ISO 17123-1:2010

INTERNATIONAL ISO
STANDARD 17123-1
Second edition
2010-10-15

Optics and optical instruments — Field
procedures for testing geodetic and
surveying instruments —
Part 1:
Theory
Optique et instruments d'optique — Méthodes d'essai sur site pour les
instruments géodésiques et d'observation —
Partie 1: Théorie




Reference number
ISO 17123-1:2010(E)
©
ISO 2010

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ISO 17123-1:2010(E)
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ISO 17123-1:2010(E)
Contents Page
Foreword .iv
Introduction.v
1 Scope.1
2 Normative references.1
3 Terms and definitions .1
3.1 General metrological terms.1
3.2 Terms specific to this International Standard .3
3.3 The term “uncertainty” .5
3.4 Symbols.7
4 Evaluating uncertainty of measurement.8
4.1 General .8
4.2 Type A evaluation of standard uncertainty.9
4.3 Type B evaluation of standard uncertainty.16
4.4 Law of propagation of uncertainty and combined standard uncertainty .18
4.5 Expanded uncertainty.19
5 Reporting uncertainty .20
6 Summarized concept of uncertainty evaluation .20
7 Statistical tests .21
7.1 General .21
7.2 Question a): is the experimental standard deviation, s, smaller than or equal to a given
value σ?.21
7.3 Question b): Do two samples belong to the same population? .22
7.4 Question c) [respectively question d)]:Testing the significance of a parameter y .22
k
Annex A (informative) Probability distributions .24
2
Annex B (normative) χ distribution, Fisher's distribution and Student's t-distribution.25
Annex C (informative) Examples .26
Bibliography.35

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ISO 17123-1:2010(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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
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.
ISO 17123-1 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee SC 6,
Geodetic and surveying instruments.
This second edition cancels and replaces the first edition (ISO 17123-1:2002), which has been technically
revised.
ISO 17123 consists of the following parts, under the general title Optics and optical instruments — Field
procedures for testing geodetic and surveying instruments:
⎯ Part 1: Theory
⎯ Part 2: Levels
⎯ Part 3: Theodolites
⎯ Part 4: Electro-optical distance meters (EDM instruments)
⎯ Part 5: Electronic tacheometers
⎯ Part 6: Rotating lasers
⎯ Part 7: Optical plumbing instruments
⎯ Part 8: GNSS field measurement systems in real-time kinematic (RTK)
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ISO 17123-1:2010(E)
Introduction
This part of ISO 17123 specifies field procedures for adoption when determining and evaluating the
uncertainty of measurement results obtained by geodetic instruments and their ancillary equipment, when
used in building and surveying measuring tasks. Primarily, these tests are intended to be field verifications of
suitability of a particular instrument for the immediate task. They are not proposed as tests for acceptance or
performance evaluations that are more comprehensive in nature.
The definition and concept of uncertainty as a quantitative attribute to the final result of measurement was
developed mainly in the last two decades, even though error analysis has already long been a part of all
measurement sciences. After several stages, the CIPM (Comité Internationale des Poids et Mesures) referred
the task of developing a detailed guide to ISO. Under the responsibility of the ISO Technical Advisory Group
on Metrology (TAG 4), and in conjunction with six worldwide metrology organizations, a guidance document
on the expression of measurement uncertainty was compiled with the objective of providing rules for use
within standardization, calibration, laboratory, accreditation and metrology services. ISO/IEC Guide 98-3 was
first published as an International Standard (ISO document) in 1995.
With the introduction of uncertainty in measurement in ISO 17123 (all parts), it is intended to finally provide a
uniform, quantitative expression of measurement uncertainty in geodetic metrology with the aim of meeting
the requirements of customers.
ISO 17123 (all parts) provides not only a means of evaluating the precision (experimental standard deviation)
of an instrument, but also a tool for defining an uncertainty budget, which allows for the summation of all
uncertainty components, whether they are random or systematic, to a representative measure of accuracy, i.e.
the combined standard uncertainty.
ISO 17123 (all parts) therefore provides, for defining for each instrument investigated by the procedures, a
proposal for additional, typical influence quantities, which can be expected during practical use. The customer
can estimate, for a specific application, the relevant standard uncertainty components in order to derive and
state the uncertainty of the measuring result.

