**1. Introduction**

Technological advancements within the last few decades have served to digitalise many aspects of metrology. For example, instruments can be programmed to undertake time-consuming measurements with little or no need for human interaction, while the availability of greater computer processing power allows complex systems to be modelled increasingly accurately. There is one aspect of metrology, viz. the provision of calibration services, to which digital transformation has been applied in a much more modest way. Many calibration service providers continue to disseminate calibration information using paper-based certificates. Some organisations have moved to providing certificates in electronic form, for example, in archiveable Portable Document Format (PDF-A) [1]. While the provision of electronic certificates brings obvious benefits such as decreased use of paper and the potential for storage within dedicated document management systems, one undesirable property persists—a lack of machine-readability, i.e., information is not presented in a form that can be processed by computer. Currently, information on a paperbased or electronic calibration certificate can only be used if it is transcribed manually. Such a process is inevitably prone to error.

Recent initiatives have looked at how paper-based or electronic calibration certificates can be replaced by fully machine-readable certificates. The European Metrology Programme for Innovation and Research (EMPIR) [2] has funded the Joint Research Project 'Communication and validation of smart data in IoT-networks' (short name 'SmartCom') [3,4]. One objective of the SmartCom project has been to develop a framework for what are referred to as 'digital calibration certificates', abbreviated hereafter in this paper to 'DCCs'. From the perspective of the SmartCom project, the critical property of DCCs is that they are fully machine-readable. It is noted that the term 'digital calibration certificate' has been and is used by other authors to refer to calibration certificates that take the form of electronic files but are not machine-readable.

**Citation:** Smith, I.; Luo, Y.; Hutzschenreuter, D. The Storage within Digital Calibration Certificates of Uncertainty Information Obtained Using a Monte Carlo Method. *Metrology* **2022**, *2*, 33–45. https://doi.org/10.3390/ metrology2010003

Academic Editor: Simona Salicone

Received: 30 September 2021 Accepted: 10 January 2022 Published: 18 January 2022

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When presenting the outcome of calibration, measurement data must be accompanied by associated uncertainty information. On a calibration certificate, it is common for a quantity value to be provided along with an associated expanded uncertainty (defined in the International Vocabulary of Metrology (VIM) [5], clause 2.35) and coverage factor (VIM [5], clause 2.38), or a coverage interval (VIM [5], clause 2.36), corresponding to a specified coverage probability (VIM [5], clause 2.37). A standard uncertainty may also be provided.

The focus of this paper is on the storage of uncertainty information obtained using the numerical approach described in the supporting document to the 'Guide to the expression of uncertainty in measurement' (GUM) [6] known as Supplement 1 to the GUM (GUMS1) [7]. The approach is a Monte Carlo method (MCM) for the propagation of probability distributions and is based on repeated random sampling. A key aspect underpinning the approach is the provision of a measurement model that describes mathematically how a quantity of interest (the measurand or output quantity) depends on other quantities (input quantities) to which probability distributions can be assigned. The output of an implementation of MCM (a 'Monte Carlo calculation') provides rich information in the form of (often hundreds of thousands of) sampled values of the measurand. The sampled values can be used to define an approximation to the probability distribution for the measurand. Summary information can be calculated using those sampled values. For example, the expectation and the standard deviation provide, respectively, an estimate of the measurand and its associated standard uncertainty, while a coverage interval for the measurand corresponding to a specified coverage probability can also be determined. The provision of additional summary information has been considered, e.g., in [8].

For calibration services where MCM is used to undertake uncertainty evaluation, it is common for only summary information to be provided on calibration certificates. The reasons for not including the sampled values of the measurand on the certificate are understandable, e.g., the number of pages could increase significantly, and the effort required to transcribe the sampled values would make it highly unlikely that they would ever be used in practice. It is possible for the sampled values to be made available in an electronic file. When doing so, consideration must be given to aspects including the provision of additional information such as units of measurement and appropriate metadata, while the electronic file must also be transmitted using a suitably secure means that ensures the file cannot be corrupted.

Should information about the measurand be required as input to a subsequent calculation, it is common, in the absence of any other information, for a Gaussian (normal) distribution, with expectation and standard deviation given, respectively, e.g., by the estimate and standard uncertainty quoted on the calibration certificate, to be assigned to the measurand. Such an assignment is often made even though the true probability distribution may be significantly different. The quality of the result of the subsequent uncertainty calculation may be significantly influenced by the assumption of normality. Were the sampled values generated by MCM available, one could instead implement MCM for the subsequent calculation by drawing randomly from those values. The SmartCom project has developed a data model that allows measurement data and associated uncertainty information to be stored in digital form. The model builds upon the International System of Units (SI) [9], the globally-agreed system of measurement units that has at its heart the seven base units of kilogram, metre, second, ampere, kelvin, mole and candela. The data model, referred to as the 'Digital SI' (frequently shortened simply to 'D-SI') [10], allows the representation of quantities that are real or complex, and univariate or multivariate.

This paper focuses on how the D-SI allows uncertainty information, including the complete set of results of a Monte Carlo calculation, to be provided within a DCC. While the GUMS1 approach to uncertainty evaluation is well-established, DCCs are a much more recent development and the potential overlap between GUMS1 and DCCs has not previously been discussed. Consideration is given to the cases where the measurand is real and univariate, i.e., a single real quantity, and real and multivariate, i.e., comprises more than one real quantity. Section 2 provides a brief summary of uncertainty evaluation undertaken using MCM for both cases. Section 3 introduces the main components of the DCC and outlines how measurement data and associated uncertainty information for real quantities can be encapsulated in the D-SI. Section 4 describes two examples, the first relating to the measurement of a univariate real quantity and based on an example in GUMS1, the second relating to the measurement of a multivariate real quantity and based on an example in Supplement 2 to the GUM (GUMS2) [11]. Concluding remarks are presented in Section 5.

Note that this paper does not discuss technical and legal aspects associated with the generation, delivery and use of DCCs. Such aspects are considered in, e.g., [12].
