Annotated Bibliography

Uncertainty Analysis and its Application to\
Hydraulic Measurements and Experimentation

# Uncertainty Analysis -- General Information

Abernethy, A.B., Benedict, R.P., and Dowdell, R.B., 1985. ASME
measurement uncertainty. Journal of Fluids Engineering, Transactions of
the ASME, June 1985. **{Wahl PDF and Wahlin}**

Available on the web at
<http://www.barringer1.com/drbob-bio_files/ASME83.pdf>

This paper gives history, background, and discussion of committee
activities leading to the development of the new (in 1985) ASME
measurement uncertainty methodology, which is the basis for two
ASME/ANSI standards (ASME PTC 19.1 Measurement Uncertainty among them).

AIAA, 1999. Assessment of Experimental Uncertainty with Application to
Wind Tunnel Testing, Standard S-071A-1999. American Institute of
Aeronautics and Astronautics. 84 pp. **{Wahl}**

The 1995 version of this standard has been proposed by Muste for
adoption by the hydraulic engineering community. This is the latest
version.

ASME. (1991).

-   September 1991 Editorial on Experimental Uncertainty. **{Wahl
    [PDF](../annotated_bibliography_files/Experimental%20Uncertainty.pdf)}**

-   Journal of Fluids Engineering Policy on Reporting Uncertainties in
    Experimental Measurements and Results. **{Wahl
    [PDF](../annotated_bibliography_files/Experimental%20Uncertainty.pdf)}**

-   Journal of Fluids Engineering Editorial Policy Statement on the
    Control of Numerical Accuracy. **{Wahl
    [PDF](../annotated_bibliography_files/Numerical%20Accuracy.pdf)}**

Guidelines for presenting uncertainty analysis results for papers
submitted to the Journal of Fluids Engineering. JFE has had some
requirement for uncertainty analysis in submitted papers since 1975.

ASME. (????). "Policy on reporting uncertainties in experimental
measurements and results." *J. Heat Transfer*. **{Wahlin, PDF}**

Guidelines for presenting uncertainty analysis results for papers
submitted to the Journal of Heat Transfer.

Belter, D.L. "Application of uncertainty methodology at the Boeing
Aerodynamics Laboratory." 19^th^ AIAA Advanced Measurement and Ground
Testing Technology Conference, New Orleans, LA, June 17-20, 1-6, AIAA
Paper 96-2215. {**Wahlin, PDF}**

This paper is a brief overview of how the Boeing Aerodynamics Laboratory
is applying the following standards: 1) Advisory Group for Aerospace
Research and Development AGARD-AR-304 "Quality assessment for wind
tunnel testing" and 2) American Institute of Aeronautics and
Astronautics AIAA S-071-1995 "Assessment of wind tunnel data
uncertainty."

Brown, K.K., and Coleman H.W. (1996). *Determination of uncertainties
for the new SSME model.* NASA contract number NAS8-38609 D.O. 140.
Propulsion Research Center, University of Alabama, Huntsville, AL.
**{Wahlin}**

This work presents uncertainty information for measurements
(temperature, pressure, rotation speed, valve positions, etc.). It has a
good overview of uncertainty analysis and a good section on how to
assess the uncertainty in linear regression. SSME stands for Space
Shuttle Main Engine. Delivery Order (D.O.) 106 dealt with experimental
uncertainty in venturi meter flow rates (this is summarized in Brown et
al. (1998)).

Brown, K.K., Coleman, H.W., Steele, W.G., and Taylor, R.P. (1996).
"Evaluation of correlated bias approximations in experimental
uncertainty analysis." *AIAA Journal*, 34(5), 1013-1018. **{Wahlin}**

The authors present a method for approximating the effect of correlated
bias errors in experimental uncertainty analysis. This is a very
interesting paper.

Brown, K.K., Coleman, H.W., and Steele, W.G. (1998). "A methodology for
determining experimental uncertainties in regressions." *J. of Fluids
Engineering*, 120, 445-456. **{Wahlin}**

The authors apply uncertainty propagation techniques to the linear
regression analysis equations. They demonstrate their methodology on the
calibration of a venturi meter. This is a good paper.

Cameron, J.M., 1974. *The Use of the Method of Least Squares in
Calibration,* National Bureau of Standards, Report NBSIR 74-587. **{Wahl
PDF}**

Available on the web at
[http://patapsco.nist.gov/mel/div821/Publications/NBSIR
74-587.pdf](http://patapsco.nist.gov/mel/div821/Publications/NBSIR 74-587.pdf)

Very mathematical discussion of the use of least squares regression.

Castrup, H. (2000). "Estimating bias uncertainty." Integrated Sciences
Group, Bakersfield, CA. **{Wahlin}**

Technical paper discussing methods for estimating bias of reference
parameters. Has a section that talks about drift.

Coleman, H.W., and Steele, W.G. (1995). "Engineering application of
experimental uncertainty analysis." *AIAA Journal*, 33(10), 1888-1896.
**{Wahlin}**

The authors present the assumptions and approximations of ISO's **Guide
to the expression of uncertainty in measurement** (1993). They also
compare this method with previously published uncertainty analysis
approaches.

Coleman, H.W., and Steele, W.G. 1999. *Experimentation and Uncertainty
Analysis for Engineers*, 2^nd^ Ed., John Wiley & Sons, New York, 275 pp.
**{Wahl}**

The authors of this text have been two of the leading developers of
modern uncertainty analysis techniques and have been involved in the
preparation of many of today's most commonly referenced guides on the
topic. The book addresses basic concepts and statistics as well as
applications to experimental planning, design, debugging, execution,
data analysis, and reporting. Appendices provide theoretical background
and a comparison of alternative and previously used uncertainty analysis
methods. This is an excellent resource.

DeLaurentis, D.A., and Mavris, D.N. (2000). "Uncertainty modeling and
management in multidisciplinary analysis and synthesis." American
Institute of Aeronautics and Astronautics. **{Wahlin PDF}**

This paper outlines a formal approach for modeling uncertainty in
multidisciplinary design problems. A new method for propagating the
uncertainty to find robust design solutions is developed and described.

Doiron, T., and Stoup, J., (1997). Uncertainty and Dimensional
Calibrations. *Journal of Research of the National Institute of
Standards and Technology,* Vol. 102, No. 6. 30 pp. **{Wahl PDF}**

Available on the web at
[http://patapsco.nist.gov/mel/div821/Publications/NIST J Res Nov-Dec 97
102-6.pdf](http://patapsco.nist.gov/mel/div821/Publications/NIST J Res Nov-Dec 97 102-6.pdf)

Describes implementation by NIST of measurement uncertainty standards
for dimensional calibrations of several types of gages, including
sieves.

