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Computation of Discharge Using the IndexVelocity Method in Tidally Affected Areas

By Catherine A. Ruhl and Michael R. Simpson

In cooperation with the Interagency Ecological Program

Scientific Investigations Report 2005-5004

U.S. Department of the Interior

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

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2005

For sale by U.S. Geological Survey, Information Services

Box 25286, Denver Federal Center

Denver, CO 80225

For more information about the USGS and its products:

Telephone: 1-888-ASK-USGS

World Wide Web: http://www.usgs.gov/

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Government.

Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation:

Ruhl, C.A., and Simpson, M.R., 2005, Computation of discharge using the index-velocity method in tidally affected 

areas: U.S. Geological Survey Scientific Investigations Report 2005-5004, 31 p.
v

Contents

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

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

Purpose and Scope ................................................................................. 1

Acknowledgements ................................................................................. 1

Principles of Operation ................................................................................. 1

Ultrasonic Velocity Meters ........................................................................ 1

Acoustic Doppler Velocity Meters ............................................................... 3

Point Velocity ....................................................................................... 3

Single Bin .......................................................................................... 3

Profiler ................................................................................................ 3

Methods ................................................................................................... 3

Calculating Area on the Basis of Stage ......................................................... 3

Calculating Mean Velocity on the Basis of the Index Velocity ................................. 5

Calculating Discharge .............................................................................. 5

Calculating Daily Flow .............................................................................. 5

Importance of High-Quality Data .................................................................. 7

Summary ................................................................................................... 7

References ................................................................................................ 8

Appendix A. Sample Computation of the Cross-Sectional Area Relation ........................ 9

Introduction .......................................................................................... 9

Step 1. Channel-Bank Survey ..................................................................... 9

Horizontal Location Correction ............................................................ 10

Vertical Elevation Correction ............................................................... 10

Channel-Bank Survey Correction Summary ............................................. 11

Step 2. Bathymetry Survey ........................................................................ 11

Horizontal Location Correction ............................................................ 12

Vertical Elevation Correction ............................................................... 12

Bathymetry Survey Correction Summary ................................................ 13

Step 3. Synthesis of Field Data .................................................................. 13

Step 4. Calculating the Stage-Versus-Area Relation .......................................... 14

Appendix B. Sample Computation of the Mean Velocity .......................................... 15

Introduction .......................................................................................... 15

Step 1. Collection of Discharge Measurements ................................................ 15

Step 2. Synthesis of Field and Gage Data to Compute the Mean Velocity .................. 16

Step 3. Computation of the Relation between the Mean Velocity and the

Index Velocity .............................................................................. 16
vi

Appendix C. Sample Discharge Calculation ......................................................... 18

Introduction .......................................................................................... 18

Step 1. Assemble Stage and Index-Velocity Data Collected at the 

Gaging Station .............................................................................. 18

Step 2. Calculate Discharge ..................................................................... 18

Linear or Polynomial Rating .................................................................. 18

Loop Rating .................................................................................... 20

Bimodal Rating .............................................................................. 22

Appendix D. Calculation of Daily Discharge ......................................................... 23

Introduction .......................................................................................... 23

Step 1. Apply a Low-Pass Butterworth Filter ................................................... 23

Step 2. Calculate the Daily Average of the Filtered Values .................................... 23

Appendix E. High-Quality Data and Examples of Common Mistakes .............................. 24

Introduction .......................................................................................... 24

Data Synchronization .............................................................................. 24

Channel-Bottom Movement ........................................................................ 26

Index-Velocity Transducer Alignment ............................................................ 27

Discharge Measurement Configuration Files and General Boat Operation .................. 28
vii

Figures

Figure 1. Schematic of standard acoustic stream-gaging stations .............................. 2

Figure 2. Schematic of bathymetry and channel-bank survey field method. .................. 4

Figure 3. Examples of index velocity versus mean velocity ratings .............................. 6

Figure 4. Example of discharge record ............................................................... 7

Figure A1. Measured channel cross section ......................................................... 10

Figure A2. Screen capture of channel cross section from AreaComp.exe ........................ 15

Figure B1. Relation between index velocity and mean velocity .................................... 18

Figure C1. Discharge calculated as a result of the stage versus cross-sectional area

 relation and the index velocity versus mean velocity relation ........................ 20

Figure C2. Sample loop rating ........................................................................... 21

Figure C3. Implementation of loop rating ............................................................ 22

Figure C4. Sample bi-modal rating ..................................................................... 23

Figure D1. Example of discharge record ............................................................... 24

Figure E1. Comparison of index-velocity versus mean-velocity relations when 

 synchronization problems exist ............................................................ 26

Figure E2. Impacts of using bottom-track in moving-bed conditions on the resulting index 

 velocity versus mean velocity relation ................................................... 27

Figure E3. Impacts of misaligned instrumentation on the resulting index velocity versus 

 mean-velocity relation ..................................................................... 28

Figure E4. Impacts of improper boat speed on the resulting index-velocity versus 

 mean-velocity relation ..................................................................... 29

Figure E5. Impacts of poor vessel-mounted downward-looking acoustic Doppler velocity 

 meter configuration and poor boat operation techniques on the resulting index 

 velocity versus mean velocity relation ................................................... 30

Figure E6. Comparison of the difference between subsequent velocity measurements as 

 recorded by the index velocity instrumentation and the vessel-mounted 

 downward-looking acoustic Doppler velocity meter .................................... 31
viii

Tables

Table A1. Summary of field notes collected during channel-bank survey ........................ 10

Table A2. Summary of stage data recorded at gaging station .................................... 10

Table A3. Summary of field data and horizontal location corrections ........................... 10

Table A4. Summary of field data and location and elevation corrections ........................ 11

Table A5. Summary of corrected channel-bank survey data ....................................... 11

Table A6. Summary of field notes associated with bathymetry survey ........................... 11

Table A7. Summary of stage data recorded at gaging station .................................... 12

Table A8. Summary of field data and location corrections ....................................... 12

Table A9. Summary of field data and elevation corrections ....................................... 12

Table A10. Summary of corrected bathymetry survey data .......................................... 13

Table A11. Summary of synthesized field data ...................................................... 13

Table B1. Summary of selected downward-looking vessel-mounted acoustic

 Doppler velocity meter discharge measurements ....................................... 15

Table B2. Summary of transect data and gaging station data .................................... 16

Table C1. Tabular summary of recorded gaging station data and calculations .................. 18

Table E1. Comparison of differences between calculated flow data with a

 5-minute synchronization offset ............................................................ 24
ix

Conversion Factors, and Abbreviations

CONVERSION FACTORS 

Multiply By To obtain

foot (ft) 0.3048 meter

foot per second (ft/s) 0.3048 meter per second

square foot (ft2) 929.0 square centimeter

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

inch (in.) 2.54 centimeter

inch per hour (in/hr) 2.54 centimeter per hour

ABBREVIATIONS

ADVM acoustic Doppler velocity meter

Bay San Francisco Bay

bins range gated sample volume

Delta Sacramento–San Joaquin River Delta 

GPS geographic positioning system

< less than

PST Pacific Standard Time

UVM ultrasonic velocity meter

 mean velocity

Vi index velocity

 Vp water velocity along the acoustic path

ORGANIZATIONS

USGS U.S. Geological Survey
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Abstract

Computation of a discharge time-series in a tidally 

affected area is a two-step process. First, the cross-sectional 

area is computed on the basis of measured water levels and the 

mean cross-sectional velocity is computed on the basis of the 

measured index velocity. Then discharge is calculated as the 

product of the area and mean velocity. Daily mean discharge 

is computed as the daily average of the low-pass filtered 

discharge. The Sacramento-San Joaquin River Delta and San 

Francisco Bay, California, is an area that is strongly influenced 

by the tides, and therefore is used as an example of how this 

methodology is used.

Introduction

The U.S. Geological Survey (USGS) has operated a 

network of flow-monitoring stations in the Sacramento−San 

Joaquin River Delta (Delta) since 1987. Additionally, equipment may be deployed to intensively study specific areas for 

short (3-to 9-month) periods. Tides from the Pacific Ocean 

enter the San Francisco Bay (Bay) and Delta system through 

the Golden Gate and cause twice-daily variations in stage and 

velocity throughout the region. Because water level cannot be 

uniquely related to discharge in tidally affected areas, standard 

stream-gaging techniques cannot be used in the Bay and Delta 

(Smoot and Novak, 1969). To overcome these challenges, a 

wide range of acoustic instrumentation has been used successfully to measure discharge. Instrumentation currently in use 

by the USGS include: ultrasonic velocity meters (UVM) and 

acoustic Doppler velocity meters (ADVM). 

