Issue 
Int. J. Metrol. Qual. Eng.
Volume 9, 2018



Article Number  3  
Number of page(s)  24  
DOI  https://doi.org/10.1051/ijmqe/2017026  
Published online  10 April 2018 
Research Article
A new approach to the analysis of Type 1 nonuniqueness of the ITS90 above 0 °C
^{1}
Formerly: National Institute of Metrology (INM),
Bucharest, Romania
^{2}
Formerly: Institut National de Métrologie – Laboratoire National de métrologie et d'essais/ Conservatoire National des Arts et Métiers,
La Plaine SaintDenis, France
^{*} Corresponding author. sonia.gaita@temperature.ro
Received:
15
March
2017
Accepted:
15
November
2017
The Type 1 nonuniqueness (NU1) is the difference between interpolated values at the same temperature in the resistance thermometer subranges of the International Temperature Scale of 1990 (ITS90) that overlap. The paper argues for a method of evaluating the NU1 at a given temperature which considers all subranges of the Scale that contain the respective temperature, not only combinations of two, and it proposes mathematical models to determine the values of NU1 for temperatures above 0 °C. The paper demonstrates that NU1 is not the right contributor to the uncertainty associated with the realisation of the ITS90. Therefore, a new concept of Correction for the Type 1 nonuniqueness of the Scale, C_{NU1}, is introduced and its mathematical model is established. Also, the estimate of C_{NU1} and its standard uncertainty are defined and they are assessed through statistical analysis. The values of standard uncertainty determined by the novel methodology do not exceed 0.26 mK and they are smaller than the values given in the specific Guides developed by the Consultative Committee for Thermometry. The proposed models allow authors to single out and analyse the factors that generate Type 1 nonuniqueness of the Scale and influence its value.
Key words: International Temperature Scale of 1990 (ITS90) / Type 1 nonuniqueness of the ITS90 / measurement uncertainty / standard platinum resistance thermometer (SPRT)
© S. Gaita and G. Bonnier, published by EDP Sciences, 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The Type 1 nonuniqueness, hereafter noted as NU1, emerged as a major concern of thermometry community during the process of establishing the International Temperature Scale of 1990 (ITS90) [1]. Unlike its predecessors, the current Scale comprises several subranges that overlap, each of them with a distinct definition of temperature T_{90}. At a given temperature, the interpolated values using the specified equations for the overlapping subranges may or may not be consistent with each other. The numerical difference between these values [1] is currently called the Type 1 nonuniqueness[2,3].
Understanding of the concept, hereafter called, indicatively, Type 1 nonuniqueness of the Scale, is important when calibrations according to the ITS90 are performed using standard platinum resistance thermometers (SPRTs). The Type 1 nonuniqueness of the Scale is one of the factors that influence the results of interpolations made between fixed points in the SPRT subranges that overlap. Therefore, the correction of its effect must be one of the input quantities in the mathematical models of interpolations and the uncertainty of this correction must be one of the components of the uncertainty in realising the ITS90 between fixed points. In their turn, the interpolated values − along with the values determined at the fixed points − and their uncertainties are the sources of traceability for all measurements made between 14 K and 962 °C; in other words, they are the references for the calibration hierarchies, where each measurement result is related to the previous result and the measurement uncertainty increases gradually.
The first study available on Type 1 nonuniqueness (formerly called subrange inconsistency) is the pioneering work of Hill and Bedford [4] performed during the development process of the ITS90. In this study, a procedure for minimizing the internal inconsistency of the Scale by adjusting the temperatures assigned to the fixed points is described. Subsequently, Crovini revealed [5] the final adjustments made by the designers of the ITS90 to obtain the highest level of agreement between the definitions of subranges that overlap.
Starting in the early 90's, the thermometry community was very interested in Type 1 nonuniqueness above 0 °C. The studies were mainly concerned [6–8] with determining the values of the Type 1 nonuniqueness for combinations of two overlapping subranges, using a larger or a smaller set of SPRTs calibrated at the fixed points. For each such combination, the NU1 values were calculated as differences between the interpolated temperatures within the two subranges that form the pair. In addition, Strouse [6] presented descriptive statistics (but not the average and the sample standard deviation) for all of the 15 combinations of two overlapping subranges above 0 °C.
In a paper that brought significant contributions to the topic [9], Zhiru Kang et al. studied the subranges pair 0 °C to 420 °C and 0 °C to 660 °C using various methods, from statistical analysis to Lagrange interpolation. The authors have derived a simple formula for calculating the NU1 values based on the c coefficient of the deviation function in the subrange 0 °C to 660 °C. Moreover, they have developed the first method to evaluate the standard uncertainty associated with NU1. This paper was followed by the work of White and Strouse [10], who investigated the same pair of two subranges, 0 °C to 420 °C and 0 °C to 660 °C. In their study, a polynomial curve was derived of the standard deviations calculated for the differences between the two subranges. The formula was taken in the Document CCT/0819/rev [11] and the Guide to the Realization of the ITS90 [3] to estimate standard uncertainties of NU1 between 0 °C and 420 °C. Also, the authors have derived [10] mathematical expressions of NU1 for combinations of two subranges between 84 K and 660 °C. Later, Zhiru Kang et al. [12,13] extended the investigation of Type 1 nonuniqueness initiated in [9] to the other 14 pairs of overlapping subranges above 0 °C. The authors have derived simple formulas for the calculation of the maximum NU1 values in the case of 11 of the subranges pairs under study.
