*1.3. A Case Study of Comparison Linking*

The intention of this case study is to look for possible benefits from digitalization of the reporting and analysis of data. A future scenario is envisaged, where participants submit results in a digital format that contains more information than today's calibration certificates. This allows data processing to be handled more directly and in a straightforward and intuitive manner. The scenario offers a glimpse into the future where, once the digitalization of the international measurement system is complete, digital reporting (DCCs) will be ubiquitous.

The study applies the methodology prescribed in [6], but a digital format called an "uncertain number" is used to represent the data [8]. An uncertain number is a data type that encapsulates information about the measured value of a quantity and the components of uncertainty in that value. Software supporting uncertain numbers greatly simplifies data processing, because calculations simultaneously evaluate the value and the components of uncertainty. Mathematical operations are expressed in terms of a value calculation, but the results include a complete uncertainty budget. Furthermore, uncertain-number results are transferable, which is extremely important in this work. (The *Guide to the expression of uncertainty in measurement* (GUM) identifies transferability and internal consistency as the desirable properties of an ideal method of expressing uncertainty ([7], §0.4). Although we refer here only to transferability for simplicity, uncertain numbers provide both transferability and internal consistency.) An uncertain number obtained as the result of some calculation may be used immediately as an argument in further calculations (exactly as one can do with numerical results). When this happens, the components of uncertainty are rigorously propagated, from one intermediate result to the next, according to the LPU. Transferability in this case study allows the CIPM and RMO comparisons, with linking, to be processed as a single, staged, measurement (the importance of adequately linking the stages of a metrological traceability chain was discussed in [9,10]). It is this aspect of digitalization that delivers the benefits we describe below.

A software tool called the GUM Tree Calculator (GTC) that implements the uncertainnumber approach was used. GTC is an open-source Python package [11,12]. A recent publication described GTC and its design in some detail [13]. A dataset containing the data and code used in the current study is available [14]. The snippets of code shown below are extracts from this dataset.

#### *1.4. Digital Records*

To create digital records for the participant and pilot measurements in this study, a small subset of data was taken from a CIPM comparison of transmittance and a subsequent RMO comparison [15,16]. In Sections 2 and 3, we describe the structure of these comparisons. Participants were required to submit an uncertainty budget for each measurement and to identify the systematic and random influence factors in that budget. The systematic factors are considered constant. For NMIs that participated in both CIPM and RMO comparisons, the systematic factors do not change. They are characterized as components of uncertainty, because the actual values of residual error are not known. The random factors are considered to be unpredictable effects that arise independently in each measurement. The nature of the components of uncertainty—systematic or random—must be known in order to account for correlations in the data.

We used the uncertainty budgets reported by participants to construct digital records for this study. In doing so, some assumptions were made about the data and some of the data were changed to resolve minor inconsistencies, so we do not identify actual participants with these records. The intention here is to present a future scenario where DCC formats have been widely adopted. The assumption was made that these formats are self-contained, with more detailed information than is available in today's calibration certificates, so there is no longer a need to request additional data for the comparison analysis. Were such a future to become reality, the processes leading to the production of DCCs would not resemble the steps taken here to artificially create the scenario. Therefore, the detail of how digital records were assembled for this study is not discussed.

#### *1.5. Mathematical Notation*

Mathematical expressions use the notation adopted in [6]. Most details are explained when the notation first appears in the text. However, the reader should note that we distinguish between quantities and estimates of quantities with upper and lower case symbols, respectively. For instance, the uncertainty in a value *y*, obtained by measuring a quantity *Y*, will be expressed as *u*(*y*)—the standard uncertainty of *y* as an estimate of *Y*. When GTC code is used to implement mathematical expressions, uncertain-numbers are associated with quantity terms (upper case terms). The corresponding estimates and uncertainties are the properties of these uncertain-number objects.