Purpose and Scope

This report describes the index-velocity method for computing discharge in tidal environments and goes on to describe 

an approach for determining daily discharge by using a tidal 

filter. The main body of the report summarizes the techniques; 

appendices at the end of the report cover the detailed procedures and calculations. In addition, a section identifying 

some potential mistakes has been included to assist those who 

develop calibration relations to identify possible explanations 

for anomalous data.

Acknowledgements

The data presented in this report were collected thanks 

to the long-term support provided by the Interagency 

Ecological Program, the California Department of Water 

Resources, the U.S. Bureau of Reclamation, the USGS Place 

Based Program, the USGS Federal Cooperative Program, 

CALFED, the California State Water Resources Control 

Board, Contra Costa Water District, and the City of Stockton. Over the past 18 years, many people from the USGS 

have contributed to the collection and analysis of these data: 

Rick Adorador, Jon Burau, Jay Cuetara, Jim DeRose, Steve 

Gallanthine, Jim George, Robert Hilditch, Rick Oltmann, 

Scott Posey, Greg Smith, and Jon Yokomizo. In addition, 

many thanks to the reviewers and production staff who have 

helped with this report: Jon Burau, Phil Contreras, Charlie 

Kaehler, Victor Levesque, Robert Meyer, Scott Morlock, Carol 

Sanchez, Dave Uyematsu, and Jeremy Woods from the USGS 

and Shawn Mayr from the California Department of Water 

Resources.

Principles of Operation

Ultrasonic Velocity Meters

UVMs work on a “time of travel” principle. The UVM 

system is comprised of two acoustic transducers that are aimed 

at each other and are mounted at the same depth diagonally 

across a channel (fig. 1A). Both transducers are connected to a 

central processing unit by underwater cables. To measure the 

water velocity, an acoustic pulse is transmitted between the 

transducers in both directions: first an acoustic pulse travels 

from A to B then a pulse travels from B to A (fig. 1A). An 

acoustic signal that has a component traveling in the same 

direction as the water (from A to B, fig. 1A) will arrive earlier 

than an acoustic signal that is traveling against the water 

velocity (from B to A, fig. 1A). The water velocity along the 

acoustic path (Vp) is proportional to the difference in time it 

takes the acoustic signal to travel between the two transducers 

(ADS Corporation, 2003). Index velocity (Vi) is determined 

by measuring the difference in time required for an acoustic 

signal to travel between the two transducers and a knowledge 

of transducer configuration (specifically the distance between 

transducers and the angle of the acoustic path, with respect to 

Computation of Discharge Using the Index-Velocity 

Method in Tidally Affected Areas

By Catherine A. Ruhl and Michael R. Simpson

Principles of Operation 1
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the principal streamflow direction) (fig. 1A). A UVM system 

can have more than one acoustic path; for example, there 

can be multiple paths in the vertical with pairs of transducers mounted at different elevations in the water column; or a 

“cross-path” configuration with pairs of transducers mounted 

so that their acoustic paths create an “X” pattern across the 

channel.

Figure 1. Schematic of standard acoustic stream-gaging stations. (A) ultrasonic velocity meter; (B) horizontal acoustic Doppler velocity meter; and (C) upward-looking acoustic Doppler velocity meter.

2 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Acoustic Doppler Velocity Meters

ADVMs utilize monostatic transducers, or transducers 

that both send and receive an acoustic pulse. An acoustic pulse 

of a known frequency is sent out into the water column along 

the acoustic beam. A fraction of that acoustic pulse is reflected 

by small particles in the water, returning to the transducer at a 

frequency that has been shifted due to the Doppler effect. The 

index velocity (Vi) is the water velocity within the acoustic 

beam and is determined on the basis of the change in the transmitted acoustic frequency and the geometric configuration of 

the transducers (SonTek Corporation, 2000) (figs. 1B, C). 

There are three general classifications for ADVMs: point 

velocity, single bin, and profiler. Each system uses the Doppler 

shifts of sound waves reflected off of particles moving with 

the water, however, implementation varies among systems.

Point Velocity

Point velocity ADVMs use converging beams to measure 

velocity in a small sample volume. These ADVMs are used 

both in the laboratory and in the field to measure point velocities but generally are not used for index-velocity measurements.

Single Bin

Single bin ADVMs use divergent beams to sample larger 

sections of the velocity field. The sample volume can be 

manipulated by range gating the received signal, or programming the start distance and end distance of the acoustic beam 

over some portion of the instrument range. The measured 

velocity is proportional to the magnitude of the Doppler frequency shift and is spatially averaged over the sample volume. 

The sample volume can vary from the maximum instrument 

range to a few centimeters. Single bin ADVMs are used primarily for index velocity measurements and can be mounted 

in downward-looking, upward-looking, and sideward-looking 

configurations. 

Profiler

ADVM profilers use diverging beams for velocity 

measurement, but contain sophisticated, high-speed, signal 

processing software that can calculate multiple velocities from 

numerous range-gated sample volumes (bins) along the beam 

path. Both the size and number of these bins can be controlled 

from the ADVM firmware and usually are spaced evenly along 

the main beam axis. ADVM profilers can be used to measure 

index velocities using upward-looking, downward-looking, 

and side-looking configurations. With bottom-tracking software or satellite geographic positioning system (GPS) integration, they also can be mounted on mobile, downward-looking 

platforms to gather velocity profiles or to collect moving-boat 

discharge measurements (Simpson, 2001). 

Methods

Discharge is a function of both area and velocity. The 

equation used to calculate discharge is:

where 

(eq. 1)

Q is discharge; 

A is the cross-sectional area (area); and

is the cross-sectionally averaged velocity (mean 

velocity) (Munson and others, 1990).

Because direct measurement of the area and mean velocity is difficult, easily measured parameters are used as surrogates. Calibration relations are used to calculate the area and 

the mean velocity using the stage and index velocity measurements collected at the gage location.

Calculating Area on the Basis of Stage

Stage or water level is recorded as a time-series at the 

gage location (fig. 1). Stage can be measured using various 

techniques (Barron, 1963; Rantz and others, 1982; Kennedy, 

1988; Latkovitch and Leavesley, 1992). Techniques currently 

used at long-term stations in the Sacramento-San Joaquin 

Delta include upward looking acoustic beams, bubble-gage 

sensors, and stilling wells equipped with a float tape and 

shaft encoder. Water-level records for the short-term sites can 

be obtained either through internally logging probes associated with the instrument packages (fig. 1C), or a stage record 

from a nearby gaging station. The measured stage is related 

to cross-sectional area based on detailed channel surveys. 

At long-term stations, these relations are confirmed approximately every 3 years, or whenever rating discrepancies are 

identified. 

The stage versus cross-sectional area relation is determined from a detailed channel survey. Channel surveys can 

be conducted using a variety of techniques such as sounding 

weights, fathometers, or downward-looking ADVM profilers 

to capture the submerged features, and standard surveying 

techniques to characterize the bank profile (fig. 2). Water 

levels change rapidly in the Bay and Delta area: reaching 

11.5 inches per hour (in/hr) at the Golden Gate; 7.3 in/hr near 

the confluence of the Sacramento and San Joaquin Rivers; 

and 0.5 in/hr near the upstream Delta boundaries on the 

Sacramento and San Joaquin Rivers (Tidelog, 2004). Due to 

rapidly varying water levels in tidally affected environments, 

close synchronization between the time of the bank surveys, 

bathymetric surveys, and water-level measurements at the gage 

Q = AV

Methods 3
must be maintained so that survey data can be related directly 

to stage. At short-term monitoring stations the approach is 

similar, though the bank elevations often are estimated rather 

than surveyed.

The final relation between stage and area is developed 

for the expected stage range at the gage location. There are a 

number of approaches that can be used to develop this relation 

including the Channel Geometry Analysis Program (Regan 

Figure 2. Schematic of bathymetry and channel-bank survey field method.

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4 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
and Schaffranek, 1985) or AreaComp.exe (Rehmel, 2002). 

In general, a quadratic equation is sufficient to characterize 

the channel area (Rantz and others, 1982). The details of how 

the stage versus area relation is determined are contained in 

Appendix A.

