Evaluating Measurement Uncertainty Using Measurement Models with Arguments Subject to a Constraint

Abstract

Measurement models that have a chemical composition as one of the arguments require special attention when used with the law of propagation of uncertainty from the Guide to the expression of uncertainty in measurement. The constraint that the amount fractions in a composition add exactly to unity not only affects the covariance matrix associated with the composition, but also impacts the differentiation of the measurement model to obtain the expressions and values of the sensitivity coefficients. Differentiating the measurement model with respect to each variable individually is not possible as it involves evaluating the model for infeasible inputs, leading to an undefined output. In this work, a numerical method for constrained partial differentiation is presented, enabling the use of the law of propagation of uncertainty for measurement models with compositions as one of their arguments. The numerical method enables treating the measurement model as a black box and using it with measurement models in the form of an algorithm. The numerical method is demonstrated by showing how the uncertainty associated with composition, temperature and pressure can be propagated through an equation of state, in this case, the GERG-2008 equation of state. It is shown that this differentiation can be completed in a few simple steps, requiring only a valid implementation of the measurement model that provides an output value for given input quantities. The numerical differentiation method applies in principle to all differentiable functions of a composition.

1. Introduction

Constrained variables occur frequently in models in many fields in physics  [1] and chemistry [2]. They arise due to, e.g., conservation laws for mass, energy and amount of substance [1], the definition of mathematical shapes such as triangles, squares and hexagons [3] or from the definition of proportions, such as those used in probability theory [4] and in chemical compositions [5,6]. Consequently, such relationships between variables may also be part of measurement models [7,8]. Models with chemical compositions are common practice in (physical) chemistry [2] and many of these are used in engineering [9] and measurement [10,11,12].

Generally, thermodynamic state functions like enthalpy (H) and entropy (S), as well as fluid properties like the speed of sound and the compressibility factor of mixtures, are functions of the composition [2,12]. Equations of state, used to describe the pressure–volume–temperature–behavior of fluids, contain coefficients that are component-specific. For mixtures, mixing rules [9] are used to compute the coefficients of the equation of state of a mixture. The composition in these models is generally expressed in amount fractions, and by their very definitions [5], these amount fractions are subject to a constraint [6].

Equations of state find their application in, e.g., the metering of natural gas. This metering serves a variety of purposes, including billing, custody transfer, and assessing conformity with contractual and legislative requirements [13,14,15]. For such assessments, it is paramount to know the uncertainty of the measurement results [16]. To obtain the measurement uncertainty for the total mass, volume or energy, it is necessary to duly propagate the measurement uncertainty [17,18,19] of, e.g., the flow rate [20,21,22,23,24,25], composition [26,27], density [28], or calorific value [29,30]. The Guide to the Expression of Uncertainty in Measurement (GUM) [8,16,17,18,19,31] is the authoritative reference for propagating uncertainty in measurement. One of the mechanisms it provides to propagate measurement uncertainty from the input to the output quantities of a measurement model is the law of propagation of uncertainty (LPU) [17,19], which is based on Gauss’ error propagation formula [32].

A property of a composition is that its amount fractions sum to unity exactly is an example of a mathematical constraint [3,33], and it is well known from mathematics that a function with constrained arguments behaves differently from the same function while ignoring the constraint. The latter case, thus ignoring the constraint, can lead to physically infeasible output. For instance, when calculating a calorific value of natural gas with a composition that adds to, say, 101 cmol ${\mathrm{mol}}^{-1}$, has no physical meaning, yet the model of ISO 6976 [30] permits at least mathematically to do so. When using an equation of state such as GERG-2008 [11,34] or AGA8-DC92 [10,35,36], it is less obvious whether it is mathematically possible to evaluate these functions while ignoring the constraint. Most valid software implementations of these models refuse the input when the constraint on the composition is not met.

Measurement uncertainty is propagated using a measurement model [8]. The LPU uses the first-order partial derivatives of the measurement model with respect to the input quantities [37]. The presence of a constraint in the input variables of a measurement model gives rise to two different sets of derivatives: constrained and unconstrained partial derivatives [33]. The GUM [17,19] does, however, not specify which partial derivatives are to be used.

