Instrumentation & Measurement Magazine 26-4 - 7

Instrument Calibration
Let us consider again the simple case of an alcohol thermometer,
and suppose for the moment that a scale of temperature
has been somehow already defined and made public, so that
everybody can in principle relate given temperatures to given
values on the scale, thus agreeing for example which etched
mark on the thermometer scale corresponds to the temperature
of 20 °C, which corresponds to 21 °C, and so on. This is
the first precondition for the thermometer to be operated as
a measuring instrument, and not only to perform a pre-measurement
as shown above. The second precondition is an
operationalization of the first one: some of such (temperature,
value of temperature) pairs must have been reliably realized
in suitably chosen objects o1
, o2, ..., whose temperatures T(o1
) = v1
, T(o2
) = v2
),
,
T(o2), ... are sufficiently stable and known in their values v1
v2, ... on the given scale, i.e., T(o1
, ... Objects
with these features are called measurement standards, where the
concept 'measurement standard' is in fact defined by the International
Vocabulary of Metrology (VIM) as the " realization of the
definition of a given quantity, with stated quantity value and
associated measurement uncertainty, used as a reference " [6]
(note that the correct phrase, " realization of the definition of
a quantity " , is often shortened as " realization of a quantity " ).
The process that transforms a temperature sensor into a
measuring instrument is called calibration, and is functionally
aimed at identifying how public values of temperature
relate to local values of instrument positions, called " indications "
in the VIM. In the ideal, and formally simplest, case,
this relation is a {temperature values}→{position values}
function. Again in the simplest case, such a calibration function
is invertible - i.e., no two distinguishable temperature
values are mapped to the same position value - and is the
informational counterpart of the transduction function {temperatures
T}→{positions p} realized by the instrument. This
is why calibration can be intended as aimed at characterizing
the transduction behavior of the sensor at the core of a measuring
instrument [7].
If no hypotheses are made on the sensor behavior, the calibration
function is built by making the instrument interact
with some suitable " working " measurement standards, that in
fact are " used routinely to calibrate or verify measuring instruments
or measuring systems " , according to the VIM definition
[6]. The result of each interaction is a (value of the temperature
realized by the standard, value of the position reached by
the alcohol as the result of the interaction) pair, and the list of
these pairs provides a partial extensional definition of the calibration
function, that can be completed by means of suitable
interpolations. An alternative, and more usual, strategy assumes
that the transduction function has a known parametric
analytical form, so that the calibration is aimed at estimating
the values of the function parameters. For example, if the instrument
behavior is supposed to be linear - i.e., its sensitivity
June 2023
defined as the derivative or the finite difference of the transduction
function dp/dT or Δp/ΔT is constant - only two
parameters have to be estimated, and therefore the interaction
of the instrument with two measurement standards, each realizing
a different temperature, is sufficient.
In fact, the description above is a simplification of what an
instrument calibration is. The broader picture is given, once
again, by the VIM definition of 'calibration': " operation that,
under specified conditions, in a first step, establishes a relation
between the quantity values with measurement uncertainties
provided by measurement standards and corresponding indications
with associated measurement uncertainties and, in
a second step, uses this information to establish a relation for
obtaining a measurement result from an indication " [6]. The
key point here is the acknowledgment that the instrument
behavior is not perfectly repeatable, and then that a single temperature
could be transduced to several different positions,
and then that, conversely, one observed position could correspond
to several different temperatures. Hence, in the more
realistic case, the outcome of a calibration is not a function, but
" the strip of the plane defined by the axis of the indication and
the axis of measurement result, that represents the relation between
an indication and a set of measured quantity values. A
one-to-many relation is given, and the width of the strip for a
given indication provides the instrumental measurement uncertainty. "
[6].
Indeed, instrumental measurement uncertainty is a core
component of the quality of the information produced by
a measurement, but surely not the only one [3]. The measurement
standards required to calibrate any measuring
instrument are in turn non-ideal devices obtained by nonideal
processes, and these non-idealities affect the quality of
measurement results with respect to their non-complete intersubjectivity.
To the whole scenario of metrological systems we
devote our attention now.
Metrological Traceability: Generating
Intersubjectivity by Defining Units and
Disseminating Measurement Standards
As already discussed, the fundamental condition of intersubjectivity
for measurement results is to guarantee that the same
measured value, as reported in different places and times and
produced by means of different measuring instruments, corresponds
to the same quantity, so that, for example, from
temperature of body A here and now = 23.4 °C
and
temperature of body B there and then = 23.4 °C
one can reliably infer that
IEEE Instrumentation & Measurement Magazine
7
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