI-measured CXS-SAND are included in figure 10. As a 

possible result of the theoretical RUTS having a steeper slope 

over the 0.09 to 0.25-mm range in D50-XS-SAND at the 1-MHz 

frequency than at the 2-MHz frequency, the agreement 

between the measured and theoretical values of the RUTS is 

better at the 1-MHz frequency than at the 2-MHz frequency. 

For the same reason, the measured RUTS at the 2-MHz

frequency is much less affected by changes in D50-XS-SAND than 

is the measured RUTS at the 1-MHz frequency over the 0.09 to 

0.25-mm range in D50-XS-SAND.

The above derivation of UTS and RUTS allows the 

commonly used Thevenot and others’ (1992) simplification
30 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

of the sonar equation to be re-derived in a convenient form 

for cases of varying grain size; a form that we employ in this

study to use backscatter measured at multiple frequencies to 

solve for both CXS-SAND and D50-XS-SAND. In their uniform-grainsize simplification of the active sonar equation, Thevenot and

others (1992) showed that, when grain size is constant:

C K B = − + 10 0 1. . 0 1 (52)

where 

B is relative backscatter (replaced in our study 

by B, beam-averaged backscatter), and 

K is a constant. 

Following equations 6, 14, 20, and 22, and because B is B

averaged among beams and cells,

B T = + S S RI L b + − SFA R N O L FFSET . (53)

Following equation 46,

B C =10log10 ( ) + + UTS SL b + − SFA R N O L FFSET . (54)

Rearranging equation 54 to be in the form of Thevenot and

others (1992) indicates that, when grain size is allowed to 

vary, K is not truly constant, but rather is the sum of constant 

(SL–RLOFFSET), nearly constant bSFAN, and varying UTS. The

rearranged version of equation 54 after these substitutions is

C SL b A RL UTS B = − + ( ) − − + 10 0 1. . 0 1 0 1. SF N OFFSET . (55)

For the special case where C is the concentration of suspended 

sediment with D50=D50-XS-SED-REF,

C SL b A RL UTS B = − + ( ) − − + 10 0 1. . 0 1 0 1. SF N OFFSET REF . (56)

Substituting equation 51 into 55 allows the general case where 

D50 varies to be written as

C SL b A RL UTS RUTS B = − + ( ) − − − + 10 0 1. . 0 1 0 1. . 0 1 SF N OFFSET REF . (57)

Because UTSREF is constant at a given study site, it is 

convenient to define another new constant,

K SL b A RL UTS 1 = −0 1. ( ) + − SF N OFFSET R + EF , (58)

which finally allows for

C K B RUTS = + − ( ) 10 1 0 1. . (59)

If only one frequency ADP is present, it is therefore 

theoretically impossible to solve for both C and RUTS for a 

given measured B. However, if two or more frequencies are 

present, it is possible to use equation 59 to iteratively solve for 

C and RUTS. 

men16-3130_fig09

D50, in millimeters

–50

–40

–30

–20

–10

0

10

20

0.01 0.1 1

Silt Sand

RUTS, in decibels

2 MHz

kaS

 = 1

1 MHz

kaS

 = 1

600 kHz

kaS

 = 1

600 kHz

1 MHz

2 MHz

Figure 9. Relative unit target strength associated with the D50

of a suspended-sediment grain-size distribution with σG=0.63φ

at acoustic frequencies of 600 kHz, 1 MHz, and 2 MHz. The 

reference median grain size, D50-XS-SED-REF, used to derive these 

values of the RUTS is 0.125 mm. The break between silt and 

sand is indicated, as are the values of D50 at the three acoustic 

frequencies where kaS=1, the boundary between Rayleigh and 

geometric scattering.
Theoretical Framework 31

men16-3130_fig10

–10

–5

0

5

10

0.10 0.15 0.20 0.25 0.30

A. 1 MHz

0.10 0.15 0.20 0.25 0.30

B. 2 MHz

RUTS, in decibels

EWI-measured D50-XS-SAND, in millimeters

CXS-SAND, in milligrams per liter

<30

30 to 100

>100

Relation from theory

Figure 10. Comparisons of measured and theoretical values of the RUTS at the CR30 study site, at (A) 1-MHz and (B) 2-MHz 

acoustic frequencies. The measurements of RUTS at each frequency were determined by (1) fitting a linear regression relating 

the independent variable log10(CXS-SAND) and the dependent variable B for all sand-size sediment within 1⁄4φ of D50-XS-SAND-REF (that 

is, sand with D50 ranging from 0.105 to 0.149 mm at this study site), and (2) detrending the data by subtracting the values of B

calculated using this relation from the measured values of B associated with each equal width increment (EWI) measurement 

of CXS-SAND and D50-XS-SAND. Measured values of the relative unit target strength (RUTS) are segregated into three different 

concentration ranges in these plots to allow evaluation of dependence on concentration; as indicated in these plots, there is 

no discernable dependence of the measured RUTS on concentration. D50-XS-SAND error bars are 95-percent-confidence-level 

error bars that include both field and laboratory-processing errors in the EWI measurements of D50-XS-SAND; RUTS error bars are 

95-percent-confidence-level error bars that include (1) both field and laboratory-processing errors in the EWI measurements 

of CXS-SAND, (2) a 2 percent estimated error in the acoustic-Doppler-profiler-calculated values of B, and (3) the 95-percentconfidence-level error in the mean value of B , time-averaged over the 1-hour interval centered on the temporal midpoint of the 

time of each EWI measurement. Ninety-three measurements made between February 22, 2007, and February 18, 2013, are plotted 

in A; 89 measurements made between August 24, 2007, and February 18, 2013, are plotted in B; the discrepancy in dates between 

A and B is because a 2-MHz ADP was not deployed at this study site until August 2007.

Development of Base-Backscatter-Calibration 

Relations

To allow accurate acoustical measurements of CXS-SAND

to be made when even large values of CXS-SILT-CLAY are present 

(and a considerable amount of B arises from the amount of 

silt and clay in suspension and not from the amount of sand in 

suspension), we developed a data-processing method to allow 

separation of the part of B arising from sand-size sediment 

from the part of B arising from silt-and-clay-sized sediment. 

This method utilizes the differing theoretical behaviors of UTS

and αS under different combinations of suspended silt and clay 

and suspended sand. An early empirical version of this method 

was described in Topping and others (2007b).

The basis for the UTS part of this method is the 

development of a base-backscatter-calibration (BBC) relation 

between B and log10(CXS-SAND) for conditions where the 

suspended sediment is dominated by sand-size sediment with 

D50 within 1⁄4φ of D50-XS-SAND-REF and assumed constant sorting 

(fig. 11). Relations are then developed using both theory 

and empirical analysis to account and correct for the excess 

backscatter relative to this relation for conditions where the 

amount of silt and clay greatly exceeds the amount of sand in 

suspension. Ideally, the BBC relation is developed using only 

the more-accurate EDI or EWI measurements (Edwards and 

Glysson, 1999; Topping and others, 2011) and not using the

less-accurate cross-section-calibrated pump measurements 

(Edwards and Glysson, 1999), although this is not always 
32 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

men16-3130_fig11

60 70 80 90 100 110

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

60 70 80 90 100 110

log10[CXS-SAND, in milligrams per liter]

A B

C

log10(CXS-SAND) = –8.27 + 0.119 B R2= 0.97 log10(CXS-SAND) = –6.81 + 0.105 B R2

= 0.98 

log10(CXS-SAND) = –7.18 + 0.110 B R2

= 0.99

B, in decibels

B, in decibels

Figure 11. Base-backscatter-calibration (BBC) relations, depicted as red lines and equations, at the CR87 study site 

for acoustic-Doppler profilers (ADPs) operating at (A) 600-kHz, (B) 1-MHz, and (C) 2-MHz frequencies. To ensure that 

these relations are developed for cases where backscatter results mainly from suspended sand with a consistent 

grain-size distribution that is typical for this study site, only those measurements where the equal-discharge-increment 

(EDI) measured CXS-SILT-CLAY is less than twice the EDI-measured CXS-SAND, and the EDI-measured D50-XS-SAND is within 1/4φ of 

D50-XS-SAND-REF= 0.125 mm are included in the least-squares linear regressions used to develop these relations. B error bars are 

95-percent-confidence-level error bars that include a 2-percent estimated error in the ADP-calculated values of B and the 

95-percent-confidence-level error in the mean value of B time-averaged over the 1-hour interval centered on the temporal 

midpoint of the time of each EDI measurement; CXS-SAND error bars are 95-percent-confidence-level error bars that include 

both field and laboratory-processing errors in the EDI measurements of CXS-SAND. Fifty EDI measurements made between 

June 20, 2007, and November 23, 2012, with either a US D-96 or US D-96-A1 sampler are plotted in A; 50 EDI measurements 

made between August 25, 2005, and March 9, 2008, with either a US D-96 or US D-96-A1 sampler are plotted in B; 39 EDI 

measurements made between February 10, 2006, and March 9, 2008, with either a US D-96 or US D-96-A1 sampler are 

plotted in C.
Theoretical Framework 33

possible. Because many EDI or EWI measurements are 

available under low silt-and-clay concentration conditions, 

BBC relations at the Colorado River study sites are developed 

for each ADP using only EDI or EWI measurements. In 

other rivers, such as the Rio Grande at the RG-CAS and 

RG-RGV study sites, low silt-and-clay-concentration 

conditions are rare and it is often difficult to make EDI or EWI

measurements when the amount of silt and clay in suspension 

does not greatly exceed the amount of sand in suspension, 

and both EDI or EWI and calibrated-pump measurements 

must be included in these BBC relations, which sometimes 

leads to less-accurate calibrations than if only EDI or EWI 

measurements could be used. 

For accurate BBC relations to be developed, two 

conditions must be met. First, either CSAND along the beam 

must be constant or any systematic variation in CSAND along 

the beam must average out. This first condition requires that

slowly varying spatial patterns in the cross-stream distribution 

of CSAND, observed by Topping and others (2011) to be stable

for many hours, must average out along the length of the 

beam. As shown below, this first condition is likely rarely

met in a strict sense. Second, the average CSAND along the 

beam must systematically relate to the value of CXS-SAND in the 

calibration cross section. Ideally, but not always, this second 

condition is met without any dependence of this relation on 

water discharge.

Because of how the flow and suspended-sedimentconcentration fields interact with the local channel geometry

at the locations of ADP deployments, there are typically 

differences between the average CSAND or D50-SAND along the 

ADP beams and the values of CXS-SAND or D50-XS-SAND in the 

calibration cross section. These differences (ideally small)

lead to differences between the theoretically predicted 

and empirically determined slopes and y-intercepts of the 

BBC relations. Because the grain-size distribution of the 

suspended sand is constant for the data used to develop a BBC 

relation, if the spatial distribution of suspended sand in the 

river cross section were uniform, equation 56 suggests that 

the slope of the BBC relation should be 0.1. By extension, 

equation 56 then also suggests for a 1- or 2-MHz ADP with 

SL=194 dB, bSF= 0.43 dB/count, AN= 25 counts, RLOFFSET= 0

dB, and UTSREF ~ –85 dB (for D50-XS-SAND-REF= 0.125 mm and

σG=0.63φ) that the y-intercept, K1, of the BBC relation should 

be about –12. In practice, although close to these theoretically 

determined values, regression-determined slopes of the BBC 

relations typically range from about 0.07 to about 0.15, and 

regression-determined values of K1 typically range from 

about –9 to –4. These less-negative empirical values of K1

arise largely because the value of the RLOFFSET is an unknown 

positive constant (see equation 22). Using the above values 

of SL, bSF, AN, and UTSREF, a range in K1 from –9 to –4 would 

therefore lead to a range in RLOFFSET from +30 to +80 dB for 

the case where the slope of the BBC relation is equal to 0.1. 

The large variation in the regression-determined values of K1

arises both from the variation in the regression-determined 

slopes of the BBC relations and because RLOFFSET varies 

between ADPs. 

Non-0.1 slopes of BBC relations at particular study 

sites result from a variety of physical processes. The most

important of these processes is likely concentration-dependent 

changes in the relation between the average CSAND along the 

acoustic beams and CXS-SAND. By this process, BBC-relation 

slopes <0.1 are explained by there being proportionally more 

suspended sand “sampled” by the acoustic beams than is 

present on average in the entire cross section, as CXS-SAND

decreases. In other words, the ratio of ADP-sampled CSAND to 

CXS-SAND decreases as CXS-SAND increases. By this same logic, 

BBC-relation slopes >0.1 are explained by increases in the 

ADP-sampled CSAND to CXS-SAND ratio as CXS-SAND increases. 

Slopes of BBC relations will therefore only equal 0.1 when 

the proportionality relating CSAND along the acoustic beams 

to CXS-SAND is constant. For cases where the BBC-relation 

slopes are not equal to 0.1, sand-concentration dependent 

changes in the ADP-sampled CSAND to CXS-SAND ratio also 

exert influence on the y-intercept (in addition to the influence

on the y-intercept of RLOFFSET, SL, UTSREF, and kSF). Because 

regression-determined values of the slope and y-intercept 

incorporate the effects of local channel geometry and account 

for the differences in RLOFFSET, SL, and bSF between different 

ADPs—and therefore compensate for mistakenly using 

incorrect values of RLOFFSET, SL and bSF—empirical-regressiondetermined values of the slope and y-intercept result in more 

accurate BBC relations than theoretically determined values. 

To allow use of empirically determined BBC-relation

slopes and y-intercepts, equation 59 is rewritten in the 

following final BBC-relation form:

log10 ( ) C K XS-SAND-REF B = +1 2 K B ASE , (60)

where 

CXS-SAND-REF is the reference CXS-SAND with D50-XS-SAND= D50-

XS-SAND-REF,

BBASE is the beam-averaged base backscatter 

associated with CXS-SAND-REF,

K2 is the regression-determined slope of the 

BBC relation, and 

K1 becomes the regression-determined 

y-intercept of the BBC relation. 

RUTS does not appear in equation 60; it is set equal to

zero because only those measurements of CXS-SAND where 

D50-XS-SAND is within 1/4φ of D50-XS-SAND-REF are included 

in the least-squares linear regression used to develop the 

BBC relation.

At a given study site, the slope of the BBC relation tends 

to be approximately constant for a given frequency ADP;

that is, changing an instrument for maintenance purposes 
34 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

may change the y-intercept but does not greatly affect the 

slope of the BBC relation. This observation is consistent with

theory. Changing an instrument can cause large changes in the 

RLOFFSET, small changes in the SL (and to a much lesser degree, 

can also cause changes in bSF), but does not change the UTSREF

so long as the transducer diameter, transmit-pulse length, and 

receive-window length are constant. RLOFFSET is known to vary 

between identical-model ADPs (Nortek, written commun., 

September 30, 2015). When ADPs have been replaced at 

a given study site, regression-determined y-intercepts may 

change by as much as 30 percent. Because this magnitude of 

difference cannot be explained by likely variation in bSF or SL, 

substantial differences in RLOFFSET between identical-model 

ADPs made by the same manufacturer are the most likely 

explanation for large changes in the y-intercepts of BBC 

relations when ADPs are replaced. Although knowing the 

RLOFFSET is not required to use the ADP-calibration method 

described herein, it may be possible to determine the value of 

RLOFFSET for a known value of SL using the method of Stanton 

and Chu (2008).

Calculation of the Backscatter from 

Silt and Clay

Owing to the behavior of the UTSSED illustrated in 

figures 7 and 8, it is impossible to develop accurate relations 

between B and suspended-sediment concentration in rivers 

where (1) the grain-size distribution of the suspended silt 

and clay is broader than the grain-size distribution of the 

suspended sand and (or) (2) a large range in the concentration 

of suspended silt and clay is possible for any given 

concentration of suspended sand. As shown in figures 7 and 

8, over the range in grain size from the 0.004-mm boundary 

between clay and silt to the 0.5-mm boundary between 

medium and coarse sand (a range encompassing grain sizes 

that are common in suspension), UTS can vary by as much 

as 57 dB for 600-kHz ADPs, 52 dB for 1-MHz ADPs, and 

44 dB for 2-MHz ADPs when the suspended sediment is 

reasonably well sorted with σG = 0.63φ. Thus, when a wide

range in CXS-SILT-CLAY is possible for any given CXS-SAND, the 

variation in suspended-sediment concentration at any given 

value of B can be large, although not typically as large as the 

44- to 57-dB range in UTS suggests. In practice, the variation 

in suspended-sediment concentration at any given value of 

B is much less than the factor of 10,000 to 100,000 variation 

implied by this range in UTS because (1) the difference in 

UTS across the 0.004- to 0.5-mm grain-size range is reduced 

as the grain-size distribution of the suspended sediment 

broadens, and (2) the UTS is dominated by the median grain 

size of the sand, silt, and clay mixtures in suspension and does 

not reflect the entire range of all grain sizes in suspension.

For example, at the CR87 study site, the observed variation in 

suspended-sediment concentration at any given value of Bis 

about a factor of 100 at 1- and 2-MHz frequencies (fig. 12). In 

this example, the D50 of the suspended sediment ranges from 

about 0.009 to 0.2 mm, with σG decreasing from ~3φ to 0.63φ

as the D50 increases over the 0.009- to 0.2-mm range. This

coupled change in D50 and sorting corresponds to a ~28-dB 

range in UTS for 600-kHz ADPs, a ~23-dB range in UTS for 

1-MHz ADPs, and a ~19-dB range in UTS for 2-MHz ADPs. 

These ranges in UTS correspond to predicted factors of 600, 

200, and 80 variations in suspended-sediment concentration, 

respectively, on 600-kHz, 1-MHz, and 2-MHz ADPs at any 

given value of B . Thus, neglecting the contribution of silt and

clay to backscatter will result in errors in CXS-SAND calculated 

on the basis of B that can equal a factor of 100, a magnitude 

of error that is unacceptably high.

The additional beam-averaged backscatter resulting from

the presence of suspended silt and clay at a given value of 

CXS-SAND is referred to as the excess backscatter, B ́. Excess 

backscatter is calculated as

B B ′ = − BBASE . (61)

Following equation 54, equation 61 can be rewritten as

B T ′ = + ( ) S S SED S L T − + ( ) S S AND-REF L , (62)

and simplified to

B T ′ = − S TS SED SAND-REF (63)

where 

TSSED is the range-independent target strength of a 

sand, silt, and clay mixture in suspension, 

and 

TSSAND-REF is the range-independent target strength of 

CXS-SAND-REF. 

By definition, when all the suspended sediment is composed

of sand with D50-XS-SAND=D50-XS-SAND-REF, then B ́=0.

B ́ is well predicted on the basis of the target-strength 

theory described above. As the amount of silt and clay varies 

in the presence of a constant concentration of suspended sand, 

B ́ is affected by both changes in the grain-size distribution 

and density of the sand, silt, and clay mixture. As the amount 

of silt and clay in suspension increases in the presence of 

suspended sand with a constant concentration and grain-size 

distribution, the D50 of the suspended sediment decreases. 

Moreover, as the grain-size distribution of the suspended 

sediment evolves through the addition of silt and clay, the 

form function also changes, in response to both the decrease 

in D50 and the decrease in sorting. Finally, if the density of the 

clay is substantially different than the density of the silt and 

sand, and the silt and clay fraction of the suspended sediment 

is dominated by clay, addition of silt and clay will result in 
Theoretical Framework 35

men16-3130_fig12 log10[CXS, in milligrams per liter]

B, in decibels

60 65 70 75 80 85 90 95 100

B

0

1.0

2.0

3.0

4.0

5.0

60 65 70 75 80 85 90 95 100

A

Calibrated-pump measurements

EDI measurements 

Figure 12. A factor of 100 variation in the velocity-weighted suspended-sediment concentration in the river 

cross section (CXS) is possible at the CR87 study site for any given value of B at (A) 1-MHz and (B) 2-MHz acoustic 

frequencies. This large variation arises from large changes in the relative amounts of sand and relative amounts of 

silt and clay comprising the suspended sediment. To prevent clutter given the large number of measurements, error 

bars are not plotted. One hundred seventy-five EDI measurements and 1,191 calibrated-pump measurements made 

between August 25, 2005, and November 25, 2012, are plotted in A; 163 EDI measurements and 1,022 calibrated-pump 

measurements made between February 10, 2006, and November 25, 2012, are plotted in B.

4

Because the density of chlorite-group clays exceeds the density of quartz, 

the wet density of kaolin-group clays are slightly less than the density of 

quartz, and the wet densities of illite- and especially smectite-group clays can 

be much less than the density of quartz (DeWit and Arens, 1950; Deer and

others, 1966; Gartner and Carder, 1979), the density of a sand, silt, and clay

mixture in suspension can be substantially different than the density of only 

suspended sand and silt, depending on the type of clay present.

either an increase or decrease in the density of the sand, silt, and clay mixture4, as described previously in the derivation of αUNIT. 

The combined effect from the increase in suspended sediment concentration, decrease in D50, modification of the form function,

and possible change in sediment density is first a gradual increase and then a more rapid increase in the target strength of the

suspended sediment. As the concentration of silt and clay increases and greatly exceeds the concentration of sand, the increase 

in the concentration of the suspended sediment dominates the behavior of TS, albeit with a D50, form function, and density of the 

sand, silt, and clay mixture that all evolve to ultimately equal the D50, form function, and density of the silt and clay. 

To derive the theoretically based value of B ́ at constant sand concentration and D50, it is useful to first rewrite equation 46

as 

TS C UTS C f D rS

SED S = + ED = +





10 10 10  3

8 10 10 10

2

50 0

2 log ( ) log ( ) log

π ρ





. (64)
36 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

For cases where the slope of the BBC relation equals the theoretical slope of 0.1, equation 63 can 

then be written as

′ = 



 



B  + + f

D

10 10 10 C C

2

10 log log SED

50-XS-SED SED

XS-SAND-REF X ρ S-SILT-CLAY

REF

0-XS-SAND-REF SAND

( ) − 



 





−

10

10

10

2

5

log

lo

f

D ρ

g10 ( ) CXS-SAND-REF

 (65)

and, after rearrangement and substitution of quartz density (2,650 kg/m3) for the density of sand 

(ρSAND) and replacement of the 0.1 theoretical BBC-relation slope with K2, finally written as

 ′ = 



 



 



 





 B

K

f

f

D

D

1

2

10

2

5

5

log SED

REF

0-XS-SAND-REF

 0-XS-SED

 









 



 +





 













 2650 1

ρSED

XS-SILT-CLAY

XS-SAND-REF

C

C 





, (66)

where 

D50-XS-SED is the velocity-weighted D50 of the bimodal suspended-sand, silt, and clay mixture 

in the river cross section, 

f

SED 	 is the value of the Thorne and Meral (2008) form function calculated for the

bimodal grain-size distribution of the sand, silt, and clay mixture, 

ρSED is the density of the sand, silt, and clay mixture calculated on the basis of 

quartz density for the sand-size sediment and a specified density for the siltand-clay-sized sediment (ranging from 1,800 to 3,300 kg/m3

), and 

f

REF 	 is the value of the Thorne and Meral (2008) form function associated with

the reference D50-XS-SAND (that is, D50-XS-SAND-REF) for the unimodal grain-size 

distribution of the suspended sand in the river cross section.

B ́ is most sensitive to changes in the D50 and sorting of the suspended silt and clay, and less 

sensitive to both changes in the grain-size distribution of the suspended sand and changes in the 

density of the suspended silt and clay (fig. 13). Factor of two changes in the σG of the silt and clay 

are similar to factor of 10 changes in the D50 of the silt and clay in their effect on B ́. As either the 

D50 of the suspended silt and clay coarsens or the σG of the silt and clay increases, B ́ for a given 

value of S (defined as the ratio of CXS-SILT-CLAY to CXS-SAND) increases substantially. Under higher 

values of S and constant suspended-sand grain-size-distribution conditions, factor of 10 increases 

in the D50 of the silt and clay can result in increases in B ́ exceeding 20 dB, and, depending on the 

D50 of the silt and clay, factor of 2 increases in the σG of the silt and clay can result in increases in 

B ́ exceeding 30 dB (fig. 13A). 

In contrast to this high sensitivity of B ́ to changes in the silt and clay grain-size distribution, 

decreases in the D50 of the suspended sand result in relatively small (<5 dB) increases in B ́, and 

reasonable increases in the σG of the suspended sand result in negligible decreases in B ́ under 

constant suspended-silt-and-clay grain-size-distribution conditions (fig. 13B). Furthermore, 

changes in the density of the silt and clay in the absence of substantial flocculation results in only

minor changes in B ́. A fifty percent change in the density of the sand, silt, and clay mixture (a

large change) will only lead to biases in B ́ of ~2 dB. For example, because the density of the sand, 

silt, and clay mixture could be as low as ~1,800 kg/m3 if the suspended sediment were dominated 

by montmorillonite, neglecting the effect of changing sediment density (for example, setting 

ρSED= 2,650 kg/m3) results in negative biases in the calculated value of B ́ of up to only about 

–2 dB (fig. 13C). This relatively weak influence of sediment density on B ́ stands in contrast to the 

strong influence of sediment density on attenuation described above and illustrated in figure 4B. 