Eisenhart, C., Ku, H.H., and Colle, R. (1983). *Expression of the
uncertainties of final measurement results: Reprints.* National Bureau
of Standards, Washington DC. **{Wahlin}**

These papers stress the need to clearly state the uncertainties of
reported values. It has a nice qualitative discussion of uncertainty and
some tables with the "appropriate" format for reporting uncertainty
estimates of various measurements and values.

Ellison, S.L.R., Rosslein, M., and Williams, A., eds. (2000).
*Quantifying uncertainty in analytical measurement*. EURACHEM/CITAC
Guide CG 4. Available on the web at
<http://www.measurementuncertainty.org/mu/quam2.pdf> **{Wahl and Wahlin
PDF}**

Despite the general-sounding title, this guide is specific to
quantitative chemical analysis. It is of interest primarily to see how
another field has developed a guide for application of the ISO-GUM.
Contains lots of examples.

EURACHEM. (1999). *Quantifying uncertainty in analytical measurement.*
EURACHEM Workshop, Helsinki. **{Wahlin}**

Extensive overview of all aspects of uncertainty analysis. It includes
lots of examples as well as appendices with general statistical
information. It follows ISO's **Guide to the expression of uncertainty
in measurement.**

European Co-operation for Accreditation. (1999). *Expression of the
uncertainty of measurement in calibration*. Publication Reference
EA-4/02. **{Wahlin PDF}**

This publication presents a methodology for expressing uncertainty for
accredited calibration laboratories. It is very similar to ISO's **Guide
to the expression of uncertainty in measurement**. There are many
examples at the end of this publication.

Holman, J.P. (1989). *Experimental methods for engineers.* Fifth
edition, McGraw-Hill, New York, NY. **{Wahlin}**

Contains some basic uncertainty analysis stuff. Mostly concentrates on
measurement techniques for various parameters (e.g., temperature,
pressure, and flow).

Hudson, S.T., Bordelon, W.J., and Coleman, H.W. (1996). "Effect of
correlated precision errors on uncertainty of a subsonic Venturi
calibration." *AIAA Journal*, 34(9), 1862-1867. **{Wahlin}**

This paper demonstrates that correlated precision errors are not always
negligible and that they can have a significant influence on the results
of an uncertainty analysis.

Intergovernmental Panel on Climate Change. (????). *Conceptual basis for
uncertainty analysis*. IPCC good practice guidance and uncertainty in
national greenhouse gas inventories, annex 1. **{Wahlin PDF}**

Qualitative assessment of uncertainty for greenhouse gas inventories.
Relies heavily on ISO standards.

ISO. Guide to the Expression of Uncertainty in Measurement (ISO-GUM)

The ISO Guide to the Expression of Uncertainty in Measurement, or GUM,
1995, is available in U.S. (\$25) and international (\$92) editions. The
editions contain the same material, differing only in decimal marker,
spelling, and size. The ISO International Vocabulary of Basic and
General Terms in Metrology (VIM), 1993, a companion document to the GUM,
is available only in an international edition (\$71). The U.S. edition
of the GUM is entitled: American National Standard for Expressing
Uncertainty\--U.S. Guide to the Expression of Uncertainty in
Measurement, ANSI/NCSL Z540-2-1997

I do not have this publication.

Kline, S.J., and McClintock, F.A. (1953). "Describing uncertainties in
single-sample experiments." *Mechanical Engineering*, 75, 3-7.
**{Wahlin}**

This is one of the first papers on uncertainty analysis. Apparently,
this is the first paper to present the root-mean-sum equations for
uncertainty.

Kline, S.J., 1985. The purposes of uncertainty analysis. *Journal of
Fluids Engineering,* Transactions of the ASME, June 1985. **{Wahl and
Wahlin}**

This was the introductory paper in a special issue of JFE that presented
several papers from a 1983 symposium on uncertainty analysis. The paper
touches such topics as the concepts of uncertainty in experiments, the
uses of uncertainty analysis, and whether or not uncertainty analyses
are worthwhile. The second appendix to the paper presents several
persuasive case histories that illustrate the potential value of
uncertainty analysis.

Kline, S.J., 1985. Closure to 1983 Symposium on Uncertainty Analysis.
Journal of Fluids Engineering, Transactions of the ASME, June 1985.
**{Wahl and Wahlin}**

This is an excellent summary of the outcomes of the 1983 symposium on
uncertainty analysis. Kline emphasizes two important conclusions:

1.  Uncertainty analysis is an essential ingredient in planning,
    controlling, and reporting experiments. *The important thing is that
    a reasonable uncertainty analysis be done.* **ALL DIFFERENCES OF
    OPINION ABOUT APPROPRIATE METHODS ARE SUBSIDIARY TO THIS
    CONCLUSION.** (my emphasis)

2.  It is particularly important to use an uncertainty analysis in the
    planning and check out stages of an experiment!

Ku, H.H. 1988. *Statistical Concepts in Metrology --- With a Postscript
on Statistical Graphics,* National Bureau of Standards Special
Publication 747. **{Wahl PDF}**

Available on the web at
[http://patapsco.nist.gov/mel/div821/Publications/NBS Special
Publication
747.pdf](http://patapsco.nist.gov/mel/div821/Publications/NBS Special Publication 747.pdf)

A detailed review of statistical concepts that relate to uncertainty
analysis. The postscript on statistical graphics is a good source of
ideas for how to examine experimental results graphically during the
debugging phase of an experiment.

Lassahn, G.D., 1985. Uncertainty definition. Journal of Fluids
Engineering, Transactions of the ASME, June 1985. **{Wahl and Wahlin}**

From the 1983 symposium on uncertainty analysis. This short,
thought-provoking paper addresses philosophical questions related to
barriers to our understanding and use of uncertainty analysis concepts
and procedures.

Mandel, J., and Nanni, L. (1986). *Measurement evaluation*. National
Bureau of Standards Special Publication 700-2, Industrial Measurement
Series. **{Wahl PDF}**

Available on the web at
[http://patapsco.nist.gov/mel/div821/Publications/NBS Special
Publication
700-2.pdf](http://patapsco.nist.gov/mel/div821/Publications/NBS Special Publication 700-2.pdf)

This work is a good overview of basic statistical concepts and control
charts. This work was developed for health-related fields of research
and is a good presentation of familiar topics from a different
perspective.

Mays, L.W., and Tung, Y.K. (1992). *Hydrosystems engineering and
management*. McGraw-Hill, New York, NY. **{Wahlin}**

Some basic uncertainty analysis. Concentrates mostly on reliability
issues.

