SIST EN ISO 25377:2023

Hydrometric uncertainty guidance (HUG) (ISO 25377:2020)

This document provides an understanding of the nature of measurement uncertainty and its significance in estimating the "quality" of a measurement or a determination in hydrometry.
This document is applicable to flow measurements in natural and man-made channels. Rainfall measurements are not covered.

Leitfaden zu Messunsicherheiten in der Hydrometrie (HUG) (ISO 25377:2020)

Dieses Dokument bietet einen Beitrag zum Verständnis der Natur der Messunsicherheit und deren Bedeutung bei der Abschätzung der „Qualität“ einer Messung oder einer Bestimmung in der Hydrometrie.
Dieses Dokument ist anwendbar für Abflussmessungen in natürlichen und künstlichen Gerinnen. Messungen der Niederschlagsmenge werden nicht berücksichtigt.

Lignes directrices relatives à l'incertitude en hydrométrie (ISO 25377:2020)

Le présent document fournit une compréhension de la nature de l'incertitude de mesure et son importance pour l'estimation de la «qualité» d'une mesure ou d'une détermination dans le domaine de l'hydrométrie.
Il s'applique aux mesures d'écoulement dans les chenaux naturels et artificiels. Il ne s'applique pas aux mesures de précipitations.

Navodila za določanje merilne negotovosti v hidrometriji (HUG) (ISO 25377:2020)

Ta dokument pojasnjuje naravo merilne negotovosti in njenega pomena pri ocenjevanju »kakovosti« meritev ali določanja v hidrometriji.
Ta dokument se uporablja za merjenje pretoka v naravnih in umetnih kanalih. Dokument ne zajema merjenja količine padavin.

General Information

• Status: Published
• Public Enquiry End Date: 22-Nov-2022
• Publication Date: 22-Jan-2023
• ICS: 17.120.20 - Flow in open channels
• Technical Committee: IMIN - Measurement instruments
• Current Stage: 6060 - National Implementation/Publication (Adopted Project)
• Start Date: 04-Jan-2023
• Due Date: 11-Mar-2023
• Completion Date: 23-Jan-2023
• Ref Project: EN ISO 25377:2022 - Hydrometric uncertainty guidance (HUG) (ISO 25377:2020)

Relations

• Replaces: SIST-TS CEN ISO/TS 25377:2007 - Hydrometric uncertainty guidance (HUG) (ISO/TS 25377:2007)
• Effective Date: 29-Mar-2023

Overview

EN ISO 25377:2022 - Hydrometric uncertainty guidance (HUG) provides practical guidance for understanding, estimating and reporting measurement uncertainty in hydrometry. Endorsed by CEN and identical to ISO 25377:2020, this standard focuses on flow measurements in natural and man-made channels (open-channel flow and structures). It clarifies the nature of random and systematic effects and explains how to quantify the “quality” of a hydrometric measurement. Rainfall measurements are excluded.

Key topics

• Foundations and GUM linkage: aligns with ISO/IEC Guide 98-3 (the GUM) for basic definitions, probability distributions and rules for uncertainty evaluation.
• Uncertainty modelling: treatment of random and systematic effects, combining uncertainties, and expressing results (including propagation of uncertainty).
• Open-channel flow methods: detailed guidance for velocity–area methods (mean velocity, velocity measurement, integration and perimeter flow uncertainties) and critical-depth methods (head, geometry and iterative calculations).
• Dilution and tracer methods: continuous feed and transient mass approaches for discharge estimation.
• Instrumentation and performance: performance specifications, evaluation of manufacturer claims, and a performance guide for hydrometric equipment.
• Practical tools and annexes: worked examples, Monte Carlo Simulation (MCS), Interpolated Variance Estimation (IVE), stage–discharge uncertainty analysis and velocity measurement guidance.
• Standards drafting: guidance for writing uncertainty clauses in hydrometric technical standards.

