Measurement Uncertainty Analysis Research Articles

Analysis of measurement uncertainty for verification/calibration of particulate matter measuring instruments using the gravimetric method

As demand for particulate matter (PM) monitoring increases for scientific and regulatory purposes, the need for an accurate calibration process for these instruments will also increase. This article focuses on studying the factors that cause measurement uncertainty during the calibration process with the calibration facility (ManDust) for reference gravimetric instruments. Measurement uncertainty in this article included type A and type B uncertainties. Type A uncertainty mostly included repeatability and reproducibility while volume uncertainty predominately contributed to type B uncertainty. To clearly examine how these factors contributed to the combined uncertainty, the calibration process was carried out at a high concentration range from 0.4 to 1.6 mg m−3. In general, the measurement uncertainty ranged from 1.95% to 6.65%. Further analysis showed that type A uncertainty contributed more to the total uncertainty than type B. In particular, uncertainty from volume calculation and measurement contributed over 80% of type B uncertainty. During the PM filter weighing process, the uncertainty during the conditioning process contributed the most to type B uncertainty. This article also conducted a sensitivity analysis and found that besides volume measurement, the filter weighing procedure was also sensitive to changes even though it only contributed less than 5% of type B uncertainty. Using high-quality devices and careful handling during these two procedures should be emphasized to reduce the uncertainty of the calibration process.

Paradox? What paradox?

AbstractThis article is a response to the preceding paper by Huang, who considers a recent result of Willink (Measurement: Sensors, 24:100416, 2022) and who describes the result as a paradox. The result implied that a set of information or a “state of knowledge” about a measurand cannot be identified with a unique probability distribution for the measurand, contrary to what seems suggested in the literature surrounding the revision of the Guide to the Expression of Uncertainty in Measurement. The result is restated and viewed in the context of CIPM Recommendation INC-1, which was foundational in the original development of the Guide. It is argued that the result is a proof, not a paradox, and that it will only appear paradoxical to those who have adopted an incorrect premise about probability. The idea of having “information” about the true value of a measurand is discussed and contrasted with the idea of having “belief” about it. The material supports the view that the analysis of measurement uncertainty is to be based on classical statistical principles.

A practical two-step procedure for taking into account all available information (prior and current) about influence quantities in measurement uncertainty analysis

A practical two-step procedure for taking into account all available information (prior and current) about influence quantities in measurement uncertainty analysis

Measurement uncertainty and behaviour analysis of indoor air quality variables of thermal zones of an in-use mechanically ventilated building

Most monitoring systems and models in buildings assume homogeneous measurements of indoor variables such as temperature, relative humidity (RH), CO2 concentration or water vapour content (Pv) throughout a thermal zone. However, these indoor variables may have different values within a thermal zone, depending on the physical location of the sensor. This spatial variability, which is not an error per se, depends on factors that cannot be controlled or eliminated, mainly due to air stratification. Consequently, if a hypothetical homogeneous value of the variable is assumed, the measurement may contain random errors. In this research, we use a statistical methodology to analyse and quantify the Overall Uncertainty (UX) when measuring these indoor variables. The UX comprises the inherent inaccuracies of the monitoring system, indicated as Systematic Uncertainty (UX(s)), and the random errors associated with the spatial variability of the variables, evaluated as Spatial Uncertainty (UX(sp)). We have found that the UX(sp) of the temperature and RH measurements are considerably higher than their UX(s). However, the spatial variabilities of the CO2 concentration and Pv measurements are similar to the systematic uncertainty. Furthermore, we have compared the indoor and outdoor variable measurements. Spearman's correlation coefficient between the indoor and outdoor Pv indicates a strong positive correlation. This, together with the uncertainty analysis, confirms that the Pv (or RH) is not a suitable variable to detect presence in mechanically ventilated buildings. Conversely, as expected, there is no correlation between the indoor and outdoor CO2 concentrations, confirming the advantages of this variable for occupancy detection.

Length measurement based on multi-wavelength interferometry using numerous stabilized frequency modes of an optical comb

Abstract The electro-optic comb (EO comb) with a relatively wide mode spacing of 25 GHz can be resolved into individual frequency modes with a commercially available high-resolution spectrometer. The EO comb has numerous discrete frequency modes, which can serve as a light source for a monochromatic laser interferometer to realize the meter. In this study, a method for measuring the absolute distances based on the multi-wavelength interferometer principle is proposed and demonstrated by simultaneously implementing 102 monochromatic laser interferometers using an EO comb. A phase shifting technique was used to determine the phase for each frequency mode by precisely translating a reference mirror with a constant interval. The phases of the interference signals for 102 stabilized individual frequency modes were measured by applying the model-based analysis on the phase-shifted interference signals. The absolute distance can be determined using phase values of a wavelength set corresponding to five to seven randomly selected frequency modes. In this study, the absolute distances for round-trip distances of 166 mm and 1316 mm were measured and the measurement uncertainty of each distance was evaluated. Through the uncertainty analysis of the distance measurement, the combined uncertainties of the measured distances in a short and long ranges were evaluated to be 30.1 nm and 211.1 nm, respectively. In addition, for each distance, the consistent measurement results of absolute distances were obtained through four different wavelength sets, which show the flexibility of wavelength selection in this work.