Because the grain-size distribution of the silt and clay is not precisely knowable without in situ 

physical measurements, the small potential bias in B ́ arising from use of incorrect sediment 

densities may be acceptable in some cases. 
Theoretical Framework 37

men16-3130_fig13

–30

–20

–10

0

10

20

30

40

A B

–30

–20

–10

0

10

20

30

40

10-1 100 101 102 103 104 105 106 107

C

S

10-1 100 101 102 103 104 105 106 107

S

B’, in decibels

Silt and clay D50 = 0.0001 mm, σG = 1.5φ

Silt and clay D50= 0.0001 mm, σG = 3φ

Silt and clay D50 = 0.001 mm, σG = 1.5φ

Silt and clay D50 = 0.001 mm, σG = 3φ

Silt and clay D50 = 0.01 mm, σG = 1.5φ

Silt and clay D50 = 0.01 mm, σG = 3φ

Silt and clay D50 = 0.01 mm, ρS

 = 2,650 kg/m3

Silt and clay D50 = 0.01 mm, ρS = 1,800 kg/m3

Silt and clay D50 = 0.03 mm, ρS = 2,650 kg/m3

Silt and clay D50 = 0.03 mm, ρS = 1,800 kg/m3

Silt and clay D50 = 0.05 mm, ρS = 2,650 kg/m3

Silt and clay D50 = 0.05 mm, ρS = 1,800 kg/m3

Sand D50 = 0.105 mm, σG = 0.63φ

Sand D50 = 0.105 mm, σG = 1.0φ

Sand D50 = 0.125 mm, σG = 0.63φ

Sand D50 = 0.125 mm, σG = 1.0φ

Sand D50 = 0.210 mm, σG = 0.63φ

Sand D50 = 0.210 mm, σG = 1.0φ

Figure 13. Theoretical behavior of B ́ under different silt and clay grain-size distributions, sand grain-size distributions, 

and silt and clay wet densities at the 1-MHz acoustic frequency. A, B ́ plotted as a function of the ratio of suspendedsilt-and-clay concentration to suspended-sand concentration, S, for a constant sand grain-size distribution and for silt 

and clay grain-size distributions with D50 values of 0.0001 mm, 0.001 mm, and 0.01 mm, and σG values of 1.5φ and 3φ. The 

suspended-sand grain-size distribution used in this example is held constant with D50= 0.125 mm and σG= 0.63φ. B, B ́ 

plotted as a function of S for a constant silt and clay grain-size distribution and sand grain-size distributions with D50

values of 0.105 mm, 0.125 mm, and 0.210 mm, and σG values of 0.63 φ and 1φ. The suspended-silt-and-clay grain-size 

distribution used in this example is held constant with D50=0.001 mm and σG=3φ. The sediment density used in the examples 

in A and B is 2,650 kg/m3. C, B ́ plotted as a function of S for a constant sand grain-size distribution and silt and clay 

grain-size distributions with D50 values of 0.01 mm, 0.03 mm, and 0.05 mm, and σG=1.5φ, and silt and clay wet densities (ρS) 

of 2,650 kg/m3 and 1,800 kg/m3. These seemingly large D50 values for silt and clay were chosen because, as the wet density 

of clay decreases, it tends to form larger flocs (see discussion in “Physical Basis for the Sediment Attenuation Coefficient” 

section of the text).
38 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Figure 14. (following page) Comparison of empirical and theoretical values of B ́ on the Colorado River and Rio Grande. A, B ́ plotted as 

a function of S, the ratio of CSILT-CLAY to CSAND, for the 600-kHz acoustic-Doppler profiler (ADP) at the CR87 study site. The base-backscattercalibration (BBC) relation used to calculate values of B ́ for the equal-discharge increment (EDI) and calibrated pump measurements 

is the relation plotted in figure 11A that has a slope of 0.119. The mean value of S among the measurements used to develop the BBC 

relation is 0.5. B, B ́ plotted as a function of S for the 1-MHz ADP at the CR87 study site. The BBC relation used to calculate values of B ́ 

for the EDI and calibrated pump measurements is the relation plotted in figure 11B that has a slope of 0.105. The mean value of S among 

the measurements used to develop the BBC relation is 0.6. C, B ́ plotted as a function of S for the 2-MHz ADP at the CR87 study site. The 

BBC relation used to calculate values of B ́ for the EDI and calibrated pump measurements is the relation plotted in figure 11C that has a 

slope of 0.110. The mean value of S among the measurements used to develop the BBC relation is 0.6. In A–C, the theoretical relations for 

B ́ are derived using a sand grain-size distribution with D50-XS-SAND-REF= 0.125 mm, σG= 0.63φ, and quartz density and a silt and clay grain-size 

distribution with D50= 0.0008 mm and σG=3φ and a density of 2,500 kg/m3. These are the same properties of the silt and clay used to derive 

the theoretical relations for αS for these ADPs in figure 6. D, B ́ plotted as a function of S for the 2-MHz ADP at the RG-RGV study site. 

The BBC relation used to calculate values of B ́ for the equal-width increment (EWI) and calibrated pump measurements has a slope 

of 0.078 and is plotted in figure 15B. The theoretical relation for B ́ is derived using a sand grain-size distribution with D50-XS-SAND-REF=0.105 

mm, σG=0.65φ , and quartz density, and a silt and clay grain-size distribution with D50=0.002 mm, σG=2.7φ, and a density of 3,200 kg/cm3. 

Because low S values in the Rio Grande are rarer than in the Colorado River, the mean value of S among the measurements used to 

develop the BBC relation in this example is 6, a factor of 10 higher than in the example plotted in A–C; this higher mean value of S leads 

to the slightly negative values of B ́ discernable in this plot at values of S less than 6. Eighty-eight EDI measurements and 572 calibrated 

pump measurements made between March 22, 2007, and November 23, 2012, are plotted in A; 96 EDI measurements and 659 calibrated 

pump measurements made between August 25, 2005, and March 12, 2008, are plotted in B; 76 EDI measurements and 484 calibrated pump 

measurements made between February 10, 2006, and March 12, 2008, are plotted in C; 21 EWI measurements and 349 calibrated pump 

measurements made between July 20, 2012, and October 5, 2014, are plotted in D.

Changes in the density and grain-distribution of clay 

cannot necessarily be separated. As the wet density of clay 

decreases, it tends to form larger flocs (for example, Dyer

and Manning, 1999; Hill and others, 2001; Curran and others,

2007). Evidence of some flocculation is evident in historical

USGS suspended-sediment samples that were analyzed for 

silt-and-clay grain-size distribution in both native river water 

and after addition of a deflocculating agent. Although unlikely

to be common in rivers, as flocs get extremely large (that

is, with D ≥1 mm), densities of flocs decrease to approach

the 1,000 kg/m3 density of water (Curran and others, 2007). 

Comparison of the results plotted in figures 13A-C indicates, 

however, that neglecting the influence of decreases in silt and

clay density from flocculation on B ́ would result in a much 

smaller bias in B ́ than would neglecting the increase in silt 

and clay D50 that would be associated with an increase in 

flocculation. To simplify equation 66, ρSED can therefore be 

set equal to 2,650 kg/m3 in many situations, allowing any bias 

from making the constant-sediment-density assumption to be 

compensated by relatively small biases in the estimated silt 

and clay grain-size distribution. 

Estimation of the Grain-Size Distribution and 

Wet Density of the Silt and Clay: Backscatter 

Constraint 

As stated above in the section called “Estimation of the 

Grain-Size Distribution and Wet Density of the Silt and Clay: 

Attenuation Constraint,” the second constraint used to estimate 

the properties of the silt and clay in suspension is to use 

multiple acoustic frequencies to solve for the values of the D50, 

σG, and ρS for the silt and clay that, in combination with the 

known values of the D50, σG, and ρSAND for the suspended sand, 

best predict the observed values of the excess backscatter, B ́, 

associated with higher relative concentrations of suspended 

silt and clay. In addition to providing information on the silt 

and clay grain-size distribution, requirement of this second 

(backscatter) constraint, in combination with the requirement 

of the first (attenuation) constraint depicted in figure 6, 

allows a fairly robust theoretical method for calculating the 

backscatter arising from higher relative concentrations of 

suspended silt and clay. Once the excess backscatter from 

silt and clay is calculated using the properties of the silt and 

clay that satisfy both the attenuation (fig. 6) and backscatter 

(fig. 14) constraints, it can then be separated from the 

backscatter arising from sand-size sediment.

As shown in figure 14, the theoretical behavior of B ́ 

illustrated in figure 13 agrees well with the empirical behavior 

of B ́ observed at study sites on the Colorado River and 

Rio Grande. In addition, it is evident from the comparison 

between theoretical and empirical B ́ in figure 14 that when 

S >>2, the backscatter from silt-and-clay-sized sediment 

cannot be ignored if unbiased acoustical measurements of 

CXS-SAND are to be made. This observation forms the basis of

our preference to use only those measurements of CXS-SAND

associated with S ≤2 in the development of BBC relations.

Details of the grain-size distributions used to develop the 

theoretical B ́ relations depicted in figure 14 are provided in 

appendix 5. Finally, calculation of B using BBC relations 

and the above theoretical values of B ́ agree well with the 

measured values of B for different ranges in S at both the 

CR87 and RG-RGV study sites (fig. 15). 
Theoretical Framework 39

men16-3130_fig14

B

-10

0

10

20

30

40

A

-10

0

10

20

30

40

0.01 0.1 1 10 100 1,000 10,000 100,000

C

S

B’, in decibels

0.01 0.1 1 10 100 1,000 10,000 100,000

D

Relation from theory

Calibrated-pump measurements

EDI measurements

Figure 14.
40 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

men16-3130_fig15

–1

0

1

2

3

4

5

60 65 70 75 80 85 90 95 100

A

log10[CXS-SAND, in milligrams per liter]

B, in decibels

60 65 70 75 80 85 90 95 100

B

Measurements Theory Measurements Theory

S <2

2 < S <10

10 < S <20

20 < S <100

100 < S <200

200 < S <1000

S = 0.6

S = 5

S = 15

S = 50

S = 150

S = 500

S < 10

10 < S <20

20 < S <100

100 < S <200

200 < S <1000

1,000 < S <2,000

2,000 < S <10,000

10,000 < S < 20,000 

S = 6

S = 15

S = 50

S = 150

S = 500

S = 1,500

S = 5,000

S = 15,000

Figure 15. Comparison of empirical and theoretical relations between B and the logarithm of CXS-SAND in the Colorado River 

and Rio Grande for different ranges of S. A, Log10(CXS-SAND) plotted as a function of B for 6 different ranges of S for the 1-MHz 

acoustic-Doppler profiler (ADP) at the CR87 study site. Measurements plotted are the same as in figure 14A; equal-discharge 

increment (EDI) and calibrated-pump measurements are not segregated in this plot to prevent clutter. Base-backscatter-calibration 

(BBC) relation is fit to EDI measurements with S≤2 shown as solid black line (BBC relation same as that in figure 11B with mean S

among measurements used to develop this relation being 0.6); theoretical relations between B and log10(CXS-SAND) at higher values 

of S calculated using the theoretical B ́ relation in figure 14A. B, Log10(CXS-SAND) plotted as a function of B for 8 different ranges of 

S for the 2-MHz ADP at the RG-RGV study site. Measurements plotted are the same as in figure 14B; equal-width increment (EWI) 

and calibrated-pump measurements are not segregated in this plot to prevent clutter. BBC relation fit to EWI and calibrated pump 

measurements with S≤10 shown as solid black line (mean S among measurements used to develop this relation is 6); theoretical 

relations between B and log10(CXS-SAND) at higher values of S calculated using the theoretical B ́ relation in figure 14B.

Suspended-Sediment-Grain-Size-Distribution 

Effects on Relations between B, αS, log10(S), and 

log10(CXS-SAND)

The theoretical behaviors of backscatter and attenuation

under different suspended-sediment grain-size distributions 

control the shapes of the relations between B, αS, log10(S) and 

log10(CXS-SAND). As a result of these behaviors, backscatter 

resulting from extremely high concentrations of suspended 

silt and clay can effectively mask the backscatter from sand. 

Examples of relations between B and log10(CXS-SAND) at 

different values of αS and relations between aS and log10(S) 

at different values of B are depicted in figure 16. Although 

the acoustic frequency used in these examples is 1 MHz, 

the general behaviors of the relations in these examples 

are unaffected by acoustic frequency. These relations were

developed using BBC relations with a slope of 0.1 (the case 

where the proportionality relating CSAND along the acoustic 

beams to CXS-SAND is constant), and the typical “washload” case 

where CSILT-CLAY ≈ CXS-SILT-CLAY. The theoretical relation between

log10(S) and B ́ was then calculated on the basis of a rearranged 

version of equation 66. B values associated with different 

combinations of CXS-SAND and CXS-SILT-CLAY were then calculated, 

with αS determined for each concentration combination. 
Theoretical Framework 41

men16-3130_fig16

–1

0

1

2

3

4

5

B

–1

0

1

2

3

4

A

–1

0

1

2

3

4

5

0.001 0.01 0.1 1 10 100 1,000

D

–1

0

1

2

3

4

50 60 70 80 90 100 110

C

log10[CXS-SAND, in milligrams per liter]

B, in decibels

S

αs , in decibels per meter

S = 10,000

S = 300

αS = 0.1 dB/m

αS

 = 1 dB/m

αS

 = 10 dB/m

BBC relation

BBC relation minus 1.09 

where S ~ 10,000

αS

 = 0.1 dB/m

αS

 = 1 dB/m

αS

 = 10 dB/m

BBC relation

BBC relation minus 2.06 

where S ~ 300

B = 75 dB

B = 85 dB

B = 95 dB

B = 75 dB

B = 85 dB

B = 95 dB

Figure 16. Behavior of theoretical relations between B and log10(CXS-SAND) for different values of αS and behavior of theoretical relations 

between αS and log10(S) for different values of B for different grain-size distributions of suspended silt and clay at the 1-MHz acoustic 

frequency. The gray shaded regions indicate the regions of “backscatter masking” resulting from relatively high concentrations of 

suspended silt and clay. The suspended-sand grain-size distribution used to develop these relations has D50= 0.125 mm and σG= 0.63φ; 

the suspended-silt-and-clay grain-size distributions used to develop these relations all have a density of 2,650 kg/m3 and σG= 3φ. The 

BBC relation has a slope of 0.1 and a y-intercept of –6 in all cases. A, Log10(CXS-SAND) plotted as a function of B for the case where the 

D50 of the silt and clay is 0.0001 mm. Shown are the BBC relation and the relations between B and log10(CXS-SAND) for the cases where 

αS=0.1, 1.0, and 10 dB/m. These relations between B and log10(CXS-SAND) are near vertical when S exceeds ~10,000. B, Log10(S) plotted as 

a function of αS for the case depicted in A. Shown are the relations between αS and log10(S) for the cases where B =75, 85, and 95 dB. 

These relations are also near vertical when S exceeds ~10,000. C, Log10(CXS-SAND) plotted as a function of B for the case where the D50 of 

the silt and clay is 0.001 mm. Shown are the BBC relation and the relations between B and log10(CXS-SAND) for the cases where αS=0.1, 1.0, 

and 10 dB/m. These relations between and log10(CXS-SAND) are near vertical when S exceeds ~300. D, Log10(S) plotted as a function of αS

for the case depicted in C. Shown are the relations between αS and log10(S) for the cases where B=75, 85, and 95 dB. These relations are 

also near vertical when S exceeds ~300.
42 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

–2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

B

–1

0

1

2

3

4

–1 0 1 2 3 4 5 6 7 8 9 10

A

log10(S)

αs , in decibels per meter

86 < dB <88 calibrated-pump 

measurements

86 < dB <88 EDI measurements

86 dB relation from theory

87 dB relation from theory

88 dB relation from theory

Figure 17. Examples of theoretical and measured relations between αS and log10(S) for the 86- to 88-dB range in B at the 

CR87 study site for the (A) 1-MHz and (B) 2-MHz acoustic-Doppler profilers (ADPs). A 2-dB range in B was chosen for these 

comparisons because of the ~2 percent estimated error in measurements of B . Measurements depicted are from the same 

period used to develop the BBC relations in figure 11. Theoretical relations between αS and log10(S) were derived using 

(1) regression-determined slopes and intercepts of the BBC relations, (2) measured typical suspended-sand grain-size 

distribution with D50-XS-SAND-REF=0.125 mm and σG=0.63φ, (3) estimated typical suspended-silt-and-clay grain-size distribution 

with D50=0.0008 mm, σG=3φ, and density of 2,500 kg/m3, and (4) αS estimated using the empirical relations between αS and 

CXS-SILT-CLAY in figure 6.

As a result of “backscatter masking” by relatively 

high CXS-SILT-CLAY, relations between B and log10(CXS-SAND) 

at constant αS become extremely steep at lower values of 

log10(CXS-SAND), making it problematic to accurately solve for 

log10(CXS-SAND) when S is relatively high. The “steepness”

transition in these relations occurs at increasing values of S

as the D50 of the silt and clay decreases. For the 0.0001-mm 

silt-and-clay-D50 case depicted in figures 16A–B, relations 

between B and log10(CXS-SAND) at constant αS and relations 

between αS and log10(S) at constant B become near vertical 

when S exceeds ~10,000. In figure16A, S exceeds 10,000 

when log10(CXS-SAND) is less than that predicted by the BBC 

relation minus ~1.09. Similarly, for the 0.001 mm silt-andclay-D50 case depicted in figures 16C–D, relations between B

and log10(CXS-SAND) at constant αS and relations between αS and 

log10(S) at constant B become near vertical when S exceeds 

~300. In figure 16C, S exceeds 300 when log10(CXS-SAND) is less 

than that predicted by the BBC relation minus ~2.06. When 

relatively strong correlation exists between the discharge of 

water and CXS-SAND, application of discharge-weighting factors 

may allow estimation of log10(CXS-SAND) under the conditions of 

backscatter masking at high values of S (see appendix 6).

Theoretical relations between aS and log10(S) at constant 

B typically agree well with measurements when the BBC 

relation does not shift with changing water discharge (a 

problem described in appendix 6). Examples of good 

agreement between theoretical and measured relations 

between aS and log10(S) over the 86-88-dB range in B at the 

CR87 study site are provided in figure 17. This particular 2-dB

range in B was chosen for this example because, at both the 

1- and 2-MHz frequencies, this range contains a relatively 

large number of both EDI and cross-section-calibrated pump 

measurements over a relatively large range in αS.
Results 43

Procedure for Applying Method

Calibration Procedure for Each 

Single-Frequency ADP in an Array

Development of cross-section calibrations for each 

single-frequency ADP in the multi-frequency ADP arrays 

is a 13-step process described in detail in appendix 7. The

outcomes of this process are single-frequency acoustical 

estimates of CXS-SILT-CLAY and CXS-SAND. Because of how changes 

in suspended-sediment concentration and grain size combine 

to affect TS at each frequency, the single-frequency estimates 

of CXS-SAND will be grain-size biased. These biases will be

minimized by the RUTS method (described in the next section) 

that is used to combine single-frequency CXS-SAND estimates 

into multi-frequency measurements of CXS-SAND and D50‐XS‐SAND. 

The essence of the calibration procedure is as follows:

(1) equation 27 serves as the basis for relating the acoustic 

attenuation measurements to CXS-SILT-CLAY, and (2) equations 

27, 60, and 66 are solved together to allow subtraction of B ́ 

from B thus allowing single-frequency acoustic-backscatter 

measurements to be used to estimate the value of CXS-SAND that 

would be present if D50-XS-SAND were equal to D50-XS-SAND-REF.

Multi-Frequency RUTS-Based Calculation 

of CXS-SAND and D50-XS-SAND Using the 

Single-Frequency Estimates of CXS-SAND

The RUTS associated with each frequency forms the basis 

for calculating CXS-SAND and D50-XS-SAND. These calculations

draw on the theoretical result that, for a given sorting of 

suspended sand and at the 600-kHz to 2-MHz range of 

frequencies used in this study, acoustic backscatter at lower 

frequencies is more affected by changes in D50-XS-SAND than is 

backscatter at higher frequencies. Inputs to these calculations 

are the single-frequency estimates of CXS-SAND produced by the 

procedure described in the previous section, and the theoretical 

RUTS at each frequency (for example, fig. 10). The difference

in the values of B measured at two frequencies (corrected for 

B ́), the theoretical RUTS relations, and the BBC relations for 

the two different-frequency ADPs are then used to calculate 

values of D50-XS-SAND and CXS-SAND that satisfy the constraint 

that the B measured at each frequency (corrected for B ́) is 

associated with the same CXS-SAND. The detailed procedure for

the multi-frequency RUTS-based calculation of CXS-SAND and 

D50-XS-SAND is provided in appendix 8.

Results

The two-frequency attenuation-based acoustical

measurements of CXS-SILT-CLAY and the two-frequency 

RUTS-based acoustical measurements CXS-SAND and 

D50-XS-SAND compare well with the physical measurements of 

these quantities at the Colorado River and Rio Grande study 

sites, with better agreement occurring at those study sites 

with less distance between the calibration cross section and 

the ADP arrays. In figures 18A–C, the acoustical and physical 

measurements are plotted in a format showing predicted 

(acoustical) versus observed (physical), segregated by 

river, in-sample versus out-of-sample data, and longitudinal 

distance between the calibration cross section and the ADP 

array (Colorado River study sites with this distance ≤200 m

are grouped as “Colorado River”). Data from the one study 

site, CR61, with a ~750-m longitudinal distance between the 

calibration cross section and ADPs are plotted separately. 

Additional versions of figures 18A–C are provided in 

appendix 9 in which these measurements are also segregated 

by study site. We define “in-sample” data as data that were

used to calibrate either one or both of the ADPs at a study site. 

Similarly, we define “out-of-sample” data as data that were

not used in any calibration. The out-of-sample data depicted in

figures 18A–C are all from the CR87 study site. The in-sample

and out-of-sample data from the CR87 study site are analyzed 

in the following error-analysis section of this report to evaluate 

whether there is a significant difference between in- and outof-sample relative errors. Also plotted in figures 18A–C are the 

lines of perfect agreement between the physical and acoustical 

measurements and the best-fit log‐linear regressions fit to all

of the data plotted in each figure panel (for the study sites

with ≤200-m longitudinal distance between the calibration

cross section and the ADP array). For brevity, time-series plots 

comparing the acoustical and physical measurements are not 

provided in this report; user‐interactive versions of such plots

can readily be made at http://www.gcmrc.gov/discharge_

qw_sediment/ or http://cida.usgs.gov/gcmrc/discharge_qw_

sediment/. 

In the comparisons of acoustical and physical 

measurements of CXS-SILT-CLAY and CXS-SAND in figures 18A–B, 

the best-fit log-linear regressions plot extremely close to the

lines of perfect agreement. This result suggests that, over the

four orders of magnitude in sediment concentration plotted 

in figures 18A–B, the methods described in this report result 

in unbiased acoustical measurements of both CXS-SILT-CLAY and 

CXS‐SAND (a result to be further tested in subsequent erroranalysis sections of this report). In figure 18A, the acoustical 

and physical measurements of CXS-SILT-CLAY are in good 

agreement over the range from ~100 mg/L to ~20,000 mg/L;

in figure 18B, the acoustical and physical measurements of 

CSAND are in good agreement over the range from ~2 mg/L to 

~5,000 mg/L. Though not shown in figure 18A because the 

physical measurements are calibrated-pump measurements, 

acoustical and calibrated-pump measurements of CXS-SILT-CLAY

have been found to agree well at concentrations as high as 

~30,000 mg/L on both the Colorado River and Rio Grande. 

Although the closeness of the lines of perfect agreement 

and best-fit log-linear regressions suggests strongly that

the acoustical measurements are unbiased, the presence 
44 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Figure 18. (following page) Predicted versus observed plots for the equal-discharge increment (EDI) or equal-width increment 

(EWI) and acoustical measurements of CXS-SILT-CLAY, CXS-SAND, and D50-XS-SAND. In each figure panel, the solid black line is the line of perfect 

agreement; n is the number of observations; horizontal error bars indicate the 95-percent-confidence-level combined field and 

laboratory-processing error in the EDI or EWI measurement. A, Predicted versus observed plot for CXS-SILT-CLAY. Solid green line is the 

best-fit log-linear regression fit to the data from all study sites in A–C with ≤200-m longitudinal distance between the calibration cross 

section and the acoustic-Doppler profiler (ADP) array. Colorado River data are segregated into in-sample and out-of-sample data 

in plots A–C. B, Predicted versus observed plot for CXS-SAND. C, Predicted versus observed plot for D50-XS-SAND. Dashed green line is the 

best-fit log-linear regression fit to the data from only the CR30 study site (the study site with the largest range in EDI- or EWI-measured 

D50-XS-SAND). D, Predicted versus observed plot for CXS-SAND for the standard two-frequency RUTS method and the three-frequency RUTS

method at the CR87 study site for the 132 measurements common to both methods. Solid blue line is the best-fit log-linear regression fit 

to the two-frequency data, dashed red line is the best-fit log-linear regression fit to the three-frequency data in D–E. E, Predicted versus 

observed plot for D50-XS-SAND for the standard two-frequency RUTS method and the three-frequency RUTS method at the CR87 study site 

for the 132 measurements common to both methods.

of reasonably large variance about the lines of perfect 

agreement indicates that these measurements are subject to 

reasonably large time-varying error (evaluated in subsequent 

error-analysis sections of this report). In addition, the variance 

about the lines of perfect agreement is larger, especially for 

CXS-SAND (fig. 18B) and D50-XS-SAND (fig. 18C) in the case where 

the longitudinal distance between the calibration cross section 

and the ADP array is much greater than 200 m (for example, 

at the CR61 study site where the calibration cross section 

is ~750 m downstream of the ADPs). This result suggests

strongly that time-varying error in acoustical measurements of 

both CXS-SAND and D50-XS-SAND increases with increasing distance 

between the calibration cross section and the ADP array, a 

point that will be revisited below.