Applications

EN ISO 25377 is designed for professionals who need rigorous, transparent uncertainty estimates for flow and discharge measurements:

• Hydrologists and water-resource managers assessing monitoring network data quality.
• Gauging-station operators calculating uncertainty for routine discharge measurements.
• Environmental and regulatory bodies requiring documented uncertainty for compliance and reporting.
• Consultants and engineering firms performing flow measurements for design, licensing or flood-risk studies.
• Instrument manufacturers and testing laboratories validating performance claims.
• Standards developers and researchers implementing uncertainty methods (including Monte Carlo approaches).

Practical uses include calibrating rating curves (stage–discharge relations), verifying measurement systems, informing flood forecasting models, water allocation decisions, and compiling defensible monitoring reports.

Related standards

• ISO/IEC Guide 98-3 (GUM) - Evaluation of measurement uncertainty
• ISO 25377:2020 (original international text) - adopted as EN ISO 25377:2022 by CEN

Keywords: hydrometric uncertainty, HUG, ISO 25377, EN ISO 25377:2022, flow measurement, open channel flow, velocity–area method, Monte Carlo simulation, stage–discharge uncertainty.

Standards Content (Sample)

SLOVENSKI STANDARD
01-marec-2023
Navodila za določanje merilne negotovosti v hidrometriji (HUG) (ISO 25377:2020)
Hydrometric uncertainty guidance (HUG) (ISO 25377:2020)
Leitfaden zu Messunsicherheiten in der Hydrometrie (HUG) (ISO 25377:2020)
Lignes directrices relatives à l'incertitude en hydrométrie (ISO 25377:2020)
Ta slovenski standard je istoveten z: EN ISO 25377:2022
ICS:
17.120.20 Pretok v odprtih kanalih Flow in open channels
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EN ISO 25377
EUROPEAN STANDARD
NORME EUROPÉENNE
December 2022
EUROPÄISCHE NORM
ICS 17.120.20
English Version
Hydrometric uncertainty guidance (HUG) (ISO
25377:2020)
Lignes directrices relatives à l'incertitude en Leitfaden zu Messunsicherheiten in der Hydrometrie
hydrométrie (ISO 25377:2020) (HUG) (ISO 25377:2020)
This European Standard was approved by CEN on 20 December 2022.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this
European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references
concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN
member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by
translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management
Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2022 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 25377:2022 E
worldwide for CEN national Members.

Contents Page
European foreword . 3

European foreword
The text of ISO 25377:2020 has been prepared by Technical Committee ISO/TC 113 "Hydrometry” of
the International Organization for Standardization (ISO) and has been taken over as EN ISO 25377:2022
by Technical Committee CEN/TC 318 “Hydrometry” the secretariat of which is held by BSI.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by June 2023, and conflicting national standards shall be
withdrawn at the latest by June 2023.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
Any feedback and questions on this document should be directed to the users’ national standards body.
A complete listing of these bodies can be found on the CEN website.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of
North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and the
United Kingdom.
Endorsement notice
The text of ISO 25377:2020 has been approved by CEN as EN ISO 25377:2022 without any modification.

INTERNATIONAL ISO
STANDARD 25377
First edition
2020-12
Hydrometric uncertainty guidance
(HUG)
Lignes directrices relatives à l'incertitude en hydrométrie
Reference number
ISO 25377:2020(E)
©
ISO 2020
ISO 25377:2020(E)
© ISO 2020
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address
below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved

ISO 25377:2020(E)
Contents Page
Foreword .v
Introduction .vi
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 1
5 ISO/IEC Guide 98-3 (GUM) — Basic definitions and rules . 3
5.1 General . 3
5.2 Uncertainty of sets of measurements . 3
5.3 Random and systematic effects . 4
5.4 Uncertainty models — Probability distributions . 5
5.5 Combining uncertainties — Law of propagation of uncertainties . 5
5.6 Expressing results . 6
6 Open channel flow — Velocity area methods . 6
6.1 General . 6
6.2 Mean velocity, V . 7
x
6.3 Velocity-area method for discharge calculation . 8
6.4 Measurement of velocity . 9
6.5 Uncertainty associated with the velocity-area method . 9
6.5.1 General. 9
6.5.2 Random and systematic effects .11
**
 
