LECTURES ON MEASUREMENT SCIENCE

The epistemic requirement that measurement be an objective and intersubjective evaluation is empirically fulfilled by adopting measuring systems that include selective and repeatable sensors and traceable standards; any measurement is then performed as a (direct or indirect) comparison to a chosen standard. From the instrument output the measurement result has to be inferred by means of a process based on the information gathered from the instrument calibration and a measurement system model. Such a process is only plausible in its results, that must be expressed specifying both a measurand value and its estimated uncertainty. 1.1. Measurement as a comparison model Measurement is an operation of data acquisition and presentation, aimed at expressing in symbolic form the information empirically obtained on a system about a quantity, the measurand (we accept the common ambiguities of calling “measurand” both the system under measurement and the measured quantity, and the latter in both its general and specific forms, e.g., length and length of a given object in a given time). Peculiar to measurement is the requirement of being objective and intersubjective, where objectivity implies that measurement results convey information only related to the system under measurement and not its environment, and intersubjectivity requires that measurement results convey the same information to different subjects. As such, these properties appear an ideal target, justifying the efforts to constantly enhance measurement devices and procedures. To achieve an acceptable degree of objectivity and intersubjectivity measuring systems are adopted, including selective and repeatable sensors and traceable standards. Indeed: • although human beings are able to directly sense a fair amount of quantities and are well trained to express in linguistic form their perception (e.g., “it is rather cold”, “this is heavier than that”), their statements are affected by subjectivity, i.e., they report information on both the sensed system and the perceiver state; to avoid the influence of the latter, and thus to enhance the objectivity of the operation, the measurand is transduced by a sensing system whose output ideally depends only on the measurand and is unaffected by influence quantities and internal imperfections; • while related to the measurand, the quantity provided by sensors still depends on their specific behavior; as a consequence, distinct sensors, even if perfectly repeatable, produce different outputs from the same input; furthermore, in many cases the sensor output quantity, appropriate for signal conditioning and for driving presentation devices, is not dimensionally homogeneous to the measurand. The sensor output must be then dealt with as an instrument reading, not a measurand value. To make the information obtained by the measurement intersubjective a common reference must be adopted, so that measurand values are expressed in comparison to such a standard. Critical is therefore the possibility Luca Mari, Lectures on measurement science 3 to trace the readings to the agreed standard, a condition operatively ensured by instrument calibration. The requirement of empirical comparison to traceable standards is so fundamental that can be assumed as distinctive of measurement; if generic scale-preserving evaluations can be formalized as homomorphisms from empirical to symbolic relational systems: Figure 1 – A generic scale-preserving evaluation. in the case of measurement such mappings are not direct but mediated by the comparison to standards: Figure 2 – Measurement as a scale-preserving evaluation obtained by the comparison to a standard. Finally, when primary standards are not directly available: Figure 3 – Measurement as a scale-preserving evaluation obtained by the comparison to a standard derived by a primary standard. Operations 1 and 2 are usually carried out before measurement: nevertheless measurement cannot be completed without them and therefore such operations play an essential role for the definition itself of measurement. As a consequence, measurement results must state a measurand value in reference to the adopted standard, usually expressed in the form of a measurement unit. 1.2. The output/input black box model It is a well-known fact that different methods of measurement exist, each of them corresponding to a specific technique to perform the comparison between the measurand and the standard. While some methods require the synchronous presence of the measurand and the standard (e.g., Liuc Papers n. 157, suppl. a novembre 2004 4 following the paradigm of the two arm balance provided with a set of standard weights: a direct comparison) many others are based on the usage of devices acting as serializers of the comparison, so that a measurement involves (at least) two interactions: standard-instrument and measurand-instrument. Figure 4 – The different usages of the measuring systems as comparators. In