Moffat, R.J. (1982). "Contributions to the theory of single-sample
uncertainty analysis." *Journal of Fluids Engineering,* Vol. 104,
250-260. **{Wahl and Wahlin}**

Moffat presents three levels of uncertainty analysis which he calls
zeroth, first, and Nth order. In zeroth order analysis the experiment is
assumed rock-steady in time, so that the only errors are due to
interpolation uncertainty (i.e., errors in reading experimental
displays). This is used mainly for preliminary planning. A first order
analysis includes process unsteadiness effects and interpolation
effects, but instrument properties (i.e., their uncertainty
characteristics) are assumed fixed in time. This is useful during the
developmental stage of an experiment. The Nth order analysis includes
uncertainties due to interpolation, unsteadiness, and instrument
calibration variations during the experiment. It is used for reporting
the final results. This paper was developed in connection with heat
transfer and fluid mechanics research experiments. The concepts
introduced here have been incorporated into modern uncertainty analysis
methodology. The discussions of this paper by Dowdell and Abernethy are
also very interesting (published in same volume).

Moffat, R.J. (1985). "Using uncertainty analysis in the planning of an
experiment." *Journal of Fluids Engineering*, Vol. 107, 173-178.
**{Wahlin}**

Moffat outlines a procedure for using uncertainty analysis in the
planning of experiments. This paper is from the 1983 symposium on
uncertainty analysis.

Moffat, R.J. (1988). "Describing the uncertainties in experimental
results." *Experimental Thermal and Fluid Science*, 1, 3-17.
**{Wahlin}**

This paper presents a general description of the sources of errors in
engineering measurements and the relationship between error and
uncertainty. The basic mathematics of both single-sample and
multiple-sample analysis are also presented. Geared for heat transfer
and fluid mechanics.

National Engineering Laboratory. (2002). *Flowmetering current practice
and uncertainty case studies*. A report for NMSD Department of Trade and
Industry, London, UK. **{Wahlin PDF}**

A British publication that discusses the issues which can affect the
quality of flow data and considers data analysis techniques which can be
used to indicate flow meter performance. This publication has some good
examples.

Nielsen, H.S. 1997. Using the ISO Guide to the Expression of Uncertainty
in Measurements to determine calibration requirements. National
Conference of Standards Laboratories Workshop & Symposium.
<http://www.hn-metrology.com/isogum.htm> **{Wahl}**

More information on how the measurement standards industry is applying
the ISO standards for expressing uncertainty.

Phillips, S.D., and Eberhardt, K.R. (1997). "Guidelines for expressing
the uncertainty of measurement results containing uncorrected bias." *J.
of Research of the National Institute of Standards and Technology*, 102,
577-585. **{Wahlin}**

Extends ISO's **Guide to the Expression of Uncertainty in Measurement**
by accounting for uncorrected systematic errors.

Taylor, B.N., and Kuyatt, C.E. 1994. Guidelines for Evaluating and
Expressing the Uncertainty of NIST Measurement Results. National
Institute of Standards and Technology Technical Note 1297. **{Wahl
PDF}**

Available on web at <http://physics.nist.gov/Document/tn1297.pdf> and
<http://physics.nist.gov/Pubs/guidelines/contents.html>

An example of one industry's guidelines for applying the ISO standards.
Very comprehensive, with lots of definitions, appendices, and
references.

Taylor, J.R. (1997). *An introduction to error analysis: the study of
uncertainties in physical measurements.* University Science Books,
Sausalito, CA. **{Wahlin}**

Very good introduction to uncertainty analysis. It is written in plain
English and is very easy to understand. The 1982 edition of this text
was recommended by S.J. Kline in his closure to the 1983 ASME Symposium
on Uncertainty Analysis.

Tripp, J.S., and Tcheng, P. (1999). *Uncertainty analysis of instrument
calibration and application.* NASA Technical Paper 1999-209545.
**{Wahlin}**

Uses confidence and prediction intervals to determine the uncertainty in
the calibration procedure.

Tung, Y.K. (1999). "Risk/reliability-based hydraulic engineering
design." *Hydraulic design handbook*. Ed. Mays, L. McGraw-Hill, New
York, NY. **{Wahlin}**

Tung gives an overview of uncertainty analysis, including some
techniques that I have never heard of such as Mellin Transformation and
Rosenblueth's Probabilistic Point Estimation Method. Most of the chapter
concentrates on reliability analysis.

USACE. (1996). *Risk-based analysis for flood damage reduction studies.*
USACE Engineering Manual 1110-2-1619. **{Wahlin PDF}**

This manual has chapters on the uncertainty of discharge-probability
functions, stage-discharge functions, and stage-damage functions.

Yen, B.C., and Tung, Y.K. (1993). *Reliability and uncertainty analysis
in hydraulic design*. ASCE, New York, NY. **{Wahlin}**

This book is a collection of papers on reliability and uncertainty
analysis. It covers everything from new advances in uncertainty analysis
to dam safety to coastal floods. There is one article by Tung and Yen
about new advances in uncertainty analysis. I have never heard of some
of these new methods.

# Web Resources On Uncertainty Analysis

<http://physics.nist.gov/cuu/> -- The NIST Reference on Constants, Units
and Uncertainty.

<http://physics.nist.gov/cuu/Uncertainty> -- Uncertainty of Measurement
Results: Essentials of Expressing Measurement Uncertainty.

[http://www.gum.dk](http://www.gum.dk/) -- A web site dedicated to
information on the ISO-GUM. Includes English and Danish pages, and
commercial software (GUM Workbench) for performing uncertainty analysis.

<http://www.measurementuncertainty.org/> -- A European source of
information about uncertainty analysis. Apparently sponsored by an
organization called Eurachem. This site provides a guidance document for
analytical chemistry applications (the EURACHEM/CITAC Guide), example
applications, a glossary, and a relatively active discussion forum.
***At this time, this site only works with Internet Explorer.***

<http://patapsco.nist.gov/mel/div821/Publications/Publications.asp> --
Engineering Metrology Toolbox: Selected Publications. Provides links to
several NIST publications related to uncertainty.

<http://web.usna.navy.mil/~ratcliff/EM375/Errors/Errors.pdf> --
*Uncertainty in measurements and the propagation of errors.* Apparently
a set of course notes.

# Applying Uncertainty Analysis to Hydraulic Engineering

Muste, M., and Stern, F. (2000). "Proposed uncertainty assessment
methodology for hydraulic and water resources engineering." *Joint ASCE
Conference on Water Resources Engineering and Water Resources Planning
and Management*, Minneapolis, MN, July 30 -- August 2. **{Wahl and
Wahlin}**

This conference paper advocates the use of AIAA standard S-071-1995 as
the standard that ASCE adopts for uncertainty analysis. It also gives a
brief overview of S-071-1995. There is an updated version of this
standard (S-071A-1999).