6.6 Integration uncertainties uF ,uF .11
()
()
yz
 
6.6.1 General.11
6.6.2 Vertical scanning uncertainties .11
6.6.3 Horizontal scanning uncertainties .12
6.7 Perimeter flow uncertainties, uQ .12
()
p
7 Open channel flow — Critical depth methods .13
7.1 General .13
7.2 Head and geometry determination .13
7.3 Iterative calculation .14
7.4 Evaluating uncertainty .14
8 Dilution methods .15
8.1 General .15
8.2 Continuous feed .15
8.3 Transient mass .17
9 Hydrometric instrumentation .18
9.1 Performance specifications .18
9.2 Validity of uncertainty statements .18
9.3 Manufacturer’s specifications .19
9.4 Performance guide for hydrometric equipment for use in technical standard examples.20
10 Guide for the drafting of uncertainty clauses in hydrometric standards .21
10.1 General .21
10.2 Equipment, methods and measurement systems .21
10.2.1 General.21
10.2.2 Equipment .21
10.2.3 Methods .21
10.2.4 Systems .22
ISO 25377:2020(E)
11 Examples .22
11.1 General .22
11.2 Uncertainty in water level measurement .22
11.2.1 Example 1: Float/shaft encoder sensor installed in stilling well at gauging
station.22
11.2.2 Example 2: Pressure transmitter installed in tube .23
11.3 Uncertainty in flow measurement using flow measurement structures .23
11.4 Uncertainty in flow measurement by current meter .26
Annex A (informative) Introduction to hydrometric uncertainty .31
Annex B (informative) Introduction to Monte Carlo Simulation (MCS) .48
Annex C (informative) Interpolated Variance Estimation (IVE) method .53
Annex D (informative) Performance guide for hydrometric equipment for use in technical
standard examples .57
Annex E (informative) Uncertainty analysis of stage-discharge relation .60
Annex F (informative) Measurement of velocity .64
Bibliography .69
iv © ISO 2020 – All rights reserved