The work is presented as follows. In Section 2, the LPU is revisited. Constrained differentiation is described in Section 3, starting with analytic differentiation (Section 3.1), followed by numerical differentiation (Section 3.2) and a description of some key features of the method (Section 3.3). In Section 4, the numerical accuracy of the algorithm is explored (Section 4.1), followed by an example of propagating the measurement uncertainty associated with the input quantities when computing a compressibility factor using the GERG-2008 equation of state (Section 4.2). The conclusions are presented in Section 5.

2. Guide to the Expression of Uncertainty in Measurement

In the industry and elsewhere, the LPU is the de facto standard for propagating measurement uncertainty. The LPU takes the estimates, associated standard uncertainties, any covariances and the first-order partial derivatives as input [17]. It is widely used in standards [20,21,22,23,25,26,27,30], regulatory documents [14,38] and industry guidance documents [39,40].

The GUM specifies in ISO/IEC Guide 98-3  [17], clause 5.1.3, that “the partial derivatives [...], often called sensitivity coefficients, describe how the output estimate y varies with changes in the values of the input estimates $x_1,x_2,\dots ,x_N$.” In NOTE 2 of the same clause, it is assumed that each of the input variables can be varied independently, which is not true for constrained variables [33]. Hence, it is fair to say that the GUM does not consider constrained input variables of measurement models, which is an omission.

If (a subset of) the input quantities $x_1,\dots ,x_N$ cannot be varied independently because of a relationship between them, then usually also the way in which the output quantity y responds to changes in the input quantities changes [33]. For the propagation of measurement uncertainty, the behaviour of the function, including any constraints on the input quantities, is relevant for propagating the measurement uncertainty. The constrained partial derivatives express this behaviour, and by virtue of the description of the desired properties of the sensitivity coefficients in ISO/IEC Guide 98-3  clause 5.1.3 [17], it would be expected that these constrained partial derivatives are needed for using the LPU. Moreover, the constraint on (some of) the arguments of a measurement model is an integral part of that measurement model, a notion missing in JCGM GUM-6 [8].

Measurement models with constrained arguments can be differentiated. Depending on whether the constraint is respected, the gradient or constrained partial derivatives are obtained. The fact that two kinds of partial derivatives can be obtained raises the question of which ones are appropriate for use with the LPU. This question is answered properly in Section 3. A further consequence of the constraint on (some of) the arguments of a measurement model is that the LPU for dependent quantities is used (Equation (13) of [17]), as arguments subject to a constraint are dependent. Consequently, it is assumed that the covariance matrix associated with the arguments subject to the constraint is developed in a fashion that is consistent with that constraint. A covariance matrix associated with a composition has specific features, one of them being that the elements of each row (and for reasons of symmetry, each column) sum to zero. For duly propagating measurement uncertainty, the covariance matrix, including the covariances, should be used [17,19,41]. In this work, such a covariance matrix will be called ‘properly formed’. The use of an improperly formed covariance matrix, as permitted by, e.g., ISO 6976 [30], ISO 6142 [42] and ISO 14912 [43], is discussed in Appendix A.

If the Monte Carlo method (MCM) [18,19] is used to obtain values for the sensitivity coefficients, for example, for constructing an uncertainty budget, it is affected in a similar manner as using the LPU, for it is generally not possible to vary just one quantity while keeping all others constant, as described in Annex B.1 of ISO/IEC Guide 98-3/Supplement 1  [18].

3. Methods

3.1. Constrained Differentiation

In many cases, the expressions for the sensitivity coefficients are obtained by analytic differentiation. That method also works for situations with constrained input quantities [33].

For the amount fractions forming a composition, the boundaries are that for any amount fraction x, $0<x<1$ [6] (situations where $x=0$ or $x=1$ require a dedicated mathematical treatment and are not relevant for calculating properties of mixtures). The constraint on the other hand, formulated as a function g, which is equal to zero is given by [33]

$$g\left(\mathit{x}\right)=\sum _{i=1}^N{x}_i-1=0,$$

(1)

where $\mathit{x}={\left[x_1,\dots ,x_N\right]}^{\top }$ denotes the composition, a vector holding the amount fractions of the components in the mixture. In the case of an equation of state, the function f has two more arguments, namely the pressure p and temperature T. These are not constrained, but have bounds (both are non-negative).