Over twentyfive years have passed since the ITS90 was adopted, but the issue of its intrinsic Type 1 nonuniqueness has not been fully clarified yet. There is limited coverage in the specific literature regarding the definition and quantification of its effect on interpolated values. All of the papers published so far [4–18] have considered and studied exclusively combinations of two subranges that overlap, the differences associated with each pair being regarded as estimates of the effect.
This article presents a different approach. A distinction is made between the general concept of Type1 nonuniqueness of the Scale and its measure defined in [1–3]: Type 1 nonuniqueness. The paper argues for a method of evaluating the Type 1 nonuniqueness between all subranges that overlap, not only between two of them. Consequently, the notions of SimpleType 1 nonuniqueness and Combined Type 1 nonuniqueness are advanced and mathematical models for combinations of two and, respectively, more subranges that overlap are developed. Analytical expressions of the Simple Type 1 nonuniqueness for the 15 pairs of subranges that overlap above 0 °C are derived.
The paper demonstrates that Type 1 nonuniqueness is not the right contributor to the uncertainties of the interpolated values in the SPRT subranges of the ITS90. For this reason, the new concept of Correction for the Type 1 nonuniqueness of the Scale,C_{NU1}, is introduced and the analytical expressions for the 15 possible cases are derived. The best estimate available of the effect of the Type 1 nonuniqueness of the Scale on the interpolated values and its standard uncertainty are defined. In addition, they are evaluated through statistical analysis of the data derived from the results of the key comparison CCTK3 [19].
A basic characteristic of all the proposed mathematical models in this study is the expression of the output quantity as function of the deviations determined at the fixed points. This form of the equations allows one to single out and analyse the factors that generate the Type 1 nonuniqueness of the Scale and influence its value.
The article is organised as follows. Starting from the interpolation equations defined in ITS90, the mathematical models for the calculation of the Simple Type 1 nonuniqueness and the Combined Type 1 nonuniqueness are proposed in Section 2. The concept of Correction for the Type 1 nonuniqueness of the Scale, along with its mathematical model, are introduced in Section 3. Section 4 concentrates on the data analysis and the discussing the results, including their comparison with results of other studies. In Section 5, a set of remarks and comments regarding the sources of Type 1 nonuniqueness of the Scale are presented. Section 6 gathers certain concluding remarks.
2 Sorts of Type 1 nonuniqueness
Within the temperature range 0 °C to 961,78 °C, which is the scope of this article, ITS90 defines the temperature T_{90} using SPRTs calibrated at specified sets of defining fixed points [1]. T_{90} is determined in terms of the resistances ratio: (1) by means of a continuous reference function W_{r} (T_{90}), hereafter denoted by W_{r}, with the coefficients provided in the ITS90, and by means of a deviation function [1] (2)
The form of the deviation function, hereafter designated by ΔW, is specified for each subrange [1].
There are six SPRT subranges of ITS90 that overlap above 0 °C (including the subrange −38.8344 °C to 29.7646 °C). They are presented in Table 1 together with the sets of fixed points and the deviation functions used to define the temperature T_{90}. (The full range 0 °C to 961.78 °C is excluded from analysis because it has the same definition for T_{90} as the subrange 0 °C to 660.323 °C in their region of overlap.) Although the analytical expressions of certain deviation functions are identical (see Table 1), the values of their coefficients are not equal because they are determined by calibration at different fixed points.
Definition of T_{90} in the SPRT subranges of the ITS90 that overlap above 0 °C.
2.1 The Simple Type 1 nonuniqueness
Let us first consider the straightforward case of Type 1 nonuniqueness between two overlapping subranges, hereafter called Simple Type 1 nonuniqueness. The six subranges will be designated in the suggestive manner used in the specific literature [6,10,12] by means of the symbol of the metal whose fixed point temperature is the upper limit of the subrange. The exception is the subrange −38.8344 °C to 29.7646 °C, for which the symbol “Hg” is used. The subrange symbols will be written between brackets so they may not be mistaken for the symbols that designate fixed points. Also, a pair of subranges (S_{j}) and (S_{h}) that overlap will be indicated very concisely by the symbol (S_{j}−S_{h}).
The number of possible combinations of two subranges that overlap above 0 °C varies as follows (Table 2): (a) 15, between 0 °C and 29.7646 °C (Region 1); (b) 6, between 29.7646 °C and 156.5985 °C (Region 2); (c) 3, between 156.5985 °C and 231.928 °C (Region 3); (d) 1, between 231.928 °C and 419.527 °C (Region 4).
For any pair of overlapping subranges (S_{j}) and (S_{h}), the calculating the Simple Type 1 nonuniqueness can be done in two ways, starting from equation (2):