For both CXS-SILT-CLAY and CXS-SAND, the log-transformed 

variance about the lines of perfect agreement decreases 

significantly with increasing concentration. In the case of

CXS-SAND, this decrease in variance is gradual, whereas in the 

case of CXS-SILT-CLAY, this decrease in variance is rapid between 

concentrations of 1 and 100 mg/L and more gradual between 

concentrations of 100 and 20,000 mg/L. These negative

correlations between concentration and the log-transformed 

variance about the lines of perfect agreement indicate 

that the time-varying relative errors in both the acoustical 

measurements of CXS-SILT-CLAY and CXS-SAND decrease with 

increasing concentration. 

Although still reasonably good, the agreement between 

the acoustical and physical measurements of D50-XS-SAND

depicted in figure 18C is not as good as the agreement 

between the acoustical and physical concentration 

measurements depicted in figures 18A–B. Unlike in the cases 

of CXS-SILT-CLAY or CXS-SAND, the best-fit log-linear regression fit

to the D50-XS-SAND data from all study sites cross-cuts the line 

of perfect agreement at a low angle. This result suggests the

presence of a small grain-size-dependent bias in the acoustical 

measurements of D50-XS-SAND (evaluated in subsequent 

error-analysis sections of this report). At only the CR30 study 

site does the best-fit log-linear regression plot close to the line

of perfect agreement. Because the CR30 study site is the site 

with the largest EWI-measured range in D50-XS-SAND, however, 

the detection of the apparent grain-size-dependent bias in the 

acoustical measurements of D50-XS-SAND at the other study sites 

could possibly be an artifact of the relatively small measured 

range in D50-XS-SAND at these sites. Although the variance about 

the line of perfect agreement in figure 18C appears larger than 

in figures 18A–B, this is a visual artifact of differences in scale 

between the figure panels; five orders of magnitude are plotted

in figures 18A–B, whereas less than one order of magnitude 

is plotted in figure 18C. In reality, the time-varying relative 

error associated with the variance about the line of perfect 

agreement in figure 18C is smaller than in figures 18A–B, a 

result further evaluated below. 

Comparison of the two- and three-frequency acoustical 

measurements with the physical measurements of CXS‐SAND

in figure 18D indicate that use of the three-frequency RUTS

method does not substantially improve the results relative 

to those obtainable through use of the two-frequency 

RUTS method, likely because of the small difference 

between the RUTS at 600 kHz and 1MHz in figure 9 for the 

suspended-sand grain-size distribution with assumed constant 

sorting (in this case, constant σG= 0.63φ). The data associated

with each version of the RUTS method essentially plot on top 

of each other (except in a few cases) and the best-fit log-linear

regressions associated with each version of the RUTS method 

plot on top of each other in figure 18D. In slight contrast 

to the CXS-SAND results, addition of the third frequency does 

result in some improvement in the acoustical measurements 

of D50‐XS‐SAND in figure 18E.These results indicate only

minor improvement in either bias or time-varying error 

associated with adding the third frequency to the RUTS-based 

calculations. If the σG of the suspended-sand grain size 

distribution and, therefore, the form function and RUTS were 

allowed to vary, iterative solution for CXS-SAND, D50-XS-SAND, 

and σG of the suspended sand using three acoustic frequencies 

could possibly improve the results, but this possibility has not 

yet been tested. 
Results 45

EDI measurement of D ,

 in millimeters

50-XS-SAND 

EDI or EWI measurement of C ,

 in milligrams per liter

XS-SAND

EDI or EWI measurement of D ,

 in millimeters

50-XS-SAND EDI measurement of C ,

 in milligrams per liter

XS-SAND

EDI or EWI measurement of C ,

 in milligrams per liter

XS-SILT-CLAY

Acoustical measurement of C , in milligrams per liter

XS-SILT-CLAY

Acoustical measurement of C , in milligrams per liter

XS-SAND Acoustical measurement of C ,

 in milligrams per liter

XS-SAND

Acoustical measurement of D ,

in millimeters

50-XS-SAND Acoustical measurement of D ,

in millimeters

50-XS-SAND

1

10

100

1,000

10,000

100,000

1

10

100

1,000

10,000

100,000

1

10

100

1,000

10,000

100,000

CR61 (~750-m distance) in-sample (n = 175)

Colorado River in-sample (n = 430)

Colorado River out-of-sample (n = 63)

Rio Grande in-sample (n = 32)

A

1 10 100 1,000 10,000 100,000 1 10 100 1,000 10,000 100,000

1 10 100 1,000 10,000 100,000

CR61 (~750-m distance) in-sample (n = 175)

Colorado River in-sample (n = 431)

Colorado River out-of-sample (n = 63)

Rio Grande in-sample (n = 32)

B

0.08

0.10

0.20

0.30

0.40

0.50

0.08 0.10 0.20 0.30 0.40 0.50

CR61 (~750-m distance) in-sample (n = 175)

Colorado River in-sample (n = 431)

Colorado River out-of-sample (n = 63)

Rio Grande in-sample (n = 23)

C

Two-frequency RUTS method (n = 113)

Three-frequency RUTS method (n = 113)

D

0.08

0.10

0.20

0.30

0.40

0.50

0.08 0.10 0.20 0.30 0.40 0.50

Two-frequency RUTS method (n = 113)

Three-frequency RUTS method (n = 113)

E

Figure 18.
46 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Introduction to the Analyses of 

Bias and Error

Measurements of suspended-sediment concentration, by 

whatever means, are often used to quantify sediment loads 

and sediment budgets for rivers and estuaries (for example, 

Topping and others 2000a, 2010; Wright and Schoellhamer,

2005; Grams and others 2013; Shellenbarger and others

2013). Suspended-sediment flux through a cross section (that

is, sediment load) is the product of measurements of water 

discharge and suspended-sediment concentration, and the 

difference in flux between the upstream and downstream

cross sections bounding a river reach forms the basis for the 

sediment budget for that reach. Suspended-sediment fluxes

through the cross sections bounding a river reach are typically 

much larger than the changes in sediment storage in that reach 

(especially over years to decades). Therefore, a sedimentbudget calculation often involves computing relatively small 

differences in flux (the change in sediment storage) between

two relatively large numbers (the flux at the upstream and

downstream sites). Thus, knowing the amount of bias and

error in sediment-flux calculations at individual river cross

sections is critical for evaluating sediment-budget uncertainty.

Several processes contribute to the overall bias and 

(or) time-varying error in acoustical suspended-sediment 

measurements made using side-looking ADPs. On the basis 

of the work presented in the previous sections of this report, 

two of the largest sources of bias and time-varying error are 

likely: (1) the contribution of silt- and clay-sized sediment 

to backscatter, and (2) changes in the grain-size distribution 

of suspended sand. In addition, bias and time-varying error 

arises from the fact that side-looking ADPs collect data in only 

a part of a river cross section, and that embedded within the 

ADP-calibration process is the requirement of a stable relation 

between the concentration and grain-size distribution in this 

ensonified part of the cross section and the concentration

and grain-size distribution in the entire river cross section. 

Topping and others (2011) showed that such relations are not

necessarily unique and observed that, at some of the same 

sites used in this study, cross-stream spatial structures in both 

suspended-silt-and-clay concentration and suspended-sand 

concentration could persist in a river cross section for many 

hours before evolving into other spatial structures that would 

then persist for many additional hours. Because this process 

of change in the lateral distribution of suspended sediment 

was observed to occur at essentially constant discharge, this 

process produces short-term shifts in calibration relations that 

are not correlated with water discharge, and therefore cannot 

be corrected using the methods described in appendix 6. Given 

the length of time that such structures can persist and their 

effect on suspended-sediment concentrations and grain-size 

distributions in the ensonified part of the cross section,

the time-varying errors in acoustical suspended sediment 

measurements made using side-looking ADPs can be 

autocorrelated. Moreover, these changes in the spatial 

structure of concentration and grain size are increasingly 

important in cases where the longitudinal distance between the 

ADP array and the calibration cross section is large. 

In the remaining sections of this report, we evaluate 

the biases and errors associated with the single- and 

two-frequency acoustical methods developed above with 

respect to these three processes, that is, silt-and-clay 

contributions to backscatter, changes in sand grain size, and 

temporal variability in the spatial structure of concentration 

and grain size in the cross section. In addition, we compare 

the biases associated with suspended-sand concentrations 

measured using the two-frequency RUTS method with 

suspended-sand concentrations estimated using simpler singlefrequency methods, including those that do not take into 

account the backscatter from silt and clay.

Error Analysis

To quantify biases and time-varying errors in the

ADP-calibration methods, analyses were conducted to:

	(1)	 compare in- and out-of-sample relative errors;

(2) compare the relative bias in CXS-SAND calculated by the 

two-frequency RUTS method to the relative biases in 

single-frequency estimates of CXS-SAND, with and without 

inclusion of the effects of silt and clay on backscatter;

(3) detect any dependence on concentration or grain size

of the relative errors in the two-frequency acoustical 

measurements of CXS-SILT-CLAY, CXS-SAND, and D50-XS-SAND;

(4) remove the errors associated with EDI or EWI 

measurements (combined field and laboratory) to thereby

calculate the “true” errors associated with only the 

acoustical measurements; and

(5) address the likely importance of changes in sedimentconcentration relations between the ensonified part and

the entire cross section (that is, spatial structure error).

Because the errors associated with EDI and EWI 

measurements are much smaller than those associated with 

calibrated-pump measurements, the errors associated with 

acoustical suspended-sediment measurements were calculated 

using only EDI and EWI measurements. In these analyses, 

the relative error, given in percent, in each acoustical 

measurement of concentration or D50 is therefore defined as

relative error =100 ( ) a b

b

− (67)

where

a is the acoustical measurement, and 

b is the paired EDI or EWI measurement. 

Because it neglects the fact that the EDI or EWI measurements 

are not without error (Topping and others, 2011), the error
Error Analysis 47

calculated by equation 67 is referred to as an apparent error 

that is slightly larger than the true error associated with the 

acoustical method only. 

As used in this report, the terms “error” or “absolute 

error” have magnitude and sign, and are reported in units of 

concentration or grain size. In this usage, the “absolute error” 

is not equivalent to the “absolute value of the error” because 

absolute error can be positive or negative. The term “relative

error” has relative magnitude and sign, and is reported in 

units of percent (as in equation 67). The central tendency of

the distribution of relative error is referred to as the relative 

bias. If the distribution of relative error is symmetric, the 

best measure of the central tendency is the mean value; if the

distribution of relative error is skewed (which may be the 

case, as described in the next section), the best measure of 

the central tendency is the median value. If the distribution 

of relative error in either concentration or grain size varies as 

a function of either concentration or grain size (for example, 

larger error at lower concentration), then the relative bias 

could vary with concentration or grain size, as well. The part

of the relative error that varies over time about the relative 

bias is referred to as the time-varying relative error. The

time-varying error is the relative error minus the relative 

bias. Thus, for an unbiased measurement, which would

have a relative bias of zero, the relative error calculated by 

equation 67 is the time-varying relative error associated with 

that individual measurement. If a distribution of relative error 

can be approximated as normally distributed about the relative 

bias, then the 68-percent-confidence-level time-varying

relative error associated with an individual measurement is, 

by definition, the standard deviation of the distribution of the

relative error. By extension, the 95-percent-confidence-level

time-varying relative error is then, by definition, 1.96 times

the standard deviation of the distribution of relative error. 

These time-varying relative errors may or may not be random.

If a time-varying relative error is random, then it will decay 

to zero as a function of 1 n , where n is the number of 

observations. However, if a time-varying error is positively 

autocorrelated, then it will decay at a slower rate.

Comparison of In-Sample and Out-of-Sample 

Relative Errors

For any instrument that requires field calibration, out-ofsample errors tend to be greater than in-sample errors because 

the range of conditions included in a calibration dataset is 

typically not as broad as the full range of conditions that 

occurs in nature. In this section, analyses are conducted to 

determine whether significant differences exist between the

in- and out-of-sample relative errors in the two-frequency 

acoustical measurements of CXS-SILT-CLAY, CXS-SAND, and 

D50‐XS‐SAND. Then, for any case where a significant difference

between the in- and out-of-sample errors is detected, a second 

analysis is conducted to estimate the relative magnitude of 

the difference between these errors. For these analyses to be 

meaningful, relatively large numbers of concurrent acoustical 

measurements and EDI or EWI measurements must be 

present in both the (in-sample) calibration and (out-of-sample) 

calibration-verification periods at a study site. Unfortunately,

the only longer-term study site that met this criterion was the 

CR87 study site. Data from the other longer-term study sites 

(that is, CR30, CR61, and CR225) could not be segregated 

into in-sample and out-of-sample data with sufficient numbers

of observations in each category for several reasons. First, 

repairs and (or) replacements of ADPs at two of these study 

sites (CR61 and CR225) required modifications to BBC

relations. Second, large changes in the relations between 

the suspended-sand conditions in the ensonified part of the

cross section and the suspended-sand conditions in the entire 

calibration cross section at two of these study sites (CR30 and 

CR61) also required modifications to BBC relations. Both

of these reasons prevented data collection over time periods 

of sufficient duration to calculate meaningful out-of-sample

errors for CXS-SAND or D50-XS-SAND at these study sites.

 Distributions of relative error in suspended-sediment 

concentration and (or) grain size are typically not normal, 

and, in many cases, are right skewed. This right, or positive,

skewness arises because distributions of relative error have 

a lower bound of –100 percent but have no upper bound 

(meaning >100 percent positive relative error is possible). 

Relative errors in acoustical measurements of CXS-SILT-CLAY

and CXS-SAND calculated by equation 67 can never be less than 

–100 percent (because concentration is always positive), 

and relative errors in acoustical measurements of D50-XS-SAND

calculated by equation 67 are additionally limited because 

acoustical measurements of D50-XS-SAND are never allowed to 

be smaller than 0.074 mm by the RUTS method. The better

measure of the central tendency in such a skewed distribution 

is the median and not the mean. Thus, herein we typically

use the median value to characterize the central value of 

distributions of relative error (relative bias). In addition, 

because distributions of error are typically not normal, a 

statistical test that does not assume normality is used to 

evaluate whether distributions of in-sample and out-of-sample 

relative error are significantly different; the test used is the

Wilcoxon-Mann-Whitney test (Mann and Whitney, 1947). 

At the CR87 study site, Wilcoxon-Mann-Whitney tests 

indicate that the in-sample and out-of-sample distributions 

of relative error in the acoustical measurements of CXS-SAND

and D50-XS-SAND are not significantly different at the pSIG= 0.05

critical level, but that the in-sample and out-of-sample 

distributions of relative error in the acoustical measurements 

of CXS-SILT-CLAY are significantly different. The in- and out-ofsample relative error distributions for acoustically measured 

CXS-SAND and D50-XS-SAND are indistinguishable at the pSIG= 0.75

and 0.72 levels of significance, respectively. In contrast, for

acoustically measured CXS-SILT-CLAY, the in- and out-of-sample 

relative error distributions are significantly different at only

the pSIG= 0.0022 level, with most of this difference occurring

at CXS-SILT‐CLAY<50 mg/L. n is 159 in each of the in-sample 

relative-error datasets and n is 63 in each of the out-of-sample 

relative-error datasets. 
48 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Estimation of the magnitude of the statistically significant

difference between the in- and out-of-sample relative errors 

in the acoustical measurements of CXS-SILT-CLAY is possible 

through comparison of these distributions of relative error. 

Both of these distributions are highly right-skewed (in-sample 

skewness = 5.6, out-of-sample skewness = 2.1) with large

positive kurtosis (in-sample kurtosis = 39.4, out-of-sample

kurtosis = 4.0). Thus, both of these distributions of relative

error are not normal. In a Gaussian normal distribution of 

relative error with zero bias, the 68-percent-confidencelevel relative error is equivalent to the standard deviation 

of that distribution. However, as these two distributions of 

relative error are not at all normal, the standard deviations 

of these distributions can only be taken as an index of error 

and not assigned a confidence level. Therefore, because the

standard deviation of the out-of-sample relative errors in 

CXS-SILT-CLAY is ~32 percent larger than the standard deviation 

of the in-sample relative errors, and most of this difference 

occurs at lower CXS‐SILT-CLAY, it is likely that the 68-percentconfidence-level out-of-sample relative error is slightly larger

than the 68-percent-confidence-level in-sample relative error.

Unfortunately, it is impossible to generalize on the basis of 

this result from a single study site because the difference in 

magnitude between in-sample and out-of-sample relative 

errors at any given study site will depend on the breadth of 

the range of suspended-sediment conditions included in the 

development of the calibrations for the different-frequency 

ADPs. It is, however, likely prudent to generally assume that 

out-of-sample relative errors will be at least slightly larger 

than in-sample relative errors.

Because no statistically significant differences between

the in- and out-of-sample relative errors in the acoustical 

measurements of CXS-SAND and D50-XS-SAND were detected at the 

CR87 study site, and the statistically significant difference

between the in- and out-of-sample relative errors in the 

acoustical measurements of CXS-SILT-CLAY at this study site is 

relatively small (~32 percent), the below analyses of relative 

error proceed by combining the in- and out-of-sample error 

datasets into a single error dataset at each study site. One 

benefit in this approach is that the single datasets have greater

numbers of observations than if in- and out-of-sample datasets 

were analyzed separately. The other benefit in this approach

is that combining the relative error data from all of the study 

sites allows the development of generalized error relations.

Comparison of Relative Biases in Acoustical 

Measurements of Suspended-Sand 

Concentration Made Using Different Methods

To determine the effect on relative bias of using different

acoustical methods, the median values of the relative errors 

in the acoustical measurements of CXS-SAND were compared 

at each study site using five methods: (1) the two-frequency

RUTS method, (2) the 2-MHz single-frequency method 

neglecting B ́ (that is, neglecting the backscatter from silt 

and clay), (3) the 2-MHz single-frequency method including 

B ́, (4) the 1-MHz single-frequency method neglecting B ́, 

and (5) the 1-MHz single-frequency method including B ́. 

For simplicity, all data at each study site were combined in 

this analysis and the effects of any dependencies of relative 

bias on either CXS-SAND or D50-XS-SAND were excluded. Thus,

the method associated with the lowest absolute value of the 

median relative error in this analysis may not ultimately be 

the most accurate method because the relative error could 

vary as a function of CXS-SAND or D50-XS-SAND. This issue is

addressed below. 

When any dependencies of relative error on concentration 

or D50-XS-SAND are excluded, the methods that are generally 

associated with the least relative bias are the two-frequency 

RUTS method, and the 2- and 1-MHz single-frequency 

methods including B ́ (table 2). As shown in table 2, the 

method that is subject to the greatest relative bias is the 1-MHz 

single-frequency method neglecting B ́. We recommend 

that this method never be used unless it can be definitively

established that silt-and-clay does not contribute to backscatter 

at the site of interest. In order of decreasing absolute value, 

the average relative bias from the Colorado River and 

Rio Grande study sites (EDI and EWI cases only) depicted in 

table 2 are: the 1-MHz single-frequency method neglecting 

B ́ (+112.8 percent bias), the 2-MHz single-frequency 

method neglecting B ́ (+69.9 percent bias), the 2-MHz 

single-frequency method including B ́ (–10.6 percent bias), the 

1-MHz single-frequency method including B ́ (+9.6 percent 

bias), and the two-frequency RUTS method (–3.4 percent 

bias). Among only the Colorado River study sites (which have 

many more observations than the Rio Grande study sites), 

the rankings for the EDI and EWI cases are identical (but 

with smaller absolute values of average relative bias): the 

1-MHz single-frequency method neglecting B ́ (+13.4 percent 

bias), the 2-MHz single-frequency method neglecting B ́ 

(+6.1 percent bias), the 2-MHz single-frequency method 

including B ́ (–2.7 percent bias), the 1-MHz single-frequency 

method including B ́ (1.2 percent bias), and the two-frequency 

RUTS method (+1.0 percent bias).

Inclusion of the effect of the backscatter from silt and 

clay results in substantial reductions in the relative errors 

in the single-frequency acoustical estimates of CXS-SAND at 

all study sites on the Colorado River and Rio Grande. As 

expected on the basis of theory, this reduction in error is 

greatest for the cases with the largest positive relative errors 

in the single-frequency acoustical estimate of CXS‐SAND. For 

example, among the 99 paired acoustical and calibrated-pump 

measurements at the RG-RGV study site, the maximum 

relative error in the acoustical measurements of CXS-SAND

when using the 1-MHz single-frequency method neglecting 

B ́ is +66,600 percent, which decreases to +364 percent 

upon inclusion of the effects of B ́. Furthermore, among 

the 173 paired acoustical and EWI measurements at 

the CR30 study site, the maximum relative error in the 
Error Analysis 49

Table 2. Effect of using different acoustical methods on the relative bias in acoustical measurements of CXS-SAND.

Study site 

and physical 

measurement type

n

Range in

D50-XS-SAND

(in mm)

Median relative error, that is, relative bias, in 

acoustical measurements of CXS-SAND (in percent) 

2-frequency 

RUTS method

2-MHz frequency 1-MHz frequency

B ́ neglected B ́ included B ́ neglected B ́ included

CR30 EWI 173 0.09–0.23 –5.1 8.3 –3.0 30.7 11.5

CR61 EWI 175 0.09–0.14 6.5 7.1 –1.1 2.8 –7.7

CR87 EDI 222 0.08–0.21 1.3 0.0 –5.8 6.8 –0.8

CR225 EDI 99 0.09–0.15 1.1 9.1 –1.0 13.1 1.7

RG-CAS EWI 25 0.07–0.17 –9.3 251.5 –9.3 469.4 35.6

RG-CAS pump 243 0.07–0.21 –4.5 138.9 1.8 347.8 –4.2

RG-RGV EWI 7 0.10–0.14 –14.7 143.5 8.6 153.8 17.0

RG-RGV pump 99 0.08–0.18 –67.1 644.0 –59.5 1,703.4 –17.4

Average values among all six 

EDI and EWI cases

–3.4 69.9 –10.6 112.8 9.6

Average values among only the 

EDI and EWI cases at the 

Colorado River study sites

1.0 6.1 –2.7 13.4 1.2

acoustical measurements of CXS-SAND when using the 2-MHz 

single-frequency method neglecting B ́ is +427 percent, which 

decreases to +157 percent upon inclusion of the effects of B ́. 

These large reductions in the relative errors on the right tail

of the error distribution are so great that they also generally 

cause reductions in the central tendencies of the distributions 

of relative error; that is, they generally cause reductions in the

relative bias. In most of the cases depicted in table 2, at both 

the 2-MHz and 1-MHz frequencies, inclusion of the effect 

of the backscatter from silt and clay results in a substantial 

reduction in relative bias. Although the reduction in relative 

bias is greater at the two Rio Grande study sites, where the 

average value of S (that is, the ratio of suspended-silt-andclay concentration to suspended-sand concentration) is much 

greater than at the four Colorado River study sites, inclusion 

of the effect of the backscatter from silt and clay typically 

causes reductions in the relative biases at the Colorado River 

study sites as well. Thus, it may be necessary, in general, to

account for the contribution of silt and clay to backscatter 

when using side-looking ADPs to measure suspended-sand 

concentrations. At minimum, the contribution of silt and 

clay to backscatter must be evaluated at a given site before a 

particular acoustical method can be selected.

Additional analyses were conducted to determine 

whether the relative biases reported in table 2 were constant 

or varied as a function of D50-XS-SAND. In these analyses, 

the effects of B ́ were included in all single-frequency 

measurements of CXS‐SAND. At the two study sites with the 

largest ranges in D50-XS-SAND, the CR30 and CR87 study sites, 

only the two-frequency RUTS method was found to produce 

measurements of CXS-SAND that were unbiased as a function 

of changing D50-XS-SAND (fig. 19). Both the 1- and 2-MHz 

single-frequency methods resulted in relative errors in the 

acoustical measurements of CXS-SAND that were significantly

and positively correlated with D50-XS-SAND. The best-fit linear

regressions fit to the relative errors depicted in figure 19

approximate the behavior of the relative bias in the acoustical 

measurements of CXS-SAND as a function of D50-XS-SAND. At both 

the CR30 and CR87 study sites, the relative bias in 1-MHz 

single-frequency measurements of CXS-SAND varied much more 

strongly as a function of D50-XS-SAND than did the relative bias 

in 2-MHz single-frequency measurements of CXS-SAND. These

results are expected on the basis of the theory and analyses 

presented in the “Theoretical Framework” section above;

that is, the higher frequency instrument is less sensitive to 

grain-size effects. 