ISO 25377:2020(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/ patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/
iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 113, Hydrometry.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
ISO 25377:2020(E)
Introduction
The management of a natural environment requires knowledge, by measurement, of what is happening.
Only then can effective action be taken and the effectiveness of the action assessed. Much depends on
the quality of the knowledge itself.
The quality of measurable knowledge is stated in terms of measurement uncertainty. The
internationally agreed method for assessing measurement quality is the guide to the estimation of
uncertainty in measurement (GUM). Without this uniformity of measurement standards, equitable
sharing of the environment is not possible and international obligations to care for the environment
would be weakened.
The essential purpose of the GUM is for a statement of the quality of a measurement result to be
presented with all measurements described in technical standards. Without this, no two measurements
can be compared, or standards set. Whereas the GUM is a reference document serving the universal
requirements of metrology, the Hydrometric uncertainty guidance (HUG) document is specific to
hydrometry, i.e. to the measurement of the components of the hydrological cycle. It borrows from
the GUM the methods that are the most applicable to hydrometry and applies them to techniques and
equipment used in hydrometry.
In the past, error analysis has provided an indication of measurement quality, but such statements
cannot properly convey the quality of the result because it presupposes knowledge of a true, error-free,
value against which the measured result can be compared. The true value can never be known and
uncertainty remains. For this reason, the GUM uses the concept of uncertainty and uses it for all stages
and components of the measurement process. This ensures consistency.
The GUM defines standard uncertainty of a result as being equivalent to a standard deviation. This can
be the standard deviation of a set of measured values or of probable values. This is broadly similar to the
approach used in error analysis that preceded the uncertainty technique. However, the GUM provides
additional methods of estimating uncertainty based on probability models. The two approaches are
equivalent, but uncertainty requires only a knowledge or estimate of the dispersion of measurement
about its mean value, and not the existence of a true value. It is assumed that a careful evaluation of the
components of measurement uncertainty brings the mean value close to a probable true value, at least
well within its margin of uncertainty.
In more general terms, uncertainty is a parameter that characterizes the dispersion of measurable
values that can be attributed to their mean value.
By treating standard deviations and probability models as if they approximated to Gaussian (or normal)
distributions, the GUM provides a formal methodology for combining components of uncertainty in
measurement systems where several input variables combine to determine the result.
Within this formal framework, the GUM can be consistently applied to a range of applications and,
thereby, be used to make meaningful comparisons of results.
The HUG seeks to promote an understanding of the nature of measurement uncertainty and its
significance in estimating the ‘quality’ of a measurement or a determination in hydrometry.
Hydrometry is principally concerned with the determination of flow in rivers and man-made channels.
This includes:
— environmental hydrometry, i.e. the determination of the flow of natural waters (largely concerned
with hydrometric networks, water supply and flood protection);
— industrial hydrometry, i.e. the determination of flows within industrial plants and discharges into
the natural environment (largely concerned with environment protection and also irrigation).
Both are the subject of international treaties and undertakings. For this reason, measured data is
intended to conform to the GUM to assure that results can be compared.
vi © ISO 2020 – All rights reserved