Ogden, F.L. (1996). "Experimental uncertainty and measurement errors in
hydraulic engineering." *Proc. ASCE North American Water and Environment
Congress*, Anaheim, CA. **{Wahl and Wahlin PDF}**

This paper outlines the objectives of a previous formulation of this
task committee.

Stern, F., Muste, M., Beninati, M.L., and Eichinger W.E. (1999).
"Summary of experimental uncertainty assessment methodology with
example." *IIHR Technical Report No. 406*, University of Iowa, Iowa
City, IA. **{Wahlin PDF}**

This report proposes using AIAA standard S-071-1995 for uncertainty
analysis. They also use a lot of stuff from Coleman and Steele. The
example they present is on the uncertainty of the measurement of density
and kinematic viscosity. There is an updated version of this standard
(S-071A-1999).

# Hydraulic Parameters - Miscellaneous

Cesare, M.A. (1991). "First-order analysis of open-channel flow."
*Journal of Hydraulic Engineering,* ASCE, 117(2), 242-247. **{Wahlin}**

This paper uses first-order reliability methods to incorporate
uncertainties involved in open-channel hydraulics and compares the
results to the traditional method.

Johnson, P.A. (1996). "Uncertainty of hydraulic parameters." *Journal of
Hydraulic Engineering*, ASCE, 122(2), 112-114. **{Wahlin}**

This paper presents a summary of uncertainty work that has been done in
the past. Parameters examined include: Manning's n, channel slope,
particle size, friction slope, sediment specific weight, and flow
velocity.

Lee, H.L., and Mays, L.W. (1986). "Hydraulic uncertainties in flood
levee capacity." *Journal of Hydraulic Engineering*, ASCE, 112(10),
928-934. **{Wahlin}**

The hydraulic uncertainty is evaluated through a first-order analysis of
uncertainties of Manning's equation. This study shows that the roughness
coefficient and the friction slope dominate the variation in hydraulic
uncertainty in that they account for 95% of the hydraulic uncertainty.

Tung, Y.K. (1990). "Mellin transform applied to uncertainty analysis in
hydrology/hydraulics." *Journal of Hydraulic Engineering*, ASCE, 116(5),
659-674. **{Wahlin}**

The author introduces a mathematical technique called the Mellin
transform. He gives two examples to demonstrate the application of the
Mellin transform to uncertainty analysis of hydrologic and hydraulic
problems.

Yeh, K.C., and Tung, Y.K. (1993). "Uncertainty and sensitivity analysis
of pit-migration model." *Journal of Hydraulic Engineering*, ASCE,
119(2), 262-283. **{Wahlin}**

This paper analyzes the uncertainties of a pit migration model using
three methods including the first-order variance estimation method, the
point estimation technique, and Latin hypercubic sampling.

# Discharge Measurement -- General Information

Abernethy, A.B., 1985. Fluid Flow Measurement Uncertainty. ISO/DIS 5168,
10th draft. 54 pp. **{Wahl
[PDF](../annotated_bibliography_files/ISO-DIS-5168%20Draft%2010.pdf). I
have assembled the 4 PDF files at this site into a single file.}**

Available on the web at <http://www.barringer1.com/drbob-bio.htm>.

According to a handwritten note from Abernethy included in the PDF file,
this standard was approved by unanimous Committee vote in 1987, and by
world vote in 1988, (17 for, France and Italy against), but was never
published because the French delegation controlled the ISO TC30
Secretariat.  Dr. Abernethy believes this earlier 10^th^ draft was
better than the 12^th^ draft, which was the one approved by vote. It was
apparently later published, as a 1998 version is for sale from ISO.
Abernethy says this was the best standard he ever wrote.

I have also seen references in some papers to a 2001 version. We do not
have the official ISO publication.

ISO/TR 7066-1:1997   Assessment of uncertainty in calibration and use of
flow measurement devices \-- Part 1: Linear calibration relationships

We do not have this publication

ISO 7066-2:1988   Assessment of uncertainty in the calibration and use
of flow measurement devices \-- Part 2: Non-linear calibration
relationships

We do not have this publication

# Streamflow -- Indirect Estimation Methods

Hardison, C.H., and Moss, M.E. (1972). "Accuracy of low-flow
characteristics estimated by correlation of base-flow measurements."
*Manual of hydrology: Part 2. Low-flow techniques*, USGS Water-Supply
Paper 1542-B. **{Wahlin}**

The authors present equations and graphs to evaluate the accuracy of low
flow measurements at ungaged sites. The method is based on the
regression equation and the length of record used. There is an appendix
at the end where E.J. Gilroy outlines the derivations of the equations.

Moss, M.E. (1972). *Serial-correlation structure of discretized
streamflows.* USGS Open-File Report 72-262, Fort Collins, CO.
**{Wahlin}**

Moss presents a model to estimate the serial-correlation structure of
discretized streamflows on a monthly and annual basis. The model
accounts only for that component of the correlation that is caused by
baseflow. Direct runoff is treated as random noise. **Not particularly
useful.**

Murdock, R.U., and Gulliver, J.S. (1993). "Prediction of river discharge
at ungaged sites with analysis of uncertainty." *J. Water Resources,
Planning, and Management*, 119(4), 473-487. **{Wahlin}**

The authors outline a technique for assigning uncertainty to the
estimation of the cumulative probability distribution functions
(flow-duration curves) for river discharge at ungaged sites. **Not
particularly useful.**

# Current-Meter Discharge Measurements

Fenton, J.D. (2002). "The application of numerical methods and
mathematics to hydrography." *Proc. 11^th^ Australasian Hydrographic
Conference*, Sydney, Australia, July 3-6. **{Wahlin PDF}**

This article concentrates on the uncertainty introduced due to the
assumed velocity profile (mean velocity in a vertical).

Herschy, R.W. (1978). "Accuracy." *Hydrometry: Principles and
practices*. Ed. Herschy, R.W., John Wiley and Sons. **{Wahlin}**

This chapter presents methods for estimating the uncertainty of
current-meter discharge measurements, weirs and flumes, and
stage-discharge relations. This is very similar to the ISO stuff.

Herschy, R.W. (2002). "The uncertainty in a current-meter measurement."
*Flow Measurement and Instrumentation*, 13, 281-284. **{Wahlin PDF}**

This is a short summary of the uncertainty stuff from Herschy's
Hydrometry book.