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INTERNATIONAL STANDARD ISO 17123-1:2010(E)

Optics and optical instruments — Field procedures for testing
geodetic and surveying instruments —
Part 1:
Theory
1 Scope
This part of ISO 17123 gives guidance to provide general rules for evaluating and expressing uncertainty in
measurement for use in the specifications of the test procedures of ISO 17123-2, ISO 17123-3, ISO 17123-4,
ISO 17123-5, ISO 17123-6, ISO 17123-7 and ISO 17123-8.
ISO 17123-2, ISO 17123-3, ISO 17123-4, ISO 17123-5, ISO 17123-6, ISO 17123-7 and ISO 17123-8 specify
only field test procedures for geodetic instruments without ensuring traceability in accordance with
ISO/IEC Guide 99. For the purpose of ensuring traceability, it is intended that the instrument be calibrated in
the testing laboratory in advance.
This part of ISO 17123 is a simplified version based on ISO/IEC Guide 98-3 and deals with the problems
related to the specific field of geodetic test measurements.
2 Normative references
The following referenced documents are indispensable for the application 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/IEC Guide 98-3:2008, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in
measurement (GUM:1995)
ISO/IEC Guide 99:2007, International vocabulary of metrology — Basic and general concepts and associated
terms (VIM)
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO/IEC Guide 99 and the following
apply.
3.1 General metrological terms
3.1.1
(measurable) quantity
property of a phenomenon, body or substance, where the property has a magnitude that can be expressed as
a number and a reference
EXAMPLE 1 Quantities in a general sense: length, time, temperature.
EXAMPLE 2 Quantities in a particular sense: length of a rod.
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ISO 17123-1:2010(E)
3.1.2
value
value of a quantity
quantity value
number and reference together expressing the magnitude of a quantity
EXAMPLE Length of a rod: 3,24 m.
3.1.3
true value
true value of a quantity
true quantity value
value consistent with the definition of a given quantity
NOTE This is a value that would be obtained by perfect measurement. However, this value is in principle and in
practice unknowable.
3.1.4
reference value
reference quantity value
quantity value used as a basis for comparison with values of quantities of the same kind
NOTE A reference quantity value can be a true quantity value of the measurand, in which case it is normally
unknown. A reference quantity value with associated measurement uncertainty is usually provided by a reference
measurement procedure.
3.1.5
measurement
process of experimentally obtaining one or more quantity values that can reasonably be attributed to a
quantity
NOTE Measurement implies comparison of quantities and includes counting of entities.
3.1.6
measurement principle
phenomenon serving as the basis of a measurement (scientific basis of measurement)
NOTE The measurement principle can be a physical phenomenon like the Doppler effect applied for length
measurements.
3.1.7
measurement method
generic description of a logical organization of operations used in a measurement
NOTE Methods of measurement can be qualified in various ways, such as “differential method” and “direct
measurement method”.
3.1.8
measurand
quantity intended to be measured
EXAMPLE Coordinate x determined by an electronic tacheometer.
3.1.9
indication
quantity value provided by a measuring instrument or measuring system
NOTE An indication and a corresponding value of the quantity being measured are not necessarily values of
quantities of the same kind.
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ISO 17123-1:2010(E)
3.1.10
measurement result
result of measurement
set of quantity values attributed to a measurand together with any other available relevant information
NOTE A measuring result can refer to
⎯ the indication,
⎯ the uncorrected result, or
⎯ the corrected result.
A measurement result is generally expressed as a single measured quantity value and a measurement uncertainty.
3.1.11
measured quantity value
quantity value representing a measurement result
3.1.12
error
error of measurement
measurement error
measured quantity value minus a reference quantity value
3.1.13
random measurement error
random error
component of measurement error that in replicate measurements varies in an unpredictable manner
NOTE Random measurement errors of a set of replicate measurements form a distribution that can be summarized
by its expectation, which is generally assumed to be zero, and its variance.
3.1.14
systematic error
systematic error of measurement
component of measurement error that in replicate measurements remains constant or varies in a predictable
manner
NOTE Systematic error, and its causes, can be known or unknown. A correction can be applied to compensate for a
known systematic measurement error.