Similar analyses conducted at the other study sites with 

much smaller ranges in D50-XS-SAND found little dependence of 

bias in the acoustical measurements of CXS-SAND, regardless of 

whether the 2-frequency RUTS method or the 1- or 2-MHz 

single-frequency methods were used. Thus, because the

relative biases in acoustical measurements of CXS-SAND reported 

in table 2 are similar for the two-frequency RUTS method 

and 2-MHz single-frequency method including B ́, either 

of these methods can likely be used to produce acoustical 

measurements of CXS-SAND with acceptably low levels of bias 

if the range in D50-XS-SAND is known independently beforehand 

at a given study site to be small (that is, approximately 

≤0.5φ). However, if the range in D50-XS-SAND at a given study 

site is relatively large (that is, approximately ≥1φ), two

acoustic frequencies are likely required to produce acoustical 

measurements of CXS-SAND that are not biased by changes 

in D50-XS-SAND.
50 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

The choice of one versus two-frequency ADP

deployments for measuring CXS-SAND depends on site-specific

sand grain-size information, as well as the acceptable levels of 

bias for a given study. Analyses of the suspended-sand grainsize distributions in the database of 22 rivers (from all parts 

of the United States) assembled by Wright and others (2010) 

indicate that 36 percent of the rivers (n = 8) fall within the

-1σG to +1σG range in D50-XS-SAND between 0.5φ and 1φ, where 

it is unclear if one or two acoustic frequencies are required 

to produce unbiased acoustical measurements of CXS-SAND;

32 percent (n = 7) of the rivers had ranges in D50‐XS‐SAND<0.5φ;

and 32 percent (n = 7) of the rivers had ranges in D50-XS-SAND

exceeding 1φ. These results suggest that, for the majority of

rivers, single-frequency ADP deployments will be subject 

to some level of bias in measurements of CXS-SAND owing 

to changes in the suspended-sand grain-size distribution. 

Because the average relative biases reported at the bottom 

of table 2 for the 2-frequency RUTS method, and the 2-MHz 

single-frequency method that includes B ́ are much smaller 

than the absolute values of the relative errors analyzed in the 

next section of this report, both of these methods do provide 

relatively unbiased acoustical measurements of CXS-SAND so 

long as the range in D50-XS-SAND is small. In cases where the 

range in D50-XS-SAND is not small or not known independently 

beforehand, two acoustic frequencies are required for unbiased 

acoustical measurements of CXS-SAND.

Dependence of Relative Errors in Acoustical 

Measurements of Concentration and Median 

Grain Size on Either Concentration or Grain Size

Analyses were conducted to determine whether the 

errors in the two-frequency acoustical measurements are 

independent of changes in concentration or D50-XS-SAND. These

analyses consisted of conducting F-tests on least-squares 

linear regressions relating CXS-SILT-CLAY, CXS-SAND, and D50-XS-SAND

to the relative errors in the acoustical measurements of these 

quantities to determine whether significant trends were present

in the relative errors as a function of either concentration or 

grain size. The results from these analyses should be taken

as somewhat approximate because most of the analyzed 

data are heteroscedastic, with much greater variance owing 

to larger right skewness in the lower part of the domain in 

either concentration or grain size, especially in the case of the 

relative errors in the acoustical measurements of CXS‐SILT‐CLAY

men16-3130_fig19

0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

EWI-measured D50-XS-SAND, in millimeters

B

– 100

0

100

200

300

400

500

A

Relative error in acoustical measurement of 

CXS-SAND,

in percent

1-MHz frequency (pSIG < 1.0 x 10-16)

2-MHz frequency (pSIG = 1.3 x 10-5)

Two-frequency RUTS method (pSIG = 0.53) 

1-MHz frequency (pSIG < 1 x 10-16)

2-MHz frequency (pSIG = 9.6 x 10-7)

Two-frequency RUTS method (pSIG = 0.59) 

Figure 19. Relative errors in single-frequency and two-frequency acoustical measurements of CXS-SAND at the 

(A) CR30 and (B) CR87 study sites plotted as a function of D50-XS-SAND. F-tests conducted on the least-squares linear 

regressions fit to these relative errors indicate significant positive correlations (at the pSIG =0.05 critical level) 

between D50-XS-SAND and error for both the 1- and 2-MHz single-frequency measurements of CXS-SAND, but no significant 

correlation between D50-XS-SAND and error for the two-frequency measurements. At both study sites, the significant 

relation between D50-XS-SAND and error is much steeper at the 1-MHz frequency than at the 2-MHz frequency. These 

results indicate the presence of grain-size-driven bias in the single-frequency measurements of CXS-SAND that is 

larger at the lower frequency, as expected on the basis of the theoretical behavior of the RUTS depicted in figures 9

and 10.
Error Analysis 51

Table 3. Levels of significance (pSIG) associated with F-tests conducted to determine whether relative errors in the 

acoustical suspended-sediment measurements depend significantly on concentration or sand median grain size (D50-XS-SAND).

[Dependencies that are significant at the pSIG= 0.05 critical level are indicated in bold type; the positive or negative trend of each significant

relation is indicated in parentheses. The number of observations, n, associated with each pSIG value is indicated in parentheses. There are fewer

observations associated with D50-XS-SAND at the RG-CAS and RG-RGV study sites because some suspended-sediment samples had insufficient sand

mass for laboratory grain-size analysis. EWI, equal-width increment; EDI, equal-discharge increment; CXS-SAND, velocity-weighted suspended-sand 

concentration in river cross section]

Study site 

and physical 

measurement 

type

pSIG value associated with...

Dependence of 

relative error in 

acoustical 

CXS-SILT-CLAY

on physical

CXS-SILT-CLAY

Dependence of 

relative error in 

acoustical 

CXS-SAND

on physical

CXS-SAND

Dependence of 

relative error in 

acoustical 

CXS-SAND

on physical

D50-XS-SAND

Dependence of 

relative error in 

acoustical 

D50-XS-SAND

on physical

CXS-SAND

Dependence of 

relative error in 

acoustical 

D50-XS-SAND

on physical

D50-XS-SAND

CR30 EWI 0.59 

(n = 172)

0.12

(n = 173)

1

0.53

(n = 173)

0.00054(–)

(n = 173)

0.018(–)

(n = 173)

CR61 EWI 0.74

(n = 175)

0.090

(n = 175)

1.5×10-7(–)

(n = 175)

0.31

(n = 175)

0.23

(n = 175)

CR87 EDI 0.063

(n = 222)

0.12

(n = 222)

1

0.59

(n = 222)

0.99

(n = 222)

<1×10-16(–)

(n = 222)

CR225 EDI 0.41

(n = 99)

0.13

(n = 99)

0.52

(n = 99)

0.75

(n = 99)

3.4×10-9(–)

(n = 99)

RG-CAS EWI 0.11

(n = 25)

0.25

(n = 25)

0.61

(n = 17)

0.28

(n = 17)

0.0070(–)

(n = 17)

RG-RGV EWI 0.21

(n = 7)

0.42

(n = 7)

0.39

(n = 6)

0.12

(n = 6)

0.89

(n = 6)

1

These analyses are depicted in figure 19.

(as discussed above, in the “Results” section). F-tests were 

first conducted to determine whether the relative errors in

the acoustical measurements of CXS-SILT-CLAY significantly

depend on the physically measured CXS-SILT-CLAY. F-tests were 

then conducted to determine whether the relative errors 

in the acoustical measurements of CXS-SAND and D50-XS-SAND

significantly depend on either the physically measured CXS-SAND

or D50-XS-SAND. Detection of a significant trend by these analyses

would indicate that the central tendency of the distribution 

of relative error in the acoustical measurements varies 

systematically over the domain in concentration or grain size. 

Because the central tendency of the distribution of relative 

error is the relative bias, the presence of significant trends thus

indicate the presence of a bias in the acoustical measurements 

driven by either changes in concentration or grain size. Results 

from these F-tests are reported in table 3.

In general, the results from the F-tests summarized in 

table 3 indicate that the central tendencies of the distributions 

of relative error in the acoustical measurements of CXS-SILT‐CLAY

and CXS-SAND do not significantly depend on concentration

or grain size, and the distributions of relative error in the 

acoustical measurements of D50-XS-SAND do not significantly

depend on concentration. These results therefore indicate that

changes in CXS-SILT-CLAY do not cause a concentration-driven 

relative bias in the acoustical measurements of CXS-SILT-CLAY, 

and that changes in CXS-SAND do not cause a concentrationdriven relative bias in the acoustical measurements of CXS-SAND. 

Furthermore, these results indicate that changes in D50-XS-SAND

do not generally cause a grain-size-driven relative bias in 

the acoustical measurements of CXS-SAND, and that changes in 

CXS‐SAND do not generally cause a concentration-driven relative 

bias in the acoustical measurements of D50-XS-SAND made using 

the two-frequency RUTS method. 

The results in table 3 do not indicate, however, that the 

distributions of relative error in the acoustical measurements 

of CXS-SILT-CLAY and CXS-SAND do not vary systematically as 

a function of concentration. As shown in figure 20, the 

distributions of relative error in the acoustical measurements 

of both CXS-SILT-CLAY and CXS-SAND are highly heteroscedastic, 

with much larger variance and right skewness in the lower 

parts of the domains in CXS-SILT-CLAY and CXS-SAND. Only 

as CXS-SILT-CLAY exceeds ~40 mg/L and CXS-SAND exceeds 

~15 mg/L is the right skewness reduced such that the 

distributions of relative error in the acoustical measurements 

in CXS‐SILT‐CLAY and CXS-SAND become approximately symmetric 

about zero. Thus, owing to the large right skewness at

lower concentrations, the acoustical measurements of 

CXS‐SILT‐CLAY are only unbiased when CXS-SILT-CLAY>~40 mg/L 

and two frequency RUTS measurements of CXS-SAND are only 

unbiased when CXS-SAND>~15 mg/L. Because the variance (and 
52 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

therefore the standard deviation) in both of these cases 

decreases with increasing concentration, these results 

indicate that the 68-percent- and 95-percent-confidence-level

time-varying relative errors in the acoustical measurements 

of CXS-SILT-CLAY and CXS-SAND significantly decrease with

increasing concentration. These negative correlations

between concentration and the time-varying relative error 

in concentration likely arise from the process described by 

equation 11 in Thorne and Hanes (2002), where increases

in suspended-sediment concentration result in a greater 

number of randomly distributed particles generating the 

acoustic return, thus reducing the error in the estimation of 

suspended-sediment concentration during a single acoustical 

measurement made over a fixed duration. The 68-percent- and

95-percent-confidence-level time-varying relative errors in

the acoustical measurements of CXS-SILT-CLAY and CXS-SAND are 

estimated in the next section of this report. 

In addition to the negative correlation between 

concentration and the variance in the distributions of relative 

error in the acoustical measurements of CXS-SAND, the variance 

in the relative error in the acoustical measurements of CXS-SAND

at the CR61 study site is generally larger across the entire 

domain than at the other study sites on the Colorado River 

and Rio Grande. This larger variance in relative error, and

therefore larger time-varying relative error, likely arises from 

the fact that the longitudinal distance between the calibration 

men16-3130_fig20

0

500

1,000

1,500

2,000

1 10 100 1,000 10,000 100,000

A

1 10 100 1,000 10,000

B

EDI- or EWI-measured D50-XS-SAND,

in millimeters

EDI- or EWI-measured CXS-SAND,

in milligrams per liter

EDI- or EWI-measured CXS-SILT-CLAY,

in milligrams per liter

Relative error in acoustical measurement of

CXS-SAND, in percent

Relative error in acoustical measurement of

CXS-SILT-CLAY , in percent

–100

0

100

200

300

400

500

600

0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

C

–100

0

100

200

300

400

500

600

Relative error in acoustical measurement of

CXS-SAND, in percent

CR30

CR61

CR87

CR225

RG-CAS

RG-RGV 

CR30

CR61

CR87

CR225

RG-CAS

RG-RGV 

Figure 20. Relative error in acoustical measurements of 

(A) CXS-SILT-CLAY at all six study sites plotted as a function of 

CXS-SILT-CLAY, measured by equal-discharge increment (EDI) or 

equal-width increment (EWI) methods (B) CXS-SAND at all six study 

sites plotted as a function of EDI- or EWI-measured CXS-SAND, and 

(C) CXS-SAND at all six study sites plotted as a function of EDI- or 

EWI-measured D50-XS-SAND. CXS-SAND Dashed black horizontal line 

through zero relative error is the line of zero bias. Errors from CR61 

study site are depicted using a lower-visibility symbol because of 

the much greater distance between the calibration cross section 

and the acoustic-Doppler profiler array at that study site.
Error Analysis 53

1 10 100 1,000 10,000

B

Relative error in acoustical measurement of

D50-XS-SAND, in percent

EDI- or EWI-measured D50-XS-SAND,

in millimeters

EDI- or EWI-measured CXS-SAND,

in milligrams per liter

–100

–50

0

50

100

150

0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

A

Lower bound imposed by 0.074-mm minimum 

CR30

CR61

CR87

CR225

RG-CAS

RG-RGV

Figure 21. Relative error in acoustical measurements of D50-XS-SAND at all six study sites plotted as a function of

(A) D50-XS-SAND measured by equal-discharge increment (EDI) or equal-width increment (EWI) methods, and (B) EDI- or 

EWI-measured CXS-SAND. Dashed black horizontal line through zero relative error is the line of zero bias. Also shown in A

is the lower bound on relative error in acoustical measurements of D50-XS-SAND imposed by the condition that acoustical 

measurements of D50-XS-SAND <0.074 mm are precluded. Errors from CR61 study site are depicted using a lower-visibility 

symbol because of the much greater distance between the calibration cross section and the acoustic-Doppler profiler 

array at that study site.

cross section and the ADP array is much greater at the CR61 

study site than it is at any other study site. At the CR61 

study site, the ADPs are located ~750 m upstream from the 

calibration cross section. This distance is much smaller at

all of the other study sites, where it ranges from 0 to 200 m. 

Because of the large distance between the calibration cross 

section and the ADPs at the CR61 study site, there is a risk 

that, during some time periods, the suspended-sediment 

characteristics within the ensonified volume could be less

correlated with those in the calibration cross section, thus 

leading to the much larger variance in the relative errors in the 

acoustical measurements of CXS-SAND at this study site. 

Although the relative bias in the acoustical measurements 

of D50-XS-SAND does not generally depend on CXS-SAND, in 

two-thirds of the cases summarized in table 3 it does depend 

on D50-XS-SAND. In all cases where these dependencies exist, 

the correlation is negative, meaning that as the EDI- or 

EWI-measured values of D50-XS-SAND increase, the relative bias 

in the acoustical measurements of D50-XS-SAND becomes more 

negative (fig. 21A). In these cases, there is a tendency for the 

acoustical measurements of D50-XS-SAND to be slightly high at 

low values of the EDI or EWI-measured D50-XS-SAND, and a 

tendency for the acoustical measurements of D50-XS-SAND to 

be slightly low at high values of the EDI or EWI-measured 

D50-XS-SAND. When the acoustical measurements of D50-XS-SAND

approach the value of D50-XS-SAND-REF used in the development 

of the BBC relations, the slight positive or negative bias in the 

acoustical measurements of D50-XS-SAND vanishes. 

The negative correlation between D50-XS-SAND and the 

relative bias in the acoustical measurements of D50-XS‐SAND

is strongest at the RG-CAS, CR87, and CR225 study sites 

(in decreasing order of correlation strength). This negative

correlation is extremely weak at the CR30 study site 

(R2 = 0.032), and no significant correlation is present at the

CR61 and RG-RGV study sites. A potential cause of this 

negative correlation could be the imposed lower bound on 

error. Because the acoustical measurements of D50-XS‐SAND

are never allowed to be smaller than 0.074 mm by the 

RUTS method, large negative relative errors in D50-XS-SAND

are precluded at lower D50-XS-SAND values (fig. 21A). Because 

of this imposed lower bound, there is right skewness in the 

distribution of relative error at lower values of D50-XS-SAND such 

that there is a much greater likelihood that the relative errors 

in the acoustical measurements of D50-XS-SAND will be positive at 

lower values of the EDI- or EWI-measured D50-XS-SAND. In any 

case, because the negative correlation is not always present, 
54 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

and when present, it tends to be relatively small, we make 

no attempt to remove it. Thus, its effects are included within

the 68-percent- and 95-percent-confidence level time-varying

relative errors in the acoustical measurements of D50-XS-SAND

estimated in the next section of this report.

Unlike the high degree of heteroscedasticity in the 

distributions of relative error in the acoustical measurements 

of CXS-SILT-CLAY and CXS-SAND over their respective domains, the 

distribution of relative error in the acoustic measurements 

of D50-XS-SAND is almost homoscedastic over the domain in 

CXS-SAND, especially when data from the CR61 study site are 

excluded (fig. 21B). Thus, in addition to the relative bias in

the acoustical measurements of D50-XS-SAND not being correlated 

with CXS-SAND, there is also minimal dependence of the 

time-varying relative errors in the acoustical measurements of 

D50-XS-SAND on CXS-SAND. Furthermore, because the distribution 

of relative error in the acoustical measurements of D50‐XS‐SAND

is approximately symmetric about zero when plotted as 

a function of CXS-SAND, the acoustical measurements of 

D50‐XS‐SAND can be treated as relatively unbiased as a function 

of concentration. These results provide further justification

for not including the sometimes-present, grain-size-correlated 

relative bias in the acoustical measurements of D50-XS-SAND

within the estimations of the 68-percent- and 95-percentconfidence level time-varying relative errors in the next

section of this report.

Estimation of the 68-Percent- and 95-PercentConfidence-Level Time-Varying Relative Errors 

in Individual Acoustical Measurements

Data from the five study sites that have calibration cross

sections within 200 m of the ADP arrays were used to estimate 

the generalized 68-percent- and 95-percent-confidence

level true relative errors in the acoustical measurements of 

CXS‐SILT‐CLAY, CXS-SAND, and D50-XS-SAND. In essence, this analysis 

was conducted to provide an estimate of the magnitude of 

the error bar associated with each acoustical measurement. 

The errors calculated in this analysis are referred to as true

relative errors because field and laboratory-processing errors

associated with the EDI or EWI measurements are accounted 

for. All other errors reported herein, referred to as apparent 

relative errors, were calculated with the assumption of zero 

error in the EDI or EWI measurements. This assumption is

acceptable for the other analyses because errors in the EDI 

and EWI measurements are typically much smaller than those 

in the acoustical measurements. However, because error is 

present in the EDI and EWI measurements (for example, 

Topping and others, 2010, 2011), the error associated with the

EDI or EWI measurements must be accounted for to yield true 

error in the acoustical measurements. 

As described in the “Introduction to the Analyses of 

Bias and Error” section, time-varying errors in the acoustical 

measurements arise from two processes: (1) error in the 

acoustical measurements of the suspended sediment within 

the volume ensonified by the acoustic beams, and (2) error

arising from slow variation in relations between the suspended 

sediment within just the ensonified part of the cross section

and the suspended sediment in the entire cross section. The

first source of error is likely random to the extent that biases

in the acoustical measurements produced by changes in the 

suspended-sediment grain-size distribution are minimized by 

the RUTS method. The second source of error may either be

random or non-random, as discussed in the next section of 

this report. The 200-m longitudinal-distance limit was chosen

so that the generalized errors estimated in this section of the 

report would be based on the errors at only those study sites 

where the influence of the second part of the two-part error is

minimized; that is, at study sites where the volume ensonified

by the acoustic beams is located either within or, at least, near 

the calibration cross section. In the best-case scenario where 

the ADPs are located within the calibration cross section (for 

example, the CR30 and RG-RGV study sites), the volume 

ensonified by the acoustic beams is an actual subsample of

the calibration cross section. In this case, the correlation 

between the suspended-sediment characteristics within just the 

ensonified volume and within the entire cross section should

be the highest possible and vary the least over time. The study

sites therefore chosen for inclusion in this analysis are the 

CR30 (0 m from calibration cross section), CR87 (200 m from 

calibration cross section), CR225 (130 m), RG-CAS (50 m), 

and RG-RGV (0 m) study sites. 

For relative errors to be calculated using equation 67, 

the denominator, b, must be known. An inconvenient fact that 

needs to be addressed before proceeding, therefore, is that 

although relative error equations must include a dependence 

on CXS-SILT-CLAY, CXS-SAND, or D50-XS-SAND, these parameters cannot 

be known independently before any acoustical measurement. 

The solution of this problem is that, because acoustical and

EDI or EWI measurements of CXS-SILT-CLAY, CXS-SAND, and 

D50-XS‐SAND are strongly correlated (for example, figure 18), 

equation 67 can be approximated for this analysis as 

proxy relative error =100 ( ) a b

a

− , (68)

where 

a is the acoustical measurement, and 

b is the EDI or EWI measurement. 

In other words, to solve this problem, we have used the strong 

correlation between the physical and acoustical measurements 

to allow approximating the unknown b (the EDI or EWI 

measurement) in the denominator of equation 67 with the 

known a (the acoustical measurement) in the denominator 

of equation 68. Making this approximation allows relative 

errors to be calculated using equations fit to the data from the

CR30, CR87, CR225, RG-CAS, and RG-RGV study sites. 

These equations allow estimation of relative errors on the

basis of only the acoustical measurements, without requiring 

additional information.
Error Analysis 55

In cases where measurements are unbiased and the 

proxy relative errors calculated by equation 68 comprise a 

Gaussian normal distribution, the 68-percent-confidence-level

time-varying relative error will be equal to the absolute value 

of the standard deviation of the distribution of relative errors 

calculated by equation 68, and the 95-percent-confidencelevel time-varying relative error will be equal to 1.96 times 

this quantity. In reality, because relative errors calculated 

by equation 67 tend to be right-skewed because of imposed 

lower bounds (described above), proxy relative errors 

calculated by equation 68 tend to be left-skewed. Because 

of the skewness (which tends to be more prevalent at lower 

values of concentration and D50-XS-SAND), the Gaussian-normal 

approximation for calculating the 68-percent-confidence-level

time-varying relative error is not quite accurate, but for the 

purposes of this analysis is used anyways. The ramifications

of making this Gaussian-normal approximation will be 

described below. 

The time-varying relative errors in the acoustical

measurements, and their relation to concentration, were 

evaluated following a seven-step procedure:

(1) Proxy relative errors in the acoustical measurements of 

CXS-SILT-CLAY and CXS-SAND were calculated using equation 

68 for the pooled data from the CR30, CR87, CR225, 

RG-CAS, and RG-RGV study sites.

	(2)	 These relative errors were segregated into CXS-SILT‐CLAY

or CXS-SAND bins centered on 1-mg/L increments over the 

entire range of the acoustical measurements of CXS‐SILT‐CLAY

or CXS-SAND associated with concurrent EDI or EWI 

measurements in the pooled dataset. The bounds on these

bins were ±20 percent of the central concentration in each 

bin. 

(3) In each bin where 20 or more observations were present, 

the sample standard deviation (σ) was calculated relative 

to an imposed mean of zero. Twenty observations

was determined to be the minimum number needed to 

calculate a sample standard deviation that is a reasonably 

accurate approximation of the population standard 

deviation. For example, when n = 20 and data are

randomly selected from a Gaussian-normal distribution, 

the 95-percent confidence interval associated with the

population standard deviation ranges from 0.76σ to 1.46σ

(Sheskin, 2003). 

	(4) 	Power-law fits were applied relating either CXS-SILT-CLAY

or CXS-SAND (as appropriate) to σ of the binned error 

distributions for the cases that satisfy the condition in step 

3 (fig. 22). 

	(5) 	 68-percent-confidence-level true relative errors in the

acoustical measurements of CXS-SILT-CLAY and CXS-SAND

were estimated by subtracting out the EWI/EDI errors in 

quadrature using the following equation:

E E 68 E 2

68

2

-ABS = − FIT-ABS E( ) DI-EWI -ABS , (69)

where 

E68-ABS is the absolute form (in mg/L) of the 

68-percent-confidence level true error

in the acoustical measurements of either 

CXS‐SILT-CLAY and CXS-SAND, 

E68-FIT-ABS is the absolute form (in mg/L) of the 

power law fit to σ of the proxy relative 

error in step 4, and

E(EDI-EWI)68-ABS is the absolute form (in mg/L) of the 

68-percent-confidence-level combined

field and laboratory-processing error in the

concurrent EDI or EWI measurement (after 

Topping and others, 2010, 2011).

 E68-ABS was then converted to the proxy relative form (E68) 

by dividing by the acoustical measurement of CXS‐SILTCLAY or CXS-SAND, as appropriate, and multiplying by 100 

percent.