International Organization for Standardization. (1985). *Liquid flow
measurement in open channels -- velocity-area methods -- collection and
processing of data for determination of errors in measurements.* ISO
1088, Geneva, Switzerland. **{Wahlin}**

ISO's take on estimating the uncertainty of a current-meter discharge
measurement. Similar to the approach taken by Pelletier (1998) and Sauer
and Meyer (1992).

Lintrup, M. (1989). "A new expression for the uncertainty of a current
meter discharge measurement." *Nordic Hydrology*, 20, 191-200.
**{Wahlin}**

The author presents a method for estimating the uncertainty of a
current-meter discharge measurement. It is not nearly as useful and
Sauer and Meyer's paper.

Pelletier, P.M. (1988). "Uncertainties in the single determination of
river discharge: a literature review." *Can. J. of Civ. Engr.*, 15,
834-850. **{Wahlin}**

This publication presents a summary of over 140 publications on
determining the uncertainty of current-meter discharge measurements. It
is set up similar to Sauer and Meyer's paper, but it is not as good.
Still, it's much better than Lintrup's paper.

Sauer, V.B., and Meyer, R.W. (1992). *Determination of error in
individual discharge measurements*. USGS Open-File Report 92-144,
Norcross, GA. **{Wahlin}**

The authors present a very good procedure for estimating the uncertainty
of a current-meter discharge measurement.

Wahlin, B.T., Clemmens, A.J., and Replogle, J.A., (2001) Procedure for
estimating measurement accuracy for surface water flows, Journal of
Irrigation and Drainage Engineering. **{Wahlin}**

This paper presents a methodology for determining the accuracy of
measured flow rates and computed water volumes for surface water flows
measured with current meters, as is typical in rivers and large canals.

Whalley, N., Iredale, R.S., and Clare, A.F. (2001). "Reliability and
uncertainty in flow measurement techniques -- some current thinking."
*Phys. Chem. Earth (C)*, 26(10-12), 743-749. **{Wahlin PDF}**

This is an article from the UK. It presents some qualitative, rather
than quantitative, aspects of current-meter discharge measurements. It
concentrates on calibration.

# Stage-Discharge Relationships

Bailey, J.F., and Ray, H.A. (1966). "Definition of stage-discharge
relation in natural channels by step-backwater analysis." USGS
Water-Supply Paper 1869-A. **{Wahlin}**

This is a comparison of stage-discharge relations developed from
current-meter discharge measurements and from step-backwater analysis.
The agreement between the two methods is good (i.e., within ± 20%).

Dymond, J.R., and Christian, R. (1982). "Accuracy of discharge
determined from a rating curve." *Hydrological Sciences Journal*, 27(4),
493-504. **{Wahlin}**

An error analysis shows that three types of errors influence the random
error of a single discharge measurement determined from a rating curve.
They are rating curve error, water level measurement error, and an error
caused by ignoring all physical parameters, other than water level, that
affect discharge. Methods in the literature for evaluating the first two
types of errors are reviewed and a method for evaluating the third type
is given. The error of average discharge for an arbitrary period is also
considered.

Freeman, G.E., Copeland, R.R., and Cowan, M.A. (1995). "Quantifying
stage discharge uncertainty at gaging stations." *Proc., 1^st^
International Conference sponsored by Water Resources Engineering,
ASCE*, San Antonio, TX, 14-18 August, 1779-1783. **{Wahlin}**

The authors analyzed the uncertainty associated with stage-discharge
relations for more than 100 sites in the U.S.

Freeman, G.E., Copeland, R.R., and Cowan, M.A. (1996). "Uncertainty in
stage-discharge relationships." *Proc., 7^th^ IAHR International
Symposium*, MacKay, Queensland, Australia, 29-31 July. **{Wahlin}**

The authors evaluated the uncertainty of 116 gage locations. The
uncertainty was broken into three parts: 1) natural, 2) measurement, and
3) modeling. They also developed relationships to estimate the
uncertainty of the stage-discharge relationship for ungaged streams and
for the estimation of Manning's n values.

Herschy, R.W. (1994). "The analysis of uncertainties in the
stage-discharge relation." *Flow Meas. Instrum.*, 5(3), 188-190.
**{Wahlin}**

This is a very short paper that estimates the uncertainty of a
stage-discharge relation. It essentially is a procedure outlining how to
determine the uncertainty of a curve fit.

International Organization for Standardization. (1996). *Measurement of
liquid flow in open channels -- Part 1: Establishment and operation of a
gauging station.* ISO 1100-1, Geneve, Switzerland. **{Wahlin}**

This standard does not talk about uncertainty, but it's related to Part
2, which does.

International Organization for Standardization. (1998). *Measurement of
liquid flow in open channels -- Part 2: Determination of the
stage-discharge relation.* ISO 1100-2, Geneve, Switzerland. **{Wahlin}**

This standard has a brief section on estimating the uncertainty of the
stage-discharge relation as well as the uncertainty of the daily mean
discharge, the monthly mean discharge, and the annual mean discharge.
This methodology is very similar to the one outlined by Herschy.

Schmidt, A.R. (2002). *Analysis of stage-discharge relations for
open-channel flows and their associated uncertainties*. Ph.D.
Dissertation, University of Illinois at Urbana-Champaign, Department of
Civil and Environmental Engineering. **{Wahlin PDF}**

I'm waiting for Schmidt to send me his entire dissertation. Right now, I
only have chapter 3, which has a section summarizing previous efforts to
describe the uncertainty in stage-discharge relationships. His
description is very thorough. I also have his bibliography which pointed
out many other references we should try to obtain.

USACE. (1996). "Uncertainty of stage-discharge function." Chapter 5,
*Risk-based analysis for flood damage reduction studies*. EM
1110-2-1619. **{Wahlin}**

The uncertainty was broken into three parts: 1) natural, 2) measurement,
and 3) modeling. They also developed relationships to estimate the
uncertainty of the stage-discharge relationship for ungaged streams and
for the estimation of Manning's n values.

Venetis, C. (1970). "A note on the estimation of the parameters in
logarithmic stage-discharge relationships with estimates of their
error." *Bulletin of the International Association of Scientific
Hydrology*, International Union of Geodesy and Geophysics, XV(2),
105-111. **{Wahlin}**

The author estimates the uncertainty of the parameters of a
stage-discharge relationship using regression analysis.