3.2 Terms specific to this International Standard
3.2.1
accuracy of measurement
closeness of agreement between a measured quantity value and the true value of the measurand
NOTE 1 “Accuracy” is a qualitative concept and cannot be expressed in a numerical value.
NOTE 2 “Accuracy” is inversely related to both systematic error and random error.
3.2.2
experimental standard deviation
estimate of the standard deviation of the relevant distribution of the measurements
NOTE 1 The experimental standard deviation is a measure of the uncertainty due to random effects.
NOTE 2 The exact value arising in these effects cannot be known. The value of the experimental standard deviation is
normally estimated by statistical methods.
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ISO 17123-1:2010(E)
3.2.3
precision
measurement precision
closeness of agreement between measured quantity values obtained by replicate measurements on the same
or similar objects under specified conditions
NOTE Measurement precision is usually expressed by measures of imprecision, such as experimental standard
deviation under specified conditions of measurement.
3.2.4
repeatability condition
repeatability condition of measurement
condition of measurement, out of a set of conditions
NOTE Conditions of measurement include
⎯ the same measurement procedure,
⎯ the same observer(s),
⎯ the same measuring system,
⎯ the same meteorological conditions,
⎯ the same location, and
⎯ replicate measurements on the same or similar objects over a short period of time.
3.2.5
repeatability
measurement repeatability
measurement precision under a set of repeatability conditions of measurement
3.2.6
reproducibility conditions of measurement
condition of measurement, out of a set of conditions
NOTE Conditions of measurement include
⎯ different locations,
⎯ different observers,
⎯ different measuring systems, and
⎯ replicate measurements on the same or similar objects.
3.2.7
reproducibility
measurement reproducibility
measurement precision under reproducibility conditions of measurement
3.2.8
influence quantity
quantity, which in a direct measurement does not affect the quantity that is actually measured, but affects the
relation between the indication of a measuring system and the measurement result
EXAMPLE Temperature during the length measurement by an electronic tacheometer.
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ISO 17123-1:2010(E)
3.3 The term “uncertainty”
3.3.1
uncertainty
uncertainty of measurement
measurement uncertainty
non-negative parameter characterizing the dispersion of quantity values attributed to a measurand, based on
the information used
NOTE Measurement uncertainty comprises, in general, many components. Some of these components can be
evaluated by a Type A evaluation of measurement uncertainty from the statistical distribution of the quantity values from
series of measurements and can be characterized by an experimental standard deviation. The other components, which
can be evaluated by a Type B evaluation of measurement uncertainty, can also be characterized by an approximation to
the corresponding standard deviations, evaluated from assumed probability distributions based on experience or other
information.
3.3.2
Type A evaluation
Type A evaluation of measurement uncertainty
evaluation of a component of measurement uncertainty (standard uncertainty) by a statistical analysis of
quantity values obtained by measurements under defined measurement conditions
NOTE For information about statistical analysis, see 4.1 and ISO/IEC Guide 98-3.
3.3.3
Type B evaluation of measurement uncertainty
evaluation of a component of measurement uncertainty (standard uncertainty) determined by means other
than a Type A evaluation of measurement uncertainty
EXAMPLE The component of measurement uncertainty can be based on
⎯ previous measurement data,
⎯ experience with, or general knowledge of, the behaviour and property of relevant instruments or materials,
⎯ manufacturer's specifications,
⎯ data provided in calibration and other reports,
⎯ uncertainties assigned to reference data taken from handbooks, and
⎯ limits deduced through personal experiences.
NOTE For more information see 4.3 and ISO/IEC Guide 98-3.
3.3.4
standard uncertainty
standard uncertainty of measurement
standard measurement uncertainty
measurement uncertainty expressed as a standard deviation
NOTE Standard uncertainty can be estimated either by a Type A evaluation or by a Type B evaluation.
3.3.5
combined standard uncertainty
combined standard measurement uncertainty
standard (measurement) uncertainty, obtained by using the individual standard uncertainties (and covariances
as appropriate), associated with the input quantities in a measurement model
NOTE The procedure for combining standard uncertainties is often called the “law of propagation of uncertainties”
and in common parlance the “root-sum-of-squares” (RSS) method.