Let $\mathrm{d}\mathit{x}$ denote a small change (perturbation) of the constrained input quantities $\mathit{x}$, then the change in the output of a function f, $\delta y$ is given by [33]

$$\delta y=\nabla fP\mathrm{d}\mathit{x},$$

(2)

where P is the orthogonal projection matrix that depends solely on the constraint g, for a composition given in Equation (1). The projection P is for a scalar function g given by [33]

$$P=I-\mathit{n}{\mathit{n}}^{\top },$$

(3)

where I denotes the $N\times N$ identity matrix and $\mathit{n}$ the unit normal vector ${(\nabla g)}^{\top }/|\nabla g|$. For the constraint in Equation (1),

$$\mathit{n}={\displaystyle \frac{1}{\sqrt{N}}}{[1,\dots ,1]}^{\top },$$

(4)

and the projection matrix equals $P=I-{\displaystyle \frac{1}{N}}\mathbf{1}$ where $\mathbf{1}$ is an $N\times N$ matrix with ones. Equation (2) shows how the constrained partial derivatives are different from their unconstrained counterparts. A change in the function value equals

$$\delta y=\nabla f\mathrm{d}\mathit{x},$$

(5)

in the unconstrained case. Equation (5) is the foundation of the LPU [17,19], when only the first terms of the Taylor expansion of the measurement model f are considered [37].

To illustrate the difference between the two sets of partial derivatives, consider the calculation of the calorific value of a natural gas, as described in ISO 6976. The formula for calculating the calorific value ($H_{\mathrm{mix}}$) takes the form [30]

$$H_{\mathrm{mix}}=\sum _{i=1}^N{x}_i{H}_i,$$

(6)

where $H_i$ denotes the calorific value of component i ($i=1,\dots ,N$), $x_i$ its amount fraction and N is the number of components in the gas mixture. The calorific values of the components are unconstrained in this measurement model, whereas the amount fractions of all components in the mixture are constrained as discussed previously. The constrained partial derivatives with respect to $x_i$, formally denoted by the operator ${\displaystyle \frac{{\partial }^{*}}{\partial x_i}}$, are [33]

$${\displaystyle \frac{{\partial }^{*}{H}_{\mathrm{mix}}}{\partial x_i}}=H_i-\stackrel{-}{H},$$

(7)

where

$$\stackrel{-}{H}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{H}_i.$$

Both the constrained partial derivatives of Equation (7) and their counterparts ignoring the constraint lead to exactly the same standard uncertainty $u\left(H_{\mathrm{mix}}\right)$ as the calculations in ISO 6976 [30].

The proof that both kinds of partial derivatives can be used rests on the properties of the covariance matrix associated with the composition [6,41]. Let the Jacobian J be defined as the gradient of the multivariate, differentiable function $\mathit{f}$, i.e., [44,45]

$$J_{ij}={\displaystyle \frac{\partial f_i}{\partial x_j}},$$

then the covariance matrix associated with the output vector $\mathit{y}$ is obtained as [19] (clause 6.2.1)

$$V_{\mathit{y}}^{\left(0\right)}=J{V}_{\mathit{x}}{J}^{\top }.$$

(8)

The proof that instead of J a sensitivity matrix can be used composed of constrained partial derivatives can be given as follows. From the equality constraint g, it follows that the constraint is in the form of an $(N-1)$-dimensional hyperplane $\mathcal{H}$. For compositions, the constraint is given in Equation (1) and the normal vector $\mathit{n}$ in Equation (4). Let the matrix D be defined as

$$D=\left[{\mathit{d}}_1,\dots ,{\mathit{d}}_{N-1},\mathit{n}\right],$$

(9)

where ${\mathit{d}}_i$ denote $(N-1)$ independent directions in the hyperplane $\mathcal{H}$, each perpendicular to the normal vector of the constraint (Equation (4)).

