**4. Discussion**

This paper provides some insight into the usefulness of uncertain numbers, which have the distinctive feature of providing an abstract representation for measured quantities, allowing uncertainty calculations to be automated. Recently, uncertain numbers have been applied to a goniometric measurement system for optical reflectance. The fouraxis goniometric system has many configuration errors that must be considered in the measurement model to account for final measurement uncertainties [7]. The application of GTC was carefully compared with alternative computational methods, using Monte Carlo and direct mathematical analysis. GTC was found to be the preferred choice. Using the information provided by uncertain numbers, the authors were able to obtain a better understanding of the measurement system and the inherent correlations between significant measurement errors. This enabled them to significantly improve the accuracy of certain measurements.

The inherent support for metrological traceability is perhaps the most important quality of uncertain numbers. This aspect is implemented in data structures and storage formats used by GTC, which is a particular choice but other formats would be possible. One can easily imagine a more heterogeneous situation, where processing at various stages would be carried out using different software tools. To support this, the format for exchange of data between stages would need to be standardised. That is, there would need to be agreed formats for representing uncertain numbers, which would be used in digital reporting documents such as calibration reports [13].

Digitalisation should offer benefits that are not currently available. The GUM recommends that detailed information about influence quantities be reported at each stage. However, this rarely happens, because calibration certificates and other measurement reports are intended to be read by people; thus, handling the additional data would be difficult. As a consequence, information about common influence factors is rarely shared. A simple situation where this might arise is the scenario of a batch of sensors that are calibrated using a more accurate reference device. If the common reference is ignored, the accuracy of results obtainable from a survey of the sensors' readings is compromised [14]. However, uncertain-number calibration factors can track common effects and account for them when comparing readings from different sensors. This was illustrated in Section 2.4, where E\_off contributed a common offset to single voltage readings but nothing to uncertainty in the voltage difference. It is also worth noting that a 'smart' sensor capable of reporting

uncertain-number results would not need to process a lot of information. As was the case of the simple voltmeter, a model of the sensor measurement might only require a few influence factors and the calculations would be simple.

The various stages of a traceable measurement often occur in different locations (national metrology institute, second-tier calibration laboratory, etc.) but they may also happen at different times in the same location. For example, a working standard might be calibrated in-house against an externally calibrated transfer standard. The working standard would then be used repeatedly to calibrate different instruments at different times. Importantly, measurement errors realised when at the time the working standard is calibrated should be treated as systematic effects in subsequent instrument calibrations. Performing this would allow any bias, or correlation, in downstream measurement results using those instruments to be accounted for correctly. This could be easily handled by digitalisation if uncertain-number storage and retrieval mechanisms are used to save calibration data for the working standard and later retrieve it for reuse when instruments are calibrated.

During formal international measurement comparisons, national metrology institutes (NMIs) go to much greater pains when reporting measurement data than they do for regular calibration work. These international comparisons assess the competence of NMIs in performing specific types of measurement. The more detailed reporting requirements in comparisons align with the GUM's recommendations in this case. A recent study, which explored a future scenario where an uncertain-number reporting format was used by all participants, showed that using uncertain numbers would not only provide the information required, but they would also simplify comparison analysis and comparison linking and provide additional insights into the results [6].

Measurement models are needed in order to use uncertain numbers effectively. The close correspondence between quantity terms in a model and uncertain numbers in data processing routines makes software development and testing more robust and reliable and avoids the need to explicitly derive expressions for the components of uncertainty from a model, which GTC handles automatically. However, although modelling lies at the heart of the GUM's approach, skilled metrologists are often confident in their ability to assess measurement uncertainty heuristically and frequently elide the formal modelling step. This presents a problem for digitalisation, because digital systems need a rigorous formal problem definition for autonomous operation. Some tutorial guidance on developing measurement models has been provided in a recent booklet [15] and is also the subject of another paper [16]. There is also a new supplement to the GUM, which deals with modelling [17].

One common conceptual difficulty when modelling is the omission of influence quantities estimated as zero. These terms would not be needed in conventional data processing; however, they must be modelled, because the actual (unknown) values affect the final measurement result, and so they contribute to uncertainty. Influence quantities with trivial estimates are often called residual errors. The electrical network example, in Section 2.4, included three residual errors that were all estimated as zero. These terms were represented by uncertain numbers and modelled imperfect voltmeter behaviour.

## **5. Conclusions**

GTC is a software tool for data processing with automatic evaluation of measurement uncertainty. It follows international best-practice, described in the GUM, and offers useful extensions to those methods for important special cases. The use of uncertain numbers is a distinctive feature of GTC. The uncertain-number data-type facilitates data processing, which can be performed in a piece-wise and open-ended manner. This allows calculations to be more easily matched to the models of a measurement performed in stages. The automation of uncertainty calculations allows measurement data processing to be made more rigorous, which can lead to accuracy enhancement in some cases. The uncertainnumber format significantly exceeds current paper-based practices that support traceability. Therefore, GTC and the data structures used to implement uncertain numbers are a useful example of software that meets the requirements of a fully functional digital infrastructure for metrological traceability.

**Funding:** This is work was funded by the New Zealand Government.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The author is grateful to Peter Saunders and Joseph Borbely for careful reviews of the manuscript.

**Conflicts of Interest:** The author declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Additional Details**

This Appendix includes details about a number of other aspects of GTC. Support for problems involving finite degrees of freedom with correlated inputs is discussed in Appendix A.1 and support for complex quantities is briefly covered in Appendix A.2. Appendix A.3 presents an unusual case, where counter-intuitive results are obtained due to the relationship between Python variables and underlying uncertain-number objects. This provides further insight into computational mechanisms. Appendix A.4 briefly describes the validation of GTC and Appendix A.5 compares GTC with some similar software projects.