ISO 25377:2020(E)
Hydrometry is also concerned with the determination of rainfall, the movement/diffusion of
groundwater and the transport by water flow of sediments and solids. This version of the HUG is
concerned with flow determination only.
The results from hydrometry are used by other disciplines to regulate and manage the environment.
If knowledge is required of biomass, sedimentary material, toxins, etc., the concentration of these
components is determined and their uncertainty estimated. The uncertainty of mass-load can then be
determined from the uncertainty of flow determination. The components of this calculation are made
compatible through compliance with the GUM.
For practitioners of hydrometry and for engineers, the GUM is not a simple document to refer to. The
document has been drafted to provide a legal framework for professional metrologists with a working
knowledge of statistical methods and their mathematical representation. A helpful document, see
Reference [2], is an abbreviated version of the GUM written to be more accessible to engineers and to
specialists in fields other than metrology.
The HUG, although simplifying the concepts, in no way conflicts with the principles and methods
of the GUM. Accordingly, the HUG interprets the GUM to apply its requirements to hydrometry in a
practical way, and, hopefully, in a way accessible to engineers and those responsible for managing the
environment.
In addition, the HUG introduces and develops the Monte Carlo Simulation, a complementary technique,
which has benefits for hydrometry, insomuch as complex measurement systems can be represented
realistically.
The HUG summarizes basic hydrometric methods defined in various technical standards. The
HUG develops uncertainty estimation formulae from the GUM for these basic methods. The basic
hydrometric methods described in the HUG might not be identical to those recited in the published
technical standards. In such cases, the methods described in these standards are to be taken as
authoritative. However, clauses in technical standards that concern uncertainty should be adapted to
be in accordance with the HUG.
NOTE 1 There is no unified definition of space coordinates within the hydrometric standards. The textbook
conventional axes are adopted in this document when describing open channel flow: the x axis being horizontal
and positive in the mean flow direction, the y axis being orthogonal to the x axis in the horizontal plane and the z
axis being vertical positive.
NOTE 2 For a complete appreciation of the scope of definitions used in measurement uncertainty, the reader is
[1] [2]
referred to the GUM or to NIST Technical Note 1297 .
INTERNATIONAL STANDARD ISO 25377:2020(E)
Hydrometric uncertainty guidance (HUG)
1 Scope
This document provides an understanding of the nature of measurement uncertainty and its
significance in estimating the "quality" of a measurement or a determination in hydrometry.
This document is applicable to flow measurements in natural and man-made channels. Rainfall
measurements are not covered.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 772, Hydrometry — Vocabulary and symbols
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 772 apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
4 Symbols
Symbols Explanations Units
b
α coefficient representing the effects of non-uniform energy (velocity)
in a channel
angles between boat axes and the x axis rad
γγ,,γ
xx xy xz
a
σ standard deviation
a
ΔΔ′′xy, dispersion of measurement from the mean value of the set of x,y
measurements for a symmetric distribution:
′
Δ xx=−05,,()x etc.
maxmin
a
+− ± dispersion about the mean value, x , for an asymmetric distribution
ΔΔ′′xx,
of measurements where
+−
ΔΔ′xx=−xxand ′ =−xx
() ()
maxmin
a
Δ
small difference in a measured quantity ΔΔQh,,ΔT ,etc.
ΔΔyz,
notional small distances in the y and z directions at a cross-section m
in the channel
Dc in the dilution method, the downstream mixed change (c − c ) mg/l
2 m b
of concentration of the tracer
A, A(z), A(h) cross-section area (in the y, z plane) of the flow m
a  Dimensional order depends on its meaning in context.
b  Non-dimensional quantity.
ISO 25377:2020(E)
Symbols Explanations Units
B channel width m
b contracted channel width or flume throat width m
c dilution method, the background concentration of tracer mg/l
b
c dilution method, the feed concentration of tracer mg/l
T
c dilution method, the downstream mixed concentration of the tracer mg/l
m
b
C discharge coefficient
b
C velocity coefficient
v
a
d deviation of a measurement (the ith measurement of a series)
i
from the mean value of that series
E datum elevation of a range measuring device m
a
f(h) general function of parameter h
b
F , F multiplying factors to be applied to the summation of velocity-area
x y
elements to account for the approximation of a summation process
to a true integration of continuously varying parameters
g gravitation acceleration m/s
h head of water relative to a defined datum level in the channel m
H total head relative to a defined datum level in the channel m
a
i,j indices of a count i = 1 to n, or j = 1 to m of a series
b
J false measurement detection factor
b
K constant of a flow determination formula for a weir or flume
b
k , k constants for the determination of flow by the dilution method
1 2
M dilution method, the mass of tracer introduced into the stream g
b
n exponent of a flow determination formula for a weir or flume
a
n, m number of measurement in a series
b
p(x) probability function
Q flow m /s
Q estimated flow passing close to boundaries or any region where m /s
p
measurement cannot be determined by the primary means
Q dilution method, the flow of tracer into the stream m /s
T
a
S sampling standard deviation of a set of measurements
b
t factor to be applied to small numbers of samples to enable the standard
e
deviation to be representative of large numbers of samples (see Annex A)
t , t in the dilution method, the interval during which a change in concentra- s
1 2
tion is detectable
T absolute temperature, in Kelvin °C
T Grubbs’ test parameter °C
n
a
U(x), u( y) uncertainty of measured variables x, y, etc.
a
u (p), u (q) the combined uncertainty of determined results p, q, etc.
c c
a
*
the percentage uncertainty of a measurement of any quantity x
ux
()
a
U measurement uncertainty expanded to the 95 % level of confidence
mean velocity through a yx plane intersecting a channel cross-section of m/s
V
x
the channel
velocity in the x direction at point y, z in the channel m/s
Vy,z
()
x

water velocity vector relative to channel m/s
V
a  Dimensional order depends on its meaning in context.
b  Non-dimensional quantity.
2 © ISO 2020 – All rights reserved