Peak power measurement device of terawatt femtosecond pulsed laser and measurement uncertainty analysis

Peak power measurement device of terawatt femtosecond pulsed laser and measurement uncertainty analysis

A minimum entropy criterion for distribution selection for measurement uncertainty analysis

This paper presents a minimum entropy criterion for selecting the best probability distribution among a set of candidate distributions based on available information for measurement uncertainty analysis. We consider two cases that are most commonly encountered in practice: A and B. In Case A, the available information is a series of observations. In Case B, the available information is the maximum permissible error according to manufacturer’s specification. Three candidate distributions are considered in Case A: the scaled and shifted z-distribution (i.e. normal distribution), the scaled and shifted t-distribution, and the Laplace distribution. Five candidate distributions are considered in Case B: rectangular, triangular, quadratic, raised cosine, and half-cosine. According to the proposed minimum entropy criterion, the scaled and shifted z-distribution is the best distribution in Case A, and the raised cosine distribution is the best distribution in Case B. A case study is presented to demonstrate the effectiveness of the proposed minimum entropy criterion.

Sub-micron inline thickness measurement of cold-rolled metal strips by multi-wavelength interferometry and laser triangulation.

Thin metal foils of thicknesses below 100 µm are finding increasing use in high-tech applications. For such foils it is essential that production be controlled inline with sub-micron accuracy in highly challenging environments. An optical thickness gauge combining laser triangulation with multi-wavelength interferometry has now been developed for this purpose. Modulation-based 2f-3f-interferometry was used to realize a compact and robust sensor. A thorough measurement uncertainty analysis of the complete thickness measurement process yielded an expanded measurement uncertainty of U=(0.30μm)2+4π R a2, which is dependent on the roughness average Ra. The influence of oil remnants on measurement results is significantly weaker in the interference measurement than in geometric optical systems. Verification measurements against tactile reference measurements support the derived measurement uncertainty, and initial measurements in actual rolling mill environments have proven the real-world capability of this measurement technique over relevant process time scales at metal strip speeds of 200 m/min.

Uncertainty Analysis of Particle Backscattering Coefficient Measurement for Multiple Highly Turbid Water Areas in Ocean Color Remote Sensing

Uncertainty Analysis of Particle Backscattering Coefficient Measurement for Multiple Highly Turbid Water Areas in Ocean Color Remote Sensing

Non-contact ultra-precision metrology of superfine cylinders with a developed two-dimensional coordinate measuring device

To address the issue of measuring force of a stylus that can move and/or hook up the measured workpiece with a small dimension in the contact measurement, a spectral confocal displacement sensor and an ultra-precision linear stage are employed to develop a two-dimensional coordinate measuring machine instead of the conventional stylus contact measurement to avoid the measuring force problem, by which the non-contact ultra-precision metrology can be achieved. A coordinate modeling-dividing method is proposed for the diameter and roundness measurement of the small cylindrical workpieces with a diameter of 0.5 mm and length of 50 mm, whose cross-sectional circle is divided into several equal arcs to be scanned by the developed two-dimensional coordinate measuring machine. Thus, a set of arcs, each of which consists of large numbers of coordinate points, can be obtained. The radius of each arc can be fitted by the least square method, and the arc profile can be characterized by a series of data processing. The diameter and roundness of the measured workpiece can be obtained by averaging these fitted arcs’ radii and stitching these arc profiles together, respectively. A preliminary measurement uncertainty analysis is carried out to verify the developed device and proposed method.

In-line measurement of fiber mass fraction using Terahertz spectroscopy for a function-oriented quality assurance of glass fiber sheet molding compound

Glass fiber sheet molding compounds are a popular fiber-reinforced polymer because of their good processability. Both fiber mass fraction (FMF) and fiber orientation distribution significantly determine the mechanical properties of manufactured components. Local random variations in both properties occur during prepreg production and molding. This contribution expands a non-destructive measurement model for local FMF using Terahertz spectroscopy towards multiple specimen thicknesses. A thorough measurement uncertainty analysis according to the Guide to the Expression of Uncertainty in Measurement (GUM) shows an uncertainty reduction by 70 % compared to the uncertainty resulting from the manufacturing process. Measurement times of 0.5 s qualify the methodology for an in-line application. Integrating the measurement results into a parametrized, component-specific finite element model reduces the root mean squared error between physical experiments and simulations by 33 and 54 % compared to simulations with nominal fiber mass fractions.