Calculating Mean Velocity on the Basis of the 

Index Velocity

The index velocity is recorded as a time-series at the gage 

location (fig. 1). Over the last several decades, many advances 

have been made in the field of hydroacoustics and now a 

variety of instruments are available to measure index velocities 

(Rantz and others, 1982; Morlock and others, 2002). Examples 

of equipment that are currently in use at long-term monitoring 

stations in the Sacramento-San Joaquin Delta include UVMs 

and sideward-looking ADVMs (both single-bin and profiling). Short-term deployments use internally logging, upwardlooking ADVMs and are calibrated in the same manner as the 

long-term stations. 

The collection of discharge measurements using acoustic 

techniques has improved significantly and now is employed 

frequently (Morlock and others, 2002). The specifics of how 

individual discharge measurements are collected are described 

in detail in Simpson and Oltmann (1993), Morlock (1994), and 

Simpson (2001). In tidally affected environments, it is important to collect discharge measurements that adequately characterize the high frequency variability of the tides as well as 

the seasonal variability associated with the annual hydrologic 

cycle (Simpson and Bland, 1999). Tidal variability is captured 

by collecting between 50−120 discharge measurements over 

a 12- to 13-hour period; the seasonal variability is captured 

by collecting a smaller set of data (10–20 measurements) 

periodically during periods of hydrologic interest (most often 

high-flow events). At many locations in the Delta, the influence of the rivers is minimal compared to the influence of the 

tides. Periodic discharge measurements must be collected over 

the life of the station to ensure that the calibration relation is 

stable. Changes in transducer alignment and channel geometry 

can change the rating.

The mean velocity during each transect is calculated by 

dividing the discharge, measured using a boat-mounted downward-looking ADVM profiler, by the channel area, calculated 

based on the water level measured at the gage. The time of 

the measurement is taken as the mid-point of the duration of 

the discharge measurement. If a water-level reading was not 

recorded at that time, the values are interpolated linearly to get 

an estimate of the area at the time the transect was conducted. 

The index velocity measured at the gage is related 

directly to the mean velocity. If the recorded index velocity 

is an average over a time interval, the time must be shifted to 

the midpoint of the interval to ensure proper synchronization 

with the boat measurements. The relation between the index 

velocity and mean velocity is developed by identifying the 

index velocity measured at the time of the midpoint of each 

transect. If the transect occurred between two data points 

recorded at the gaging station, the resulting index velocity 

is calculated based on linear interpolation to ensure that all 

values are on the same time-base. The final relation is based 

on a least-squares regression between the index velocity and 

mean velocity.

A wide range of relations have been developed in the Bay 

and Delta region (fig. 3) most of which are linear (fig. 3A). 

However, more complex ratings also are possible. In this 

system, we have documented several higher-order polynomial 

ratings (fig. 3B); loop ratings that are indicative of ebb-flood 

asymmetries in the current structures at the measurement 

location causing a different relation between the flood-to-ebb 

transition versus the ebb-to-flood transition (fig. 3C); and 

occasionally we have found a bimodal relation (fig. 3D). A 

sample calculation of the mean velocity based on the index 

velocity, is given in Appendix B. 

Methods 5
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Figure 3. Examples of index velocity versus mean velocity ratings. (A) simple linear rating; (B) quadratic rating; (C) loop rating; 

and (D) bi-modal rating.

6 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Calculating Discharge

Once the cross-sectional area is calculated from the stage 

time-series data and the mean velocity is calculated from the 

index velocity time-series data, discharge can be calculated 

(fig. 4A, blue line). Discharge is the product of these two 

values: (eq. 1). A sample discharge time-series based 

on a linear velocity calibration is given in Appendix C. In 

addition, Appendix C presents the methodology for calculating discharge at locations that have more complex loop and 

bimodal velocity ratings.

Calculating Daily Flow 

Calculating daily discharge in a tidally influenced environment cannot be accomplished simply by averaging all of 

the values collected during that 24-hour period. Simple averaging causes cyclical variations, or aliasing, in the data that are 

spurious and are a function of the averaging scheme, not the 

data. Therefore, a low-pass filter is used to remove frequencies that have periods less than 30 hours. The most energetic 

variations removed in this process are the astronomical tides 

(typically with periods at or around 12 and 24 hours); however, other variations (meteorological, hydrologic, or operational) that have periods less than 30 hours also are removed. 

A number of filters are available including the Godin filter 

(Godin, 1972), a Fourier transform filter (Walters and Heston, 

1982; Burau and others, 1993), or a Butterworth filter (Roberts 

and Roberts, 1978). All of these filters are used by the San 

Francisco Bay/Delta Hydrodynamics Program for a variety of 

purposes, however, published daily discharge values are calculated using a Butterworth filter with a 30-hour stop period and 

a 40-hour pass period (fig. 4A, red line; and fig. 4B, red line). 

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Figure 4. Example of discharge record. (A) tidal discharge, and filtered discharge; and (B) filtered discharge, and daily 

average discharge.

Q = AV

Methods 7
Note that tidal variations with periods greater than 30 hours, 

such as spring/neap cycle effects, will remain in the resulting 

tidally averaged data (Roberts and Roberts, 1978). In addition, 

approximately 2 days of filtered data at the beginning and end 

of the time-series or adjacent to any gap in the time-series are 

erroneous due to filter ringing and are not used. The daily discharge is calculated as the 24-hour daily average of the tidally 

filtered data (fig. 4B, black line with open circles). A sample 

calculation of daily discharge time-series is given in 

Appendix D. 

Importance of High-Quality Data

Collecting high-quality data is important in any system. 

It is particularly critical in tidal systems where the tidally 

averaged flows are desired. Often tidally averaged flows are 

several orders of magnitude smaller than the tidal flows; therefore, a relatively small bias in the tidal flows can become a 

substantial error in the tidally averaged data. Clock synchronization, channel-bottom movement, boat positioning, discharge 

measurement duration (too fast or too slow), configuration file 

settings, and equipment positioning all can affect the resulting data. Attention to detail is critical in minimizing problems 

during data collection. A summary of how different types of 

problems are manifest in the final calibration data sets is given 

in Appendix E.

Summary

The index-velocity method used by the U.S. Geological Survey for calibrating flow-monitoring stations in tidally 

influenced environments is described. Discharge is computed 

as a three step process: (1) calculating the cross-sectional area 

based on a stage time-series; (2) calculating the mean crosssectional velocity based on a measured index velocity time 

series, and (3) calculating discharge as the product of the area 

and mean velocity: Q=VA. A daily mean discharge value is 

computed based on a daily average of the low-pass filtered 

discharge. 

References

ADS Corporation, 2003, Accusonic Technologies, Multipath 

flowmeter systems theory and operating principles: 

East Falmouth, Mass., 4 p. http://www.accusonic.com/

technology.html

Barron, E.G., 1963, New instruments for surface-water investigations, in Mesnier, G.N., and Iseri, K.T., Selected techniques in water-resources investigations: U.S. Geological 

Survey Water-Supply Paper 1669-Z, 64 p. 

Burau, J.R., Simpson, M.R, and Cheng, R.T., 1993, Tidal and 

residual currents measured by an acoustic Doppler current 

profiler at the west end of Carquinez Strait, San Francisco 

Bay, California, March to November 1988: U.S. Geological 

Survey Water-Resources Investigations Report 92-4064, 

76 p.

Godin, P., 1972, The analysis of tides: University of Toronto 

Press, 264 p.

Kennedy, E.J., 1988, Levels at streamflow gaging stations: 

U.S. Geological Survey Open-File Report 88-710, 42 p.

Latkovitch V.J., and Leavesley, G.H., 1992, Automated data 

acquisition and transmission, in D.R. Maidment (ed.), 

Handbook of Hydrology, chap. 25.

Lipscomb, S.W., 1995, Quality assurance plan for discharge 

measurements using broad band acoustic Doppler current 

profilers: U.S. Geological Survey Open-File Report 95-701, 

7 p.

Morlock, S.E., 1994, Evaluation of acoustic Doppler current 

profiler measurements of river discharge: U.S. Geological 

Survey Water-Resources Investigations Report 95-4218, 

37 p.

Morlock, S.E., Nguyen, H.T., and Ross, J.H., 2002, Feasibility 

of acoustic Doppler velocity meters for the production of 

discharge records from U.S. Geological Survey StreamflowGaging Stations: U.S. Geological Survey Water-Resources 

Investigations Report 01-4157, 56 p.

Munson, B.R., Young, D.F., and Okiishi, T.H., 1990, 

Fundamentals of fluid mechanics: New York: John Wiley & 

Sons, 843 p.

Rantz, S.E., and others, 1982, Measurement and computation 

of streamflow: vol. 2. Computation and discharge: U.S. 

Geological Survey Water-Supply Paper 2175, 2 v., 631 p.