at a given W in the region of overlap: NU1 is the difference between the reference functions and, respectively, determined at W in accordance with the two definitions of the ITS90:

at a given temperature T_{90} in the region of overlap (meaning a given W_{r}): NU1 is the difference between the resistances ratios W^{ (Sj)} and, respectively, W^{ (Sh)} determined at T_{90} in accordance with the two definitions of the ITS90:
Note: Hereafter we shall refer only to Case 1, because Case 2 can be easily derived from the first by changing the sign, see (3) and (4).
It follows that Type 1 nonuniqueness for the (S_{j}S_{h}) pair can be expressed by the difference between the deviation functions and specific to the two overlapping subranges. But each deviation function ΔW can be expressed in terms of its values obtained directly from the calibration of the thermometer at the fixed points (FP) in the respective subrange; these values are hereafter termed deviations and denoted by ΔW_{FP}. Thus, for the subrange (S_{j}), the deviation function can be written in the form (5) where are polynomial functions of W and of the ratios of resistances W_{FPi} determined at the N points of calibration in the respective subrange, except for the triple point of water (N ≤ 3).
The expressions of the functions can be derived through elementary algebra: by solving a linear equation with one variable or by solving systems of linear equations with 2 or 3 variables. The functions will be called hereafter functions of propagation, because, through them, the deviations ΔW_{FPi} are propagated from the temperatures of fixed points to intermediary temperatures. The formulae^{1} of are given in Appendix A, Section A.1.
For the sake of simplicity and concreteness, let us consider one of the subranges above 0 °C of the ITS90, say (Zn). After rearranging the expressions obtained for the coefficients a and b of the deviation function ΔW (Table 1) in terms of the deviations ΔW_{Sn} and ΔW_{Zn}, the interpolating equation becomes (6) where and are functions of propagation, and their expressions are given in Appendix A, Section A.1.
Similarly, for the subrange (Sn), the deviation function is (7) with and given in Appendix A, Section A.1.
Type 1 nonuniqueness between subranges (Zn) and (Sn) is obtained from (3) in combination with (6) and (7) (8) where , , and .
Coming back to the general case now, the Simple Type 1 nonuniqueness between any 2 subranges (S_{j}) and (S_{h}) follows from (3) and (5) (9) where M and N represent the number of calibration points in subrange (S_{h}) and, respectively, in subrange (S_{j}) (except for the triple point of water). If there are fixed points common to both subranges that overlap (such as the freezing point of Sn in the example (ZnSn) above), then the similar terms in the corresponding deviations are combined into one single term and (9) becomes (10)P is the sum of the calibration points M and N, where the fixed points common to (S_{j}) and (S_{h}) are included once.
Equation (10) represents the mathematical model of the Simple Type 1 nonuniqueness. It describes the relation between the output quantity − the Simple Type 1 nonuniqueness − and the input quantities − the functions and the deviations ΔW_{FPk}. The expressions of the Simple Type 1 nonuniqueness for the 15 pairs of subranges that overlap above 0 °C are presented in Appendix A, Section A.2^{2}.
are polynomial functions of W and of the ratios of resistances W_{FPi} at all the P points of calibration in the subranges S_{j} and S_{h} . It is important to note that the functions are either identical to the propagation functions and , or they are algebraic sums of the latter (Appendix A, Section A.2). For the sake of simplicity, the functions shall be named hereafter combined functions of propagation or, shortly, functions of propagation.
If we substitute W_{FPk} = W_{r,FPk}, where W_{r,FPk} are constants − substitution that generates negligible errors in NU1 (less than 1 µK) −, then each function in (10) becomes a function of a single variable, namely of W. As a result, at a W given, becomes constant and the only variables remaining in the expression (10) are the deviationsΔW_{FPk}. So, the proposed model (10) provides an extremely simple and rapid manner to calculate the values of the Simple Type 1 nonuniqueness.
Yet, in its exact meaning, the Type 1 nonuniqueness at a given W is to be considered in reference to all subranges that include this W value, not only to two of them. The number of subranges that overlap differs from one region to another (Table 2): (a) six, in Region 1; (b) four, in Region 2; (c) three, in Region 3; (d) two, in Region 4.
Pairs of subranges that overlap above 0 ^{°}C.
2.2 The Combined Type 1 nonuniqueness
The text of the ITS90 [1] states the Type 1 nonuniqueness of the Scale:
For measurements of the very highest precision there may be detectable numerical differences between measurements made at the same temperature but in accordance with differing definitions.
But which of the different values determined “at the same temperature” is closer to the value of the measurand^{3}? Is it the one obtained through the calibration in the subrange (Al) or the one obtained through the calibration in the subrange (Zn), (Sn), (In), (Ga) or (Hg) − if we refer, for instance, to Region 1 (Table 2)? Obviously, we can not know. What we do know for sure is that the 6 values may be affected by the nonuniqueness of the Scale and that they should then be corrected.
Under the current approach, the values of NU1 are calculated as the differences between two subranges that overlap. But the values of the Type 1 nonuniqueness thus calculated at a given W (or a given T_{90}) for the different combinations of two subranges may also significantly differ from one another. If, for instance, the calibration has been made in the subrange (Al), then which is the value of NU1 at W: the difference calculated for the pair (AlZn) or for the pair (AlSn), (AlIn), (AlGa) or (AlHg)? We can not know that either. For this reason, we have proposed a new approach where NU1 at W is evaluated between all subranges that contain the respective W, not only for combinations of two.
Specifically, the Type 1 nonuniqueness at a given W between more than two overlapping subranges, hereafter called Combined Type 1 nonuniqueness, can be calculated in two ways:

as the difference between the value of the reference function W_{r} determined at W in calibration subrange and the arithmetic mean of the values of W_{r} determined at the same W in the other subranges that it overlaps.
For example, if the calibration was performed in the subrange 0 °C to 660.323 °C, then, in one of the regions of overlap, say Region 1 (R1), the Combined Type 1 nonuniqueness between subrange (Al) and the other 5 subranges that it overlaps shall be given by: (11) where the superscript “R1” is used to indicate the region of overlap under analysis.

as the arithmetic mean of the Simple Type 1 nonuniqueness values for the pairs of overlapping subranges, that, for the example above, are (AlZn), (AlSn), (AlIn), (AlGa) and (AlHg):
Equation (12) is identical to equation (11) and it represents the mathematical model of the Combined Type 1 nonuniqueness between the subrange (Al) and the other 5 that it overlaps in Region 1.
Similar equations can also be written for the other regions of overlap and calibration subranges. The equations can be expressed explicitly in terms of the propagation functions and the deviations ΔW_{FP}, using only a few elementary algebra operations. But these formulas are of limited interest for the subject of our article. The correction applied to the interpolated value in order to compensate for the effect of the Type 1 nonuniqueness of the Scale can take neither the form of Simple Type 1 nonuniqueness (10), nor the form of Combined Type 1 nonuniqueness (11) or (12). Instead, the new concept of Correction for the Type 1 nonuniqueness of the Scale will be introduced, which plays an important role in the proposed approach.
3 The correction for the Type 1 nonuniqueness of the Scale and its standard uncertainty
For any pair of overlapping subranges, (S_{j}) and (S_{h}), the mathematical model of the Simple Type 1 nonuniqueness is given by (10). Therefore, the (combined) standard uncertainty of the Simple Type 1 nonuniqueness between (S_{j}) and (S_{h}) subranges, denoted by , is obtained by combining the standard uncertainties associated with the deviations ΔW_{FPk}, which are the only input quantities in the model (10). But ΔW_{FPk} = W_{FPk} − W_{r,FPk}, where W_{r,FPk} are constants, so that, in fact, is obtained by combining the standard uncertainties associated with the resistances ratios determined at the fixed points, u(W_{FPk}). The subject is developed in [16], but the authors neglect the correlations among the input quantities because of the complexity of the expression for the combined standard uncertainty. There is a simple way to eliminate the necessity to evaluate the covariance when calculating . It consists [20–22] of the expression of the resistances R_{FP} and R_{TPW} of the ratio W_{FP} = R_{FP}/R_{TPW} in terms of independent input quantities, based on the physical phenomena involved in the measurement process. Furthermore, a direct link between the accuracy of measurement and the manageable physical factors involved can thus be traced. Also, an analysis of different cases of correlation and a study on their influence on the combined standard uncertainty are presented in [23].
Here, we shall not develop the evaluation method for the standard uncertainty of the Simple Type 1 nonuniqueness because it can not be a component of uncertainty in realising the ITS90. For two reasons.