Now, Equation (8) can be written as

$$V_{\mathit{y}}^{\left(0\right)}=\left(JD\right)\left(D^{-1}{V}_{\mathit{x}}{D}^{-T}\right){\left(JD\right)}^{\top }.$$

(10)

Now, let $V_D=D^{-1}{V}_{\mathit{x}}{D}^{-T}$. Furthermore, let L denote the Cholesky factor of $V_{\mathit{x}}$, ${\mathit{e}}_N={[0,\dots ,0,1]}^{\top }$ and $\mathbf{0}={[0,\dots ,0,0]}^{\top }$. The constraint ${\mathbf{1}}^{\top }\mathit{x}=\kappa$, where $\kappa$ is some exactly known constant [46] and $\mathbf{1}={[1,\dots ,1,1]}^{\top }$, implies that ${\mathit{n}}^{\top }\mathit{x}=\kappa /\sqrt{N}$ and

$$\begin{array}{c}\hfill 0=\mathrm{var}\left({\displaystyle \frac{\kappa }{\sqrt{N}}}\right)=\mathrm{var}\left({\mathit{n}}^{\top }\mathit{x}\right)={\mathit{n}}^{\top }\mathrm{var}\left(\mathit{x}\right)\mathit{n}={\mathit{n}}^{\top }{V}_{\mathit{x}}\mathit{n}={\mathit{n}}^{\top }{L}^{\top }L\mathit{n}={\parallel L\mathit{n}\parallel }^2,\end{array}$$

and thus $L\mathit{n}=\mathbf{0}$. It then follows that and

$$V_{\mathit{x}}\mathit{n}=L^{T}L\mathit{n}=\mathbf{0}.$$

(11)

From the definition of D (see Equation (9)), it is evident that

$$D^{\top }\mathit{n}={\mathit{e}}_N\mathrm{and}{D}^{-\top }{\mathit{e}}_N=\mathit{n}.$$

So,

$$V_D{\mathit{e}}_N=\left(D^{-1}{V}_{\mathit{x}}{D}^{-\top }\right){\mathit{e}}_N=\mathbf{0}.$$

This result implies that the last column, and by symmetry the last row of $V_D$ contain only zeros.

Let a directional derivative $f_{\mathit{d}}^{\prime }$ at $\mathit{x}$ be defined as [47]

$$f_{\mathit{d}}^{\prime }\left(\mathit{x}\right)=\underset{\epsilon \to 0}{lim}{\displaystyle \frac{f(\mathit{x}+\epsilon \mathit{d})-f\left(\mathit{x}\right)}{\epsilon }},$$

(12)

where $\mathit{d}$ denotes the direction. Now, it is shown that the method does not rely on a particular choice of vectors ${\mathit{d}}_1,\dots ,{\mathit{d}}_{N-1}$ by looking at the product $J_D=JD$. From calculus, for the directional derivative $f_{\mathit{d}}^{\prime }$ of f in direction $\mathit{d}$ it holds that (Section 12.1, [47])

$$\begin{array}{cc}\hfill f_{\mathit{d}}^{\prime }& =J\mathit{d},\hfill \end{array}$$

(13)

so that

$$J_D=JD=(f_{{\mathit{d}}_1}^{\prime },\dots ,f_{{\mathit{d}}_{N-1}}^{\prime },f_{\mathit{n}}^{\prime }).$$

(14)

Note that all columns of $J_D$ except the last one can be calculated if $\mathit{x}$ is limited to values in $\mathcal{H}$. However, as the last row and column of $V_{\mathit{d}}$ are zero, knowledge about the last column of $J_D$ is not needed. Let

$${\tilde{J}}_D=(f_{{\mathit{d}}_1}^{\prime },\dots ,f_{{\mathit{d}}_{N-1}}^{\prime }).$$

(15)

The result of the generalized algorithm equals

$$V_{\mathit{y}}^{\left(1\right)}={\tilde{J}}_D{\tilde{V}}_D{\tilde{J}}_D^{\top }.$$

(16)

Shortly, the correspondence with the algorithm presented in Section 3.2 will be explained in more detail. Using the fact that the last row and column of $V_D$ only contain zeros, it becomes clear that

$$V_{\mathit{y}}^{\left(1\right)}={\tilde{J}}_D{\tilde{V}}_D{\tilde{J}}_D^{\top }=J_D{V}_D{J}_D^{\top }=V_{\mathit{y}}^{\left(0\right)},$$

(17)

which shows that the constrained partial differentiation yields the same covariance matrix as using the LPU (see Equation (8)) with the Jacobian. As no assumptions on the ${\mathit{d}}_i$, $1\le i\le N-1$ were made except for their independence and perpendicularity to $\mathit{n}$, the result of the algorithm is independent on the particular choice for the ${\mathit{d}}_i$. The choice between using the unconstrained and constrained partial derivatives does however matter for improperly formed covariance matrices, see also Appendix A.