Regan, R., and Schaffranek, R.W., 1985, A computer program 

for analyzing channel geometry: U.S. Geological Survey 

Water-Resources Investigations Report 85-4335, 49 p.

Rehmel, M.S., Stewart, J.A., and Morlock, S.E., 2003, 

Tethered acoustic Doppler current profiler platforms for 

measuring streamflow: U.S. Geological Survey Open-File 

Report 03-237, 15 p.

Rehmel, M., 2002, Area comp.exe, version 1.1, 

U.S. Geological Survey software http://hydroacoustics.usgs.

gov/software/

Roberts, J., and Roberts, T.D., 1978, Use of the Butterworth 

low-pass filter for oceanographic data: Journal of 

Geophysical Research, v. 83, no. C11, p. 5510–5514.

8 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Simpson, M.R., 2001, Discharge measurements using a broadband acoustic Doppler current profiler: U.S. Geological 

Survey Open-File Report 01-1, 123 p.

Simpson, M.R., and Bland, Roger, 1999, Techniques for 

accurate estimation of net discharge in a tidal channel: IEEE 

Sixth Working Conference on Current Measurement, San 

Diego, Calif., March 11−13, 1999, Proceedings p. 125–130.

Simpson, M.R., and Oltmann, R.N., 1993, Discharge-measurement system using an acoustic Doppler current profiler with 

applications to large rivers and estuaries: U.S. Geological 

Survey Water-Supply Paper 2395, 32 p.

Smoot, G.F., and Novak, C.E., 1969, Measurement of discharge by the moving-boat method: U.S. Geological Survey 

Techniques of Water-Resources Investigations, chap. A11, 

in book 3, Applications of hydraulics, 22 p. [reprinted in 

1977].

SonTek Corporation, 2000, SonTek ADVM-series instruments 

technical documentation: San Diego, Calif., 77 p.

Tidelog, 2004, Bolinas, Calif., Pacific Publishers. 

U.S. Geological Survey Office of Surface Water, 2002a, 

Configuration of acoustic profilers (RD Instruments) 

for measurement of streamflow: OSW Technical Memo 

2002.01, 6 p. http://il.water.usgs.gov/adcp/policy/OSW2002-

01.pdf

U.S. Geological Survey Office of Surface Water, 2002b, 

Policy and technical guidance on discharge measurements 

using acoustic Doppler current profilers: OSW Technical 

Memo 2002.02, 4 p. http://il.water.usgs.gov/adcp/policy/

OSW2002-02.pdf

Walters, R.A., and Heston, Cynthia, 1982, Removing tidal 

period variations from time-series data using low-pass digital filters: Journal of Physical Oceanography, v. 12, no. 1, 

p. 112–115.

References 9
Introduction

This appendix describes the standard procedure used to 

compute the cross-sectional area. In this example, standard 

survey techniques were used to characterize the riverbanks 

and a downward-looking vessel-mounted ADVM was used 

to collect bathymetry data. The cross section is presented in 

figure A1. This is an example of one approach for collecting 

and processing the data—-this information may be collected 

and analyzed using a variety of techniques.

The cross-sectional area is a function of the stage. In tidal 

systems, water levels vary significantly over short periods of 

time; therefore, it is important to accurately document the time 

of the channel survey to ensure that all data can be corrected 

to the same vertical datum and related to the concurrently 

recorded gaging-station data.

Establishing the stage-versus-area relation is a four-step 

process: (1) channel-bank survey; (2) bathymetry survey; 

(3) synthesis-of-field data; and (4) determination of stageversus-area relation.

Step 1. Channel-Bank Survey

Standard surveying techniques are used to measure the 

elevations of the channel banks from the crest of the river 

bank to the water’s edge. Both horizontal location and vertical 

elevation information must be recorded during the field survey. 

It is critical to document the time that elevations at the water’s 

edge are measured; this allows the application of an accurate 

correction to the survey elevation data ensuring that all of the 

information used to develop the relation has the same vertical 

datum.

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���

��

�

�

��

��

��

��

�����������������

���������������������������������������������

�

�

�

�

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�

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Figure A1. Measured channel cross section. Standard survey techniques used on the channel banks, and a vessel-mounted 

downward-looking acoustic Doppler velocity meter used in the submerged portion of the channel.

Appendix A. Sample Computation of the Cross-Sectional Area Relation

10 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
of the vertical elevation data so that the datum is consistent 

between the stage gage and the survey data.

Summaries of the field notes associated with the channelbank survey and the stage data recorded at the gaging station 

are presented in tables A1 and A2. For the purposes of the 

channel-bank survey, the data recording interval was set to a 

1-minute interval rather than the standard 15-minute interval.

Horizontal Location Correction

The horizontal location correction is used to zero the 

cross section at the crest of either the left or right bank, 

depending on where the channel bank survey is started. The 

horizontal location measurements collected in the field all are 

relative to the initial position of the surveying instrument. In 

this particular case, the horizontal location correction is +19 

feet (ft) to account for the placement of the instrument 19 ft 

from the crest of the left bank (table A3).

Vertical Elevation Correction

A vertical elevation correction is necessary if the channel-bank survey is not referenced to the same vertical datum as 

the stage gage. Discrepancies between the measured elevation 

at the water’s edge (positions “e” and “p” on fig. A1) and the 

concurrently recorded stage also may occur due to hydrodynamic processes causing lateral variations in water level. For 

the purposes of developing the stage-versus-cross-sectional 

area relation, we assume that the water surface is level across 

the channel and that the measured stage at the gaging station 

is an accurate reflection of the elevation of the entire water 

surface. The correction applied to the measured elevations is 

calculated as follows:

(eq. 2)

where 

Hcorr is applied to each of the measured elevations to ensure 

that the measured elevations are directly related to 

the datum at the gaging station; 

stg is the stage recorded at the gaging station at the time 

the elevation at the water’s edge is measured by the 

survey crew; and

H is the elevation measured by the survey crew at the 

water’s edge. 

The subscripts refer to the time that the measurements 

were collected so they can be related to the values recorded at 

the gaging station.

In this example, the calculation is determined as follows:

= 9.89 (eq. 3) 

Therefore, 9.89 ft is added to each surveyed elevation 

(table A4).

Table A1. Summary of field notes collected during channel-bank 

survey. [All units in feet]

Time Stage

(in feet)

Comments

1438 11.85

1439 11.86

1440 11.87

1441 11.88 Left bank water’s edge elevation collected

1442 11.89

1443 11.90

1444 11.91

1445 11.92

1446 11.93

1447 11.94 Right bank water’s edge elevation collected

1448 11.95

1449 11.96

Table A3. Summary of field data and horizontal location corrections [All units in feet]

Label on 

figure A1

Horizontal

 location

Vertical elevation Field notes

a −19 14.41

b 0 12.60

c 18 8.02

e 44 2.00 time = 1441

p 660 2.04 time = 1447

r 684 12.20

Label 

on 

figure 

A1

Horizontal 

location

Corrected

horizontal 

location

Vertical 

elevation

Field 

notes

Remarks

a −19 0 14.41 (1) Location 

correction is 

+19 feet to 

account for 

the initial 

placement 

of the 

surveying 

equipment.

b 0 19 12.60

c 18 37 8.02

e 44 63 2.00 time = 

1441 

p 660 679 2.04 time = 

1447 

r 684 703 12.2

2

[( ) ( )] 1 1 2 2 t t t t

corr

stg H stg H

H

+ =

2

[( ) ( )] 1441 1441 1447 1447 = + = stg H stg H Hcorr

2

[(11.88 2.00) + (11.94 2.04)]

Appendix A. Sample Computation of the Cross-Sectional Area Relatio 11

Table A2. Summary of stage data recorded at gaging station 
Channel-Bank Survey Correction Summary

The corrected channel-bank survey data (table A5) is 

used with the bathymetry survey data to compute the area 

relation. 

Step 2. Bathymetry Survey

For the purposes of illustration, this example uses the 

mean beam depth output from a downward-looking, vesselmounted, ADVM profiling system. A fathometer or a sounding weight also can be used to obtain bathymetry data along 

the cross-section. The time of the bathymetry survey must be 

recorded so that an accurate correction can be applied to make 

sure all of the data are referenced to the same vertical datum.

Similar to the channel-bank survey, both the horizontal 

location and the vertical elevation must be corrected. The horizontal location is corrected so that the zero horizontal location 

of the bathymetry survey is the water’s edge, allowing for a 

user-defined measured edge distance. The horizontal location must be shifted a second time when the bank-survey and 

bathymetry-survey data are synthesized (Appendix A, Step 3). 