# Weirs and Flumes

Abt, S.R., Florentin, C.B., Genovez, A., and Ruth, B.C., 1995.
Settlement and submergence adjustments for Parshall flume. Journal of
Irrigation and Drainage Engineering, Vol. 121, No. 5. **{Wahl}**

A large set of experiments were conducted on small Parshall flumes (2-ft
throat width or less) to develop a procedure for adjusting measurements
to correct for effects of submergence and differential settlement which
produces lateral and/or longitudinal slope of the flume. Several years
ago Dr. Abt provided me with a computer program to perform the
calculations. I have noted some inconsistencies in the data tables of
this paper, and was told by Dr. Abt that a 1998 version of the paper
corrects the problems, but I do not have a copy of it and do not know
the exact reference.

Abt, S.R., and Ruth, B.C., 1997. Flume condition assessment in Colorado.
Journal of the American Water Resources Association, Vol. 33, No. 1,
February 1997. **{Wahl}**

Field installations of Parshall flumes were evaluated, and serious
problems were found with more than half of the installations.

Clemmens, A., Wahl, T., Bos, M., and Replogle, J. (2001). *Water
measurement with flumes and weirs*. International Institute for Land
Reclamation and Improvement Publication 58, Wageningen, The Netherlands.
**{Wahl}**

Authors give some basic uncertainty concepts related to flow measurement
flumes.

Dodge, R.A., 1990. Effects of Mountain Stream Topography on the Accuracy
of Small Parshall Flumes, U.S. Dept. of the Interior, Bureau of
Reclamation, Research Report R-90-03. **{Wahl}**

This laboratory study of 6- and 9-inch Parshall flumes investigated
approach flow problems with installations in small mountain streams. The
study developed upstream pool modifications to improve approach flow
conditions and measurement accuracy. The focus of the study is on
eliminating bias errors and the flow conditions that contribute to
excessive random uncertainty.

Jones, R.W. (2002). "A method for comparing the performance of open
channel velocity-area flow meters and critical depth flow meters." *Flow
Measurement and Instrumentation*, 13, 285-289. **{Wahlin PDF}**

Jones assumes that the uncertainty of critical depth flow meters (e.g.,
weirs and flumes) is well defined, and he uses them as a bench mark to
compare the performance of velocity-area flow meters. This work only
applies to rectangular channels.

Peck, H., 1988. Submerged flow in Parshall flumes. Model-Prototype
Correlation of Hydraulic Structures, International Symposium, Colorado
Springs, CO Aug. 9-11, 1988. **{Wahl}**

This laboratory study of 1-ft Parshall flumes revealed a discontinuity
in the discharge-head relationship originally developed by Parshall.
Parshall's original data was too sparse to reveal the problem. For a
range of submergences, two different discharges are possible. Errors as
large as 12 percent can occur between actual discharge and that
calculated from Parshall's rating equation. Improved submergence
corrections were developed, and a submergence limit of 86 percent was
recommended to avoid the zone of discontinuity.

Thomas, C.W. (1959). "Errors in measurement of irrigation water." *ASCE
Transactions*, 124, 319-340. **{Wahlin}**

This is a very old work that attempts to identify sources of error in
Parshall flumes and weirs. It points out many possible sources of error
(e.g., sloping weir crest, ignoring the velocity of approach, rounding
of the crest of the weir, etc.) and gives a quantitative approximation
of the error for each identified source.

# Annual Discharge

Anning, D.W. (2002). "Uncertainty in annual streamflow and change in
reservoir content data from selected surface-water gaging stations on
the Lower Colorado River streamflow-gaging station network 1995-99."
USGS Fact Sheet 108-01, Tucson, AZ. **{Wahlin PDF}**

Qualitative outline of procedure used to estimate the standard error of
annual discharges at various sites along the Lower Colorado River.

Anning, D.W. (2002). "Standard errors of annual discharge and change in
reservoir content data from selected stations in the Lower Colorado
River streamflow-gaging station network, 1995-99." USGS Water-Resources
Investigation 01-4240, Tucson, AZ. **{Wahlin}**

Updated version of Moss and Gilroy's method. Applied to selected sites
on the Lower Colorado River.

Burkham, D.E., and Dawdy, D.R. (1970). Error analysis of streamflow data
for an alluvial *stream.* USGS Professional Paper 655-C. **{Wahlin}**

The authors computed the uncertainty of a computed discharge at a site
by randomly choosing a group of discharge measurements for use in a
rating analysis and using the remaining elements as a control group. The
variance of the computed discharge was obtained by subtracting the
variance of between the measured and true discharge from the variance
between the measured and computed discharge. I don't like this approach
very much.

Clarke, R.T. (1999). "Uncertainty in the estimation of mean annual flood
due to rating-curve indefinition." *J. of Hydrology*, 222, 185-190.
**{Wahlin PDF}**

The author assumes the stage-discharge relation can be described as Q =
γ(h + α)^β^. He takes into account the uncertainty in the parameter α.
The annual maximum discharges are all estimated from the same
stage-discharge relations, so they are correlated. This correlation is
not usually accounted for. Clarke presents a method to estimate the
uncertainty of the mean annual flood flow that takes into account this
correlation as well as the uncertainties in the stage-discharge
relationship.

Clarke, R.T., Mendiondo, E.M., and Brusa, L.C. (2000). "Uncertainties in
mean discharge from two large South American rivers due to rating curve
variability." *Hydrological Sciences*, 45(2), 221-236. **{Wahlin}**

The authors assume that the sequence of mean annual discharges is a
stationary time series with uncertainty expressed by the standard
deviation of annual discharge. They use the uncertainty of the curve-fit
stage-discharge relationship and the correlation between estimated mean
annual discharges to obtain the uncertainty of the mean annual
discharge.

Fontaine, R.A. (1983). Uncertainties in records of annual mean discharge
in Maine. USGS Water-Resources Investigations 83-4025, Augusta, ME.
**{Wahlin}**

Moss and Gilroy's uncertainty analysis applied to rivers in Maine.

Fontaine, R.A., Moss, M.E., Smath, J.A., and Thomas, W.O. (1984). Cost
effectiveness of the *stream-gaging program in Maine -- a prototype for
nationwide implementation.* USGS Water-Supply Paper 2244. **{Wahlin}**

A more detailed account of Fontaine's earlier publication.

Gilroy, E.J., and Moss, M.E. (1981). Cost-effective streamgaging
strategies for the Lower *Colorado River Basin.* USGS Open-File Report
81-1019. **{Wahlin}**

Application of Moss and Gilroy's uncertainty analysis to the Lower
Colorado River Basin.

Matsuoka, I., Lee, R., and Thomas, W.O. (1985). Cost-effectiveness of
the stream-gaging program in the Hawaii District. USGS Water-Resources
Investigation Report 84-4126, Honolulu, HI. **{Wahlin}**

Moss and Gilroy's uncertainty analysis applied to rivers in Hawaii.