	(6) 	 95-percent-confidence level true relative errors in the

acoustical measurements of CXS-SILT-CLAY and CXS-SAND were 

estimated using a version of equation 69 in which all 

three 68-percent-confidence-level errors were replaced by

95-percent-confidence-level errors.

	(7)	 Power-law fits relating CXS-SILT-CLAY or CXS-SAND (as 

appropriate) to E68 and E95 were then calculated to result 

in estimations of the true time-varying relative errors in 

the acoustical measurements that can be calculated on the 

basis of the acoustical measurements of either CXS‐SILTCLAY or CXS-SAND (fig. 22). As a result of the binning and 

filtering processes in steps 2 through 6, the R2 values 

associated with these power-law fits are exceptionally

high.

Because the 68-percent- and 95-percent-confidence-level

relative errors in the acoustical measurements of D50-XS-SAND

are invariant as a function of D50-XS-SAND, a simpler version 

of the above 7-step process was used to estimate the true 

time-varying relative errors in the acoustical measurements 

of D50-XS-SAND. In this simpler process, steps 1–3 in the 

above procedure were combined into one step where σ was 

calculated on the basis of all proxy relative errors in the 

pooled dataset (instead of binning the errors), step 4 in the 

above procedure was not needed, and steps 5–7 were followed 

as described above. 

As shown in figure 22, at low concentrations, true 

time-varying relative errors in individual acoustical 

measurements of CXS-SILT-CLAY and CXS-SAND are extremely large, 

whereas at high concentrations, the errors in the acoustical 

measurements of CXS-SILT-CLAY and CXS-SAND are similar to those 

associated with EDI and EWI measurements. The true errors

in the acoustical measurements of D50-XS-SAND are roughly 
56 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Figure 22. (folllowing page) Six plots showing 68-percent- and 95-percent-confidence-level relative errors in individual acoustical 

suspended-sediment measurements and in individual equal-discharge increment (EDI) or equal-width increment (EWI) measurements 

plotted as a function of the acoustical suspended-sediment measurements. These errors can have positive or negative sign. E68 is 

the power-law relation of step 7 (of the analysis of time-varying relative errors, described above in text) that is fit to the calculated 

68-percent-confidence-level true proxy relative errors in the acoustical measurements; E95 is the power-law relation fit, in step 7, to 

the calculated 95-percent-confidence-level true proxy relative errors in the acoustical measurements. As described in the text, the 

exceptionally high R2 values associated with the E68 and E95 power-law fits result from the binning and filtering processes in steps 2 

through 6 of the 7-step error-calculation procedure. E68-FIT is the power-law relation fit, in step 4, to the standard deviation (σ) of the 

proxy relative errors in the acoustical measurements. Solid red lines indicate the largest possible negative value of these errors. A, 

68-percent-confidence-level proxy errors in CXS-SILT-CLAY plotted as a function of acoustical measurements of CXS-SILT-CLAY. B, 95-percentconfidence-level proxy errors in CXS-SILT-CLAY plotted as a function of acoustical measurements of CXS-SILT-CLAY. C, 68-percent-confidencelevel proxy errors in CXS-SAND plotted as a function of acoustical measurements of CXS-SAND. D, 95-percent-confidence-level proxy errors 

in CXS-SAND plotted as a function of acoustical measurements of CXS-SAND. E, 68-percent-confidence-level proxy errors in D50-XS-SAND plotted 

as a function of acoustical measurements of D50-XS-SAND. F, 95-percent-confidence-level proxy errors in D50-XS-SAND plotted as a function 

of acoustical measurements of D50-XS-SAND. So that their magnitudes could be directly compared with the magnitudes of the proxy 

relative errors in the acoustical measurements, the 68-percent- and 95-percent-confidence-level relative errors in the EDI and EWI 

measurements depicted in these plots were converted to errors calculated relative to their associated acoustical measurement. This 

operation was done by dividing the 68-percent- and 95-percent-confidence level absolute errors in the EDI and EWI measurements by 

the concurrent acoustical measurement of concentration or median grain size, and then multiplying by 100 percent.

2 to 3 times larger than those associated with EDI and EWI 

measurements. Even though the 68-percent- and 95-percentconfidence-level errors in the acoustical measurements plotted

in figure 22 may seem large, it is important to note that these 

errors are associated with single acoustical measurements of 

CXS-SILT-CLAY, CXS-SAND, or D50-XS-SAND, just as the errors associated 

with the EDI and EWI measurements are the errors associated 

with single EDI or EWI measurements. If these time-varying 

errors are random (investigated in the next section of this 

report), they will decay as a function of 1 n , where n is the 

number of acoustical measurements. Thus, the random error

in a quantity measured at 15-minute intervals can effectively 

decrease tenfold over the course of one day such that, under 

approximately constant sediment conditions, a ±100-percent 

random error will decay to roughly a ±10-percent error in the 

daily average after 96 measurements at 15-minute intervals 

have been made. 

Skewness in the distributions of relative error 

arising from the imposed lower bounds produces 

two important effects in the results plotted in figure 22. 

First, the true proxy errors based on a normal distribution 

plotted in figure 22 are slight overestimates of the error on one 

tail of the distribution, but slight underestimates of the error 

on the other tail of the distribution. This result is acceptable

because one does not know independently whether the 

time-varying error associated with any individual acoustical 

measurement of CXS-SILT-CLAY, CXS-SAND, or D50-XS-SAND is positive 

or negative. Second, if the distribution of relative error 

in concentration were truly a normal distribution with no 

imposed bounds, relative errors exceeding ±100 percent would 

be impossible because the distribution would be symmetric, 

and it is impossible to have errors less than –100 percent. 

Because the skewness is largest at the lowest concentrations, 

68-percent- and 95-percent-confidence-level relative errors

exceeding ±100 percent are predicted at low concentrations 

(fig. 22A–D). When errors exceeding ±100 percent are 

predicted, the largest negative error allowed is capped at 

–100 percent. The lower bound on the negative errors in

the acoustical measurements of D50-XS-SAND arising from the 

imposed constraint of D50‐XS‐SAND≥0.074 mm has a more

complicated influence on the allowed range of negative errors,

which is best described graphically (see fig. 22E–F).

Importance of Errors from Slow Changes in the 

Spatial Structure of Suspended Sediment in the 

River Cross Section

Spatial structures in both suspended-sand and suspendedsilt-and-clay concentration can evolve slowly over hours to 

days in a river cross section (Topping and others, 2011). At

both the CR30 and CR61 study sites, the lateral distribution 

of suspended silt and clay and suspended sand were both 

observed by Topping and others (2011) to be stable for

large fractions of a day before evolving into different lateral 

distributions that were then sometimes stable over similar 

timescales. These slow changes in the spatial structure of

the suspended sediment were observed to occur at relatively 

constant discharge. On the basis of Rubin and others (2001), 

Topping and others (2011) interpreted this behavior to arise

from relatively slow changes in the lateral distribution of 

suspendable bed sediment, which is, in turn, produced by 

the relatively slow downstream migration of irregular dunes 

within patches of sand on the gravel bed. It is also possible 

that these slow changes in the lateral distribution of suspended 

sediment were produced by the downstream migration of 

the patches of sand on the bed. The existence of such slow

changes in the lateral distribution of suspended sediment in 

a river cross section indicates that CSILT-CLAY and CSAND will 

likely not be constant within the ensonified part of the cross
Error Analysis 57

men16-3130_fig22

0

100

200

300

400

500

600

700

800

900

1,000

0

100

200

300

400

500

600

700

800

900

1,000

0

100

200

300

400

500

600

0.1

0

100

200

300

400

500

600

0.1

0

10

20

30

40

50

60

70

80

90

100

0.26

0

10

20

30

40

50

60

70

80

90

100

Acoustical measurement of CXS-SILT-CLAY, in milligrams per liter

Acoustical measurement of CXS-SAND, in milligrams per liter

Acoustical measurement of D50-XS-SAND, in millimeters

68%-confidence-level relative error in 

CXS-SILT-CLAY, in percent

95%-confidence-level relative error in 

CXS-SILT-CLAY, in percent

68%-confidence-level relative error in 

CXS-SAND, in percent

95%-confidence-level relative error in 

CXS-SAND, in percent

68%-confidence-level relative error in 

D50-XS-SAND, in percent

95%-confidence-level relative error in 

D50-XS-SAND, in percent

1 10 100 1,000 10,000 100,000 1 10 100 1,000 10,000 100,000

1 10 100 1,000 10,000 100,000 1 10 100 1,000 10,000 100,000

0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26

A B

C D

E F

Largest possible

negative relative error 

Largest possible

negative relative error 

Largest possible

negative relative error 

Largest possible

negative relative error 

Largest possible negative relative error

Largest possible negative relative error

E68-FIT = 778 CXS-SILT-CLAY

-0.527 R2

= 0.818

E68 = 807 CXS-SILT-CLAY

-0.538 R2

= 0.9999

σ of proxy relative error in 

acoustical CXS-SILT-CLAY

True proxy relative error in 

acoustical CXS-SILT-CLAY

Combined field and laboratory 

relative error in EDI or EWI CXS-SILT-CLAY

E95 = 1580 CXS-SILT-CLAY

-0.538 R2

= 0.9999

True proxy relative error in 

acoustical CXS-SILT-CLAY

Combined field and laboratory

relative error in EDI or EWI CXS-SILT-CLAY

E68-FIT = 125 CXS-SAND

-0.247 R2

= 0.603

E68 = 126 CXS-SAND

-0.253 R2

= 0.995

σ of proxy relative error in 

acoustical CXS-SAND

True proxy relative error in acoustical CXS-SAND

Combined field and laboratory

relative error in EDI or EWI CXS-SAND

E95 = 247 CXS-SAND

-0.253 R2

= 0.995

True proxy relative error in 

acoustical CXS-SAND

Combined field and laboratory

relative error in EDI or EWI CXS-SAND

E68-FIT = 18.2

E68 = 17.5

σ of proxy relative 

error in 

acoustical 

D50-XS-SAND

E95 = 34.3

True proxy relative error in 

acoustical D50-XS-SAND

Combined field and laboratory

relative error in EDI or EWI D50-XS-SAND

True proxy relative error 

in acoustical D50-XS-SAND

Combined field and laboratory

relative error in EDI or 

EWI D50-XS-SAND

Figure 22.
58 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

section, even when the discharge of water, CXS-SILT-CLAY, and 

CXS-SAND remain constant over time. Furthermore, the existence 

of changes in the lateral distribution of suspended sediment 

over timescales of a day suggest that slower changes in the 

lateral distribution are also possible. These changes will result

in slowly varying, positively correlated errors in the acoustical 

measurements of CXS-SILT-CLAY and CXS-SAND that behave as 

biases over timescales smaller than those associated with the 

changes in the lateral distribution of suspended sediment. 

Positively correlated errors in acoustical measurements of 

CXS-SILT-CLAY or CXS-SAND arising from only slow changes in 

the lateral distribution of suspended sediment in the cross 

section will depend on the error associated with the previous 

measurement (that is, autocorrelation) and be independent of 

water discharge, suspended-sediment concentration, and the 

grain-size distribution of the suspended sand. 

Slowly varying, correlated time-varying errors can 

be distinguished from random errors through analyses of 

temporal autocorrelation (Durbin and Watson, 1950, 1951;

Anderson, 1971; Box and Jenkins, 1976). Autocorrelation

analysis provides insight into whether the sign of a 

time-varying error remains positive or negative among 

sequential measurements for durations different than those 

expected for a random error. The case where errors are

positively correlated is commonly referred to as having 

positive serial correlation. Positive serial correlation is a serial 

correlation in which the occurrence of a positive error in 

one measurement increases the likelihood of a positive error 

in the next measurement, and in which the occurrence of a 

negative error in one measurement increases the likelihood of 

a negative error in the next measurement. Conversely, the case 

where errors are negatively correlated is commonly referred 

to as having negative serial correlation. Negative serial 

correlation is a serial correlation in which the occurrence of 

a positive error in one measurement increases the likelihood 

of a negative error in the next measurement, and in which the 

occurrence of a negative error in one measurement increases 

the likelihood of a positive error in the next measurement. 

Positive serial correlation leads to errors that are larger than 

those predicted by the formula for the standard error of the 

mean, and negative serial correlation leads to errors that 

are smaller than those predicted by the standard formula 

for the standard error of the mean (Dunlop, 1994; Bence,

1995; Quinn and Keough, 2003). The ramifications of serial

correlation on error are that positively correlated errors, that 

is, positive serial correlation, will lead to errors that decay in 

absolute value toward zero more slowly than expected on the 

basis of the standard error formula as more observations are 

made. Similarly, negative serial correlation will lead to errors 

that decay in absolute value toward zero more quickly than 

expected on the basis of the standard error formula as more 

observations are made (Topping and others, 2011).

The first step in this analysis was to determine whether,

on average, the time-varying relative errors in the acoustical 

measurements of CXS-SILT-CLAY and CXS-SAND were independent 

of water discharge, suspended-sediment concentration, and 

the grain-size distribution of the suspended sand. If the 

time-varying errors in the acoustical measurements were 

found to be independent of these parameters, then the most 

likely dominant source of the time-varying error would be 

the observed slow variation in the lateral distribution of 

suspended sediment in the cross section. F-tests were used in 

these correlation analyses, as described below. Given that the 

relative errors in the acoustical measurements of CXS‐SILT‐CLAY

and CXS-SAND were found to be largely independent of water 

discharge, sediment concentration, and grain size, the second 

step in this analysis was to use autocorrelation plots (Box 

and Jenkins, 1976) to determine whether the time-varying 

relative errors exhibited serial correlation between successive 

acoustical measurements. In cases of significant positive serial

correlation, these plots will exhibit consecutive values of 

positive autocorrelation in excess of the positive 95-percentconfidence band associated with random processes for

contiguous lags >0. Likewise, in cases of significant negative

serial correlation, these plots will exhibit consecutive values of 

negative autocorrelation in excess of the negative 95-percentconfidence band associated with random processes for

contiguous lags >0. In contrast, random errors in acoustical 

measurements of CXS-SILT-CLAY or CXS-SAND will exhibit no 

significant autocorrelation for non-zero lags.

Meaningful analyses of correlation and autocorrelation 

in the relative errors require a large number of 

suspended-sediment measurements made using a method 

that is much more accurate than the RUTS acoustical method. 

The only measurement techniques that meet this criterion are

the EDI and EWI method (Topping and others, 2011), and

the only four study sites with sufficient numbers of EDI or

EWI measurements for these analyses are the CR30, CR61, 

CR87, and CR225 study sites. Ideally, the measurements 

used to calculate the error analyzed for autocorrelation would 

be made at evenly spaced intervals similar to the 15-minute 

interval of the acoustical measurements. This condition is, of

course, impossible. Several EDI measurements are typically 

made over 1–2 days every 2 months at the CR87 and CR225 

study sites, 8 EWI measurements are typically made over 

2 days every 6 months at the CR30 and CR61 study sites, 

with a larger number of EDI or EWI measurements clustered 

at all study sites during discrete high-discharge periods. If it 

were possible to make highly accurate suspended-sediment 

measurements at 15-minute intervals, there would be no need 

for this study. 

Although the EDI or EWI measurement programs at 

these study sites are nonrandom with the distribution of the 

interval between measurements being highly right-skewed, 

this nonrandom sampling design does not adversely affect 

the autocorrelation analyses. An evenly spaced, random, 

or nonrandom sampling of a time series will yield similar 

results so long as the number of observations is sufficiently

large. Autocorrelation plots appear similar when the time 

series analyzed is random regardless of whether the sampling 
Error Analysis 59

Table 4. Results from autocorrelation analyses conducted on relative errors in acoustical measurements of suspendedsediment concentration (CXS-SILT-CLAY and CXS-SAND).

Study site n

Central tendency of the interval between sequential 

EDI or EWI measurements, that is, the variable time 

step of each lag, expressed as the: 

Number of contiguous lags >0 where

the autocorrelation in the relative errors in 

acoustical measurements exceeds the positive 

95-percent-confidence band for:

Mean number

of days

Median number

of days CXS-SILT-CLAY CXS-SAND

CR30 172 13.84 0.15 3 4

CR61 175 12.57 0.13 6 5

CR87 224 15.15 0.70 3 4

CR225 99 23.34 0.76 0 0

scheme is evenly spaced, random, or nonrandom. Thus,

detection of nonrandom error behavior is possible using 

the autocorrelation-plot method regardless of whether the 

sampling scheme is nonrandom (as it is in this study). 

The results from the correlation analyses were partially

presented in table 3, with the remaining parts of these analyses 

presented in text below; the results from the autocorrelation

analyses are presented in table 4. Both the correlation and 

autocorrelation analyses were conducted on the apparent 

relative errors calculated using equation 67. F-tests were used 

to determine whether significant correlations existed between

the relative errors in the acoustical sediment-concentration 

measurements and water discharge, CXS-SILT-CLAY, CXS-SAND, or 

D50-XS-SAND (see table 3). 

The results summarized previously in table 3 indicate 

that the relative errors in the acoustical measurements of 

CXS‐SILT-CLAY are typically uncorrelated with CXS-SILT-CLAY, and the 

acoustical measurements of CXS-SAND are typically uncorrelated 

with CXS-SAND and D50-XS-SAND. F-tests used to detect significant

correlations between water discharge (Q) and the relative 

errors in the acoustical measurements of CXS‐SILT-CLAY and 

CXS‐SAND yield slightly different results, with the relative 

errors in the acoustical measurements of CXS‐SILT‐CLAY and 

CXS-SAND being slightly more dependent on Q than they are on 

concentration or grain size. However, although significant

correlations between Q and relative error in the acoustical 

measurements occur at several of the Colorado River study 

sites, these significant correlations are extremely weak,

with R2 values <0.05. Relative errors in the acoustical 

measurements of CXS-SILT-CLAY were found to depend on Q at the 

pSIG= 0.037 level at the CR30 study site, and at the pSIG= 0.0011

level at the CR87 study site. Relative errors in the acoustical 

measurements of CXS-SAND were found to depend on Q at the 

pSIG= 0.0034 level at only the CR61 study site. In these three

cases, as in the few cases of significant correlations between

CXS-SILT-CLAY, CXS-SAND, or D50-XS-SAND and the relative errors in the 

acoustical measurements of CXS-SILT-CLAY and CXS-SAND presented 

in table 3, the relative errors in the acoustical measurements of 

CXS-SILT-CLAY and CXS-SAND are slightly negatively correlated with 

Q, meaning the relative errors become slightly more negative 

with increasing Q. 

The results from the autocorrelation analyses

indicate that, of the four study sites, only at the CR225 

study site do the errors in the acoustical measurements 

of CXS-SILT-CLAY and CXS-SAND appear to be completely 

random (table 4). At the other three study sites, significant

positive autocorrelation exists in the relative errors in 

acoustically measured CXS‐SILT‐CLAY and CXS-SAND, and 

these positive autocorrelations persist for multiple lags 

(figs. 23–24). Positive autocorrelation is most persistent 

at the CR61 study site for the case of the relative error in 

the acoustical measurements of CXS-SAND (fig. 24B), perhaps 

again because of the much greater distance between the 

calibration cross section and the ADP array at that study site. 

Because the distribution of the intervals between the 

sequential EDI or EWI measurements at the study sites are 

highly right skewed (that is, there are distinct clusters of 

measurements at monthly to semi-annual time steps), the 

median value reported in table 4 is a better measure than the 

mean value for the timescale associated with each lag in the 

autocorrelation analyses. Thus, the relative errors in acoustical

measurements of CXS-SILT-CLAY are typically nonrandom for 

durations exceeding ~7 hours at CR30, ~16 hours at CR61, 

and 1.4 days at CR87, and the relative errors in acoustical 

measurements of CXS-SAND are typically nonrandom for 

durations exceeding ~14 hours at CR30, 6 hours at CR61, and 

2.1 days at CR87. 
60 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Figure 23. Autocorrelation in the relative errors associated with acoustical measurements of CXS-SILT-CLAY at the 

(A) CR30, (B) CR61, (C) CR87, and (D) CR225 study sites. Relative errors exhibit random behavior at only the CR225 

site; relative errors exhibit varying degrees of positive autocorrelation at the other 4 study sites. Red lines are the 

95-percent-confidence bands (Box and Jenkins, 1976). Lag refers to the nonuniform temporal spacing between 

successive equal-discharge-increment or equal-width-increment measurements; see text and table 4 for the 

estimated duration of a lag at each study site.

men16-3130_fig23

B

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

A

–1.0

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–0.6

–0.4

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0.6

0.8

1.0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

C

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

D

Lag

Autocorrelation
Error Analysis 61

Figure 24. Autocorrelation in the relative errors associated with acoustical measurements of CXS-SAND at the (A) CR30, 

(B) CR61, (C) CR87, and (D) CR225 study sites. Relative errors exhibit random behavior at only the CR225 site; relative 

errors exhibit varying degrees of positive autocorrelation at the other four study sites. Red lines are the 95-percentconfidence bands (Box and Jenkins, 1976). Lag refers to the non-uniform temporal spacing between successive equaldischarge-increment or equal-width-increment measurements; see text and table 4 for the estimated duration of a lag 

at each study site.

men16-3130_fig24

A B

C D

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Lag

Autocorrelation –1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0
62 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

These timescales of nonrandom behavior in the acoustical

measurements are consistent with the >4 to <24-hour 

timescale reported for the physical measurements in Topping

and others (2011) over which stable spatial structures in the 

lateral distribution of suspended silt and clay and suspended 

sand were observed to persist in river cross sections. 

Therefore, it is plausible that the positive autocorrelation in

these errors does arise from the slow evolution of the lateral 

distribution of suspended sediment in a river cross section. 

Recalculation of the 68-percent- and 95-percent-confidencelevel errors in the preceding section by segregating the data 

from the CR225 study site from the data from the CR30 and 

CR87 study sites indicates, however, that little difference 

exists in the magnitudes of these errors when random or 

nonrandom error behavior is suggested by autocorrelation 

analyses. Therefore, the presence of nonrandom error

behavior at a study site does not necessarily indicate the 

presence of larger time-varying relative error, but rather only 

that the time-varying error may require a larger number of 

observations to decay.

The presence of positive autocorrelation in the errors

indicates that a substantial number (perhaps many more than 

100) of acoustical measurements of CXS-SILT-CLAY or CXS-SAND

are required before the errors in time-averaged quantities 

decay to acceptably small values. Random errors will decay 

as a function of 1 n, whereas positively correlated errors 

will decay as a function of 1 nx where x<0.5. For example, 

in the random case, a ±50 percent 95-percent-confidence

level relative error will decay to an error of approximately 

±5 percent after 96 observations (one day of 15-minute 

measurements). However, in the nonrandom case, many hours 

to days of 15-minute observations are likely required before 

any substantial decay in the error associated with the time 

average of the measurements is realized. 

Therefore, although random behavior in the errors in

the acoustical measurements of CXS-SILT-CLAY and CXS-SAND

are possible (for example, at CR225), it is perhaps safer to 

assume nonrandom behavior of the errors at a study site (for 

example, CR30, CR61, and CR87) until proven otherwise 

by comparison with EDI or EWI measurements. Finally, it is 

important to reiterate that nonrandom behavior in the errors 

in the acoustical measurements of CXS-SILT-CLAY and CXS-SAND

most likely arises not because of error in the acoustical 

measurements themselves but, rather, because the ADP 

samples only a part of the entire river cross section. 

Conclusions

The two-frequency RUTS method presented herein 

for processing ADP data allows for reasonably accurate, 

unbiased, 15-minute acoustical measurements of CXS-SILT-CLAY, 

CXS-SAND, and D50-XS-SAND to be made over long timescales. 

Multi-frequency arrays of single-frequency ADPs, with data 

processed using the RUTS method, have been continuously 

deployed for over a decade at most of the study sites in this 

report. The power of the RUTS method arises from the fact 

that it is based on the physics of sound scattering by small 

particles. Acoustical theory forms the basis for all aspects of 

the ADP-calibration procedure. Empiricism is used only in the 

development of the relations between the suspended-sediment 

conditions within the ensonified part of the river cross section

“sampled” by the acoustic beams to the suspended-sediment 

conditions in the entire cross section. 

In most situations over the range of acoustic frequencies 

tested in our study, 600 kHz to 2 MHz, acoustic attenuation 

was caused primarily by silt-and-clay-sized sediment, whereas 

acoustic backscatter resulted from sand-, silt-, and clay-sized 

sediment. Only when CXS-SILT-CLAY is smaller than roughly 

twice CXS-SAND can acoustic backscatter be accurately related 

to only sand-size sediment. Thus, in rivers where a large range

in CXS-SILT-CLAY is possible for any given value of CXS-SAND, 

accounting for the acoustic backscatter from silt and clay is 

required to produce acoustical measurements of CXS-SAND with 

acceptably low levels of bias. Positive relative biases in the 

acoustical measurements of CXS-SAND will be produced if the 

backscatter from silt-and-clay-sized sediment is neglected 

(a common practice in the recent literature). In cases where 

the backscatter from silt and clay is incorrectly assigned to 

sand-size sediment, positive relative biases in the acoustical 

measurements of CXS-SAND of many hundreds of percent are 

common and many thousands to tens of thousands of percent 

are possible. This result is supported by the target-strength

theory for different grain-size distributions, and is independent 

of the acoustic frequencies used in this study.