#### **2. A CIPM Key Comparison**

In the initial CIPM key comparison, there were eleven participants (identified here by the letters A, B, . . . , K) and a pilot laboratory (Q). Each participant measured a particular artifact, while the pilot measured all eleven artifacts. The comparison was carried out in five stages: first, the pilot measured the artifacts; second, each participant reported a measurement; third, the pilot measured the artifacts again; fourth, each participant made a second measurement; and fifth, the pilot made one last measurement of all the artifacts (Two participants submitted only the second measurement, so these data were processed with pilot results from only Stages 1 and 3).

Listing 1 displays information about the first measurement by Participant A (Stage 2). The measured transmittance appears at the top, with the combined standard uncertainty in parentheses. Two uncertainty budgets follow: first, the individual components of uncertainty; second, the net systematic and random effects components. Component labeling uses a capital letter to identify the participant (A, B, etc.). If a component of uncertainty contributed only to a specific stage, then a stage number (1, 2, 3, 4, or 5) is appended in parentheses. A colon then precedes the participant's name for the influence quantity, and finally, the component is classified as random or systematic ((rnd) or (sys)). For example, there are both random and systematic contributions to uncertainty in the wavelength, so Listing 1 includes two terms: A:Wavelength (sys) is a systematic component that contributes to uncertainty at every stage, and A(2):Wavelength (ran) is a random component that contributes at Stage 2 (another independent component A(4):Wavelength (ran) appears in the budget at Stage 4). It is important to understand that the information shown in Listing 1 was all obtained from a single entity representing the measurement result—a single uncertain number. In the scenario we considered, this was submitted by the participant in a digital record.

When mathematical operations are applied to uncertain numbers, the components of uncertainty are handled according to the LPU. To illustrate this, we compared the results submitted by Participant A at Stages 2 and 4 by subtracting the corresponding uncertain numbers. With Y\_A\_2 and Y\_A\_4 for the results, we display the uncertain-number difference in Listing 2. Notice that the only non-zero terms in the uncertainty budget are now associated with random effects at each stage. The systematic terms from the budget of Listing 1 (the non-linearity, wavelength, stray light, and the beam size and position) contribute nothing to the combined uncertainty in the difference. This is to be expected, because each systematic term contributes a fixed (albeit unknown) amount to the combined measurement error. The influence of these constant terms on the difference is zero. Uncertain-number calculations arrive at the correct result by strictly implementing the LPU. In order to do that, information about all uncertainty components must be encapsulated in the uncertain-number data.

**Listing 1.** Data from Participant A for the Stage-2 measurement. The measured value is shown at the top, with the combined standard uncertainty in parentheses. Two uncertainty budgets follow. The first shows the individual components of uncertainty reported by the participant. The second shows total systematic and random components.

```
A(2):Transmittance = 0.919644(0.000296)
Uncertainty budget:
        A:Beam Size & Position (sys): 0.00019516
     A(2):Beam Size & Position (ran): 0.00019516
                   A(2):Type-A (ran): 0.00007800
              A(2):Instability (ran): 0.00006800
               A:Non-linearity (sys): 0.00003000
                  A:Wavelength (sys): 0.00000354
               A(2):Wavelength (ran): 0.00000354
                 A:Stray Light (sys): 0.00000300
Systematic and random:
                          A(2) (ran): 0.00022093
                             A (sys): 0.00019751
```
**Listing 2.** The difference between Participant A's results at Stage 2 and Stage 4. The difference is shown at the top, with the combined standard uncertainty in parentheses. The uncertainty budget follows. Note that all systematic components are now zero.

```
Difference:Y_A_4 - Y_A_2 = 0.000449(0.000489)
```
Uncertainty budget:

```
A(4):Type-A (ran): 0.00038400
A(2):Beam Size & Position (ran): 0.00019516
A(4):Beam Size & Position (ran): 0.00019516
             A(2):Type-A (ran): 0.00007800
        A(2):Instability (ran): 0.00006800
        A(4):Instability (ran): 0.00006800
         A(2):Wavelength (ran): 0.00000354
         A(4):Wavelength (ran): 0.00000354
         A:Non-linearity (sys): 0.00000000
            A:Wavelength (sys): 0.00000000
           A:Stray Light (sys): 0.00000000
  A:Beam Size & Position (sys): 0.00000000
```