The first reason is as simple and obvious as possible. In order to evaluate , we need to know, according to (10), the uncertainties u(W_{FPk}) both at the fixed points in the subrange where calibration was performed, let us say subrange (S_{j}), and at the fixed points in the other subrange, (S_{h}). In most cases, the data for (S_{h}) do not exist and, then, the calculation of is impossible. The situation is the more so complicated in the regions where more subranges overlap.
Under these conditions, the uncertainty in knowing the Type 1 nonuniqueness of the Scale is evaluated based on observations made outside the current calibration, and this component of uncertainty in realising the ITS90 will be treated just like the components that are directly determined in the current calibration.
The evaluation method of the uncertainty associated with the effect of the Type 1 nonuniqueness of the Scale is based on statistical analysis of data derived from a large number of calibrations made at the fixed points, in several primary thermometry laboratories and using a set of very stable SPRTs. The aim of the study is to determine, from a finite sample of data, the best estimates of parameters that describe the (infinite) population of all such possible measurements. These parameters are: the mean of the population (or expectation), μ, i.e. the value of the measurand, and the standard deviation, σ, that characterise the dispersion of the theoretically infinite number of measured values of the measurand about μ. The best estimate of μ is the average (or the arithmetic mean) of the sample, M, and the best estimate of σ is the standard deviation of the sample, called experimental standard deviation[24] and hereafter denoted by ESD or s. An additional measurement (one different from those in the sample) will fall within the individual members of the entire population. Since the sample mean M and the experimental standard deviation s are unbiased estimates of μ and σ, an additional measurement will fall within M ± 2s at a level of confidence of approximately 95 percent for a normal distribution.
Sets of data obtained from the calibration of a large group of SPRTs above 0 °C were already used in [9,10,12,13], but the characteristics under study were Simple Type 1 nonuniqueness and its uncertainty. This article proposes a substantive departure from the traditional method, and the new approach is presented below.

Here is the second reason, briefly: should not be included as a component of the uncertainty in realising the ITS90 because Type 1 nonuniqueness, as it is defined[1,2,3], is not an input quantity for the mathematical models of interpolations made between fixed points. This statement is substantiated in the next section.
Now, let us just take a moment to point out that any uncertainty evaluation process should be preceded by the elaboration of the measurement model: the development of reliable models prevents faulty evaluation of the uncertainty. Unfortunately, few studies are available that observe this imperative.
Before proceeding to the presentation of the proposed approach, we will briefly discuss the current approach.
3.1 The current approach
Let us analyse the case of uncertainty associated with T_{90} at a given W in the region of overlap of a subranges pair when a calibration according to the ITS90 is performed. The case has been dealt with in [11], where the combined variance of T_{90} is described in the equation (9.8) by the sum of the variance in “the interpolated resistance ratio”, u^{2}(W_{r}) and the variance” due to Type 1 nonuniqueness (subrange inconsistency)”, u^{2}(ΔW_{SRI}) [11]. (An identical treatment of the case is given in [3, Chapter 5].) The variance ”due to Type 3 nonuniqueness” u^{2}(ΔW_{NU}) is not the subject of this article and we shall ignore it. is characterised [11,10,3] by standard deviation of differences between two subranges that overlap.
No mathematical model is in place for equation (9.8) in [11]; but since only two input quantities, uncorrelated, are involved, and the sensitivity coefficients are equal to 1, we can develop the model here without any difficulty.
The Type 1 nonuniqueness is considered in [11,3] to be dominated by the difference between the (Al) and (Zn) subranges over most of the 0 °C to 420 °C subrange. Let us suppose that the calibration of the SPRT was performed in the subrange (Al). If we slightly modify the notation used in [11] in order to preserve the homogeneity of our article, the mathematical model corresponding to (9.8) in [11] is simply (13) where is the value interpolated in the calibration subrange (Al) and ΔW_{r,NU1} ≡ ΔW_{SRI} is the correction applied to compensate for Simple Type 1 nonuniqueness (or subrange inconsistency) between (Al) and (Zn) subranges. For ΔW_{r,NU1}, we use the model (14)where and are the values determined according to the two different definitions of the ITS90.
Then (13) becomes (15) It follows that, once we apply the correction ΔW_{r,NU−1} (i.e. ΔW_{SRI}), is simply replaced with , or, in other words, one definition of T_{90} is replaced by another. Nothing makes "truer" than , they have equal status. Therefore, the use of correction ΔW_{r,NU1} ≡ ΔW_{SRI} is intrinsically unsuited for mathematical modelling of T_{90} (or W_{r}).
3.2 The new approach
One of the basic differences from the conventional approach lies in the consideration that the effect generated by the Type 1 nonuniqueness of the Scale on the interpolated value is not equal to the difference between and , but to the difference between the W_{r} value determined in the calibration subrange and the arithmetic mean^{4} of the W_{r} values determined in all subranges that overlap.
The corrected result of the interpolation is not the value of the reference function W_{r} determined in one subrange or another, but it should be the best estimate available of the value of Wr whatever the subrange in which measurements were made. We therefore introduce the new concept of Correction for the Type 1 nonuniqueness of the Scale, C_{NU1}. The proposed mathematical model for this correction allows one to calculate the uncertainty in knowing the Type 1 nonuniqueness of the Scale at a given Wseparately for each calibration subrange that contain the respective W. Just as the corrections for the recognized systematic effects (the self heating effect, the hydrostatic head of liquid in the cell, the deviation of the pressure of gas in the fixed point cell from the Reference pressure, etc.) are included in the mathematical model of the measurement at the fixed points, C_{NU1} will be included in the mathematical model of interpolation.
Let us consider the region between 232 °C and 420 °C where only the (Al) and (Zn) subranges overlap. Assuming again that the calibration of SPRT has been performed in the subrange (Al), the correction in the Region 4 (Tab. 2) is defined by: (16) where is the arithmetic mean . The mathematical model of the interpolation then becomes (17)
The corrected result at a given W in the region of overlap of the subranges (Al) and (Zn) is equal to the arithmetic mean of the values of W_{r} determined at W in the two subranges.
The same procedure is applicable in the case of more than two subranges that overlap. For example, in the case of a temperature region where J subranges overlap and the thermometer is calibrated in the subrange (Al), the correction is also expressed by (16), but where is the arithmetic mean of the values of W_{r} determined in all of the J subranges: (18) Also in this case, the mathematical model of the interpolation is given by (17), where is defined by (18). Similar expressions are obtained in case the thermometer is calibrated in the subrange (Zn), (Sn), (In), (Ga) or (Hg).
The corrections C_{NU1} can be expressed, in their turn, in terms of the deviations ΔW_{FP} and in terms of the propagation functions; their analytical expressions are presented in Appendix A, Section A.3, for each region of overlap and each calibration subrange.
If a thermometer is not calibrated at all the fixed points in the overlapping subranges, the results are not sufficient for the calculation of C_{NU1} in (16) and its uncertainty. In this case, C_{NU1} could be estimated by the statistical analysis of the data derived from a large number K of calibrations made at all the fixed points involved, in several primary thermometry laboratories and using a large set of very stable SPRTs.
The estimate of C_{NU1} for a calibration subrange, say (S_{j}), can be then taken as the arithmetic mean of the values, k = 1,…,K, derived from the results of the K calibrations. Thus, the estimate of the input quantity C_{NU1} will not be derived directly from the current calibration, but will be brought into the interpolation model (17) from external sources [24,25]. The arithmetic mean, , is obtained from (19) where , and are the values derived from the results of the kth calibration.
But determining the estimate of is only one stage of the process of evaluating this input quantity in the model (17). The second step involves the assessment of its associated uncertainty, u(). It is characterized by the standard deviation of the observations, s(), which is obtained from (20) A somewhat similar approach concerning the evaluation of the standard uncertainty of the Type 1 nonuniqueness has been developed in [16] but, in their formulas, the authors ignore the average calculated for the entire set of values under analysis (denoted by in our study).
4 Data analysis
The data for this study were derived from the results of a key comparison, namely CCTK3 [19], in which each thermometer was calibrated in two or more laboratories. The main advantages derived from the use of the CCTK3 results are:

The calculation of the correction C_{NU1} at temperatures above 0 °C requires the knowledge of the values of ΔW_{FP} at all of the fixed points in overlapping subranges. The calibrations are usually carried out on a single subrange above 0 °C, while the calibrations performed in the interlaboratory comparisons cover the fixed points in several subranges; and, in all of these comparisons, CCTK3 is the only one that covers, in addition to the triple point of water, all of the six fixed points of interest for the subject of this article: the triple point of mercury, the melting point of gallium, and the freezing points of indium, tin, zinc and aluminium.

If the measurements are performed in a single laboratory, then the unrecognized systematic effects and the partially reduced (recognized) systematic effects are inherently present in all the results obtained in that laboratory at a fixed point and, thus, in the mean calculated for the data sample. By contrast, it is possible that some local effects affecting the results of measurements made in different laboratories can compensate for each other to some extent in the arithmetic mean of the sample. Moreover, in case of comparisons, the presence of significant local effects becomes immediately obvious by comparing the results obtained with the same SPRT in different laboratories.

In general, the measurements made by the primary laboratories participating in the comparison are of the highest accuracy.
We shall not rewrite here the results of CCTK3, which are presented in an extremely rigorous manner in [19]. We shall only remind certain general information regarding the organisation of the comparison that is relevant for this study:

The comparison involved measurements in 15 primary thermometry laboratories and the W_{FP} values at the fixed points were determined;

Only part of the participating laboratories made measurements at all of the fixed points covered by comparison;