Changes in the grain-size distribution of the suspended 

sand can greatly affect acoustic backscatter. At any given 

constant CXS-SAND, coarser grain-size distributions of suspended 

sand will result in much more backscatter than finer grain-size

distributions of suspended sand. Because of the theoretical 

behavior of the RUTS, this grain-size effect causes larger 

changes in acoustic backscatter at lower frequencies than 

at high frequencies. Therefore, in cases where the grainsize distribution of the suspended sand is not constant, two 

acoustic frequencies are required for acoustical measurements 

of CXS‐SAND that are not grain-size biased. Addition of a third 

frequency did not substantially improve the two-frequency 

measurements of CXS-SAND and only slightly improved the twofrequency measurements of D50-XS-SAND in our study. If singlefrequency measurements of acoustic backscatter are to be used 

to calculate CXS-SAND, these measurements should not be made 

at acoustic frequencies much less than ~2 MHz. Substantial 

grain-size-driven bias in single-frequency acoustical 

measurements of CXS-SAND is always present at these lower 

frequencies. Only in cases where the variation in D50-XS-SAND

is less than ~0.5φ are reasonably unbiased single-frequency

acoustical measurements of CXS‐SAND possible, but only when 

the 2-MHz frequency is used to make these measurements. 

These results are supported by the theoretical behavior of

target strength at different acoustic frequencies.
Conclusions 63

Time-varying relative errors in the acoustical

measurements of CXS-SILT-CLAY and CXS-SAND decrease as a 

function of increasing concentration, as expected on the basis 

of Thorne and Hanes (2002). Although these errors are much

larger than the combined field and laboratory‐processing

errors associated with EDI or EWI measurements at lower 

concentrations, they decrease to become similar to EDI or 

EWI errors at the higher concentrations measured in this study. 

Because of the right skewness in distributions of relative 

error, time-varying relative errors in excess of 100 percent 

are possible at the lowest concentrations. The estimated

68-percent-confidence-level time-varying relative error in

the acoustical measurements of CXS-SILT-CLAY decreases from 

±50 percent to ±10 percent as CXS-SILT-CLAY increases from 

~180 to ~3,500 mg/L. Similarly, the estimated 68-percentconfidence-level time-varying relative error in the acoustical

measurements of CXS-SAND decreases from ±50 percent to 

±15 percent as CXS-SAND increases from ~39 to ~4,500 mg/L. 

The behavior of the relative error in the acoustical

measurements of D50-XS-SAND is slightly more complicated 

than the behavior of errors in concentration. The acoustical

measurements of CXS-SAND are generally unbiased with 

respect to changes in either the concentration or the 

grain-size distribution of the suspended sand. Although the 

acoustical measurements of D50-XS-SAND are unbiased with 

respect to changes in CXS-SAND, there is a tendency for these 

measurements to be slightly biased as a function of changes 

in D50-XS-SAND. In essence, at most of the study sites there is 

a tendency for the acoustical measurements of D50-XS-SAND to 

be slightly low when the suspended sand is relatively coarse 

and for these measurements to be slightly high when the 

suspended sand is relatively fine. As D50-XS-SAND approaches 

the D50-XS-SAND-REF used in the ADP calibrations, the acoustical 

measurements of D50-XS-SAND become essentially unbiased. As 

it is impossible to know independently how coarse or fine the

suspended-sand grain-size distribution is during an acoustical 

measurement (which would allow for a bias correction), we 

chose to include this small bias within the calculation of the 

time-varying relative error. Largely as a result of the inclusion 

of this small grain-size-driven bias, the time-varying relative 

error in the acoustical measurements of D50-XS-SAND is much 

larger than the combined field and laboratory-processing errors

associated with EDI or EWI measurements. The 68-percentconfidence-level time-varying relative error in the acoustical

measurements of D50-XS-SAND is roughly ±18 percent. 

Because there is a general tendency for positive 

autocorrelation to exist in the time-varying relative errors, 

the errors in time-averaged acoustical measurements of either 

CXS-SILT-CLAY or CXS-SAND do not decay as fast as if these errors 

were random. The behaviors of the time-varying errors in

CXS-SILT-CLAY and CXS-SAND were completely random at only one 

of the four study sites where sufficient data existed to conduct

an autocorrelation analysis. When time-varying errors are 

random, these errors in a time-averaged quantity decrease 

as a function of 1 n. Thus, at the one study site where

the autocorrelation analyses indicated only random error 

behavior, the time-varying relative errors in the 15-minute 

acoustical measurements of CXS-SILT-CLAY or CXS‐SAND would 

decrease roughly tenfold over the course of a day (after 

96 measurements have been made), under conditions of 

constant concentration. At the other study sites exhibiting 

nonrandom error behavior, however, it would take much 

longer (likely at least several days depending on the timescale 

of the nonrandom errors) for these errors to decay to similarly 

small values. Recognition of whether positive autocorrelation 

exists in the time-varying errors at a study site is important 

because it provides information on the timescale over which 

the errors in sediment loads calculated using the acoustical 

measurements decay to acceptably small values, and thus sets 

limits on the timescales associated with the calculation of 

accurate sediment budgets. 

The source of positive autocorrelation in time-varying

error is not likely the acoustical measurements themselves, 

but rather, the presence of spatial structures in both 

suspended-sand and suspended-silt-and-clay concentration that 

evolve slowly over time in a river cross section (for example, 

Topping and others, 2011). This slow variation in the lateral

distribution of suspended sediment results in variation in the 

relation between the suspended sediment within the ensonified

part of the cross section and the suspended sediment in the 

entire river cross section. Because the ADP calibrations are 

constant over time, this effect produces the slower-thanrandom time-varying error behavior. Because of its likely 

source, this nonrandom error behavior is likely to be present in 

any method that relies on suspended-sediment measurements 

made within a subsample of an entire river cross section (for 

example, automatic-pump, optical, and LISST methods).

Furthermore, because the ADP beams effectively sample the 

suspended-sediment conditions within a larger part of the river 

cross section than do any of these other methods that measure 

the suspended-sediment conditions at only a point in the cross 

section, the nonrandom time-varying errors in the acoustical 

measurements of CXS-SILT-CLAY, CXS-SAND, and D50-XS-SAND are 

expected to be smaller than errors associated with any of the 

point methods. 

Finally, although both the random and nonrandom 

relative errors in the acoustical measurements decay to 

relatively small values as more measurements are made, 

sediment loads calculated on the basis of acoustical 

measurements are still subject to small potential biases 

in EDI/EWI measurements (Topping and others, 2010)

because the acoustical measurements are calibrated to the 

EDI/EWI measurements. However, because (1) it is typically 

not standard to make more than one or two EDI or EWI 

measurements on any given day, (2) it is certainly not typical 

to make many EDI or EWI measurements over consecutive 

days, and (3) the suspended-sediment conditions can change 

substantially in a river over minutes to hours, long-term 

sediment loads calculated on the basis of the acoustical 

measurements described herein will be much more accurate 

than loads calculated on the basis of a relatively sparse EDIor EWI-measurement program using interpolation to estimate 

concentrations between the times of the sparse measurements. 
64 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Acknowledgments

Funding for this study was provided by the Department 

of the Interior’s Glen Canyon Dam Adaptive Management 

Program administered by the Bureau of Reclamation; by

the U.S. Geological Survey Southwest Biological Science 

Center; by the National Park Service; by the Commission

for Environmental Cooperation (funded by the North 

American Free Trade Agreement); by the Utah State

University Watershed Sciences Department; and by the

Federal Interagency Sedimentation Project. Ted Melis played

a key role in getting this study established. Ron Griffiths and

David Dean both played substantial roles in maintaining the 

instrumentation described in this report and in processing the 

acoustical data. Ron Griffiths, Tom Sabol, Nick Voichick,

David Dean, and Hunter Edwards played critical roles in 

collecting the physical suspended-sediment samples used 

in the ADP calibration and verification process. David

Schoellhamer provided thoughtful and helpful comments on 

an earlier condensed draft of this paper. Daniel Buscombe, 

Francisco Pedocchi, and Mark Landers provided insightful 

reviews that greatly improved the quality of this report. All 

data used in this report can be obtained at http://www.gcmrc.

gov/discharge_qw_sediment/, http://cida.usgs.gov/gcmrc/

discharge_qw_sediment/, or from the authors.

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Appendixes 1–9
72 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Appendix 1. Example Illustrating how Transmit-Pulse, Blanking, and 

Receive-Window Durations Together Determine the Locations and Sizes of the 

Measurement Cells

The following hypothetical example is used to more

completely describe how the transmit-pulse, blanking, and 

receive-window durations together determine the locations 

and sizes of the measurement cells along an acoustic beam. 

This example consists of a transmit-pulse length of 0.44 m,

followed by a blanking distance of 0.22 m, and two sequential 

receive windows of 0.44 m in length. These spatial magnitudes

were chosen for this hypothetical example because they are 

the magnitudes of the transmit-pulse length, blanking distance, 

and receive-window lengths used by the 1-MHz EasyQ and 

SLD ADPs operating in diagnostics mode. In shallow, fresh 

water, at 20 oC, the speed of sound is 1,480 m/s. Under these 

conditions, the above spatial magnitudes therefore equate to 

a transmit-pulse duration of 0.00030 s, blanking duration of 

0.00030 s, and receive-window durations of 0.00059 s. In 

this example, the first 0.44-m-long receive window therefore

begins 0.00030 s after the end of the transmit pulse and 

extends to 0.00089 s after the end of the transmit pulse. 

Similarly, the second 0.44-m-long receive window begins 

0.00089 s after the end of the transmit pulse and extends to 

0.00148 s after the end of the transmit pulse. 

By virtue of the convolution of the 0.44-m-long transmit 

pulse and receive window, at the beginning of the first receive

window the transducer is able to detect sound backscattered 

from particles located between 0.22 and 0.66 m from the 

transducer. The smaller of these two distances (0.22 m) arises

from the interaction of the end of the transmit pulse with 

the beginning of the receive window, whereas the larger of 

these two distances (0.66 m) arises from the interaction of 

the beginning of the transmit pulse with the beginning of 

the receive window. At the end of the first receive window

the transducer is able to detect sound backscattered from 

particles located between 0.66 and 1.1 m from the transducer. 

The smaller of these two distances (0.66 m) arises from

the interaction of the end of the transmit pulse with the 

end of the receive window, whereas the larger of these two 

distances (1.1 m) arises from the interaction of the beginning 

of the transmit pulse with the end of the receive window. 

The cell associated with the first receive window therefore

extends from 0.22 to 1.1 m from the transducer, and most of 

the detected backscatter is from the part of the cell located 

between 0.44 and 0.88 m from the transducer. Likewise, at 

the beginning of the second receive window the transducer 

is able to detect sound backscattered from particles located 

between 0.66 and 1.1 m from the transducer; at the end of the

second receive window the transducer is able to detect sound 

backscattered from particles located between 1.1 and 1.44 m 

from the transducer. The cell associated with the second

receive window therefore extends from 0.66 to 1.44 m from 

the transducer, and most of the detected backscatter is from 

the part of the cell located between 0.88 and 1.22 m from 

the transducer. This example illustrates that, in addition to it

producing measurement cells wherein most of the backscatter 

is from particles located in the central 50 percent of each cell, 

the triangular-weighting function arising from the convolution 

of equal-length transmit pulse and receive windows also 

produces 50 percent overlap between sequential measurement 

cells (Nortek, 2013). 
Appendixes 73

Appendix 2. Calibration Measurements, ADP-Array Configurations, and the 

Distance Between the Calibration Cross Section and the ADP Arrays at 

Each Study Site

The EDI sample-collection method was used in the

ADP calibrations at the two study sites with previously 

determined, well-known lateral distribution of discharge 

within the measurement cross sections, that is, the CR87 

and CR225 study sites, whereas the EWI sample-collection 

method was used at the other study sites. Suspended-sediment 

measurements made with either method are identical when 

done properly. Depending on the river and cross section, the 

EWI method may require collection of cumulatively more 

water and more sampling verticals than the EDI method. 

Spatial and temporal field errors associated with EDI and

EWI measurements are described in Topping and others

(2011); laboratory-processing errors associated with all

physical suspended-sediment samples are based on methods 

in Topping and others (2010); field errors associated with

suspended-sediment measurements made using cross-section 

calibrated automatic pump samplers are estimated using 

an analysis currently being prepared for publication. At 

the Colorado River study sites, EDI measurements at the 

CR87 and CR225 study sites were made from a cableway, 

whereas EWI measurements at the CR30 and CR61 study 

sites were made from a boat deployed under a tagline. EWI 

measurements at the Rio Grande study sites were made under 

a tagline, from a boat at higher discharge and by wading at 

lower discharge.

At the CR87, CR225, RG-CAS, and RG-RGV study 

sites, the ADPs of different frequency are located adjacent to 

one another within the same array. At the CR30 study site, 

the 1- and 2-MHz ADPs are mounted on the same side of the 

river, 30 m apart; at the CR61 study site, because the banks

are unsuitable for attaching multiple ADPs to the same mount, 

the 1- and 2-MHz ADPs are mounted on opposite sides of the 

river; one ADP is located 60 m upstream of the other.

At the CR30 and RG-RGV study sites, the ADP arrays 

are located within the calibration cross section. At the 

RG-CAS study site, the calibration cross section is located 

~50 m downstream from the ADP array; at the CR225

study site, the calibration cross section is located ~130 m 

downstream from the ADP array; at the CR87 study site,

the calibration cross section is located ~200 m downstream 

from the ADP array; at the CR61 site, the calibration cross

section is located ~750 m downstream from the ADPs at the 

former location of a USGS cableway (Griffiths and others,

2012). This exceptionally large, and undesirable, longitudinal

distance between the ADP array and the calibration cross 

section at the CR61 site originated from the desire to continue 

making EWI measurements at the cross section where a large 

number of historical suspended-sediment measurements were 

made by the USGS. Unfortunately, no suitable mounting 

locations could be found on the riverbank for the ADPs 

near the former location of the USGS cableway because of 

poor bank material, spatially nonuniform flow in the river,

or proximity to a downstream tributary (the Little Colorado 

River) that episodically backwaters the former location of 

the cableway.
74 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Within the near-field region of a circular piston

transducer, the pressure field is extremely complicated. In the

near-field region in front of the transducer, there is a central

zone of higher pressure surrounded by multiple rings of 

slightly lower pressure; the number of these rings increases

with the ratio of aT/λ (figs. 7–8 in Lockwood and Willette,

1973), where aT is the transducer radius and λ is the acoustic 

wavelength. Along the axis of the transducer, the pressure 

amplitude is an oscillating function of range that reaches a 

maximum at the critical, that is, Fresnel, distance rC=aT

2

/λ

(Kino, 1987). Beyond this distance, the pressure decays 

monotonically into the far field as a function of range and the

spreading losses are spherical (Lockwood and Willette, 1973;

Medwin and Clay, 1997). 

On the basis of laboratory experiments in a suspension 

tank conducted using 5 circular piston transducers operating 

over the 1- to 5-MHz range in acoustic frequency, Downing 

and others (1995) developed an empirical correction to 

account for nonspherical spreading losses in the near-field

region. The form of this correction is:

ψ NF

D D

D D

= + ( ) + ( ) ( )

( ) + ( ) ( )

1 1 35 2 5

1 35 2 5

3 2

3 2

. .

. .

.

.

r r r r

r r r r

(3-1)

where

r is the axial distance, that is, range from the 

transducer in m, and

rD is a larger critical distance defined as rD= πrC.

This correction is largest at r = 0 and decreases to 1 at r≥rD. 

The larger critical distance of rD used by Downing and others 

(1995) is defined in the acoustical literature as the safe distance

beyond which there is no doubt that the pressure decays as 1/r

(Medwin and Clay, 1997). Because the Downing and others 

(1995) empirical correction applies at distances greater than rC

and less than rD, where the pressure likely decays as 1/r, there 

is a possibility that it may correct data that do not need to be 

corrected. 

The Downing and others (1995) empirical near-field

correction has gained acceptance within the literature of those 

using commercially available acoustic-Doppler current profilers

(ADCPs) and ADPs (for example, Wall and others, 2006; Wood

and Teasdale, 2013; Latosinski and others, 2014) even though

it was developed under very different conditions than those 

associated with acoustical measurements made by ADCPs and 

ADPs. Downing and others (1995) reported transmit-pulse 

durations of ~0.000014 s and measurement cells of ~0.01 m. 

Assuming a speed of sound of 1,480 m/s, the transmit-pulse 

duration of ~0.000014 s corresponds to a transmit-pulse length 

of ~0.02 m. In the Downing and others (1995) experiments, 

each time-averaged acoustical measurement consisted of 

somewhat more than 100 pings. In contrast, the transducers 

in the ADPs used in this study have transmit-pulse lengths 

that range from 0.22 to 0.55 m and nominal measurement 

cells that range in size from 0.22 to 0.55 m. Each timeaveraged acoustical measurement made by the ADPs in this 

study consists of an average of between 960 to 2,880 pings. 

Appendix 3. Effect of the Downing and Others (1995) Empirical Near-Field 

Correction

Figure 3-1. (following page) Examples of measurements of the relative backscatter, B, made with and without the Downing and others 

(1995) empirical near-field correction. Shown in each figure panel are the values of r

C

 and r

D for each ADP. The values of B shown 

are those averaged between the two horizontal beams. Because inclusion of the Downing and others (1995) correction affects the 

calculations of the sediment attenuation coefficient, and therefore affects the calculated values of B in each cell, use of the Downing 

and others (1995) correction may substantially affect the values of B in each cell when the correction of data within the region between 

r

C

 and r

n

 is particularly large. A, Measurements of B made with a Nortek 1-MHz EasyQ at the CR87 study site at 4:32 MST on December 

7, 2011; no measurement cells on this ADP fall within the region between r

C

 and r

D. B, Measurements of B made with a Nortek 2-MHz 

EasyQ at the CR87 study site at 4:35 MST on December 7, 2011; two measurement cells on this ADP fall within the region between r

C

 and 

r

D. C, Measurements of B made with a Nortek 1-MHz EasyQ at the CR87 study site at 2:35 MST on September 29, 2009; no measurement 

cells on this ADP fall within the region between r

C

 and r

D. D, Measurements of B made with a Nortek 2-MHz EasyQ at the CR87 study 

site at 4:35 MST on September 29, 2009; two measurement cells on this ADP fall within the region between rC and rD. E, Measurements 

of B made with the OTT 1-MHz SLD at the RG-RGV study site at 4:04 CST on July 16, 2011; one measurement cell on this ADP falls within 

the region between r

C

 and r

D. F, Measurements of B made with an OTT 2-MHz SLD at the RG-RGV study site at 4:04 CST on July 16, 

2011; one measurement cell on this ADP falls within the region between r

C

 and r

D. G, Measurements of B made with an OTT 1-MHz SLD 

at the RG-RGV study site at 16:04 CST on July 15, 2012; one measurement cell on this ADP falls within the region between rC and rD. 

H, Measurements of B made with a Nortek 2-MHz EasyQ at the RG-RGV study site at 16:04 CST on July 15, 2012; two measurement cells 

on this ADP fall within the region between rC and r

D. I, Measurements of B made with an OTT 1-MHz SLD at the RG-RGV study site at 

22:49 CST on July 14, 2012; one measurement cell on this ADP falls within the region between rC and rD. J, Measurements of B made with 

a Nortek 2-MHz EasyQ at the RG-RGV study site at 22:49 CST on July 14, 2012; two measurement cells on this ADP fall within the region 

between rC and r

D.
Appendixes 75

men16-3130_fig 03-1A-E

80

82

84

86

88

90

80

82

84

86

88

90

70

72

74

76

78

80

70

72

74

76

78

80

66

68

70

72

74

76

B, in decibels

r

C

r

C

rC

rC

r

C

 = 0.13 m

 = 0.83 m

 = 0.26 m

 = 0.13 m

 = 0.83 m

 = 0.26 m

 = 0.34 m

 = 1.08 m

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

r, in meters

A

B

C

D

E

 = 0.42 m

0 2 4 6 8 10 12 14 16 18 20 22 24

0 1 2 3 4 5 6 7 8 9 10 11 12

0 2 4 6 8 10 12 14 16 18 20 22 24

0 1 2 3 4 5 6 7 8 9 10 11 12

r

D

rD

rD

rD

rD

 = 0.42 m

Downing near-field correction applied

No near-field correction applied

r

C

rC

rC

r

C

r

C

rD

r

D

r

D

rD

r

D

B, in decibels

75

80

85

90

95

56

58

60

62

64

66

70

71

72

73

74

75

80

85

90

95

100

75

76

77

78

79

80

G

F

H

I

J

 = 1.08 m

 = 0.26 m

 = 0.83 m

 = 0.26 m

 = 0.83 m

 = 0.21 m

 = 0.67 m

 = 0.34 m

 = 1.08 m

 = 0.34 m

r, in meters

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 1 2 3 4 5 6 7 8 9 10 11 12

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3

Downing near-field correction applied

No near-field correction applied

Figure 3-1.
76 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Thus, the reverberating volumes sampled by the ADPs in this

study are over an order of magnitude larger than those sampled 

in the Downing and others (1995) experiments and the amount 

of averaging in the acoustical measurements made by the 

ADPs used in this study is more than two orders of magnitude 

greater than in the Downing and others (1995) experiments. 

Therefore, because of the large differences between the

reverberating volumes and the amount of averaging, it is not 

known whether the empirical near-field correction of Downing

and others (1995) should apply to the acoustical measurements 

made by ADCPs or ADPs. In addition, because none of the 

ADPs used in this study make acoustical measurements in 

cells located at distances of r<rC, use of the more conventional 

definition of the critical distance would thus indicate that no

near-field correction should need to be applied.

Comparison of data processed with and without the 

Downing and others (1995) near-field correction suggests

that the Downing and others correction is generally an 

overcorrection in the region between rC and rD, and that 

within region between rC and rD, the use of no near-field

correction (ѱNF=1) generally performs better (fig. 3-1). 

The examples shown in figure 3-1 were selected because 

they capture the typical behavior of ADP measurements at 

the various study sites with and without inclusion of the 

Downing and others (1995) correction. In all but one of the 

examples shown (that is, fig. 3-1F), use of the Downing 

and others (1995) correction degrades the measurements of 

relative backscatter, B, along the beam by overcorrecting 

the data in the cells that fall in the region between rC and rD. 

This apparent overcorrection sometimes results in values

of B in these cells that are much higher than the values of 

B in the cells located at slightly greater distances along the 

beam (see, for example, figs. 3-1B, D, E, H, and J), which is 

not physically realistic. The overcorrection has the greatest

effect at higher concentrations of silt and clay, when B can be 

measured in fewer cells (fig. 3-1J). In this case, inclusion of 

the Downing and others (1995) correction, by overcorrecting 

the data in the first cell, generally results in greater variance in

the measurements of B along the beam. This occurs because

inclusion of the Downing and others (1995) empirical 

correction in the method results in greater variance about 

the regressions used to calculate the sediment attenuation 

coefficient (method described in appendix 7). Because it 

is not clear that the Downing and others (1995) empirical 

near‐field correction is even appropriate for the types of ADPs

used in this study, and because inclusion of this correction 

either has minimal or apparently detrimental effects on the 

measurements made by these ADPs, the Downing and others 

(1995) correction is not used in this study. 
Appendixes 77

Appendix 4. Comparison of the Effects of Water Temperature, Pressure, and 

Salinity on the Water Absorption Coefficient

At the small depths and low-salinity conditions that 

generally occur in rivers, water temperature provides the 

only meaningful control on the value of the water absorption 

coefficient (αW). Figure 4-1 illustrates that the dominant 

control on the water absorption coefficient is change in water

temperature, with lesser and relatively unimportant regulation 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35 40

Water temperature, in degrees Celsius

αw, in decibels per meter

Ainslie and McColm (1998)

Schulkin and Marsh (1962)

Figure 4-1. The water absorption coefficient, αW, plotted as a function of water 

temperature in 1-degree increments over wide ranges of pressure and salinity; the 

lengths of the vertical lines at each 1-degree increment indicate the total range 

in αW at each 1-degree increment resulting from changes in water depth ranging 

from 0 to 20 m and changes in specific conductance (a measure of salinity) ranging 

from 100 to 4,000 mS/cm at 25 °C. The ranges in both water depth and specific 

conductance used in this plot are larger than those found in almost all rivers where 

an acoustic-Doppler profiler might be deployed.

of this coefficient by changes in water depth (pressure) and

salinity. At each value of water temperature, the bars plotted 

include the combined range of the effects of pressure and 

salinity. Shown in this figure are the values of αW calculated 

by the methods of Schulkin and Marsh (1962) and, for 

comparison, Ainslie and McColm (1998). 
78 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

At the CR87 study site, the grain-size distribution of 

the suspended sand under typical conditions is characterized 

by D50-XS-SAND-REF= 0.125 mm and σG= 0.63φ (on the basis of

1,770 EDI measurements made between August 21, 1999, and 

November 17, 2013). Because the sand is mostly composed 

of quartz, the density of the suspended sand is assumed to be 

equal to the density of quartz (2,650 kg/m3) at all study sites. 