The vertical elevation is corrected so the bathymetric survey 

and the stage gage use a consistent vertical datum.

A summary of the bathymetry field data and the stage 

data recorded at the gaging station are presented in tables A6

Table A4. Summary of field data and location and elevation corrections [All units in feet]

Label on 

figure A1

Horizontal 

location

(feet)

Corrected horizontal 

location (feet) 

(1)

Vertical elevation 

(feet)

Corrected vertical 

elevation (feet)

(2)

Field notes

(3)

Remarks

a −19 0 14.41 24.30 (1) Location correction is +19 feet 

to account for the initial placement of the surveying equipment.

(2) Elevation is corrected +9.89 

feet to reflect the elevation at 

the water’s edge recorded by the 

stage gage. 

(3) Stage data taken from table A2

b 0 19 12.60 22.49

c 18 37 8.02 17.91

e 44 63 2.00 11.89 time = 1441 

stage = 11.88 feet

p 660 679 2.04 11.93 time = 1447 

stage = 11.94 feet

r 684 703 12.2 22.09

Table A5. Summary of corrected channel-bank survey data 

[All units in feet]

Label on 

figure A1

Corrected 

horizontal 

location

(1)

Corrected 

elevation 

(2)

Remarks

a 0 24.30 (1) Location correction is +19 

feet to account for the initial 

placement of the surveying 

equipment.

(2) Elevation is corrected +9.89 

feet to reflect the elevation at 

the water’s edge recorded by 

the stage gage. 

b 19 22.49

c 37 17.91

e 63 11.89

p 679 11.93

r 703 22.09

and A7. In this case, the bathymetry survey was conducted 

at 1525 PST, approximately 2 hours after the channel-bank 

survey (Appendix A, Step 1) and the stage rose nearly 0.1 ft 

in that time. For the purpose of illustration, 10 points across 

the cross-section were used to characterize the bathymetry 

(fig. A1). The entire ADVM data file can be processed using 

AreaComp.exe (Rehmel, 2002); however, a subset is presented 

here to highlight the procedure. Because the downward-looking ADVM profiler cannot measure all the way to the water’s 

edge, the operator must record the edge distance that is necessary in the analysis. The distance from the downward-looking 

ADVM profiler to the shore should be measured using an 

accurate technique such as a laser range finder.

Table A6. Summary of field notes associated with bathymetry 

survey [All units in feet]

Label on 

figure A1

Horizontal 

location 

Water 

depth 

Field notes

d −34 0 Left bank edge distance: 34 

feet

f 0 18.27

g 35.87 21.49

h 54.30 19.55

i 82.01 23.98

j 206.08 20.48

k 263.16 21.15 Mid-time of transect=1525 

PST

l 383.01 20.30

m 431.59 15.43

n 504.45 18.19

o 559.10 16.76

q 587.1 0 Right bank edge distance: 28 

feet

 Channel width = 587.1 feet + 34 feet= 621.1 feet

12 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Horizontal Location Correction

The initial horizontal-location correction “zeros” the 

bathymetry survey at water’s edge of either the left or right 

bank, depending on the starting point of the survey. A second horizontal location correction must be applied when the 

channel bank survey data and the bathymetry survey data are 

synthesized (Appendix A, Step 3). The horizontal location 

measurements all are relative to the position of the ADVM 

profiler at the start of the transect. In this case, the horizontal 

location correction was +34 ft because the ADVM profiler was 

located 34 ft from the left bank water’s edge as recorded by 

the operator in the field notes (table A8, remarks column).

H = stgMP D

Table A7. Summary of stage data recorded at gaging station

Time Stage

(feet)

Remarks

1520 11.95

1521 11.96

1522 11.97

1523 11.97

1524 11.98

1525 11.99 Bathymetry survey conducted

1526 11.99

1527 12.00

1528 12.01

1529 12.02

1530 12.02

1531 12.03

Table A8. Summary of field data and location corrections

[All units in feet]

Label on 

figure A1

Horizontal 

location

Corrected 

horizontal 

location 

(1)

Water 

depth 

Remarks

d −34 0 0 (1) Location corrected 

by +34 feet to adjust 

for the initial edge 

estimate. A second 

correction will be 

applied later when 

these data are 

synthesized with 

the channel-bank 

survey.

f 0 34 18.27

g 35.87 69.87 21.49

h 54.30 88.30 19.55

i 82.01 116.01 23.98

j 206.08 240.08 20.48

k 263.16 297.16 21.15

l 383.01 417.01 20.30

m 431.59 465.59 15.43

n 504.45 538.45 18.19

o 559.10 593.10 16.76

q 587.10 621.10 0

Table A9. Summary of field data and elevation corrections [All units in feet]

Label on 

figure A1

Horizontal 

location 

Corrected horizontal

location

(1)

Water depth Corrected vertical 

elevation

(2)

Remarks

d −34.00 0 0 11.99 (1) Location corrected by +34 feet to 

adjust for the initial edge estimate. An 

additional correction is necessary to 

synthesize these data with the channelbank survey.

(2) Elevation corrected by 11.99 feet to 

account for the stage recorded at the 

gaging station at the mid-point of the 

bathymetry survey.

f 0 34.00 18.27 −6.28

g 35.87 69.87 21.49 −9.50

h 54.30 88.30 19.55 −7.56

i 82.01 116.01 23.98 −11.99

j 206.08 240.08 20.48 −8.49

k 263.16 297.16 21.15 −9.16

l 383.01 417.01 20.30 −8.31

m 431.59 465.59 15.43 −3.44

n 504.45 538.45 18.19 −6.20

o 559.10 593.10 16.76 −4.77

q 587.10 621.10 0 11.99

Vertical Elevation Correction

An elevation correction is necessary because the 

bathymetry data are measured relative to the water’s surface. 

The correction is based on the (1) stage recorded at the midpoint of the bathymetry survey and (2) a conversion from 

depth to elevation.

(eq. 4)

where 

H is the corrected elevation; 

stgMP is the stage recorded at the gaging station at the midpoint of the bathymetry survey; and

D is the depth at each location recorded during the 

bathymetry survey. 

In this example, stgMP is 11.99 ft (table A7). The correction results are summarized in table A9.

Appendix A. Sample Computation of the Cross-Sectional Area Relatio 13
1

2 1

2

[( ) ( )]

WE

WE WE b

corr L

L L W

L = +

2

[( ) ( )]

1

2 1 = + WE = WE WE b

corr L

L L W

L

63 60.5

2

[(679 63) (621)] + =

Bathymetry Survey Correction Summary

The corrected bathymetry-survey data (table A10) is used 

with the channel-bank survey data to develop the cross-sectional area relation. 

Step 3. Synthesis of Field Data

The vertical corrections are assumed to have been applied 

(Appendix A, Step 2) to combine the bank survey with the 

bathymetry survey. However, the horizontal position of the 

bathymetry survey must be shifted again so that the “centerlines” of the two channel surveys are aligned. In this example, 

the width of the water surface was measured at 616 ft during 

the channel-bank survey and at 621 ft during the bathymetric 

survey based on the edge measurements and the “distance 

made good” recorded by the profiling ADVM. Instrument 

precision and changing water elevation both contribute to this 

discrepancy. The centerline of the bathymetric survey is placed 

at the centerline of the gap between the two bank surveys.

(eq. 5)

where

Lcorr is the correction that will be applied to the 

bathymetry survey location data;.

Lwe1 and Lwe2 are the horizontal locations at the water’s edge as 

measured during the bank survey; and

Wb is the width of the submerged portion of the 

channel, as measured during the bathymetric 

survey.

In this example the calculation is as follows:

(eq. 6)

The results are summarized in table A11. In this example, 

there is some overlap between the channel-bank survey and 

the bathymetry survey near the water’s edge due to the changing tide.

Table A10. Summary of corrected bathymetry survey data

[All units in feet]

Label on 

figure A1

Corrected 

horizontal 

location

(1)

Corrected 

vertical 

elevation

(2)

Remarks

d 0 11.99 (1) Location corrected 

by +34 feet to adjust 

for the initial edge 

estimate. An additional 

correction is necessary 

to synthesize these data 

with the channel-bank 

survey.