ISO 25377:2020(E)
Symbols Explanations Units

boat velocity vector relative to the channel m/s
V
b

water velocity vector relative to boat m/s
′
V
water velocity components relative to boat along boat coordinate axes m/s
′′ ′
VV,,V
xy z
components of boat velocity relative the boat axes m/s
′′ ′
VV,,V
bx by bz
angles between boat axes and the channel x axis rad
γγ,,γ
xx xy xz
x,y,z channel coordinates m
x’,y’,z’ boat coordinates m
a
x,y measurable variables
a  Dimensional order depends on its meaning in context.
b  Non-dimensional quantity.
In this document, the term “uncertainty” refers to measurement uncertainty and the following formulae
are used to signify
n
— a sum of n values of x xx++xx++. .xx= ;
12 3 in ∑ i′
i=1
df
— a difference, df ()x in the function, fx() , due to a small change, Δx in the value x df ()x =Δx ;
dx
x
n
— a value of an integral, F, of a function, fx() , between, x =x , and x = x Ff= ()xdx.
1 n
∫
x
5 ISO/IEC Guide 98-3 (GUM) — Basic definitions and rules
5.1 General
This clause summarizes the methods described in the GUM for the expression of uncertainty in
measurement. For a general introduction to measurement uncertainty, refer to Annex A.
5.2 Uncertainty of sets of measurements
The GUM describes measurement uncertainty as a value that characterizes the dispersion of
measurements that could reasonably be attributed to the result. The GUM goes on to define standard
uncertainty as uncertainty expressed as a standard deviation, s.
From this definition, it follows that uncertainty only deals with the natural spreading in a series of
measurement results. It should, therefore, be emphasized that uncertainty does not describe any
constant (systematic) deviations of these measurements from the true value. The difference between
random and systematic effects is further elaborated upon in 5.3.
So, for a set of n measurements, uncertainty is related to the difference between each measured value,
x , from the average value, x of the set. The standard deviation, and hence the uncertainty, ux() is:
i
2 2 2 2
 
ux()==s xx− +−xx +−xx +−. xx
() () () ()
1 2 3 n
 
n−1
where component dx=− x is the deviation of the ith measurement, x , from the mean value, x .
ii i
ISO 25377:2020(E)
Or, more concisely:
n
ux()==s d (1)
∑ i
n−1
where dx=−x is the deviation of the ith measurement from the mean value, x shown as Formula (2):
ii
1 n
x= x (2)
∑
i
i=1
n
Formula (1) for the uncertainty of x only applies to steady, stationary stochastic processes, where the
mean value and the standard deviation of the sampled process remain unchanged during the whole
measurement process.
The uncertainty of the mean value, ux decreases as the number of measurements, n, increases. The
()
GUM relationship for this is shown as Formula (3):
ux()= ux() (3)
n
It should be noted that Formula (3) applies only for uncorrelated measurements, which means that
there is no mutual relationship or connection among these measurements.
5.3 Random and systematic effects
Formula (3) applies only to the random variations of the measured quantity. This random effect is
determined from the measured data and, as such, is evaluated after a set of measurements have been
taken. Random effects can be determined from analysis of the historic data or by the instrumentation
itself if it is designed to analyse the data in real time. Random effects diminish the uncertainty in the
average value of a set of n uncorrelated measurements by the factor . Random conditions often
n
exist as natural turbulence. However, random variation can sometimes occur through human
interpretation of a reading of an indicator, such as a staff gauge.
Constant deviations in the measurements that are inherent to the measurement equipment or to the
method are called systematic effects. They should be clearly distinguished from the hereto described
stochastic or random deviations as described by the term uncertainty. Systematic effects cannot be
diminished by the use of Formula (3). During each measurement session, systematic effects can usually
be taken as constant for the measurement device. Systematic components:
a) should be assessed as part of an installation or commissioning procedure, and/or
b) can be specified beforehand for the equipment by the manufacturer. Refer to Clause 9
c) can sometimes be detected and quantified by a careful comparison of a few independent measuring
techniques.
Refer to A.6 for more information on random and systematic effects.
For the evaluation of the uncertainty of a continuous stochastic process, include the presence of
any unsteady effects also as a random component. This is only allowed, however, if the physical
quantity being measured is varying slowly and in a relatively small manner during the measurement
process. Such a procedure will of course widen the dispersion of measured values and hence add to
the assessment of the random component. Such variation becomes then part of the randomness of
the measurement. If during the measurement process the rate and amount of change are such that it
significantly exceeds the natural dispersion of measurements, then the result shall be discarded.
4 © ISO 2020 – All rights reserved