Measurement Uncertainty Analysis of the Stitching Linear-Scan Method for the Measurable Dimension of Small Cylinders

A stitching linear scan method is proposed for roundness and diameter measurement of small cylindrical workpieces instead of the conventional rotary scan method due to the crucial alignments of eccentricity and inclination. To verify the reliability of the proposed method, by which the coordinates of the cross-sectional circle of a small cylinder is divided into several equal parts to be obtained and reconstructed, the diameter and roundness measurement uncertainties of the small cylinders with a diameter 1.5 mm are evaluated to be 0.047 μm and 0.095 μm, respectively, which can meet the uncertainty target of 0.1 μm. To investigate the measurable dimension by the proposed method, measurement uncertainty analysis of the small cylinders with various dimensions has been conducted according to the previous evaluation, since all the procedures are the same except for the uncertainty of X coordinates, which changes with the measured dimension’s change. The results show that the small cylinders with a diameter range from 0.01 mm to 50 mm can be measured by the proposed method when the position error θZ is reduced to the corresponding value. There is no measuring limitation set by the proposed model theoretically in the case of θX = θZ = 0.1°, while the machine has a measuring limitation.

Measurement uncertainty analysis of sound intensity using double-coupler calibration system

Currently, sound intensity instrument calibrations consist of a paired microphone phase and pressure calibrations, and the general uncertainty analysis focuses on how to calculate and express the uncertainty in the pressure lever and phase difference. However, no guidance for intensity level uncertainty is available. A significant barrier is the requirement to develop a measurement model that constitutes a relationship between the output quantities and input quantities, known to be the paired microphone pressure levels and phase difference. The Beijing Aerospace Institute of Metrology and Measurement Technology (AIM) is addressing this issue using a double-coupler calibration system, which is the incentive for the systematic evaluation of the associated measurement uncertainty in this study. A set of methodologies based on the plane-wave model is developed for estimating intensity and particle velocity uncertainty for sound intensity instruments.

Research on OTA Testing Optimization of 5G IoT Devices

Over-the-air (OTA) testing for wireless devices is crucial to guarantee actual network performance. Current OTA testing requires respectively 266 and 62 grid points for Total Radiated Power (TRP) and Total Isotropic Sensitivity (TIS) measurement. For 5G Internet of Things (IoT) devices, especially for low transmit duty-cycle devices, this number of grid points could lead to an unacceptably large amount of test time. Therefore, test time reduction is significant. The objective of this paper is to determine a suitable measurement grid for OTA testing of 5G IoT devices which balances measurement grid uncertainty/errors with test time. Two reference patterns representing a reasonable worst-case scenario of IoT devices are proposed for measurement uncertainty (MU) analysis of different grid configurations. The effects of different grid configurations on OTA testing accuracy are evaluated. The determination of associated MU term is proposed and determined based on statistical analysis. Finally, the reduction of grid points from currently required 62 (30/30 in theta/phi) to 32 (45/36) could save roughly 50% test time while requiring an uncertainty increase of less than 0.2 dB, which is acceptable for OTA testing of 5G IoT devices.

Uncertainty analysis and performance evaluation of thermophysical property measurement of liquid Au in microgravity

A new method for quantifying facility performance has been discussed in this study that encompasses uncertainties associated with thermophysical property measurement. Four key thermophysical properties: density, volumetric thermal expansion coefficient, surface tension, and viscosity of liquid Au have been measured in microgravity environment using two different levitation facilities. Levitation experiments were conducted using the Electrostatic Levitation Furnace (ELF) onboard the ISS in Argon and air, and the TEMPUS Electromagnetic Levitation (EML) facility on a Novespace Zero-G aircraft parabolic flight in Argon. The traditional Maximum Amplitude method was augmented through the use of Frequency Crossover method to identify the natural frequency for oscillations induced on a molten sample during Faraday forcing in ESL. The EML tests were conducted using a pulse excitation method where two techniques, one imaging and one non-imaging, were used to study surface oscillations. The results from both facilities are in excellent agreement with the published literature values. A detailed study of the accuracy and precision of the measured values has also been presented in this work to evaluate facility performance.

Geometric error measurement uncertainty analysis of rotary axis based on Monte Carlo method

Ballbar is widely used for geometric error measurement of the rotary axis, and the uncertainty analysis is of great significance to ensure the reliability of error identification. This study analyzes the measurement uncertainty based on the Monte Carlo method. First, this study clarifies the uncertainty mechanisms caused by tool setting error, translational axis motion error, and ballbar installation error, and proposes a measurement model comprehensively considering the uncertainty sources. In this study, the same ballbar measurement experiments were carried out 10 times in three measurement modes, and the standard deviation of the ballbar length change under different rotation angles was analyzed to ensure that the uncertainty of the geometric error can be effectively analyzed. Furthermore, the uncertainty of each error at different angles of the rotary axis is evaluated based on the measurement model. The results show that the translational axis motion error and tool setting error are important sources of uncertainty. The measurement uncertainty of the angle positioning error is the largest in A- and C-axis, and its standard uncertainty reaches 9.38″ and 2.81″. After optimizing the sine and cosine terms in the identification model, the standard uncertainties of the angular positioning errors are reduced to 6.43″ and 2.26″, and also the estimated errors are close to the actual values, which verifies the validity of the uncertainty analysis method.