3.2. Numerical Constrained Differentiation

Analytic differentiation can become prohibitively difficult for complex measurement models, such as equations of state [9,34,36] or in instances where the measurement model takes the form of an algorithm, such as in the calculation of fluid properties at the vapour–liquid equilibrium [9,48]. Numerical differentiation [44,45] is a practical alternative for analytic differentiation to obtain values for the sensitivity coefficients used in the LPU.

Numerical differentiation methods use finite differences [44,45]. These differences should be formed by appreciating any constraint(s) on the arguments to ensure physically feasible outputs of the function f. So, the finite differences for p and T should be such that the perturbed pressure and temperature are still valid in their own right. The perturbed composition should still meet the constraint in Equation (1).

At first glance, it is not straightforward to perturb a composition and still satisfy the condition given in Equation (1). Consider a normalized direction vector $\mathit{q}$. For constrained differentiation with respect to compositions, it is required that $\mathit{x}+\epsilon \mathit{q}$ is also a valid composition for a sufficiently small $\epsilon$. In this case, the constraint (1) can be expressed as $g\left(\mathit{x}\right)={\mathit{e}}^{\top }\mathit{x}-1=0$ where ${\mathit{e}}^{\top }=[1,\dots ,1]$. $\mathit{x}+\epsilon \mathit{q}$ is a valid composition if $\mathit{q}$ is orthogonal to $\mathit{e}$. From the requirement that $\mathit{x}+\epsilon \mathit{q}$ is a valid composition, it follows that the sum of the elements of $\mathit{q}$ is zero.

To illustrate how a composition $\mathit{x}$ can be perturbed by a small shift, let us consider a ternary mixture ($N=3$), $\mathit{x}={\left[x_1,x_2,x_3\right]}^{\top }$. A change in $x_1$ should be compensated by changes in $x_2$, $x_3$ or both, so that the perturbed input vector $\mathit{x}+\delta \mathit{x}$ should again be a composition. Consequently, the sum of the elements of the perturbation $\delta \mathit{x}$ should be zero. Amongst the infinitely many options to satisfy this condition, one option is to orthogonally project the vectors ${\left[0,d{x}_2,0\right]}^{\top }$ and ${\left[0,0,d{x}_3\right]}^{\top }$ onto the hyperplane $\mathcal{H}$ containing $\mathit{x}$ and with as normal vector $\mathit{n}$, the gradient of the constraint. Using the resulting projection P (see also Equation (3)) yields the normalized perturbation directions

$$\begin{array}{cc}\hfill {\mathit{q}}_2& ={\displaystyle \frac{1}{\sqrt{5}}}{\left[-1,2,-1\right]}^{\top },\hfill \\ \hfill {\mathit{q}}_3& ={\displaystyle \frac{1}{\sqrt{5}}}{\left[-1,-1,2\right]}^{\top },\hfill \end{array}$$

such that it becomes physically meaningful and mathematically feasible to assess the change in the property of interest when the composition is changed from $\mathit{x}$ to either of the compositions

$$\begin{array}{cc}\hfill {\delta \mathit{x}}_2& =\mathit{x}+{\epsilon }_2{\mathit{q}}_{\mathbf{2}},\hfill \\ \hfill {\delta \mathit{x}}_3& =\mathit{x}+{\epsilon }_3{\mathit{q}}_{\mathbf{3}},\hfill \end{array}$$

where ${\epsilon }_2$ and ${\epsilon }_3$ are some sufficiently small numbers. (The cases that ${\mathit{x}}_0$ equals ${[1,0,0]}^{\top }$, ${[0,1,0]}^{\top }$ or ${[0,0,1]}^{\top }$ would require a different definition of ${\mathit{q}}_2$ and ${\mathit{q}}_3$.)