ISO 25377:2020(E)
5.4 Uncertainty models — Probability distributions
In hydrometry, measurements are often made manually or using automated instruments. They have a
margin within which measured values can vary randomly in steady conditions. If they also have a steady
offset inherent to the measurement process, this is termed a systematic component. It is commonly
expressed as a probability distribution. Probability distributions have standard deviations about the
mean value which are equivalent to the standard deviation of discrete measurements as defined above.
The standard uncertainty equivalent to Formulae (1) and (2) are shown as Formulae (4) and (5):
Δx
xx=⋅px()dx (4)
−Δx
∫
and
Δx
ux()= dx() ⋅px()dx (5)
∫ −Δx
where
px() is a probability density function;
d is the deviation from the mean;
Δx
is the bin of discrete measurements.
Refer to Annex A for details.
5.5 Combining uncertainties — Law of propagation of uncertainties
The GUM also defines a rule for combining uncertainties from several sources. It is called "the law of the
propagation of uncertainties". For a relationship, f, between a result, y, and variables, xx .x defined
12,'n
as, yf= xx .x , the combined uncertainty, uy , of y is
()
()
12, n c
in=
∂f
2  
uy = ux
() ()
ci∑
 
∂xi
 
i=1
or
2 2 2
     
∂f ∂f ∂f
uy = ux + ux ++. ux (6)
() () () ()
     
c 1 2 n
∂x ∂x ∂x
     
1 2 n
where xx .x are independent variables. This linear approximation with the first derivatives is
12, n
only allowed, however, when the deviations in the variables x are relatively small compared with their
i
mean values.
Formula (6) applies only where the variables xx .x are uncorrelated, i.e. if variable x changes
i
12, n
value, no other x variable is affected by that change. If two or more variables x do influence each other
(i.e. they are correlated), then an additional component of uncertainty exists. Formula (6) then becomes
jn=
in= in=−1
∂f ∂f ∂f
 
uy() = ux +2 ux x (7)
()
  ()
∑∑ ∑
ci ij
∂
∂xi  xi ∂∂xj
i=1 i=1 ji=+1
Almost all hydrometric uncertainty estimations require the use of the simpler form, i.e. Formula (6).
ISO 25377:2020(E)
∂f
The components can be random or systematic. The partial derivatives are referred to as
∂x
n
“sensitivity coefficients”, and ux xx=cov x .
() ()
ij ij
5.6 Expressing results
Formula (6) expresses the final result in terms of standard uncertainty. For the Gaussian probability,
used as a model distribution for general analysis, one standard deviation covers only 68 % of the range
of possible results. This means that for a result expressed as
3 3
Flow rate = 10,8 m /s ± 0,6 m /s
or
QQ =± uQ
()
3 3
only 68 % of the measurement will lie between 10,2 m /s and 11,4 m /s. Almost one third of the
measurement can be expected to lie outside this band. Such a statement is of little value in hydrometry.
A more meaningful statement is required that will cover a larger portion of possible results.
A.9 defines expanded uncertainty. By expanding the margin of uncertainty, a greater portion of the
expected range of measurements is covered. For the Gaussian probability distribution, it can be shown
that by doubling the uncertainty margin, 95 % of expected measurements are covered. Subclause A.9
defines expanded uncertainty.
The same result expressed in the form
3 3
Flow rate = 10,8 m /s ± 1,2 m /s at the 95 % confidence level
or
QQ =± UQ()
3 3
means that 95 % of the measurements are expected to lie between 9,6 m /s and 12,0 m /s. This is a
more practical expression of the result.
In hydrometry, all measurements shall be expressed at the 95 % confidence level with a statement of
the form:
Quantity = Value ± uncertainty at the 95 % confidence level
or
Quantity = Value ± percentage uncertainty at the 95 % confidence level
Refer to A.9 for more detail.
6 Open channel flow — Velocity area methods
6.1 General
Figure 1 shows the coordinate system used in this document with orthogonal axes x, y, z. The mean
velocity is calculated in the x direction. The xy-plane is horizontal. The z axis is vertical. Note that a