Meyer, R.W. (1998). Assessment of peak discharge uncertainty in the
American River Basin, *California.* USGS Open-File Report 97-668,
Sacramento, CA. **{Wahlin}**

*Meyer uses flood-discharge data, current-meter discharge measurements,
indirect measurements,* and stage-discharge relations to estimate the
uncertainty of annual peak floods.

Moss, M.E., and Gilroy, E.J. (1980). Cost effective stream-gaging
strategies for the Lower Colorado River Basin: The Blythe field office
operations. USGS Open-File Report 80-1048, Reston, VA. **{Wahlin}**

This is a very interesting paper. Moss and Gilroy assume that the
uncertainty in the annual mean discharge at a stream gage is a function
of the number of current-meter discharge measurements that are performed
at that site. Using Kalman filtering techniques, the authors estimate
the uncertainty in the annual mean discharge as a function of the number
of current-meter discharge measurements that were performed. The
mathematics in this paper are quite complex and some of the concepts are
not explained very well. A couple of years ago, I talked to Moss and
even he couldn't clear up some of my questions. As he was looking over
his paper, he commented on how poorly he had written it. Moss and Gilroy
wrote a computer program to perform the calculations outlined in this
Open-File Report. Moss gave me a printed copy of the code (he wasn't
sure if what he gave me was the complete code). This methodology has not
been used very widely, but it is still very interesting.

Thomas, W.O., and Gilroy, E.J. (unpublished). Computer procedures for
determining cost-effective stream-gaging strategies. Unpublished USGS
Open-File Report. **{Wahlin}**

Description of the computer program used in Moss and Gilroy's
uncertainty analysis.

Wahlin, B.T., Clemmens, A.J., and Replogle, J.A. (1997). Measurement
accuracy for major *surface-water flows entering and leaving the
Imperial Valley*. USDA, ARS, WCL Report 23, Phoenix, AZ. **{Wahlin}**

My attempt at estimating the uncertainty of the annual volume. Simpler
than Moss and Gilroy's method but not as accurate. Considers both random
and systematic uncertainties (Moss and Gilroy only consider random
uncertainties). Heavily dependent on the user's ability to identify and
quantify uncertainties. I am working on a new version of this paper that
is easier to understand and follow.

# Lower Colorado River Accounting System

Lane, W.L. (1998). "Statistical analysis of the 1995 lower Colorado
River accounting system: an assessment of current procedures with
recommended improvements." Prepared for the LCRAS Team and the USBR,
contract number 1425-97-PG-30-07260. **{Wahlin}**

Statistical analysis of the LCRAS procedure. Uses standard statistical
analysis techniques to estimate the uncertainty of various components of
a mass-balance equation for the Colorado River. It has a good overview
of uncertainty with a strong emphasis on systematic errors. Some of his
uncertainty analysis is presented in a slightly different manner than
what I have normally seen. It also discusses digitization and
linearization errors, which aren't typically mentioned.

Lane, W.L. (2002). "Analysis, treatment and propagation of errors for
crop evapotranspiration for the lower Colorado River accounting system."
Prepared for the USBR, Boulder City, NV. **{Wahlin}**

An example of the uncertainty analysis described in Lane's 1998 report.

# Closed-Conduit Flow Meters

Clark, W.J. (1965). *Flow measurement by square-edged orifice plate
using corner tappings.* Pergamon Press, Oxford, UK. **{Wahlin}**

This book has a detailed section of how to properly install an orifice
meter to minimize the error. It covers things like how to tap the pipe,
how much straight approach length is needed, etc.

Hanson, B., and Schwankl, L., 1998. Error Analysis of Flowmeter
Measurements. Journal of Irrigation and Drainage Engineering, Vol. 124,
No. 5, Sept./Oct. 1998, pp. 248-256. **{Wahl}**

This paper examines the effect of flow disturbances on the performance
of several varieties of closed-conduit flow meters, including propeller
meters, pitot-tube devices, paddle wheel meters, and a Doppler acoustic
meter. The effect of straightening vanes was also investigated.

Husain, Z.D., 1995. Theoretical Uncertainty of Orifice Flow Measurement.
7 pp. **{Wahl PDF}**

[http://www.emersonprocess.com/daniel/Products/Gas/Orifice/Senior/AppNotes/Theoretical
Uncertainty of Orifice Flow Measurement
172KB.pdf](http://www.emersonprocess.com/daniel/Products/Gas/Orifice/Senior/AppNotes/Theoretical Uncertainty of Orifice Flow Measurement 172KB.pdf)

Discusses the factors that can cause measurement errors and influence
uncertainty. This report appears to be from a commercial flow meter
company or maybe a consulting firm specializing in orifice meter
applications..

International Organization for Standardization. (1991). *Measurement of
fluid flow by means of pressure differential devices -- Part 1: Orifice
plates, nozzles and Venturi tubes inserted in circular cross-section
conduits running full.* ISO 5167-1, Geneva, Switzerland. **{Wahlin}**

There is a small section on uncertainty in this standard.

Reader-Harris, M.J., Brunton, W.C., Gibson, J.J., Hodges, D., and
Nicholson, I.G. (2001). "Discharge coefficients of Venturi tubes with
standard and non-standard convergent angles." *Flow Measurement and
Instrumentation*, 12, 135-145. **{Wahlin PDF}**

This paper presents some equations for the discharge coefficients of
Venturi meters operating with water and high-pressure gas. It also gives
estimates of the uncertainty of the discharge coefficients. There is a
section on the error introduced from the pressure tap.

Replogle, J.A., 2002. Correcting Unreliable Velocity Distributions in
Short Culverts and Canal Reaches. USCID/EWRI Conference on Energy,
Climate, Environment and Water. San Luis Obispo, California, July 10-13,
2002. **{Wahl}**

This paper presents results of laboratory testing to evaluate the effect
of distorted velocity distributions on the accuracy of closed-conduit
and open channel flow meters and flumes. The majority of the paper talks
about closed-conduit devices. The influence of straightening vanes and
orifice-type flow conditioners is evaluated. Research on this topic is
ongoing.

# Ultrasonic Flow Meters

Johnson, A.L., Benham, B.L., Eisenhauer, D.E., and Hotchkiss, R.H.,
(2001). Ultrasonic Water Measurement in Irrigation Pipelines with
Disturbed Flow. Transactions of the ASAE, Vol. 44 No. 4, pp. 899-910.
**{Wahl}**

This paper investigates accuracy of USFM's at various distances
downstream from flow disturbances commonly encountered in irrigation
systems. Error correction multipliers were developed to overcome bias
errors when flow meters were installed close to flow disturbing devices.