Given a function f having as one of its arguments a composition $\mathit{x}$, the derivatives with respect to amount fractions of the components in the composition can be obtained numerically as follows. For the numerical differentiation, instead of computing the limit in Equation (12), a finite, small value for $\epsilon \mathit{q}$ will be used instead

$$f_{\mathit{q}}^{\prime }\left(\mathit{x}\right)\approx {\mathit{b}}_j={\displaystyle \frac{f(\mathit{x}+\epsilon \mathit{q})-f\left(\mathit{x}\right)}{\epsilon }}.$$

(18)

To evaluate the partial derivative exactly at the value of the estimates $\mathit{x}$ [17] (clause 5.2), the directional derivative can be approximated by

$$f_{\mathit{q}}^{\prime }\left(\mathit{x}\right)\approx {\mathit{b}}_j={\displaystyle \frac{f(\mathit{x}+\epsilon \mathit{q})-f(\mathit{x}-\epsilon \mathit{q})}{2\epsilon }}.$$

(19)

For the numerical evaluation of the constrained partial derivatives, the following algorithm can be used:

• Choose a small value for $\epsilon$ so that $0<x_i+\epsilon <1$ for all components i
• Choose an orthogonal matrix Q with columns ${\mathit{q}}_1,\dots ,{\mathit{q}}_{N-1}$, which are all perpendicular to the vector $\mathit{e}$
• Compute the directional derivatives $[f_{{\mathit{q}}_1}^{\prime },\dots ,f_{{\mathit{q}}_{N-1}}^{\prime }]=\mathit{b}$ using Equations (18), or (19) for symmetric partial derivatives
• Now,

  $$C=\mathit{b}{Q}^{\top }.$$

  (20)

  where C denotes the sensitivity matrix in the sense of ISO/IEC Guide 98-3  and ISO/IEC Guide 98-3/Supplement 2  [17,19]. (In the case of a scalar function f, the sensitivity matrix C takes the form of a row vector, or equivalently, an $N\times 1$ matrix.)

The algorithm can be readily further generalized to multivariate functions $\mathit{f}$, which can be used for propagating measurement uncertainty using the LPU from ISO/IEC Guide 98-3/Supplement 2  [19].

The matrix Q can be obtained in different ways. One such way is using a matrix derived from the Helmert matrix [49], which is available as the function ilrBase in the R package ‘compositions’ [50]. Alternatively, the (modified) Gram–Schmidt algorithm can be used [51]. For a composition, the following orthogonal matrix can be used. For each column $j=1,\dots ,N$, the first j elements are equal to

$${\displaystyle \frac{-1}{\sqrt{j(j+1)}}}$$

and the following element is equal to

$${\displaystyle \frac{j}{\sqrt{j(j+1)}}}$$

The remaining elements are zero. For $N=5$, using these formulæ,

$$Q=\left[\begin{array}{cccc}-0.70711\hfill & -0.40825\hfill & -0.28868\hfill & -0.22361\hfill \\ 0.70711\hfill & -0.40825\hfill & -0.28868\hfill & -0.22361\hfill \\ 0\hfill & 0.81650\hfill & -0.28868\hfill & -0.22361\hfill \\ 0\hfill & 0\hfill & 0.86603\hfill & -0.22361\hfill \\ 0\hfill & 0\hfill & 0\hfill & \hfill 0.89443\end{array}\right]$$

(21)

3.3. Properties of the Numerical Method

The algorithm presented in Section 3.2 is based on the consideration that if the directions ${\mathit{d}}_i$ (see Equation (9)) are chosen normalized and mutually orthogonal, the matrix $D=Q_0$ becomes orthogonal so that $D^{-1}=Q_0^{\top }$. This matrix $Q_0$ enables formulating the particularly form of the algorithm, cf. Equation (20). In particular, the matrix Q from the numerical algorithm in Section 3.2 corresponds to the first $N-1$ columns of $Q_0$ such that

$$\begin{array}{c}\hfill Q_0=\left(\begin{array}{cc}Q& \mathit{n}\end{array}\right),\end{array}$$