its interaction with the measurand the instrument generates an output; a general problem of measurement can be then stated as follows: from the output of the measuring instrument (“the reading”) its input (the state of the system under measurement and its environment) must be reconstructed, and from this state a measurand value must be inferred. To cope with this input-from-output inference problem two basic strategies can be in principle followed: • the analytical model of the measuring system behavior is identified and the obtained characteristic function is inverted, so that from the output readings the input signals are computed. Because of its complexity this approach is seldom adopted; • the system is regarded as a black box and only its input-output behavior is taken into account: the instrument is put in interaction with a set of (known) standard states and the corresponding output readings are recorded; by a suitable interpolation this collection of couples becomes the so-called calibration curve, that can be thought of as a mapping from measurand values to instrument readings; this function is then inverted, so that each instrument reading can be associated with a measurand value. The interactions standard-instrument and measurand-instrument have therefore a complementary function: while the former is aimed at creating a calibration diagram: Luca Mari, Lectures on measurement science 5 Figure 5 – A diagram with the example of a curve generated by calibration. the latter uses the inverted diagram to find the measurand value that corresponds to the obtained reading: Figure 6 – The example of an inverted calibration diagram, for usage in measurement. To enhance the user-friendliness of the measuring systems it is customary to set up their presentation component so that the data they display are expressed directly in measurand units, i.e., the calibration diagram is embedded into the systems. While always measurement requires calibration information, in these cases one can properly speak of calibrated instruments. 1.3. Set-theoretical model The sensor behavior, therefore critical for both calibration and measurement, is usually expressed as a characteristic function formalizing the input-output conversion performed by the sensor itself. The sensor input, a couple ( ) w x, where X t x x ∈ = ) ( is the measurand and W t w t w w w w n n ∈ = = ) ( ),..., ( ,..., 1 1 is a collection of further quantities influencing the sensor behavior, is transformed to its output Y y∈ . Therefore the sensor characteristic function: Y T W X f → × × : (1) Liuc Papers n. 157, suppl. a novembre 2004 6 takes the measurand ) (t x , the influence quantities ) (t w and the current time t , included to take into account possible time-dependent effects, and associate them with the output signal ) ), ( ), ( ( ) ( t t w t x f t y = to which both the measurand (“the signal”) and the influence quantities (“the noise”) contribute. This simple formalization allows us to introduce some basic parameters describing the static behavior of a sensor: • sensitivity: ideally 2 1 x x ≠ implies ) , , ( ) , , ( 2 1 t w x f t w x f ≠ , i.e., distinct measurand values always produce distinct outputs; the ratio x y ∆ ∆ / expresses the aptitude of the sensor to reproduce measurand variations to output values; • selectivity: ideally ) , , ( ) , , ( 2 1 t w x f t w x f = even if 2 1 w w ≠ , i.e., the sensor output is not affected by the variations of influence quantities; the less is the variability of y due to w the better is the sensor (therefore selectivity corresponds to non-sensitivity to influence quantities: the relative contribution of the measurand to the output can be formalized as a signal-to-noise ratio); • repeatability and stability: ideally ) , , ( ) , , ( 2 1 t w x f t w x f = even if 2 1 t t ≠ , i.e., the sensor output is not affected by short-term (fluctuations) and long-term (aging) time effects; the less is the variability of y due to t the better is the sensor (a stable sensor does not require frequent re-calibrations); • linearity: ideally b ax y + = (where a and b are given coefficients, possibly with 0 = b ), i.e., f is a straight line, the better the actual sensor behavior is approximated by this equation the better is usually considered the sensor (a linear, zero-crossing sensor is calibrated in a single operation, aimed at determining the slope a ). In addition to these static parameters, the dynamic behavior of the sensor is synthesized by parameters such as its frequency response. The technical specifications for sensors usually include some quantitative evaluation for these parameters in the nominal conditions of usage, expressed by the allowed ranges of measurand and influence quantities. 1.4. Generalized model The inference process that leads to the evaluation and the expression of a measurand value is always only plausible in its results, and in general nothing can be inferred with certainty about the measurand value. The causes of this