7 SPRTs were used from 3 different commercial sources: 4 Model M1, 2 Model M2 and 1 Model M3;

Each participating laboratory calibrated one up to three SPRTs, and the coordinating laboratory − six SPRTs.
4.1 The case of the correction C_{NU1}
We shall consider a complete set of results to be the group of values of the resistances ratio W_{FP} determined at all of the six fixed points by a participating laboratory using a SPRT. Given the definition of C_{NU1}, only one complete set of results per laboratory per thermometer in [19] has been taken into account in the process of forming the sample on which our study is based.
In the process of data analysis for the variable C_{NU1}, we first investigated whether the data contain values that are extremely different from the others − the socalled outliers. Identification of the extreme values in the tails of the distribution and decision to either keep or discard them are two essential parts of the statistical analysis. The presence of a single outlier can severely distort the values of certain statistics indispensable for our analysis, such as the mean or the standard deviation.
First, we used the Tukey’s boxplot method [26] of inner and outer ^{“}fences”, a hybrid method to identify the outliers. 5 outside [26] values − values between the inner and the outer fences − and 7 far out[26] values − extreme values beyond the outer fences − were identified. 5 of the 7 probable outliers represent the same laboratory and the same SPRT, but 5 different subranges; similarly, 2 of the 5 possible outliers represent another laboratory and SPRT and 2 different subranges.
Since the presence of the outliers influences the normality of the distribution, before applying the outlier detection test, we performed the normality evaluation using the ShapiroWilk test [27]. The test results showed, for a significance level of 0.05, a nonnormal distribution (pvalues <0.05 in the majority of the regions of overlap). We assumed that the distribution is disturbed by the presence of outliers and that it will regain normality after their elimination.
The outlier identification was then performed using Grubbs’ modelbased test [28], for a significance level of 0.05. 7 outliers were detected that belong to the 12 revealed through Tukey’s method.
There is no unanimous opinion of experts on what to do with the detected outliers. In this study, the decision process was facilitated by the possibility ensured by CCTK3 to compare each outlying value against the other values in the loop to which it belongs, all of them being determined with one and the same SPRT. Moreover, we evaluated the influence of the outliers on sample standard deviation by calculating its value with and without outliers; for each subrange, the maximum value decreased by more than 25% after the elimination of outliers.
The final decision was to eliminate the 7 outliers − in fact, the values of , , , , and calculated from the measurement results of a participating laboratory and the values of and calculated from the measurement results of another participating laboratory. The extreme values of the remaining sample were checked again using the same Grubbs' test, but no outlier has been detected.
We reinvestigated then the normality of the distributions, for a significance level of 0.05, using ShapiroWilk test. With the exception of the distribution of the variable in Region 1 (where 0.02 ≤ pvalues ≤ 0.05) and of the distribution of the variable between 15 °C and 25 °C (where 0.03 ≤ pvalues ≤ 0.05), all of the other distributions passed the test of normality this time. (The remaining values are graphically represented as a function of temperature in Appendix B, Fig. B.1.)
In order to find the best available estimates of the correction C_{NU1} and its uncertainty, we calculated the sample mean , (equation (19)) and, respectively, the experimental standard deviation (equation. (20)). Their extreme values for the investigated sample, determined for each calibration subrange, are listed in Table 3, along with the maximum and the minimum values of .
A simple graphical verification (Appendix B, Figure B.2) shows that the results obtained after the correction for the effect of the Type 1 nonuniqueness in the different subranges that overlap are consistent (they agree) within their uncertainties calculated for a level of confidence of 95%.
Wdependent polynomial curves (21) were fitted to the experimental curves, based on the method of least squares (r^{2} ≥ 0.9964). The coefficients a_{i} are given in Table 4 and they can be used to predict the values of the input quantity in the model (17) for each W of the regions of overlap. The values are graphically represented as a function of temperature in Appendix B, Figure B.1.
Wdependent polynomial curves in analytical form (22) were fitted to the experimental curves by the least squares method (r^{2} ≥ 0.99996). The coefficients b_{i} are presented in Table 5 and they can be used to predict the values of the standard uncertainty associated with for each W of the regions of overlap.
When calibrations are made according to the ITS90, the estimate of C_{NU1} at any W can be quickly and easily obtained by equation (21) (using Table 4) and then introduced in the model (17) to calculate the estimate of the measurand. Also, the standard uncertainty s(C_{NU1}) can be calculated by equation (22) (using Table 5) and then used to evaluate − by means of the same model (17) − the “uncertainty in knowing the measurand” [25].
The most important finding of this section relates to the standard uncertainty associated with the effect of the Type 1 nonuniqueness of the Scale. Its values determined for each W of the regions of overlap of the six subranges do not exceed 0.26 mK and they are up to 5 times smaller than the values given for the subrange 0 to 420 °C in the specific Guides developed by the CCT [3,11].
A notable example is the (Ga) subrange of the ITS90, which was designed to achieve “the highest accuracy thermometry in the range room temperature” [29]. It should be noted that the values of the standard uncertainty declared in Annex C of KCDB by the laboratories participating in CCTK3 are really small at the melting point of gallium: their average is about 0.12 mK, and two values are smaller even than 0.1 mK (0.035 mK, and, respectively, 0.07 mK). The standard uncertainty in knowing the Type 1 nonuniqueness at the Ga point determined by applying the method proposed here for the (Ga) subrange is also very small: 0.072 mK; the value derived by applying the method developed in the Guides of the CCT [3,11] is equal to 0,265 mK.
Another relevant example: in the region where only the (Al) and (Zn) subranges overlap − from 232 °C to 420 °C −, the ratio of the uncertainties calculated by the two methods is also equal to 1:4.
Descriptive statistics for C_{NU1.}
4.2 The case of the Simple Type 1 nonuniqueness
As shown in Section 3, the Simple Type 1 nonuniqueness is not relevant for the evaluation of the uncertainty in realising the ITS90. However, the most important statistics on NU1 derived from CCTK3 are shown in Table 6 in order to make possible their comparison with the results of the studies published so far, on the one hand and to provide a comprehensive view of the subject, on the other hand. Since the existence of the outliers has not been verified, it is highly possible that such values exist among the data presented.
It should be noted that little information is available in the specific literature concerning the estimate of Type 1 nonuniqueness. Most of it [9,10] refer to the (AlZn) pair and indicate the maximum values of the sample mean (M) and the experimental standard deviation (ESD or s). The maximum value of the ESD calculated here (0.220 mK, Table 6), is 1.4 times lower than the maximum value in [9] (0.3161 mK) and, respectively, 2 times lower than the one in [10] (0.48 mK). The maximum value of M in Table 6 (0.039 mK) is also smaller than the values in [9] (0.0575 mK) and [10] (0.12 mK).
The Table 6 also shows that the largest dispersions occur for the combinations (AlSn), (AlIn), (ZnSn) and (ZnIn), whereas, for the (AlZn) pair, the sample standard deviation is almost 3 times lower. Similar proportions can be also found between the dispersions calculated in [12] and [9].
The Simple Type 1 nonuniqueness curves determined for all of the 15 overlapping subranges pairs above 0 °C are graphically represented as a function of temperature in Appendix B, Figure B.3.
Descriptive statistics for the Simple Type 1 nonuniqueness.
5 What generates the Type 1 nonuniqueness of the Scale? Final analysis
The models proposed in previous sections allow us to single out and analyse the factors that generate Type 1 nonuniqueness of the Scale and influence its value.
As we have already shown, for a given W, the Type 1 nonuniqueness and the correction C_{NU1}– both of them hereafter in this section generically referred to as “nonuniqueness” – are explicit functions of the deviations ΔW_{FP} = W_{FP}  W_{r,FP} and of the corresponding propagation functions. We recall briefly here that the propagation functions, in their turn, depend on the resistances ratio W and on its values at fixed points, W_{FP}. If we substitute − substitution that influences the nonuniqueness values by less than 1 µK − the propagation functions will depend on a single variable, namely on W; and for a given W, any function of propagation becomes a constant. Consequently, the nonuniqueness calculated for a given W will be function of only the deviations ΔW_{FP}
The ΔW_{FP} values are, evidently, inextricably linked to all the factors that influence the W_{FP} values^{5}, that is [29,3,30]:

the construction of the SPRT. The defects or the irregularities in the crystalline structure of platinum (the impurities, first of all) that alter its electrical and thermal properties, the immersion characteristic, the internal self heating phenomenon, the mechanical strains in the platinum resistor, the partial pressure of oxygen in the thermometer, the radiation piping, the electrical leakage across the insulators, etc. can influence the value of nonuniqueness;

the realisation of the defining fixed points of the ITS90. The impurities in the fixed point substance, the isotopic composition of water, the hydrostatic head of liquid in the cell, the deviation of the pressure of gas in the fixed point cell from the Reference pressure, the stray heat transfer, the external self heating phenomenon, etc., can be also responsible for the value of nonuniqueness;