(2) Elevation corrected by 

11.99 feet to account 

for the stage recorded 

at the gaging station 

at the mid-point of the 

bathymetry survey.

f 34.00 −6.28

g 69.87 −9.50

h 88.30 −7.56

i 116.01 −11.99

j 240.08 −8.49

k 297.16 −9.16

l 417.01 −8.31

m 465.59 −3.44

n 538.45 −6.20

o 593.10 −4.77

q 621.10 11.99

Table A11. Summary of synthesized field data [All units in feet]

Label on 

Figure A1

Location Finalized location Elevation Remarks

a 0 0 24.30 The location of the bathymetry survey data was corrected by +60.5 

feet to place the bathymetry survey in the middle of the bank 

survey. Only the bathymetry data were corrected (shown in the 

shaded grey portion of the table)

b 19 19 22.49

c 37 37 17.91

e 63 63 11.89

d 0 60.50 11.99

f 34 94.50 −6.28

g 69.87 130.37 −9.50

h 88.30 148.80 −7.56

i 116.01 176.51 −11.99

j 240.08 300.58 −8.49

k 297.16 357.66 −9.16

l 417.01 477.51 −8.31

m 465.59 526.09 −3.44

n 538.45 598.95 −6.20

o 593.10 653.60 −4.77

q 621.10 681.60 11.99

p 679 679 11.93

r 703 703 22.09

14 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Step 4. Calculating the Stage-Versus-Area 

Relation

In order to establish the relation between cross-sectional 

area and stage, a table that establishes the area over the entire 

range of expected stage values is developed. This table can 

be developed manually or by a number of different software 

products. For the purposes of this example, the elevation and 

location data in table A11 are used as input into AreaComp.

exe (Rehmel, 2002) (fig. A2). By selecting the “Create Stage 

Area Rating” button at the bottom of the screen, a table with 

the range of expected stage values specified in the form (in 

this case from 5 to 20 ft with an interval of 0.01 ft) and the 

associated cross-sectional areas were calculated. These results 

were exported and simple least-squares regression analysis 

was performed to develop a quadratic equation describing the 

relation between stage and cross-sectional area. In this case, 

the resulting relation was:

(eq. 7)

where

A is the cross-sectional area, in square feet; and

S is the stage recorded at the gaging station, in feet.

In this case, the difference between the table results and 

the quadratic equation estimate is less than 0.1 percent. Other 

systems, such as the USGS automated data-processing system 

(ADAPS) use table look-up functions rather than equations.

2.75( ) 557.6( ) 4715 2 A = S + S +

Figure A2. Screen capture of channel cross section from AreaComp.exe. The expected tidal range is summarized on 

the upper left of the figure; tabular location and elevation data are summarized on the left of the figure; and graphical 

representation is presented on the right of the figure.

Appendix A. Sample Computation of the Cross-Sectional Area Relatio 15
Introduction

This appendix describes the standard procedure used 

to compute the mean velocity from the index velocity. In 

this example, a downward-looking, vessel-mounted ADVM 

profiler was used to collect discharge measurements over a 

full tidal cycle. These data were combined with additional data 

collected over the life of the station to establish the final relation. Relations are finalized when the full range of expected 

flows have been measured and are stable over time.

One of the most important aspects of establishing the 

mean-velocity relation is ensuring that the discharge measurements and the gaging-station index velocity and stage measurements are well synchronized. In highly dynamic systems, 

timing that is off by as few as 5 minutes can cause erroneous 

ratings.

Establishing the index velocity-versus-mean-velocity 

relation is a three-step process: (1) collection of discharge 

measurements; (2) synthesis of field and gage data to compute 

the mean velocity; and (3) computation of the relation between 

the mean velocity and the index velocity.

Step 1. Collection of Discharge Measurements

In this step, the discharge measurements are organized for 

use in the calibration process. This section does not address 

the mechanics of collecting discharge measurements (see 

Simpson and Oltmann, 1993; Simpson, 2001; and OSW Technical Memoranda, 2002, for details). A table summarizing the 

results of each ADVM transect is generated once the data have 

been processed (table B1). 

All of the data presented in the table below are values that 

are extracted from the ASCII summary files generated by the 

ADVM profiler software for each transect. The data in table 

B1 only are a subset of the dataset used in the rating process 

and are presented to exemplify the procedure. Discharge measurements were collected over a 13-hour period and cover the 

full range of tidal conditions that regularly occur at this station. Spring and neap tides are not specifically targeted unless 

there are site-specific conditions that suggest that specific tidal 

conditions are critical to developing the calibration. These data 

will be combined with other discharge measurements collected 

at other times to develop the final calibration.

Table B1. Summary of selected downward-looking vesselmounted acoustic Doppler velocity meter discharge 

measurements

Transect 

number

Date Start

Time

End

Time

Duration

(in seconds)

Discharge

(in cubic feet 

per second)

1 6/26/2003 5:18:11 5:25:57 465 24967

2 6/26/2003 5:26:19 5:32:03 344 24701

3 6/26/2003 5:32:26 5:40:53 507 24462

4 6/26/2003 5:41:10 5:46:46 337 24267

5 6/26/2003 6:05:20 6:12:17 418 24173

6 6/26/2003 6:12:33 6:19:02 389 24359

7 6/26/2003 6:19:23 6:27:07 464 24052

36 6/26/2003 10:21:31 10:29:57 505 6898

37 6/26/2003 10:30:09 10:36:01 351 4929

38 6/26/2003 10:39:50 10:48:35 524 1326

58 6/26/2003 13:51:53 13:58:12 378 −25012

59 6/26/2003 14:29:17 14:33:52 275 −22675

60 6/26/2003 14:34:05 14:40:15 371 −22966

78 6/26/2003 17:02:26 17:09:18 412 −3264

79 6/26/2003 17:09:31 17:15:57 386 −1957

80 6/26/2003 17:16:08 17:23:06 418 −666

81 6/26/2003 17:23:16 17:29:50 394 597

82 6/26/2003 17:30:02 17:36:52 410 1758

83 6/26/2003 17:37:02 17:42:19 317 3093

89 6/26/2003 18:18:26 18:24:51 385 9045

90 6/26/2003 18:25:02 18:31:33 391 9322

91 6/26/2003 18:31:52 18:39:46 474 9552

Appendix B. Sample Computation of the Mean Velocity 

16 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Step 2. Synthesis of Field and Gage Data to 

Compute the Mean Velocity

The mean velocity is calculated by dividing the ADVMmeasured discharge by the area calculated by the cross-sectional area relation (Appendix A) at the time of the discharge 

measurement. First, the transect time for each measured 

discharge is established as the mid-point of the transect time 

(table B2, column 3). The stage measured at the gaging station 

(table B2, column 5) is computed by linearly interpolating 

between the nearest two recorded data points. The cross-sectional area (table B2, column 6) is calculated using the equation developed in Appendix A based on the stage measurement 

(table B2, column 5). The index velocity (table B2, column 

7) is taken to be the index velocity measured at the time of 

the discharge measurement by linearly interpolating between 

the two nearest recorded data points. Finally, the mean-velocity (table B2, column 8) is calculated by dividing the measured discharge (table B2, column 4) by the calculated area 

(table B2, column 6). 

Step 3. Computation of the Relation between the 

Mean Velocity and the Index Velocity

The mean velocity is correlated to the index velocity 

using least-squares regression: relating the index velocity 

measured at the gaging station at the time of the transect 

(table B2, column 7) to the mean velocity calculated from 

the ADVM discharge measurement (table B2, column 8). 

Numerous software packages are available that can assist in 

developing this relation. In this example, a linear relation was 

established (fig. B1) between the index velocity and the mean 

velocity. There are cases where more complicated calibrations 

(fig. 3B−D) have been developed due to local hydrodynamic 

features at the station. In this example the resulting relation 

was:

(eq. 8)

where 

is the mean velocity; and

is the index velocity.