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ISO 17123-1:2010(E)
3.3.6
coverage factor
numerical factor larger than one, used as a multiplier of the (combined) standard uncertainty in order to obtain
the expanded uncertainty
NOTE The coverage factor, which is typically in the range of 2 to 3, is based on the coverage probability or level of
confidence required of the interval.
3.3.7
expanded uncertainty
expanded measurement uncertainty
half-width of a symmetric coverage interval, centred around the estimate of a quantity with a specific coverage
probability
NOTE A fraction can be viewed as the coverage probability or level of confidence of the interval.
3.3.8
coverage interval
interval containing the set of true quantity values of a measurand with a stated probability, based on the
information available
NOTE It is intended that a coverage interval not be termed “confidence interval” in order to avoid confusion with the
statistical concept. To associate an interval with a specific level of confidence requires explicit or implicit assumptions
regarding the probability distribution, characterized by the measurement result.
3.3.9
coverage probability
probability that the set of true quantity values of a measurand is contained within a specific coverage interval
NOTE The probability is sometimes termed “level of confidence” (see ISO/IEC Guide 98-3).
3.3.10
uncertainty budget
statement of a measurement uncertainty, of the components of that measurement uncertainty, and of their
calculation and combination
NOTE It is intended that an uncertainty budget include the measurement model, estimates, measurement
uncertainties associated with the quantities in the measurement model, type of applied probability density functions and
type of evaluation of measurement uncertainty.
3.3.11
measurement model
mathematical relation among all quantities known to be involved in a measurement
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ISO 17123-1:2010(E)
3.4 Symbols
Table 1 — Symbols and definitions
a Half-width of a rectangular distribution of possible values of input quantity X :a = (a − a )/2
i + −
a Upper bound or upper limit of input quantity X
+ i
a Lower bound or lower limit of input quantity X
− i
A Design or Jacobian matrix (N × n)
∂f
Partial derivates or sensitive coefficient: c = (i = 1, 2, ., N)
c
i i
∂x
i
c Vector of sensitive coefficients c (i = 1, 2, ., N)
i
e Unit vector
Functional relationship between a measurand, Y , and the input quantity, X , and between output
k j
f
k
estimate, y , and input estimates, x
k j
T
f Vector with elements f (x ) (k = 1, 2, ., n)
k
Fisher's F (or Fisher-Snedecor) distribution with degrees of freedom (v, v) and confidence level of
F (v, v)
1 − α/2
(1 − α) %
g Functional relationship between the estimate of input quantity, x , and the observables, l
j j i
Coverage factor used to calculate expanded uncertainty U = k × u (y) of the output estimate y from its
c
k
combined uncertainty u (y)
c
l Observables, random variables (i = 1, 2, ., m)
i
m Number of observations, l
i
M Number of input quantities, whose uncertainties can be estimated by a Type A evaluation
n Number of output quantities, measurands
N Number of input quantities
N − M Number of input quantities, whose uncertainties can be estimated by a Type B evaluation
N Normal equation matrix (n × n)
p Weight of the input estimates x ( j = 1, 2, ., N)
j j
P Weight matrix of p (N × N)
j
Q Cofactor of the output estimate, y
y y k
k k
Q Cofactor matrix of the output estimates, y (n × n)
y k
r Residual of input estimates, x ( j = 1, 2, ., N)
j j
r Vector of residuals, r
j
r(x , x ) Correlation coefficient between the input estimates, x and x
i j i j
s Experimental standard deviation (general notation)
s(y ) Experimental standard deviation of the output estimate y
k k
t (v) Student's t-distribution with the degree of freedom, v, and a confidence level of (1 − α) %
α
u Standard uncertainty (general notation)
u(y ) Standard uncertainty of the output estimate y
k k
u(x ) Standard uncertainty of the input estimate x
j j
u (y) Combined standard uncertainty of the output estimate y
c k k
U Expanded uncertainty (general notation)
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ISO 17123-1:2010(E)
Table 1 (continued)
x Estimate of input quantity, input estimate ( j = 1, 2, ., N)
j
x Vector of the estimates of input quantities x
j
X jth input quantity on which the measurand Y depends
j k
X Vector of input quantities X
j
y Estimate of measurand Y , output estimate; (k = 1, 2, ., n)
k k
y Vector of output estimates of measurands y
k
Y kth measurand (k = 1, 2, ., n)
k
Y Vector of measurands Y
k
α Probability of error, as a percentage
(1 − α) Confidence level
v Degrees of freedom
σ Standard deviation of the normal distribution
2
χ ()ν Chi-squared distribution with the degree of freedom, v, and a confidence level of (1 − α) %
1− α