the measurement of the electrical resistance of SPRT. In their turn, the variation of the reference resistance against the temperature, the stability of the measuring current, the nonlinearity of the bridge, the noise, and so on, can contribute to nonuniqueness.
But the dependency of the nonuniqueness on the deviations ΔW_{FP} should not be understood strictly in the sense that the nonuniqueness is larger if the values of ΔW_{FP} are larger. Nonuniqueness also depends on the relation that exists between the values of ΔW_{FP} at the fixed points involved. An illuminating example is the Simple Type 1 nonuniqueness between subranges (In) and (Ga) (Appendix A) (23) where and . As a result, between subranges (In) and (Ga) there is no Type 1 nonuniqueness only if between ΔW_{In} and ΔW_{Ga} there is the relation (24)regardless of whether the values of ΔW_{Ga} and ΔW_{In} are large or small. For a given SPRT, any change in the ratio between ΔW_{In} and ΔW_{Ga}, generated by the local imperfections in the realisation of the two fixed points and/or in the measurement of the electrical resistance, determine a change of the NU1 value.
The relationship between ΔW_{In} and ΔW_{Ga} in (24) is simple and it does not depend on W. A similar case is the Simple Type 1 nonuniqueness between subranges (Sn) and (In), where the relation between ΔW_{Sn} and ΔW_{In} for zero Type 1 nonuniqueness is given by (25) or the case of the Simple Type 1 nonuniqueness between subranges (Ga) and (Hg), where (26)for zero Type 1 nonuniqueness.
For pairs of overlapping subranges with 3 up to 5 independent variables ΔW_{FPk}, the condition of zero NU1 leads to more complicated relations, because the value of the deviation at a fixed point depends on the values of the deviation at other two, three or four fixed points simultaneously.
We can say that the nonuniqueness derives from the fact that the ΔW_{FPk}values − determined during calibrations − do not fit into the relationship between them imposed by interpolation equations (by means of the propagation functions).
It is obvious that nonuniqueness is zero if all deviations determined at fixed points involved are equal to zero, i.e. when the W_{FP} values are equal to the W_{r,FP} values (the latter being the W_{FP} values of SPRT used by the designers of the ITS90).
The advantage of an interlaboratory comparison is that it reveals similarities and differences among values and it helps in understanding the causes. An example is illustrated in Figure 1, that represents the values of the Simple Type 1 nonuniqueness at a given W between subranges (Al) and (Sn), derived from the results of the CCTK3. Each CCTK3 loop is designated by the calibrated thermometer (T) and the loops are grouped according to the thermometer model: M1, M2 and M3.
It is easy to see that:

the NU1 values in a loop differ among them, although the laboratories had calibrated the same thermometer;

the sets of NU1 values in the different loops are somewhat similar, although another SPRT had been calibrated within each loop; in each loop there is at least one laboratory with values of NU1 that differ significantly from others.
The differences between the values in the same loop could only to a small extent be due to the thermometer, because one should suppose that most of the influence quantities related to the construction of the SPRT have approximately the same impact upon the measurement results in each laboratory of the loop (see the internal self heating effect, for instance, if the same measuring current is used). It can be deduced from here that the differences are due rather to the problems related to the realisation of fixed points (most probable, they are caused by impurities in the substance fixed point) and/or to the measurement of the electrical resistance of SPRT.
The sets of values surprisingly similar of NU1 in different loops could also indicate a reduced contribution of the effects related to the construction of the thermometers compared to the other causes.
If the dominant factor is the design of the SPRT, then the values of NU1 in a loop should be close enough one to the other. Such seems to be the case with the pair (GaHg) in Figure 2: the values of the NU1 in each loop differ from one another by only several tens of microkelvins, and the differences between the NU1 values in different loops have the same order of magnitude. The explanation could lie in the very small errors that generally occur in the realisation of the two fixed points involved, Ga and Hg. Moreover, similar values in different loops might indicate that the influence of the SPRT construction upon the values of the NU1 is insignificantly different from one thermometer to another at these temperatures.
An interesting segregation of the values of ΔW_{FP} according to the SPRT models can be seen in Figure 3, where there are represented the deviations ΔW_{FP} determined by the participating laboratories using the 7 SPRTs: 4 Model M1, 2 Model M2 and 1 Model M3. The chart shows the distinct groups of values that have been formed depending on the model. This clear separation into three groups cannot be explained only through the different quality of the platinum used in the three models as it is hard to believe that the manufacturers of SPRTs use platinum so different, but also through the other construction elements and through the technology used in the manufacturing process. It should be noted the narrow range of the ΔW_{FP} values at the points Hg (≈3 mK) and Ga (≈2.5 mK), which increases gradually but substantially at the other fixed points.
Although the difference between the deviations ΔW_{Al} determined using models M1 and M3 can reach approximately 57 mK (Figure 3), the values of Type 1 nonuniqueness associated with the two models differ by much less than that. This is yet another confirmation of the fact that nonuniqueness depends on the relation that exists between the experimentally determined deviations for a given subrange (not only on the magnitude of these values) versus the relation prescribed by the Scale equation (corresponding to the reference SPRT).
Fig. 1 The values of the Simple NU1 for the pair (AlSn). T − calibrated thermometer; M − thermometer model. 
Fig. 2 The values of the Simple NU1 for the pair (GaHg). T − calibrated thermometer; M − thermometer model. 
Fig. 3 The values ΔW_{FP} determined at the temperatures of fixed points using the three models of SPRT: M1 (circles), M2 (diamonds), and M3 (squares). 
6 Conclusions
Much of the interest in the Type 1 nonuniqueness of the Scale concerns its influence on the results of calibration of the SPRTs performed according to the ITS90. The new approach described in this study diverges considerably from the current approach. Therefore, it is not surprising that the results differ significantly from those published so far. A notable example is that the values of standard uncertainty assessed by the novel methodology for each region of overlap of the six calibration subranges do not exceed 0.26 mK and they are up to 5 times smaller than the values specified in the Guides developed by the CCT [3,11].
A value diminished of an component will be reflected in a reduction of the combined standard uncertainty itself, i.e. a diminution of the uncertainty in realising the ITS90 and, thus, a diminution of the uncertainty of all measurement results that are metrologically traceable to the Scale realised.
The values determined in this study for the input quantity C_{NU1} and its uncertainty are applicable to all measurement models when calibrations according to the ITS90 are performed in any of the subranges between 0 °C and 962 °C.
The concepts and methods introduced here for the temperature subranges above 0 °C can also be developed without difficulty and used to study the Type 1 nonuniqueness of the Scale in the temperature subranges below 0 °C.
Appendix A Formulas for The Simple Type 1 nonuniqueness and the Correction C_{NU1}
A.1 The deviation functions and their propagation functions