Solution for the theoretical values of αUNIT at the 600-kHz, 

1-MHz, and 2-MHz acoustic frequencies that agree with the 

measured values of αUNIT within a few percent (fig. 6 in the 

main part of the report) indicates that the suspended silt and 

clay at the CR87 study site is best characterized as having a 

D50~0.0008 mm, σG~3φ, and ρS~2,500 kg/m3

. This lower-thanquartz density value for ρS is also justified on the basis of the

documented presence of lower-density clays in the Colorado 

River (see “Physical Basis for the Sediment Attenuation 

Coefficient” and “Estimation of the Grain-Size Distribution 

and Wet Density of the Silt and Clay: Attenuation Constraint” 

sections in the main part of report). Use of these parameters to 

characterize the typical grain-size distributions and densities 

of the suspended sand and the suspended silt and clay at 

this study site also results in the theoretical predictions of 

B ́ at the 600-kHz, 1-MHz, and 2-MHz frequencies plotted 

in figures 14A–C (in the main part of the report) that are in 

excellent agreement with the empirical values of B ́ calculated 

relative to the BBC relations plotted in figure 11 (in the main 

part of the report). 

At the RG-RGV study site, the grain-size distribution of 

the suspended sand under typical conditions is characterized 

by D50-XS-SAND-REF=  0.105 mm and σG=  0.65φ (on the basis of

35 EWI measurements made between December 11, 2010, and 

January 30, 2014). Solution for the theoretical values of αUNIT

at the 1-MHz and 2-MHz acoustic frequencies that agree with 

the measured values of αUNIT within a few percent indicates 

that the suspended silt and clay at the RG-RGV study site is 

best characterized as having a D50~0.002 mm, σG~2.7φ, and

ρS~3,200 kg/m3. This higher-than-quartz density value for

ρS suggests the dominance of higher-density chlorite-group 

clays in suspension in the Rio Grande. This prediction of

chlorite clay in suspension is reasonable given that chlorite 

has been described in the volcanic and intrusive igneous rocks 

that supply sediment to the Rio Grande in the study area 

(Udden, 1907; Stevens, 1969; West Texas Geological Society,

1972; Henry and others, 1989). Use of these parameters to

characterize the typical grain-size distributions and densities 

of the suspended sand and the suspended silt and clay at 

this study site result in the theoretical predictions of B ́ at 

2-MHz frequency that are in excellent agreement with the 

empirical values of B ́ plotted in figure 14D (in the main part 

of the report). 

Appendix 5. Details of Grain-Size Distributions Used to Develop Theoretical B ́ 

Relations at the CR87 and RG-RGV Study Sites
Appendixes 79

A key requirement for accurate acoustical measurements 

of CXS-SAND is that the BBC relation does not shift over 

time as a result of changes in the relation between CSAND

in the ensonified part of the cross section and CXS-SAND. 

Unfortunately, there are multiple processes in sand-bedded 

rivers that could cause the BBC relation to shift through 

changes in either the vertical or lateral distribution of 

suspended sand in the cross section. Examples of these 

processes are (1) bed scour and fill (Colby, 1964; Topping and

others, 2000a, b), (2) changes in the spatial distribution of the 

bed sand (Anima and others, 1998; Schmidt and others, 2007;

Buscombe and others, 2014), (3) changes in the grain-size 

distribution of the bed sand (Rubin and others, 1998, 

2010; Topping and others, 1999, 2000a, b), and (4) density

stratification (Gelfenbaum and Smith, 1986; McLean, 1991,

1992; Wright and Parker, 2004). Although these processes

may typically be correlated with water discharge, they may 

also lag discharge. For example, sand introduced to a river 

during a tributary flood will travel downstream as a sediment

wave, with a component in suspension, in the bedload, and 

in the bed (Topping and others, 2000b). As the front of this

sediment wave reaches a given cross section far downstream 

from the tributary that supplied the sand, the changes in the 

sediment conditions at this cross section will greatly lag 

the changes in water discharge associated with the tributary 

flood. These changes in sediment conditions can include

changes in bed elevation, the lateral distribution of bed sand, 

the grain-size distribution of bed sand, and perhaps also in 

density stratification (as a result of the changes in the bed‐sand

conditions). All of these changes may greatly influence the

relation between CSAND in the ensonified part of the cross

section and CXS-SAND, thus causing shifts in the BBC relations 

for the two ADPs. If at all possible, care should therefore be 

taken to not deploy ADPs at locations where these processes 

could have a large effect on the acoustical data, especially in 

cases where these processes could greatly lag changes in water 

discharge. However, in some cases, like in the Rio Grande 

at the RG-CAS and RG-RGV study sites, several of these 

processes likely occur over long reaches of the channel and are 

thus unavoidable. 

In this appendix, we provide examples of how to 

recognize and correct discharge-correlated shifts in BBC 

relations under conditions of relatively high CXS-SILT-CLAY—

conditions where it may be difficult to directly detect shifts in

BBC relations. In cases where large shifts in BBC relations 

greatly lag discharge (as in the example above), there may 

be no method to correct for these shifts. In these unfortunate 

situations, there may be no recourse but to either accept largerthan-normal errors in the acoustical measurements of CXS-SAND

or to move the ADPs to locations not affected by these shifts. 

The Rio Grande at both the RG-CAS and RG-RGV study

sites is a sand-bedded channel subject to large amounts of bed 

scour (up to several m) and large ranges in stage (up to 7 m) 

during floods. The ADPs at these study sites are deployed such

that the horizontal beams project through the flow ~1 m above

the bed at low discharge. As discharge increases, the bed 

scours at the ADP locations at both study sites. For example, 

as discharge increases from <30 m3/s to ~250 m3/s, channel 

cross sections measured during EWI measurements indicate 

that the flow depth above the ADP horizontal beams at the

RG-RGV study site increases by ~1.5 m while the bed below 

these beams scours by ~1 m. Thus, as discharge increases,

the vertical distance between the horizontal beams and the 

bed increases almost as much as the vertical distance between 

these beams and the water surface. If one assumes a simple 

Rouse-profile shape for a suspended-sand concentration profile

(for example, McLean, 1992), then, because of the substantial 

increase in distance between the horizontal beams and the 

bed, an identical suspended-sand concentration “sampled” by 

the fixed horizontal beams is likely to be related to a larger

value of CXS-SAND at higher discharge than at lower discharge. 

Therefore, large amounts of bed scour under an ADP deployed

close to the bed may result in large discharge-associated 

changes in the relation between CSAND sampled by the 

acoustic beams and CXS-SAND. As a result of the bed scour in 

this example, identical values of B and αS will therefore be 

associated with a lower value of CXS-SAND at lower discharges 

than at higher discharges. 

Another process that is likely to occur in a sand-bedded 

river and can result in similar behavior in the relation between 

CSAND in the ensonified part of the cross section and CXS‐SAND is 

density stratification. As discharge increases and the amount

of coarser sand in suspension increases, the increased vertical 

gradient in a suspended-sand concentration profile results in

the development of density stratification that will gradually

damp the turbulence (Gelfenbaum and Smith, 1986; McLean,

1991, 1992; Wright and Parker, 2004). The key result of

this density-stratification process pertinent to this study

is that, as discharge increases, the vertical gradient in the 

suspended‐sand concentration profile will increase such that,

without any bed scour required, an identical CSAND sampled 

by the fixed horizontal beams will be related to a larger value

of CXS-SAND at higher discharge than at lower discharge. As 

in the case of bed scour, this process could therefore also 

result in large discharge-correlated changes in the relation 

between CSAND in the ensonified part of the cross section and

CXS-SAND. As the effects of density stratification increase with

increasing discharge, identical values of B and αS will likely 

be associated with progressively higher values of CXS-SAND.

Appendix 6. A Method for the Detection and Correction of Discharge-Correlated 

Shifts in the BBC Relation at High Values of S
80 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

At least one of these processes (or one of the other 

processes listed at the beginning of this appendix) is likely to 

be present at an ADP-deployment site, but in many cases the 

effects of these processes will be minor and can be neglected. 

In essence, both bed-scouring and density-stratification

processes result in a downward shift in the BBC relation (that 

is, the y-intercept of the BBC relation becomes more negative) 

as water discharge decreases. Depending on how large the 

effects of these processes are at any given ADP-deployment 

site, this downward shift may be inconsequential, and 

therefore unimportant, or relatively large and important. 

Large amounts of bias will be introduced into acoustical 

measurements of CXS-SAND if a large discharge-correlated 

downward shift in the BBC relation is not recognized 

and corrected. 

In rivers where CXS-SILT-CLAY is always relatively low 

compared to CXS-SAND, discharge-correlated shifts in BBC 

relations can be detected directly by simply regressing BBC 

relations for different ranges in discharge. Unfortunately, in 

rivers with naturally high values of CXS-SILT-CLAY, this direct 

approach is usually impossible, owing to the fact that S >>2 

at higher discharges, and accurate BBC relations cannot be 

regressed under such conditions. Because of this complexity, 

we have developed the following indirect approach for 

detecting discharge-correlated BBC shifts that does not require 

relatively low CXS-SILT-CLAY. This approach utilizes F-tests to

detect relations between water discharge and log10(S) under 

conditions of constant B and αS. If no such significant relations

exist, no unique relation between B and CXS-SAND exists under 

conditions of constant αS, and the grain-size distribution of 

the suspended sand is relatively constant, then the lack of 

relation between B and CXS-SAND is likely the result of the 

backscatter masking process depicted in figure 16 in the 

main part of the report. However, if these F-tests indicate 

that there are significant relations between water discharge

and log10(S) under conditions of constant B and αS and at 

values of log10(S) less than those where backscatter masking is 

expected to occur (see, for example, fig.16 in the main part of 

the report), discharge-correlated BBC shifts are likely present 

(fig. 6-1). If further analysis indicates that these significant

relations arise because of positive correlations between water 

discharge and CXS-SAND at constant B and αS (fig. 6-2), and it 

is impossible to choose a different ADP-deployment location, 

corrections must be made to achieve acceptable accuracies 

in acoustical measurements of CXS‐SAND. The presence of

uncorrected discharge-correlated BBC shifts will result in 

increased upward scatter in measurements of log10(S) relative 

to theoretical relations between αS and log10(S), at values 

of αS less than those where backscatter masking (depicted 

in figures 16–17 in the main part of the report) becomes 

important. Thus, lack of recognition of these shifts will result

in underpredictions of log10(S) over lower and moderate ranges 

in αS and therefore result in severe underpredictions of B ́ at all 

but the highest values of αS.

Correction for the negative effect of discharge-correlated 

BBC shifts on acoustical measurements of CXS-SAND is usually 

possible through application of discharge-weighting factors 

to log10(S). In some cases, application of discharge-weighting 

factors may also allow reasonable estimation of CXS-SAND

under conditions of backscatter masking at high values 

of S. In the cases of the 1-MHz-ADP and 2-MHz-ADP at 

study site RG-RGV, shown in figures 6-1A–B and 6-2A–B, 

discharge-weighting the measured values of log10(S) reduces 

both the mean value and variance in log10(S) at constant B

and αS to be in better agreement with theoretical predictions 

of log10(S) in figure 6-3. In the 1-MHz-ADP example, the 

discharge-weighting factor that results in the best agreement 

between measurements and theory across all values of B is 

(Q/80), where Q is the water discharge (in m3/s) associated 

with each measurement of log10(S). In the 2-MHz-ADP 

example, the discharge-weighting factor that results in the best 

agreement is (Q/71). These discharge-weighting factors are

only applied when Q is less than the value in the denominator. 

Once discharge-weighting factors are determined, corrected 

values of log10(S) at any given value of B and αS can therefore 

be estimated by dividing the theoretically predicted values of 

log10(S) by these discharge-weighting factors. 
Appendixes 81

men16-3130_fig 06-1

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

300 340 380 420 460 500 540

Water discharge, in cubic meters per second

log10(S)

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

C

B

A

0 20 40 60 80 100 120

0 20 40 60 80 100 120 140

Figure 6-1. Examples of the presence and absence of significant 

relations between water discharge and log10(S) at constant B and 

αS. Regressions used in F-tests are depicted as solid black lines. A, 

Significant relation (p SIG= 0.031) between water discharge and log10(S) 

for the 92–94 dB range in B and the 4.0–4.5 dB/m range in αS for the 

1-MHz ADP at the RG-RGV study site. B, Significant relation (pSIG = 0.0018) 

between water discharge and log10(S) for the 81–83 dB range in B and 

the 33–34 dB/m range in αS for the 2-MHz ADP at the RG-RGV study site. 

C, Lack of a significant relation (pSIG = 0.10) between water discharge 

and log10(S) for the 86–88 dB range in B and the 1.6–1.8 dB/m range in αS

for the 1-MHz ADP at the CR87 study site. Gray shaded regions denote 

approximate ranges of log10(S) where backscatter masking is expected. 

Measurements of log10(S) are from the periods used to develop the 

base-backscatter-calibration relations. Further F-test analyses (fig. 6-2) 

indicate that the significant relations in A and B arise purely from 

significant relations between water discharge and CXS-SAND, and not from 

relations between water discharge and CXS-SILT-CLAY.
82 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

men16-3130_fig 06-2

0

1,000

2,000

3,000

4,000

5,000

300

0

5,000

10,000

15,000

20,000

0

1,000

2,000

3,000

4,000

5,000

6,000

340 380 420 460 500 540

Water discharge, in cubic meters per second

0

200

400

600

800

1,000

CXS-SAND, in milligrams per liter

CXS-SILT-CLAY , in milligrams per liter

0

100

200

300

400

500

600

700

800

120

0

50

100

150

140

C

B

A

0 20 40 60 80 100

0 20 40 60 80 100 120

CXS-SILT-CLAY

CXS-SAND

Figure 6-2. Examples of the presence and absence of 

significant correlations between water discharge and CXS-SAND at 

constant B and αS for the measurements depicted in figure 6-1. 

There is no significant correlation between water discharge 

and CXS-SILT-CLAY in any of these examples. A, Significant relation 

(pSIG =0.0027) between water discharge and CXS-SAND for the 

92–94 dB range in B and the 4.0–4.5 dB/m range in αS for the 

1-MHz ADP at the RG-RGV study site. B, Significant relation 

(pSIG= 9.7×10-9) between water discharge and CXS-SAND for the 

81–83 dB range in B and the 33–34 dB/m range in αS for the 

2-MHz ADP at the RG-RGV study site. C, Lack of a significant 

relation (pSIG = 0.46) between water discharge and CXS-SAND for 

the 86–88 dB range in B and the 1.6–1.8 dB/m range in αS for the 

1-MHz ADP at the CR87 study site.
Appendixes 83

men16-3130_fig 06-3

C

B

A

–1

0

1

2

3

4

log10(S) –1

0

1

2

3

4

–1

0

1

2

3

4

–1 0 1 2 3 4 5 6 7 8 9 10

αS, in decibels per meter

0 5 10 15 20 25 30 35 40 45 50 55 60

0 2 4 6 8 10

86–88 dB

81–83 dB

92–94 dB

Calibrated-pump

measurements

EWI measurements

Q -weighted

-weighted

measurements

92 dB relation

93 dB relation

94 dB relation

Calibrated-pump 

measurements 

EDI measurements

86 dB relation

87 dB relation

88 dB relation

Calibrated-pump

measurements

EWI measurements

Q

measurements

81 dB relation

82 dB relation

83 dB relation

Figure 6-3. Examples of the increased 

upward scatter in measured log10(S) relative 

to theoretical log10(S) that arises from 

discharge-correlated base-backscattercalibration (BBC) relation shifts affecting the 

examples in figures 6-1 and 6-2. Gray shaded 

regions denote the ranges in αS used in the 

examples in figures 6-1-and 6-2. Theoretical 

and measured relations between αS and 

log10(S) for the (A) 92–94 dB range in B for the 

1-MHz acoustic-Doppler profiler (ADP) at the 

RG-RGV study site; (B) 81–83 dB range in B

for the 2-MHz ADP at the RG-RGV study site, 

and (C) 86–88 dB range in B for the 1-MHz 

ADP at the CR87 study site (data in this plot 

are the same as in figure 17A in the main 

part of the report). Equal-width increment 

(EWI), equal-depth increment (EDI), and 

calibrated-pump measurements of log10(S) 

are indicated. Multiplying the measurements 

of log10(S) by the discharge-weighting factors 

in the text for the two RG-RGV examples in A

and B that exhibit discharge-correlated BBC 

shifts in figures 6-1A–B and 6-2A–B greatly 

improves the agreement between theoretical 

and measured relations between αS and 

log10(S). No discharge weighting is required 

in the CR87 example in C because dischargecorrelated BBC shifts were not detected in 

figures 6-1C and 6-2C.
84 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

In order, the 13 steps in the single-frequency calibration 

procedure are as follows:

(1) Determine which horizontal beams to use and the 

maximum number of cells to use. Factors that will 

limit the maximum cell number are: beams hitting the 

opposite bank, beams hitting or grazing an underwater 

bar, or beams grazing the water surface at low stage. 

Because suspended-sediment conditions are rarely 

uniform along the acoustic beams, cell number 

is directly related to error. Decreases in cell number 

result in increases in the variances in both αS and B, 

with the effect on B being greater than the effect on αS

(figs. 7-1–7-2) Because αS is the basis for acoustical 

measurements of CXS-SILT-CLAY and αS and B are used in 

combination for acoustical measurements of CXS-SAND, 

these increases in variance in αS and B will directly 

result in greater error in acoustical measurements of 

CXS-SILT-CLAY and CXS-SAND (fig. 7-3). In addition, once the 

maximum cell number is determined, this value cannot 

be reduced without changing the ADP calibration. 

Otherwise, substantial biases in αS and B , and therefore 

also in CXS-SILT-CLAY and CXS-SAND, may be introduced.

(2) If more than one beam is used, average the raw 

measurements of A between the corresponding cells 

in each beam (for example, see fig. 3A in main part of 

report). These average values of A among the beams are 

then converted to BF in each cell where A exceeds the 

effective noise floor (for example, fig. 3B in main part 

of report). Calculation of the effective noise floor is an

iterative process described in steps 3–5, below.

	(3)	 An initial effective noise floor is calculated as the sum of

the instrument noise floor and the noise-floor offset. The

instrument noise floor is the noise floor recorded at each

time step by the ADP. The noise-floor offset is iteratively

determined as the value added to the instrument noise 

floor to result in an effective noise floor that exceeds the

values of A in the distal part of the beam where A tends 

to asymptotically approach the instrument noise floor in

a curved manner (fig. 7-4). Because of this asymptotic 

behavior, selection of a noise-floor offset that is too low

will yield linear regressions in step 4 with slopes that are 

too low as a result of fitting a concave-up curve with a

line. As stated in step 5, selection of noise-floor offsets

that are too low will result in an incorrect dependence 

of αS and B on the number of cells where A exceeds the 

effective noise floor.

(4) A least-squares linear regression is used to regress the 

values of BF on r to solve for αS using the working 

effective noise floor determined in the previous step. αS

is equal to the slope of this regression divided by –2. 

(5) Calculate B in each cell using the values of BF and αS. 

Average the values of B first among equivalent cells

in each beam used and then among all cells along 

the beams where A exceeds the effective noise floor

to calculate the beam-averaged backscatter, B (for 

example, fig. 3C in the main part of report).

	(6)	 Time series of αS and B during conditions of varying, 

but high, CXS-SILT-CLAY are evaluated to ensure that large 

step changes in αS and B do not coincide with changes in 

the number of cells where A exceeds the effective noise 

floor. If such step changes are observed (fig. 7-5), the 

value of the noise-floor offset is increased and steps 4–5

are repeated. In our study, noise-floor offsets have been

found to range from 5 to 30 counts, depending on ADP 

and deployment location.

(7) Determine whether non-physical correlations exist 

between αS and non-sediment-related parameters, such 

as stage (figs. 7-6–7-7). If such correlations exist, the 

ADP deployment location must be moved to a region of 

more spatially uniform flow. Flow patterns in regions

of nonuniform flow, such as in lateral recirculation

eddies (Schmidt, 1990), may change substantially with 

changes in stage (Schmidt and Graf, 1990). These

stage‐dependent changes in flow pattern may greatly

alter the lateral distribution of suspended sediment 

along the acoustic beams, thus changing spatial patterns 

in A along the acoustic beams, ultimately affecting 

the calculated values of αS and B. Because accurate 

acoustical suspended-sediment measurements depend 

on αS varying mainly as a function of the characteristics 

of the suspended sediment, αS must either remain 

constant or increase slowly as stage increases. Inverse 

correlations between stage and αS are indicators of poor 

ADP deployment locations affected by stage-dependent 

changes in spatial flow patterns. A slight positive

correlation between stage and αS is acceptable, because 

suspended-sediment concentration is generally positively 

correlated with stage, although this correlation may be 

poor depending on the magnitude of temporal changes in 

the upstream sediment supply.

	(8)	 Time average αS and B and merge the time-averaged 

values of αS and B with the laboratory-processed EDI, 

EWI, and calibrated-pump measurements. Depending 

on the hydrologic flashiness of the river, time-averaging

windows ranging from 30 minutes to 1 hour centered 

on the temporal midpoint of each EDI, EWI, or 

calibrated-pump measurement give the best results.

Appendix 7. Single-Frequency ADP Calibration Procedure
Appendixes 85

(9) Regress relations between time-averaged αS and 

CXS‐SILT‐CLAY. Because calibrated-pump measurements 

of CXS-SILT-CLAY are reasonably accurate, especially at 

higher concentrations, and these measurements typically 

include higher-concentration values of CXS-SILT-CLAY than 

are typically captured during EDI or EWI measurements, 

all EDI, EWI, and calibrated-pump measurements are 

included in these regressions. Depending on ADP and 

site characteristics, one or two relations may need to 

be regressed to provide the most accurate calculation 

of CXS-SILT-CLAY over the entire range of the domain in 

αS. In many cases, where the suspended-silt-and-clay 

concentration is likely constant along the length of the 

acoustic beams, the theoretically expected linear relation 

between αS and CXS-SILT-CLAY (on the basis of equation 27) 

provides the best fit between αS and CXS-SILT-CLAY (for 

example, fig. 6 in main part of report) Although these 

empirically determined, best-fit linear relations between

αS and CXS-SILT-CLAY typically require a small positive 

or negative y-intercept, sometimes the best relation 

does pass through the origin, as expected on the basis 

of theory. In other cases, where either (a) the lateral 

distribution of suspended-silt-and-clay concentration 

may systematically change along the acoustic beams as a 

function of concentration, or (b) the clay-mineral content 

or grain-size characteristics of the suspended silt and 

clay may vary as a function of concentration, better fits

between αS and CXS-SILT-CLAY can be obtained by dividing 

the domain in αS into two parts, where a two-part 

relation between αS and CXS-SILT-CLAY is then regressed. 

In these cases, the relation between αS and CXS-SILT-CLAY

in the lower part of the domain is best described by 

either a line or second order (typically concave-down) 

polynomial, and the relation between αS and CXS-SILT-CLAY

in the upper part of the domain being best described by 

a line.

(10) Develop the BBC relation by using least-squares 

linear regression to solve for coefficients K1 and K2 in 

equation 60 (for example, see fig. 11 in main part of 

report). The values used for BBASE in this regression are 

the time-averaged values of B associated with CXS-SAND

that meet the following two criteria: (a) D50-XS-SAND is 

within 1/4φ of D50-XS-SAND-REF, and (b) suspended-sand 

measurements are associated with relatively small 

S values, preferably only those with values of S ≤2

where silt and clay contributes negligible amounts of 

backscatter (for example, see fig. 14 in main part of 

report). D50-XS-SAND-REF is usually set equal to the values 

of the nearest 1/4-φ increment. Although K1 largely 

depends on the SL associated with each ADP, both K1

and K2 will depend on the local site geometry and flow

characteristics (if the average CSAND and D50-SAND along 

the acoustic beams were directly proportional to CXS‐SAND

and D50-XS-SAND, then K2 would equal the theoretical value 

of 0.1). Empirical values of K2 may have a large range, 

but they typically fall within the range from 0.07 to 

0.15. Because EDI and EWI measurements provide 

a much more accurate measurement of CXS-SAND than 

do calibrated-pump measurements, only EDI or EWI 

measurements should be used in the development of the 

BBC relation if sufficient EDI or EWI measurements

exist at low S values. 

(11) Use equation 66 to calculate B ́. The grain-size

distribution of the sand used in this calculation is that 

used to develop the BBC relation in step 10. D50 and σG

of the suspended silt and clay are solved for iteratively 

to find the values of D50 and σG that, at all frequencies 

present at a study site, results in the best agreement 

between theoretical and empirical values of B ́ (for 

example, fig. 14 in main part of report) and in the best 

agreement between theoretical and empirical values of 

αUNIT (for example, fig. 6 in main part of report). 