Uncertainty Analysis of Spherical Joint Three-Dimensional Rotation Angle Measurement

A precision spherical joint is a type of spherical motion pair that can realize three degrees of rotation freedom. In this paper, a specific method is used to assess the uncertainty of our measurement system. The measurement system uses the codes and eddy current sensor to measure the angle. Different codes are engraved on the surface of the spherical joint. Eddy current sensors are embedded in the spherical socket to detect and recognize the spherical code. The uncertainty of the measurement system hardware and an artificial neural network are studied. Based on the Monte Carlo method, the uncertainty components are evaluated and combined, and the comprehensive uncertainty of the measurement system is obtained. The evaluation results of system measurement uncertainty are verified from specific angles. In the three-axis measurement uncertainty, the minimum is about 2′, and the maximum uncertainty is about 1°. The proposed method can be used to evaluate the uncertainty of other multiple-input multiple-output measurement systems.

A propensity-based framework for measurement uncertainty analysis

We regard the measurand (or influence quantity) as a random variable characterized by central tendency and dispersion. We define ‘propensity’ as the sum of the central tendency and dispersion. We argue that the state of propensity can be described by a probability density function (PDF) that we name ‘propensity spectrum’. This paper proposes a propensity-based framework for measurement uncertainty analysis of the measurand that is related to influence quantities through a measurement model. The proposed framework encodes the state of propensity of the measurand in the form of a PDF (propensity spectrum) based on all available information about influence quantities. This can be achieved by an “exact” method based on the principle of propagation of distributions, or a first-order approximation method (the law of propagation of uncertainty). A case study is presented to demonstrate the methods provided by the proposed framework, compared to the GUM method and Bayesian methods.

Using measurement uncertainties to detect incomplete assumptions about theory in an experiment with rolling marbles

In this lab activity, carbon copy paper is used to record the horizontal distance a marble flies off a table after rolling down an incline. The minimal scatter of the dots visually shows the high precision—i.e. the small uncertainty—of the measurements to students. The theoretical prediction of this distance will be too big if students forget to include rotational energy in the energy balance when they calculate the marble’s speed at the bottom of the incline. This results in a discrepancy between the predicted horizontal distance and the measurement result. The precision of the experiment and the absence of overlap with the theoretical prediction is evidence that the prediction has to be wrong. Including rotational energy and taking a 10% energy loss due to friction into account, makes the measurement result overlap with the theoretical prediction, bringing them into agreement. Thus, measurement uncertainties guide the process of comparing the measurement result with the prediction: overlap between the theory-based prediction and the measurement result indicates agreement, whereas no overlap implies discrepancy. The lab activity presented here is an activity where measurement uncertainties are used in a meaningful, indispensable manner. The experimental result is evidence that forces students to rethink their assumptions, in this case about the conservation of energy. This leads to the revision of their calculation, emphasizing the necessity to include rotational energy and friction. Without it, the highly precise measurement result is in disagreement with the theoretical prediction. A procedure such as this—comparing empirical data with theory—is an authentic and common practice in science and should thus find its way into the physics classroom; but it cannot be done without an analysis of measurement uncertainties.

Quantitative Assessment Method of Muzzle Smoke in a Field Environment.

The muzzle of barrel weapons produces a large amount of smoke (muzzle smoke), a major source of pollution in the battlefield. Quantitative assessment of muzzle smoke is an important support for the development of advanced propellants. However, due to the lack of effective measurement methods for field experiments, most of the previous studies were based on a smoke box, and few studies have focused on muzzle smoke in the field environment. In view of the nature of the muzzle smoke and the conditions of the field environment, the characteristic quantity of muzzle smoke (CQMS) was defined based on the Beer-Lambert law in this paper. CQMS is used to characterize the danger level of muzzle smoke produced by the propellant charge, and theoretical calculations indicated that when the transmittance is e -2, the impact of the measurement errors on CQMS can be minimized. Seven firings with the same propellant charge of a 30 mm gun were carried out in a field environment to verify the effectivity of CQMS. The measurement uncertainty analysis on the experimental results showed that the CQMS of the propellant charge used in this study was 2.35 ± 0.06 m2, which indicates that CQMS can be used to quantitatively assess muzzle smoke.