# Acoustic Doppler Current Profilers

Nystrom, E.A., Oberg, K.A., and Rehmann, C.R. (2002). "Measurement of
turbulence with acoustic Doppler current profilers -- sources of error
and laboratory results." ASCE conference on *Hydraulic Measurements and
Experimentation Methods*. Estes Park, CO, July 28 -- August 1. **{Wahlin
PDF}**

The authors identify several sources of error in turbulence measurements
made with ADCPs including: inaccuracy of Doppler-shift measurements,
poor temporal and spatial measurement resolution, and inaccuracy of
multi-dimensional velocities resolved from one-dimensional velocities
measured at separate locations.

Simpson, M.R. (2001). Discharge measurements using a broad-band acoustic
Doppler current profiler. USGS Open-File Report 01-1, Sacramento, CA.
**{Wahlin PDF}**

This Open-File Report has a section which describes the possible sources
of error in an acoustic Doppler discharge measurement.

# Laser Doppler Velocimetry

Martin, P.B., Pugliese, G.J., and Leishman, J.G. (2000). "Laser Doppler
velocimetry uncertainty analysis for rotor blade tip vortex
measurements." American Institute of Aeronautics and Astronautics.
**{Wahlin PDF}**

This paper presents a detailed uncertainty analysis of the beam
alignment and vortex measurement technique used in LDV measurements.
Sources of uncertainty are identified with optics calibration, data
acquisition, and data reduction.

# Environmental Measurements

Yoe, C.E. (1996). *An introduction to risk and uncertainty in the
evaluation of environmental investments.* Institute for Water Resources
Report 96-R-8. **{Wahlin PDF}**

This is a qualitative report designed to introduce people to the
concepts of risk and uncertainty. It has a pretty good description of
the difference between risk and uncertainty.

Yoe, C.E. (1996). *Incorporating risk and uncertainty into environmental
evaluation: an annotated bibliography.* Institute for Water Resources
Report 96-R-9. **{Wahlin PDF}**

A very detailed annotated bibliography on risk and uncertainty analysis.
I need to take a closer look at this.

# Sediment Measurements

McBean, E.A., and Al-Nassri, S. (1988). "Uncertainty in suspended
sediment transport curves." *Journal of Hydraulic Engineering*, ASCE,
114(1), 63-74. **{Wahlin}**

The uncertainties implicit in suspended sediment transport curves are
examined and the practice of using sediment load versus discharge is
shown to be misleading, since the goodness of fit implied by this
relation is spurious.

Shin, H.S., and Salas, J.D. (1996) "Uncertainty analysis of reservoir
sedimentation." *Proc. ASCE North American Water and Environment
Congress*, Anaheim, CA. **{Wahlin PDF}**

In estimating reservoir sedimentation, a number of uncertainties arise.
These are related to annual streamflow, sediment load, sediment particle
size, trap efficiency, and reservoir operation. Monte Carlo simulation
is used to quantify the uncertainty of reservoir sedimentation.

# Pressure Measurements

McKeon, B.J., and Smits, A.J. (2002). "Static pressure correction in
high Reynolds number fully developed turbulent pipe flow." *Measurement
Science and Technology*, 13, 1608-1614. **{Wahlin PDF}**

This is a new report on the error introduced in pressure measurements
from the pressure tap. This study indicates that the correction term for
the pressure tap continues to increase as the hole Reynolds number
increases, which is contrary to previous studies.

Shaw, R. (1960). "The influence of hole dimensions on static pressure
measurements." *J. of Fluid Mechanics*, 7, 550-564. **{Wahlin}**

This paper determines the error in a pressure measurement due to the
size and shape of the pressure tap.

# Numerical Computations and Modeling

Freitas, C.J., Ghia, U., Celik, I., Roache, P., and Raad, P.  (2003). 
\"ASME\'s quest to quantify numerical uncertainty.\"  41st AIAA
Aerospace Sciences Meeting and Exhibit, Reno, NV, January 6-9.  AIAA
Paper 2003-627.  **{Wahlin PDF}**

This paper reviews the formulation of methods for quantifying numerical
uncertainty in simulations.  The authors present a five-step approach
for the estimation of numerical uncertainty based on Richardson
Extrapolation and the Grid Convergence Index (whatever those are).

Oberkampf, W.L, DeLand, S.M, Rutherford, B.M., Diegert, K.V., and Alvin,
K.F. (1999). "A new methodology for the estimation of total uncertainty
in computational simulation." 40^th^ AIAA/ASME/ASCE/AHS/ASC Structures,
Structural Dynamics, and Materials Conference and Exhibit, St. Louis,
MO, April 12-15, 3061-3083, AIAA Paper 99-1612. **{Wahlin PDF}**

This paper develops a general methodology for estimating the total
uncertainty in computational simulations that deal with the numerical
solution of a system of partial differential equations. This methodology
has six phases: 1) conceptual modeling of the physical system, 2)
mathematical modeling of the conceptual model, 3) discretization of the
model, 4) computer programming of the model, 5) numerical solution of
the computer model, and 6) representation of the numerical solution.

Thompson, D.B., and Rogers, T.D. (1993). "Water surface profile
computations -- how many sections do I need?" *Proc., ASCE Conference on
Hydraulic Engineering*, San Francisco, CA, July 25-30, 791-796.
**{Wahlin PDF}**

This is a short article that has two simple examples that illustrate the
importance of using enough computational cross-sections in hydraulic
simulations. In some instances, using too few computational
cross-sections led to errors as large at 2 feet.

USACE. (1986). *Accuracy of computed water surface profiles*. Hydrologic
Engineering Center, RD 26, Davis, CA. **{Wahlin}**

This document describes the results of an investigation of the effects
of using survey and mapping technology for determining cross-sectional
coordinate geometry and the reliability of Manning's roughness
coefficient on the accuracy of computed water surface profiles.

USACE. (1987). *Accuracy of computed water surface profiles: commercial
survey guidelines for water surface profiles.* Hydrologic Engineering
Center, RD 26A, Davis, CA. **{Wahlin}**

This document presents information that can be used to select the
appropriate method of data collection for the development of water
surface profiles.

Wesolowski, E.A. (1996). *Uncertainty analysis of the simulations of
effects of discharging treated wastewater to the Red River of the North
at Fargo, North Dakota, and Moorhead, Minnesota.* USGS Water-Resources
Investigations Report 96-4015, Bismarck, ND. **{Wahlin}**

Wow! Now that's a title. In a previous work, the effects of discharging
treated wastewater into two different rivers were simulated. In this
work, Wesolowski performs a first order uncertainty analysis of the
simulated constituent concentrations and property values using the
Enhanced Stream Water Quality Model-Uncertainty Analysis, whatever that
is.