(12) Conduct comparisons between theoretically predicted 

and measured values of log10(S) over the entire range of 

αS and B associated with EDI, EWI, and calibrated-pump 

measurements to evaluate the quality of the ADP 

calibration. If the calibration is good, the agreement 

between the theoretically predicted and measured values 

of log10(S) will be good, and should look like those in 

figure 17. In cases where there is only slight systematic 

disagreement between the theoretically predicted and 

measured values of log10(S), some improvement in the 

agreement is possible through minor adjustment of the 

relation between αS and CXS-SILT-CLAY.

 If physical processes are present in the river at the ADP 

deployment site that result in substantial shifts in the 

BBC relation (see appendix 6), then the agreement 

between the theoretically predicted and measured values 

of log10(S) will be poor, exhibiting increased upward 

scatter of the measured values of log10(S) relative to 

the theoretical values of log10(S) (see, for example, 

figs. 6-3A–B in appendix 6). In such cases, F-tests 

should be conducted to determine whether significant

relations exist between Q and log10(S) or between Q and 

CXS-SAND at constant B and αS (for example, fig. 6-1 in 

appendix 6). In some cases, where significant relations

are detected, discharge-weighting factors may be 

developed and applied to the theoretical predictions of 

log10(S) to mitigate the impact of discharge-correlated 

BBC shifts, and therefore result in an ADP calibration 

with acceptably low levels of error in the acoustical 

measurements of CSAND. As described in appendix 6, the 

discharge-weighting factor takes the following form:
86 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Q Q/ ( ) THRESH (7-1)

 where

 QTHRESH	 is the threshold discharge below which

 substantial systematic shifts in the BBC

 relation are detected on the basis of the

 methods described in appendix 6.

 When discharge-weighting factors are required, the 

theoretical predictions of log10(S) are adjusted by 

dividing by the discharge-weighting factor. The need

for discharge-weighting factors at a study site depends 

on the suspended-sediment conditions in the river and 

not on the properties of the ADPs. Therefore, when

discharge-weighting factors are required, they will 

be required for both ADPs at a study site. The value

of QTHRESH may vary between different-frequency 

ADPs at the same study site, however, because 

different-frequency ADPs typically ensonify different 

volumes of water. In this study, discharge-weighting 

factors were required only at the Rio Grande study sites 

and not at any of the Colorado River study sites.

(13) Although acoustical measurements outside the linear 

range between the logarithm of the gain setting and A

should be avoided, sometimes this is impossible. To

check for this potential problem, conduct comparisons 

between theoretically predicted and measured values 

of log10(S) at values of B>~90 dB. If measured values 

of log10(S) greatly exceed the theoretically predicted 

values of log10(S) in these comparisons, then application 

of a high-dB correction to the predicted values log10(S) 

may improve the single-frequency estimate of CXS-SAND. 

This correction is only applied under the extremely high

values of B where the systematic disagreement between 

theoretically predicted and measured values of log10(S) 

is positively correlated with B. The high-dB correction

is added to the theoretically predicted values of log10(S) 

after application of any required discharge-weighting 

factor and takes the following form: 

( ) − + K B3 2 THRESH K B (7-2)

 where

K3 is the constant that results in the best 

agreement between the theoretically 

predicted and measured values of log10(S) 

at values of B B > THRESH.

 BTHRESH is typically slightly greater than ~90 dB, but 

varies somewhat among the ADPs that require a high-dB 

correction. The need for a high-dB correction at a study

site depends on the properties of an individual ADP and 

not on the suspended-sediment conditions in the river. 

Because of the floating-scale bias in the RL among 

different ADPs, identical suspended-sediment conditions 

may be associated with a measured B much less than 

90 dB on one ADP, but associated with a measured 

B much greater than 90 dB on a different ADP. Thus,

on the first ADP in this example, the value of B falls 

within the linear range between the logarithm of the gain 

setting and A, whereas on the second ADP, the value of 

B falls within the nonlinear range between the logarithm 

of the gain setting. In this study, high-dB corrections 

were only required for the 2008 redesigned version 

of the OTT SLD ADPs that were deployed at the Rio

Grande study sites. For the same suspended-sediment 

conditions, the values of B measured by the post-2008 

OTT SLDs were up to 30 dB larger than those measured

by identical-frequency Nortek EasyQ ADPs deployed at 

the same locations. 
Appendixes 87

Figure 7-1. Examples of how limiting the number of cells in the acoustic beam affects αS, the sediment attenuation coefficient.Cell 

size is not changed between examples; a lower maximum number of cells corresponds to measurements along a shorter length of the 

acoustic beam. The red solid line is the line of perfect agreement between αS calculated using different numbers of cells. As the number 

of cells decreases, αS becomes negatively biased, and variance in αS increases, especially at lower values of αS. The increase in 

variance with decreasing cell number is a general response; although the increase in bias with decreasing cell number is also a general 

response, the sign of this bias will vary between different ADPs and deployment locations because the curvature in the measurements 

along the beam will vary between different ADPs and locations. Data used in these examples are from 277,327 15-minute measurements 

made using the 1-MHz ADP at the CR87 study site between August 22, 2005, and August 1, 2013. All examples are plotted relative to 

αS calculated using the maximum number of cells above the effective noise floor (as many as 64 cells when suspended-silt-and-clay

concentrations are relatively low). A, αS calculated using 50 cells; B, αS calculated using 40 cells; C, αS calculated using 30 cells; D, αS

calculated using 20 cells; E, αS calculated using 10 cells; F, αS calculated using 5 cells; and G, αS calculated using 3 cells.
88 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Figure 7-1.—Continued

D1200.png
Appendixes 89

Figure 7-2. Examples of the effect on how limiting the number cells in the acoustic beam affects B, the beam-averaged backscatter. 

Cell size is not changed between examples; a lower maximum number of cells corresponds to measurements along a shorter length 

of the acoustic beam. The red solid line is the line of perfect agreement between B calculated using different numbers of cells. As 

the number of cells decreases, B becomes negatively biased and variance in B increases; comparison of this figure with figure 7-1

indicates that imposing cell limits affects B much more than αS, the sediment attenuation coefficient. As in the examples in figure 7-1, 

the increase in variance with decreasing cell number is a general response; although the increase in bias with decreasing cell number 

is also a general response, the sign of this bias will vary between different ADPs and deployment locations because the curvature in 

the measurements along the beam will vary between different ADPs and locations. Data used in these examples are from the same 

acoustical measurements used in figure 7-1. All examples are plotted relative to B calculated using the maximum number of cells above 

the effective noise floor (as many as 64 cells when suspended-silt-and-clay concentrations are relatively low). A, B calculated using 

50 cells; B, B calculated using 40 cells; C, B calculated using 30 cells; D, B calculated using 20 cells; E, B calculated using 10 cells; F, B

calculated using 5 cells; and G, B calculated using 3 cells.
90 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

Figure 7-2.—Continued
Appendixes 91

men16-3130_fig 07-3

0

1

2

3

4

0 10 20 30 40 50 60 70

Imposed limit on maximum number of cells

Variability introduced by imposing

limit on maximum number of cells

Bias introduced by imposing limit

on the maximum number of cells

–4

–3

–2

–1

0

1

0 10 20 30 40 50 60 70

B

A

1.96σ in α , in dB/m

1.96σ in B, in dB

s

Bias in α , in dB/m

Bias inB, in dB

s

Figure 7-3. Summary of the effects of imposing lower maximum limits 

on the number of cells on the mean bias and variability in αS andB

limits on the number of cells on the (A) bias and (B) variability in αS (the 

sediment attenuation coefficient) and B (the beam-averaged backscatter), 

depicted in figures 7-1 and 7-2. The biases depicted in A are the averages 

of the absolute errors in the values of αS and B calculated using the 

imposed lower limits on the numbers of cells specified for each panel in 

figures 7-1 and 7-2. The variability shown in B is set equal to 1.96 times 

the standard deviation among these absolute errors so that it brackets 

roughly 95 percent of all observations in a Gaussian-normal distribution. 

The effect of limiting the maximum number of cells has a much greater 

effect on B than on αS. Because B is linearly related to log10(CXS-SAND), 

the rapid increase in the negative bias in B and in the variation in B

indicates that imposing lower limits on the maximum number of cells will 

greatly increase both the negative bias and random error in acoustical 

measurements of CXS-SAND. Although the increase in the magnitude of the 

bias with decreasing cell number is a general response, the sign of the bias 

will vary between different ADPs and deployment locations because the 

curvature in the measurements along the beam will vary between different 

ADPs and locations. The increase in the variability of the absolute error 

with decreasing cell number is a general response and does not depend 

on the curvature in the measurements along the beam. This increase in 

the variability of the absolute error in αS and B will result in an increase 

in the 68-percent- and 95-percent-confidence level errors in acoustical 

suspended-sediment measurements made with fewer numbers of cells.

0

50

100

150

200

0.0 0.5 1.0 1.5 2.0 2.5

Instrument noise floor

Noise-foor offset = +5

Distance from transducer along horizontal beams (r), in meters

Acoustic signal strength (

A), in counts

Chosen effective noise floor

(Noise-floor offset = +22)

Measurement at 5:04

Measurement at 7:49

Measurement at 16:49

Figure 7-4. Example showing noise-floor offsets required to 

exclude acoustical measurements in the “curved part” of the 

acoustic profile immediately above the instrument noise floor. 

The +5 count noise-floor offset is too small to fully exclude the 

measurements in the curved part of the profile, whereas the 

+22 count noise-floor offset (chosen graphically) is large enough 

to exclude the measurements in this region. These measurements 

were made using the 2-MHz acoustic-Doppler profiler at 

the RG-RGV study site on July 28, 2012; times are in Central 

Standard Time.
92 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

MEN16-3130_fig 07-5

65

70

75

80

85

90

0

5

10

15

20

25

30

0:00 4:00 8:00 12:00 16:00 20:00 24:00

B

0

10

20

30

40

50

60

0

5

10

15

20

25

30

0:00 4:00 8:00 12:00 16:00 20:00 24:00

A

0

10

20

30

40

50

60

0

5,000

10,000

15,000

20,000

0:00 4:00 8:00 12:00 16:00 20:00 24:00

C

αS , in decibels per meter αS , in decibels per meter

B, in decibels

B, in decibels

Number of cells (n) above effective noise floorNumber of cells (n) above effective noise floor

Time (CST) on July 28, 2012 Time (CST) on July 28, 2012

Time (CST) on July 28, 2012 Time (CST) on July 28, 2012

C XS-SILT-CLAY, in milligrams per liter

C XS-SAND, in milligrams per liter

65

70

75

80

85

90

1

10

100

1,000

0:00 4:00 8:00 12:00 16:00 20:00 24:00

D

n with zero noise-floor offset

n with +5 count noise-floor offset

n with +22 count noise-floor offset

Cross-section-calibrated 

pump measurement 

Cross-section-calibrated 

pump measurement 

n

n

n

 with zero noise-floor offset

 with +5 count noise-floor offset

 with +22 count noise-floor offset

 with zero noise-floor offset

 with +5 count noise-floor offset

 with +22 count noise-floor offset

αs

αs

αs

 with zero noise-floor offset

 with +5 count noise-floor offset

 with +22 count noise-floor offset

αs

αs

αs

with zero noise-floor offset

 with +5 count noise-floor offset

 with +22 count noise-floor offset

B

B

B

with zero noise-floor offset

 with +5 count noise-floor offset

 with +22 count noise-floor offset

B

B

B

Figure 7-5. Effect of the different noise-floor offsets (depicted in figure 7-4) on calculations of αS (the sediment attenuation coefficient) and 

B (the beam-averaged backscatter) for the 2-MHz acoustic-Doppler profiler at the RG-RGV study site on July 28, 2012. A noise-floor offset 

of +22 counts above the recorded instrument noise floor is required to remove the step changes in αS and greatly reduce the step changes 

in B associated with the changing numbers of cells above the effective noise floor. A, Calculated αS using 0, +5, and +22 count noise-floor 

offsets, shown with the number of cells in the acoustic beam that are above the effective noise floor. B, Calculated B using 0, +5, and 

+22 count noise-floor offsets, shown with the number of cells in the acoustic beam that are above the effective noise floor. C, Cross-sectioncalibrated pump measurements of suspended-silt-and-clay concentration (with 95-percent confidence-level error bars) superimposed on 

the values of αS in A. D, Cross-section calibrated-pump measurements of suspended-sand concentration (with 95-percent confidence-level 

error bars) superimposed on the values of B in B. The superimposed suspended-sediment measurements in C and D agree most favorably 

with the smoother variation in αS and B produced by using the higher noise-floor offset of +22 counts. Use of lower noise-floor offsets not 

only results in large cell-number-dependent step changes in αS andB , it also results in periods during which changes in αS and B have the 

incorrect sign.
Appendixes 93

men16-3130_fig 07-6

–0.2

–0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

0 5 10 15 20 25 30

Date in December 2004

Stage, in meters

α , in decibels per meter

S

Distance (r) from transducer along 

 horizontal beam 2, in meters

Acoustic signal strength (

A), in counts

–0.2

–0.1

0.0

0.1

0.2

0.3

0.4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

8 9 10 11 12 13 14

α , in decibels per meter S

Stage, in meters

A

C

B

0:14, 12-10-2004

12:14, 12-9-2004

Bad deployment location

Good deployment location

Stage

Measurement made at bad

location at 12:14 on 12-9-2004

Measurement made at bad 

location at 0:14 on 12-10-2004 

Measurement made at good

location at 12:14 on 12-9-2004

Measurement made at good

location at 0:14 on 12-10-2004

Bad deployment location

Good deployment location

Figure 7-6. 1-MHz acoustic-Doppler 

profilers (ADP) measurements made at a 

bad deployment location in a large lateralrecirculation eddy compared to 1-MHz ADP 

measurements made at a good deployment 

location in uniform flow above a rapid at 

the CR30 study site. The bad deployment 

location corresponds to the location of the 

“primary site” in Griffiths and others (2012; 

shown in their fig. 3A); the good deployment 

location corresponds to the location of the 

“1-MHz ADP” in Griffiths and others (2012; 

their fig. 3A). Stage correlated changes in 

the calculated values of αS that arise from 

likely stage-associated changes in the 

lateral distribution of suspended-sediment 

along the acoustic beam dominate the 

data at the bad deployment location. A, αS

plotted as a function of stage at the bad 

and good ADP deployment locations. αS

calculated using all 15-minute acoustical 

measurements made using beam 2 on each 

ADP from December 8 through 14, 2004. B, 

Time series of stage superimposed on the 

calculated values of αS from both ADPs. 

Timestamps in Mountain Standard Time of 

the example raw measurements in C are 

indicated. C, Raw acoustical measurements 

from beam 2 on the two ADPs at the 

indicated timestamps in B showing the 

source of the negative correlation between 

stage and αS at the bad ADP deployment 

location in the lateral recirculation eddy.
94 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

men16-3130_fig 07-7

Acoustic signal strength (

A), in counts

–0.2

–0.1

0.0

0.1

0.2

0.3

–0.2

–0.1

0.0

0.1

0.2

0.3

0

50

100

150

200

250

Stage, in meters

Distance (r) from transducer along 

 horizontal beam 2, in meters

Stage, in meters

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4:00 8:00 12:00 16:00 20:00 0:00 4:00 8:00

0 5 10 15 20 25 30

Time (MST) on January 19–20, 2005

A

B

C

13:36

17:06

αs, in decibels per meter αs, in decibels per meter

Bad deployment location

Good deployment location

Stage

Bad deployment location

Good deployment location

Measurement made at bad

deployment location at 13:36 

Measurement made at bad

deployment location at 17:06 

Measurement made at good

deployment location at 13:36

Measurement made at good

deployment location at 17:06

Figure 7-7. Second comparison between 1-MHz acousticDoppler profiler (ADP) measurements made at the bad and good 

deployment locations at the CR30 study site. A, αS plotted as a 

function of stage at the bad and good deployment locations. αS

calculated using all 15-minute acoustical measurements made 

using beam 2 on each ADP from 6:00 MST (Mountain Standard 

Time) on January 19, 2005, through 6:00 MST on January 20, 

2005. B, Time series of stage superimposed on the calculated 

values of αS from both ADPs. Timestamps of the example 

raw measurements in C are indicated. C, Raw acoustical 

measurements from beam 2 on the two ADPs at the indicated 

timestamps in B showing the source of the negative correlation 

between stage and αS at the bad deployment location in the lateral 

recirculation eddy.
Appendixes 95

The RUTS-based calculation of CXS-SAND and D50-XS-SAND

proceeds as follows for two paired acoustic frequencies within 

the 600-kHz to 2-MHz range:

	(1) 	 The theoretical RUTS for each frequency is calculated 

using the D50-XS-SAND-REF and σG of the suspended sand 

typical at the study site. 

(2) Because backscatter at higher frequencies is less affected 

by changes in D50-XS-SAND, the estimate of CXS-SAND at 

the higher frequency is chosen as the initial estimate of 

CXS‐SAND in this calculation.

(3) In cases where the lower-frequency estimate of CXS‐SAND

is greater than the higher-frequency estimate, the initial 

estimate of CXS-SAND in step 2 is reduced using the 

theoretical RUTS relations and empirical BBC relations 

for the two frequencies. In these cases, D50-XS-SAND is 

coarser than D50-XS-SAND-REF, and the RUTS relations for 

the two frequencies are used to calculate this larger D50. 

Conversely, in cases where the lower-frequency estimate 

of CXS-SAND is less than the higher-frequency estimate, the 

initial estimate of CXS-SAND in step 2 is increased using 

the RUTS and BBC relations for the two frequencies. 

In these cases, D50-XS-SAND is less than D50-XS-SAND-REF, and 

the RUTS relations for the two frequencies are used to 

calculate this smaller D50. If the estimates of CXS-SAND

are identical at both frequencies, the initial estimate of 

CXS-SAND in step 2 is not modified, and D50-XS-SAND equals 

D50-XS-SAND-REF. 

Step 3 proceeds mathematically as follows. First, using 

data from the lower-frequency ADP, calculate the effective 

B for CXS-SAND estimated by the lower-frequency ADP, and 

correct for effects of B ́ and for any discharge-correlated BBC 

shifts. This first effective lower‐frequency beam-averaged

backscatter, B1-LOWf, is found by: 

 B C 1-LOWf = ( ) ( ) XS-SAND-LOWf − K K 1-LOWf 2-LOWf log10 (8-1)

where

 CXS-SAND-LOWf is the initial CXS-SAND estimated by the

 procedure described in appendix 7 for

 the lower-frequency ADP, 

 K1-LOWf is K1 from the BBC relation for the

 lower-frequency ADP, and 

 K2-LOWf is K2 from the BBC relation for the

 lower-frequency ADP. 

Second, calculate the effective B on the lower-frequency ADP 

that would exist for the value of CXS-SAND estimated by the 

higher-frequency ADP, and correct for effects of B ́ and any 

discharge-correlated BBC shifts (see appendix 7). This second

effective lower-frequency beam averaged backscatter, B2-LOWf, 

is found by: 

 B C 2-LOWf = ( ) ( ) XS-SAND-HIGHf − K K 1-LOWf 2-LOWf log10 (8-2)

where

 CXS-SAND-HIGHf is the suspended-sand concentration 

 estimated by the procedure described in 

 appendix 7 for the higher-frequency

 ADP, and the values of 

 K1-LOWf and 

 K2-LOWf are the same as used in equation 8-1. 

Step 3 proceeds mathematically as follows. The difference

between these two effective values of B is then referred to as 

the lower-frequency backscatter defect, where

B B LOW-DEFECT = − 1−LOWf B2-LOWf . (8-3)

Once BLOW-DEFECT has been determined, D50-XS-SAND is set 

equal to the value of D50 along the lower-frequency theoretical 

RUTS relation at RUTS = BLOW-DEFECT. The higher-frequency

backscatter defect, BHIGH-DEFECT, is then set equal to the value 

of RUTS along the higher-frequency theoretical RUTS relation 

at D50-XS-SAND. The logarithm of CXS-SAND, using both acoustic 

frequencies, is then calculated as:

 log10 ( ) C K XS-SAND - = + 1 2 HIGHf K B -HIGHf ( ) HIGHf − BHIGH-DEFECT (8-4)

where 

K1-HIGHf is K1 from the BBC relation for the 

higher-frequency ADP, and 

K2-HIGHf is K2 from the BBC relation for the 

higher-frequency ADP. 

BHIGHf in equation 8-4 is the effective B using the higherfrequency ADP that would exist for CXS-SAND estimated by the 

higher-frequency ADP, corrected for effects of B ́ and for any 

discharge-correlated BBC shifts, and is calculated by:

 B C HIGHf = ( ) ( ) XS-SAND-HIGHf − K K -HIGHf -HIGHf log10 1 2 . (8-5)

Appendix 8. Multi-Frequency RUTS-Based Calculation of CXS-SAND and 

D50-XS-SAND
96 Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Theory, Application, Bias, and Error

To prevent calculating overly high values of CXS‐SAND

by, in effect, counting silt-and-clay-sized sediment as 

sand, BLOW‐DEFECT is limited such that the predicted value of 

D50‐XS‐SAND along the lower-frequency RUTS relation can never 

be less than 0.074 mm (that is, the size class of sand 1/4φ

greater than the 0.0625-mm silt-sand break). Not limiting 

D50‐XS-SAND at 0.074 mm would result in double counting 

sediment near the silt-sand break on the basis of both acoustic 

attenuation and acoustic backscatter. 

In cases where more than two frequencies of ADPs are 

deployed at a study site (for example, the CR87 study site 

where 600-kHz, 1-MHz, and 2-MHz ADPs are all present), 

the above method is modified slightly. At the CR87 study site,

steps 1–3 are solved twice, once using the 2-MHz ADP for the 

higher-frequency measurements and the 1-MHz ADP for the 

lower-frequency measurements, and once using the 2-MHz 

ADP for the higher-frequency measurements and the 600-kHz 

ADP for the lower-frequency measurements. Three-frequency

values of CXS-SAND and D50-XS-SAND are then calculated as the 

weighted average of the values of CXS-SAND and D50-XS-SAND

calculated using each of the two paired frequencies. 
Appendixes 97

Appendix 9. Plots of Predicted Versus Observed Values of CXS-SILT-CLAY, CXS-SAND, 

and D50-XS-SAND, with Data Segregated by Study Site 

Figure 9-1. Predicted versus observed plots for the equal-discharge 

increment (EDI) or equal-width increment (EWI) and acoustical

measurements of CXS-SILT-CLAY, CXS-SAND, and D50-XS-SAND, with data 

segregated by study site. In each figure panel, the solid black line 

is the line of perfect agreement; n is the number of observations; 

horizontal error bars indicate the 95-percent-confidence-level 

combined field and laboratory-processing error in the EDI or EWI 

measurement. A, Plot of predicted versus observed values of 

CXS-SILT-CLAY. Solid red line is the best-fit log-linear regression fit to the 

data from all study sites in A–C. B, plot of predicted versus observed 

values of CXS-SAND. C, plot of predicted versus observed values of D50-XSSAND. Dashed red line is the best-fit log-linear regression fit to the data 

from only the CR30 study site (the study site with the largest range in 

EDI- or EWI-measured D50-XS-SAND).

A

B

C

10

100

1,000

10,000

100,000

10

100

1,000

10,000

100,000

0.08

0.10

0.20

0.30

0.40

0.50

EDI or EWI measurement of D50-XS-SAND,

in millimeters

Acoustical measurement of 

D50-XS-SAND,

in millimeters

EDI or EWI measurement of CXS-SAND,

in milligrams per liter

Acoustical measurement of 

CXS-SAND,

in milligrams per liter

EDI or EWI measurement of CXS-SILT-CLAY,

in milligrams per liter

Acoustical measurement of 

CXS-SILT-CLAY,

in milligrams per liter

 

1 10 100 1,000 10,000 100,000

1 10 100 1,000 10,000 100,000

0.08 0.10 0.20 0.30 0.40 0.50

1

1

CR30 (n = 172)

CR61 (n = 175)

CR87 (n = 222)

CR225 (n = 99)

RG-CAS (n = 25)

RG-RGV (n= 7)

CR30 (n = 173)

CR61 (n = 175)

CR87 (n = 222)

CR225 (n = 99)

RG-CAS (n = 25)

RG-RGV (n= 7)

CR30 (n = 173)

CR61 (n = 175)

CR87 (n = 222)

CR225 (n = 99)

RG-CAS (n = 17)

RG-RGV (n= 6)

Publishing support provided by the U.S. Geological Survey Science 

Publishing Network, Menlo Park, California and Tacoma, 

Washington Publishing Service Centers

Manuscript approved for publication March 10, 2016

Edited by Claire M. Landowski

Design and layout by Sharon L. Wahlstrom 
ISSN 1044-9612 (print)

ISSN 2330-7102 (online)

http://dx.doi.org/10.3133/pp1823

Topping and Wright—Long-Term Continuous Acoustical Suspended-Sediment Measurements in Rivers—Professional Paper 1823