From 0 °C to 660.323 °C:

From 0 °C to 419.527 °C:

From 0 °C to 231.928 °C, the deviation function is:

From 0 °C to 156.5985 °C:

From 0 °C to 29.7646 °C:

From −38.8344 °C to 29.7646 °C:
A.2 The Simple Type 1 nonuniqueness and the combined propagation functions

(AlZn)

(AlSn)

(AlIn)

(AlH g)

(ZnSn)

(ZnIn)

(ZnGa)

(ZnHg)

(SnIn)

(SnGa)

(SnHg)

(InGa)

(InHg)
(A2.19)

(GaHg)
A.3 The Corrections C_{NU1}

Region 1

Region 2

Region 3

Region 4
Appendix B Additional figures
Fig. B.1 Correction C_{NU1} versus temperature for the 6 subranges under analysis: (Al), (Zn), (Sn), (In), (Ga), and (Hg). The vertical lines on each chart mark the regions of temperature, with 6, 4, 3 and, respectively, 2 overlapping subranges (Ri, i = 1, 2, 3, 4). 
Fig. B.2 We have graphically verified the consistency of the corrected results obtained in the different subranges that overlap. The values of in each subrange (Sj) are several orders of magnitude higher than the values of the applied corrections. For this reason, before drawing the diagrams, we resorted to shifting each value by the same quantity, namely , an adjustment that does not affect the validity of the test. The corrected (and adjusted) result then becomes , where the difference is considered to be a number without uncertainty, is the sample mean, U is the extended uncertainty, U = k , which defines an interval having a level of confidence of approximately 95% for a normal distribution, with the standard uncertainty = and k = 2. The graphics created for the corrected results vs temperature demonstrate that the intervals of uncertainty overlap in all cases. By consequence, we can claim that the results obtained after the correction in the different subranges that overlap are consistent (they agree) within their uncertainties (calculated for a level of confidence of 95 %) [31,32]. The vertical lines on each chart mark the regions of temperature, with 6, 4, 3 and, respectively, 2 overlapping subranges (Ri, i = 1, 2, 3, 4). L − participating laboratory; T − calibrated thermometer. 
Fig. B.3 The Simple Type 1 nonuniqueness vs temperature for the 15 pairs of overlapping subranges under analysis: (AlZn); (AlSn); (AlIn); (AlGa); (AlHg); (ZnSn); (ZnIn); (ZnGa); (ZnHg); (SnIn); (SnGa); (SnHg); (InGa); (InHg); (GaHg). 
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Note that the analytical expressions of the propagation functions , that relate the deviation functions to the deviations at fixed points (Eq. (5)), are identical to the expressions of the “sensitivity coefficients f_{i}(W)” derived in [11], where they relate the reference functions W_{r} to their values at fixed points (which, as a matter of fact, are constants). But each set of functions f_{i}(W) in [11] contains, in addition, a function that corresponds to the triple point of water, noted as f_{H2O}, or, in other words, each interpolation equation in [11] contains a supplementary term that corresponds to the triple point of water. These elements are not necessary when we work with the deviation functions, as shown above.
One of the equations − namely, the equation of (NU1)^{(AlZn)} – is identical to the formula derived in [9] by factoring and Lagrange interpolation that we have mentioned in Introduction.
The measurand is the resistances ratio W of a given SPRT at a specified temperature between the defining fixed points. It should not be confused with the realised quantities using the values assigned to the fixed points and the equations of the ITS90: these are only approximations of the measurand due to incomplete knowledge of certain physical phenomena [24].
Cite this article as: Sonia Gaita, Georges Bonnier, A new approach to the analysis of Type 1 nonuniqueness of the ITS90 above 0 °C, Int. J. Metrol. Qual. Eng. 9, 3 (2018)
All Tables
Definition of T_{90} in the SPRT subranges of the ITS90 that overlap above 0 °C.
All Figures
Fig. 1 The values of the Simple NU1 for the pair (AlSn). T − calibrated thermometer; M − thermometer model. 

In the text 
Fig. 2 The values of the Simple NU1 for the pair (GaHg). T − calibrated thermometer; M − thermometer model. 

In the text 
Fig. 3 The values ΔW_{FP} determined at the temperatures of fixed points using the three models of SPRT: M1 (circles), M2 (diamonds), and M3 (squares). 

In the text 
Fig. B.1 Correction C_{NU1} versus temperature for the 6 subranges under analysis: (Al), (Zn), (Sn), (In), (Ga), and (Hg). The vertical lines on each chart mark the regions of temperature, with 6, 4, 3 and, respectively, 2 overlapping subranges (Ri, i = 1, 2, 3, 4). 

In the text 
Fig. B.2 We have graphically verified the consistency of the corrected results obtained in the different subranges that overlap. The values of in each subrange (Sj) are several orders of magnitude higher than the values of the applied corrections. For this reason, before drawing the diagrams, we resorted to shifting each value by the same quantity, namely , an adjustment that does not affect the validity of the test. The corrected (and adjusted) result then becomes , where the difference is considered to be a number without uncertainty, is the sample mean, U is the extended uncertainty, U = k , which defines an interval having a level of confidence of approximately 95% for a normal distribution, with the standard uncertainty = and k = 2. The graphics created for the corrected results vs temperature demonstrate that the intervals of uncertainty overlap in all cases. By consequence, we can claim that the results obtained after the correction in the different subranges that overlap are consistent (they agree) within their uncertainties (calculated for a level of confidence of 95 %) [31,32]. The vertical lines on each chart mark the regions of temperature, with 6, 4, 3 and, respectively, 2 overlapping subranges (Ri, i = 1, 2, 3, 4). L − participating laboratory; T − calibrated thermometer. 

In the text 
Fig. B.3 The Simple Type 1 nonuniqueness vs temperature for the 15 pairs of overlapping subranges under analysis: (AlZn); (AlSn); (AlIn); (AlGa); (AlHg); (ZnSn); (ZnIn); (ZnGa); (ZnHg); (SnIn); (SnGa); (SnHg); (InGa); (InHg); (GaHg). 

In the text 
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