4 Evaluating uncertainty of measurement
4.1 General
The general concept is documented in ISO/IEC Guide 98-3, which represents the international view of how to
express uncertainty in measurement. It is just a rigorous application of the variance-covariance law, which is
very common in geodetic and surveying data analysis. However, the philosophy behind it has been extended
in order to consider not only random effects in measurements, but also systematic errors in the quantification
of an overall measurement uncertainty.
In principle, the result of a measurement is only an approximation or estimate of the value of the specific
quantity subject to a measurement; that is the measurand. Thus, the result is complete only when
accompanied by a quantitative statement of its quality, the uncertainty.
The uncertainty of the measurement result generally consists of several components, which may be grouped
into two categories according to the method used to estimate their numerical values:
a) those which are evaluated by statistical methods;
b) those which are evaluated by other means.
Basic to this approach is that each uncertainty component, which contributes to the uncertainty of a measuring
result by an estimated standard deviation, is termed standard uncertainty with the suggested symbol u.
The uncertainty component in category A is represented by a statistically estimated experimental standard
deviation, s, and the associated number of degrees of freedom, v. For such a component, the standard
i i
uncertainty u = s. The evaluation of uncertainty components by the statistical analysis of observations is
i i
termed a Type A evaluation of measurement uncertainty (see 4.2).
In a similar manner, an uncertainty component in category B is represented by a quantity, u , which may be
j
considered an approximation of the corresponding standard deviation and which may be attributed an
assumed probability distribution based on all available information. Since the quantity u is treated as a
j
standard deviation, the standard uncertainty of category B is simply u . The evaluation of uncertainty by means
j
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ISO 17123-1:2010(E)
other than statistical analysis of series of observations is termed a Type B evaluation of measurement
uncertainty (see 4.3).
Correlation between components of either category are characterized by estimated covariances or estimated
correlation coefficients.
Input: Output:
vector x , U vector y and u
x
y


input quantity x and its
output quantity y and its
j
k
uncertainty u(x )
j standard uncertainty u (y )
k


Type A:
expanded uncertainty


observations, measurement
U(y )
k

data analysed by statistical
Model


methods
x, U

x of evaluation: y, u
y Final result:
x , U
A x(A)


T
y ± U(y )
y = f (x )
k k




Type B:


previous, external

Can be used as input
measurement data analysed

quantity in further
by other means


applications
x , U
B x(B)


Figure 1 — Universal mathematical model and uncertainty evaluation

4.2 Type A evaluation of standard uncertainty
4.2.1 General mathematical model
In most cases, a measurand, Y, is not measured directly, but is determined by N other quantities x , x , ., x
1 2 N
through the functional relationship given as Equation (1):
Y = f (X , X , ., X ) (1)
1 2 N
An estimate of the measurand, Y, the output estimate, y, is obtained from Equation (1) by using the input
estimates, x , x , ., x , thus the output estimate, y, which is the result of measurements, is given by
1 2 N
Equation (2):
y = f (x , x , ., x ) (2)
1 2 N
In most cases, the measurement result (output estimate, y) is obtained by this functional relationship.
But in some cases, especially in geodetic and surveying applications, the measurement result is com
...