V = 0 + .8054(Vi) 0.0265

Table B2. Summary of transect data and gaging station data

Transect 

number

Date Time Discharge (in cubic 

feet per second)

Stage

(in feet)

Area

(in square feet)

Index velocity 

(in feet per second)

Mean velocity 

(in feet per second)

1 6/26/2003 5:22 24967 11.60 11553 2.688 2.161

2 6/26/2003 5:29 24701 11.53 11510 2.713 2.146

3 6/26/2003 5:37 24462 11.46 11466 2.733 2.133

4 6/26/2003 5:44 24267 11.40 11429 2.719 2.123

5 6/26/2003 6:09 24173 11.18 11293 2.779 2.141

6 6/26/2003 6:16 24359 11.12 11256 2.767 2.164

7 6/26/2003 6:23 24052 11.05 11212 2.763 2.145

36 6/26/2003 10:26 6898 10.09 10621 0.925 0.649

37 6/26/2003 10:33 4929 10.15 10658 0.654 0.462

38 6/26/2003 10:44 1326 10.25 10719 0.248 0.124

58 6/26/2003 13:55 −25012 11.59 11547 −2.572 −2.166

59 6/26/2003 14:32 −22675 11.79 11671 −2.349 −1.943

60 6/26/2003 14:37 −22966 11.81 11684 −2.315 −1.966

78 6/26/2003 17:06 −3264 11.90 11740 −0.355 −0.278

79 6/26/2003 17:13 −1957 11.87 11721 −0.231 −0.167

80 6/26/2003 17:20 −666 11.84 11702 −0.103 −0.057

81 6/26/2003 17:27 597 11.82 11690 −0.001 0.051

82 6/26/2003 17:33 1758 11.80 11678 0.182 0.151

83 6/26/2003 17:40 3093 11.78 11665 0.275 0.265

89 6/26/2003 18:22 9045 11.65 11584 0.940 0.781

90 6/26/2003 18:28 9322 11.63 11572 1.023 0.806

91 6/26/2003 18:36 9552 11.62 11566 1.114 0.826

Appendix B. Sample Computation of the Mean Velocity 17
Figure B1. Relation between index velocity and mean velocity.

In this example, the mean and index velocity is strongly 

linearly correlated as indicated by an r-squared value of 

0.998 and a root-mean-squared error of prediction of 0.0663. 

Because the tidal stage range (2 ft) at this station is small, 

compared to the channel depth (25 ft), the effect of water 

depth on the index-velocity measurement is minimal. However, if the range in stage at a particular location is large, stage 

may be included as a regression variable as part of a multiple 

regression analysis to increase the statistical significance of 

the resultant equation. There also are other ways to include 

stage in the mean velocity calculation (Simpson and 

Bland, 1999).

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18 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Step 2. Calculate Discharge

Linear or Polynomial Rating 

Using the stage-versus-area relation,

(eq. 7), 

and the index-velocity-versus-mean-velocity relation, 

(eq. 8)

discharge readily can be calculated based on equation 1 

(table C1).

Using the same procedure and equations, a 1-month data 

record has been processed and presented graphically (fig. C1). 

These data also will be used in Appendix D to show the procedure for calculating daily discharge values.

Introduction

This appendix describes the standard procedure used to 

determine the discharge at an index-velocity station. Discharge 

calculations are straight-forward when the index velocity 

relation is either a simple linear relation (fig. 3A) or a higherorder polynomial relation (fig. 3B). A sample calculation is 

presented below using a linear relation. Once the area relation 

(Appendix A) and mean velocity relation (Appendix B) are 

established, the discharge calculation follows from equation 1. 

In the event that a more complex rating is necessary, such as a 

loop rating or a bimodal rating, then more sophisticated programming is required (discussed at the end of this appendix). 

Computing discharge is a two-step procedure: (1) 

assemble stage and index-velocity data collected at the gaging 

station, and (2) calculate results.

Step 1. Assemble Stage and Index-Velocity Data 

Collected at the Gaging Station

Assemble the quality-reviewed stage and index-velocity data collected at the gaging station (table C1). Ensure that 

the two time-series sample the same points in time. A brief, 

4-hour period is presented in tabular format (table C1) as an 

example, the graphical presentation of discharge contains 1 

month of data (fig. C1).

Appendix C. Sample Discharge Calculation

Date and time Stage 

(in feet)

Area 

(in square feet)

Index velocity 

(in feet 

per second)

Mean velocity 

(in feet 

per second)

Discharge 

(in cubic feet 

per second)

2003/01/11 10:00 13.22 12567 −2.778 −2.2109 −27784

2003/01/11 10:15 13.29 12611 −2.686 −2.1368 −26947

2003/01/11 10:30 13.34 12643 −2.578 −2.0498 −25916

2003/01/11 10:45 13.38 12668 −2.439 −1.9379 −24549

2003/01/11 11:00 13.41 12687 −2.273 −1.8042 −22890

2003/01/11 11:15 13.41 12687 −2.084 −1.6520 −20959

2003/01/11 11:30 13.40 12681 −1.878 −1.4860 −18844

2003/01/11 11:45 13.36 12655 −1.617 −1.2758 −16145

2003/01/11 12:00 13.31 12624 −1.334 −1.0479 −13229

2003/01/11 12:15 13.24 12580 −1.031 −0.8039 −10113

2003/01/11 12:30 13.16 12529 −0.704 −0.5405 −6772

2003/01/11 12:45 13.06 12466 −0.334 −0.2425 −3023

2003/01/11 13:00 12.95 12397 0.093 0.1014 1257

2003/01/11 13:15 12.84 12328 0.546 0.4662 5747

2003/01/11 13:30 12.72 12253 0.976 0.8126 9957

2003/01/11 13:45 12.59 12171 1.438 1.1847 14419

Table C1. Tabular summary of recorded gaging station data and calculations

Appendix C. Sample Discharge Calculation 19

2.75( ) 557.6( ) 4715 2 A = S + S +

V = 0 + .8054(Vi) 0.0265
Figure C1. Discharge calculated as a result of the stage versus cross-sectional area relation (see Appendix A) and the 

index velocity versus mean velocity relation (see Appendix B).

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20 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Loop Rating

In the event that a loop rating is necessary, the discharge 

calculations become more involved. A loop rating actually is 

two mean velocity ratings that have been developed for the 

same tidally affected channel: a flood-to-ebb relation (fig. C2, 

rating A) and an ebb-to-flood relation (fig. C2, rating B).

Computing discharge from a loop rating is complex 

because the computational algorithm must change between 

ratings based on the tidal phase. In general, as velocities 

increase from maximum flood (negative) to maximum ebb 

(positive), rating A is used to compute discharge; as velocities 

decrease from maximum ebb to maximum flood, rating B is 

used. At points beyond the transition points where ratings A 

and B intersect, a single relation is used: in this example rating 

A, is used at the extremes (fig. C3). 

In the event that the flows reverse before the transition 

point between ratings is reached, the algorithm becomes 

significantly more complex. Although there are many ways 

to transition between the two ratings, the approach presented 

here is a time-stepped percentage calculation. A total of 10 

data points are used to transition between the two rating 

curves. The transition begins 5 points before the local maximum velocity and ends after the time-stepped percentage algorithms have stepped through 10 data points. At the beginning 

of the transition, 100 percent of rating A is used and 0 percent 

of rating B is used; at the next point, 90 percent of rating A 

and 10 percent of rating B is used, and so on until at the tenth 

and final point used in the transition 0 percent of rating A 

Figure C2. Sample loop rating. Asymmetries in the flow distributions in the channel between flood flows and 

ebb flows cause a loop rating. These ratings are very difficult to implement and can often be reduced or 

eliminated by selecting a different gage location or different hydroacoustic instrumentation.

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Appendix C. Sample Discharge Calculation 21
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Figure C3. Implementation of loop rating. Data are broken down into four categories: (1) flood to ebb conditions; 

(2) ebb to flood conditions; (3) beyond transition-point conditions; and (4) data requiring the time-stepped percentage calculation. Each of these categories utilizes a specific index-velocity versus mean-velocity relation. (A) 

a 6-day period to show when the time-stepped percentage calculation is necessary (black asterisks); and (B) a 

24-hour period to show in greater detail how the time-stepped percentage calculation is implemented.

22 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
and 100 percent of rating B is used (fig. C3B). Once the mean 

velocity is calculated, the discharge record is determined using 

equation 1. 

Given the complexity and ambiguity in the discharge 

calculation using a loop-rating, they should be avoided, where 

possible, by using equipment that can sample a large fraction 

of the width of the channel, or place the station in locations 

that are less geometrically complex (for example, away from 

bends or junctions).

Bimodal Rating

In the event that a bimodal rating is necessary (fig. C4), 

the discharge calculations are more complicated than the 

standard polynomial relations. In order to calculate discharge, 

the measured index velocity first is compared to the intersection point. If the measured index velocity is greater than the 

rating intersection point, then the upper index velocity rating 

is applied; if the measured index velocity is less than the rating intersection point, then the lower rating is applied. Once 

the mean velocity has been calculated, equation 1 is used to 

calculate the discharge. 

Figure C4. Sample bi-modal rating.

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Appendix C. Sample Discharge Calculation 23
Introduction

This appendix describes the standard procedure used to 

determine the daily discharge at an index velocity station. This 

is a two-step process: (1) apply a low-pass Butterworth filter 

(Roberts and Roberts, 1978); and (2) calculate the daily average of the filtered values.