velocity v vector representing the mean velocity does not have to align with the x axis. The flow in the
channel can be determined from velocities passing obliquely through an intersecting yz plane.
6 © ISO 2020 – All rights reserved

ISO 25377:2020(E)
The origin of the coordinate system may be located at any point relative to the channel but is typically
located at the hydraulic datum for weirs and flumes or, for velocity-area methods, on a gauge datum
lower than the streambed.
For example, vertical measurement can be h(z), expressed from a hydraulic datum relative to the z
coordinate system origin.
The determination of flow in open channels requires the following:
a) the determination of the mean velocity V across the channel section; and
x
b) the measurement of the cross-section area A(h), in the yz plane, through which the flow passes; h is
the water depth.
The product of these two quantities is the discharge, Q.
QV= Ah
()
x
Figure 1 — Coordinate relationship at a channel cross-section
6.2 Mean velocity, V
x
The evaluation of mean velocity shall deal with the variability V y, z, to with respect to position, y,z,
x
across the channel and with respect to time, t. At the banks, friction slows the mainstream velocity to
zero which causes steep velocity gradients to occur, illustrated in Figure 2. Velocity gradients and shear
stress within the body of the flow induce vortices which causes turbulent conditions. Turbulence exists
in a moving body of water even when the water surface appears tranquil.
The evaluation shall therefore scan the cross-section while integrating and averaging the velocity
component in the x direction. The flow can be steady and hence V can be constant, but turbulence
x
causes the local value of V (y,z,t) to be unsteady.
x
ISO 25377:2020(E)
Key
1 velocity in the x direction
2 lines of equal velocity
Figure 2 — Typical current profiles and contours in rectangular channels
6.3 Velocity-area method for discharge calculation
The quantity, V is determined across the channel from instantaneous point velocities, V , y, z, t. In this
x
x
subclause, it is assumed that steady flow conditions prevail. If the flow does not vary with time, t, during
the integration process, then
Qd= Vy(),zA (8)
x
∫
A
The “arithmetic” method of integration is summation of velocity through notional stream segments of
defined area.
8 © ISO 2020 – All rights reserved

ISO 25377:2020(E)
This is typically done by dividing the cross-section into a number of vertical segments, then measuring
velocity at frequent intervals along the centreline of each segment to determine the segment mean
velocity. Flow through the segment is the mean velocity through the segment multiplied by the segment
area. The flows through each segment are summed to give the total flow in the channel. Therefore,
Formula (8) becomes
m n
QF= FV yz ΔΔ +Q (9)
()
∑∑
yz xpij zj yi
where
F is a factor, often assumed to be unity, relating the discrete summation in the y direction to an
y
ideal integration of a true continuous velocity profile;
F is a factor, often assumed to be unity, relating the discrete summation in the z direction to an
z
ideal integration of true continuous velocity profile; and
Q represents perimeter flow passing between the region of segments and the channel boundary.
p
The summation method divides the area into mn× rectangular stream tubes of height, Δz , and width,
j
Δy . A set of Δy stream segments makes up each horizontal segment, and a set of Δz , stream segments
i i j
makes up each vertica
...