Step 1. Apply a Low-Pass Butterworth Filter

Once the tidal time-series of discharge has been calculated [fig. D1A (blue line)], a low-pass Butterworth filter (Roberts and Roberts, 1978) is applied to the data to remove the 

high-frequency tidal signals [fig. D1A (red line) and fig. D1B

(red line)]. A stopband period of 30 hours and a passband 

period of 40 hours are used: signals with periods less than 

30 hours are not transmitted to the filtered data; signals with 

periods greater than 40 hours are transmitted to the filtered 

data with minimal loss; and signals that fall in the transition 

between the stopband and the passband are damped, but some 

fraction of them are transmitted to the filtered data. Filter 

ringing causes erroneous data at the beginning and end of a 

continuous data set; therefore, 2 days of data at the beginning 

and end of the time series and on either side of a data gap are 

rejected as part of this process.

Step 2. Calculate the Daily Average of the 

Filtered Values

Once the tidal time-series of filtered discharge has been 

calculated (Appendix D, step 1), the daily discharge is calculated as the daily-average of the filtered discharge data over 

the 24-hour period (fig. D1B, black line with open circles). In 

the event that the tidally averaged discharge data are incomplete for a particular day, a daily average is not calculated.

Figure D1. Example of discharge record. (A) tidal discharge, and filtered discharge; and (B) filtered discharge, 

and daily average discharge.

Appendix D. Calculation of Daily Discharge 

24 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas

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Introduction

This section describes the importance of collecting highquality data and provides examples of potential problems that 

may arise during data collection and data processing and how 

these problems affect the resulting calibrations and discharge 

calculations. The USGS Office of Surface Water (OSW) has 

released a number of policy and technical guidance memoranda to ensure that data are collected consistently throughout 

the USGS (U.S. Geological Survey Office of Surface Water, 

2002a,b). The issues discussed in this section are not meant 

to be an exhaustive list of the potential problems, but rather 

a selection of some of the common mistakes that we have 

encountered in our data analysis. These examples are meant 

to show how these problems can affect the final discharge 

calculations. The following scenarios will be discussed: data 

synchronization, channel-bottom movement, index-velocity 

Appendix E. High-Quality Data and Examples of Common Mistakes

transducer alignment, and configuration files and general boat 

operation.

Data Synchronization

Making sure the clocks at the gaging station and the vessel-mounted ADVM profiler used to collect direct discharge 

measurements are synchronized is critical. Even small errors 

(5 minutes or less) can cause significant errors in the final 

discharge calculations (table E1).

The percent errors are consistently greater on the daily 

flow calculations because the daily flows typically are orders 

of magnitude smaller than the tidal flows. Therefore, small 

errors in the tidal flows can create significant errors in the 

resulting daily flow calculations. Timing errors cause spurious 

flood-ebb asymmetry in the calibration that does not exist 

(fig. E1). 

Maximum flow 

difference 

(in cubic feet per 

second)

Minimum flow 

difference 

(in cubic feet per 

second)

Average flow 

difference 

(in cubic feet per second)

Average percent flow 

difference 

(in percent)

Station 1

Tidal Flows 225 −100 60 1.6

Daily Flows 110 42 60 3.1

Station 2

Tidal Flows 12200 −3080 3400 4.0

Daily Flows 11000 1070 3370 10.5

Station 3

Tidal Flows 4570 −2390 360 4.3

Daily Flows 780 −200 370 15.7

Station 4

Tidal Flows 14260 −13710 230 5.4

Daily Flows 3400 −1260 270 7.5

Table E1. Comparison of differences between calculated flow data with a 5-minute synchronization offset.

Appendix E. High-Quality Data and Examples of Common Mistakes 25
Figure E1. Comparison of index-velocity versus mean-velocity relations when synchronization problems exist.

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26 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Channel-Bottom Movement

Channel-bottom movement can contribute significant 

error in the discharge measurements using ADVM systems if 

a GPS system is not used. When taking discharge measurements, the ADVM profiler can use a “bottom tracking” mode 

to subtract the boat velocity from the measured velocities. 

When the bed sediments are moving, the bottom tracking algorithm will compute an erroneous boat velocity that then is used 

to erroneously correct the measured velocity for movement of 

the boat. Therefore, it is essential to confirm that the channel bottom is stable when using the “bottom-tracking” mode. 

The ADVM bottom-tracking algorithm tends to underestimate the discharge under conditions when there is significant 

bed movement (fig. E2). In this example, bottom-movement 

caused an 8.8-percent underestimation of the channel velocity. Channel bottom movement is well-documented and can 

happen under a wide variety of flow conditions. A 10-minute 

anchored boat test should be conducted whenever velocities 

exceed 1 ft/s or if bottom movement is suspected (Lipscomb, 

1995; Simpson, 2001; U.S. Geological Survey Office of Surface Water, 2002b; Rehmel and others, 2003).

Figure E2. Impacts of using bottom-track in moving-bed conditions on the resulting index velocity versus mean 

velocity relation.

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Appendix E. High-Quality Data and Examples of Common Mistakes 27
Index-Velocity Transducer Alignment

The orientation of the UVM or ADVM index-velocity 

transducers should be kept constant. A change in transducer 

orientation results in a change in the water volume sampled 

and, thus, causes a change in the relation between the index 

velocity and mean velocity. At a site with a sideward-looking ADVM, a rotation of approximately 5 degrees caused a 

shift of 4.7 percent between the predicted-mean velocity and 

the measured-mean velocity (fig. E3). The differences were 

greater at lower flows (Vi < 2.0 ft/sec), with a discrepancy 

of 7 percent. This slight rotation in the ADVM unit caused 

an over-estimation of the discharge at this location until the 

problem was identified and corrected. In general, the effect of 

changes in transducer orientation is site-specific and depends 

on the local lateral velocity structure, which, in turn, depends 

on the local geometry. 

Figure E3. Impacts of misaligned instrumentation on the resulting index velocity versus mean-velocity relation.

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28 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Figure E4. Impacts of improper boat speed on the resulting index-velocity versus mean-velocity relation.

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Discharge Measurement Configuration Files and 

General Boat Operation

There are many other aspects of data acquisition that 

can affect the quality of the calibration. Examples include: 

boat speed; boat pitch and roll; and the configuration of the 

downward-looking vessel mounted ADVM profiler, including 

transducer depth, channel depth, edge estimation, etc. 

Boat speed can affect the calibration in several ways. 

Making discharge measurements when the boat speed is too 

fast leads to decreased precision in the form of an increase in 

variability between successive measurements. If the boat speed 

is too slow, the tide can change the velocities across the channel sufficiently during the period of the discharge measurement to cause a poor correlation between the measured mean 

velocity and the index velocity (fig. E4). Most discharge measurements are being collected in 4 to 5 minutes in a channel 

approximately 600 ft wide, with tidal flows of approximately 

12,000 ft3/s. In this example, the discharge measurement 

(fig. E4, red asterisk) took 11 minutes to complete. Because 

tidal conditions change very rapidly at this location, the resulting mean velocity was 55 percent below the predicted mean 

velocity.

Appendix E. High-Quality Data and Examples of Common Mistakes 29
In a different situation, the discharge measurements collected resulted in a poor correlation with the index velocity 

data (fig. E5). A retrospective analysis of this situation showed 

that the boat speed was too fast; the estimated channel depth 

was approximately five times greater than the actual channel 

depth; the transducer depth was held constant, even though 

the boat loading was changed and the crew moved about the 

boat; and edge estimates were made without using a range 

finder. In a wide and shallow channel, the effects of erroneous 

transducer depth are much greater than in a narrow and deep 

channel. “Eyeball” estimates of edge distances tend to underestimate the distance to shore, are highly variable, and can be 

a significant source of error if the estimated areas are large 

compared to the total cross-sectional volume (Simpson, 2001).

Figure E5. Impacts of poor vessel-mounted downward-looking acoustic Doppler velocity meter configuration and poor boat 

operation techniques on the resulting index velocity versus mean velocity relation.

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30 Computation of Discharge Using the Index-Velocity Method in Tidally Affected Areas
Figure E6. Comparison of the difference between subsequent velocity measurements as recorded by the index velocity instrumentation and the vessel-mounted downward-looking acoustic Doppler velocity meter.

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It is difficult to identify a single issue that caused problems in the above example; however, the standard deviation 

between the mean velocity data is approximately double the 

standard deviation between the index velocity data (fig. E6). 

Some variability is expected because of changing field conditions, but the high variability in this calibration suggests that 

greater attention to detail in the field was necessary.

Appendix E. High-Quality Data and Examples of Common Mistakes 31