and let

$$\mathit{b}={\tilde{J}}_{Q_0}=J_Q=JQ.$$

From Equation (8), it follows that

$$\begin{array}{cc}\hfill V_{\mathit{y}}^{\left(0\right)}& =J{V}_{\mathit{x}}{J}^{\top }=J\left(\begin{array}{cc}Q& \mathit{n}\end{array}\right)\left(\begin{array}{c}{Q}^T\\ {\mathit{n}}^{\top }\end{array}\right)V_{\mathit{x}}\left(\begin{array}{cc}Q& \mathit{n}\end{array}\right)\left(\begin{array}{c}{Q}^T\\ {\mathit{n}}^{\top }\end{array}\right)J^{\top }\hfill \\ \hfill & =J\left(\begin{array}{cc}Q& \mathit{n}\end{array}\right)\left(\begin{array}{cc}{Q}^T{V}_{\mathit{x}}Q& \mathbf{0}\\ \mathbf{0}& 0\end{array}\right)\left(\begin{array}{c}{Q}^T\\ {\mathit{n}}^{\top }\end{array}\right)J^{\top }\hfill \\ \hfill & =JQ{Q}^T{V}_{\mathit{x}}Q{\left(JQ\right)}^T\hfill \\ \hfill & =\mathit{b}{Q}^T{V}_{\mathit{x}}Q{\mathit{b}}^{\top }\hfill \\ \hfill & =C{V}_{\mathit{x}}{C}^{\top }.\hfill \end{array}$$

So, the algorithm provides a sensitivity matrix C that matches the descriptions in ISO/IEC Guide 98-3 and ISO/IEC Guide 98-3/Supplement 2 [17,19]. In the case that J cannot be evaluated because the function f is not defined outside the region defined by the constraint, the covariance matrix associated with $\mathit{y}$ cannot be defined and evaluated by means of Equation (8). In this case, however, the sensitivity matrix defined in Equation (20) can be used instead. This case is currently neither covered in ISO/IEC Guide 98-3 nor in ISO/IEC Guide 98-3/Supplement 2 [17,19].

A complementary viewpoint to the propagation of measurement uncertainty is obtained by considering the orthogonal projection P, which was introduced in Equation (3). Noting that an orthogonal projection fulfills $P^{\top }=P$ and $P^2=P$, it follows from Equation (11) that for a well-formed covariance matrix [6,41], it holds that

$$V_{\mathit{x}}=P{V}_{\mathit{x}}P.$$

As a consequence,

$$\begin{array}{c}\hfill V_{\mathit{y}}=J{V}_{\mathit{x}}{J}^T=JP{V}_{\mathit{x}}P{J}^T=\tilde{C}{V}_{\mathit{x}}{\tilde{C}}^{\top },\end{array}$$

where

$$\tilde{C}=JP={\left(P{J}^{\top }\right)}^{\top }.$$

(22)

This matrix $\tilde{C}$ as an orthogonal projection of J on $\mathcal{H}$ offers an alternative definition to the matrix $C=\mathit{b}{Q}^{\top }$ as constructed in the algorithm.

4. Results

4.1. Accuracy of the Numerical Method

Numerical differentiation can be prone to poor performance due to subtractive cancellation [44,52]. This can occur if two large numbers are subtracted, resulting in a difference with (very) few significant digits. To assess the performance of the method described in Section 3.2, the analytic differentiation is compared with the output of the algorithm. Using the matrix Q from (21) and the composition in

Table 1. Amount fractions, molar masses and sensitivity coefficients with respect to the amount fractions for a five-component natural gas composition, computed numerically ($c_{\mathrm{num}}$) and analytically ($c_{\mathrm{an}}$).

Component	x	M	${\mathit{c}}_{\mathbf{num}}$	${\mathit{c}}_{\mathbf{an}}$
	mol ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$	g ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$	g ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$	g ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$
Nitrogen	0.0328	28.0134	−4.4326	−4.4326
Carbon dioxide	0.0242	44.0095	11.5635	11.5635
Methane	0.8434	16.0425	−16.4035	−16.4035
Ethane	0.0659	30.0690	−2.3770	−2.3770
Propane	0.0338	44.0956	11.6496	11.6496

, the sensitivity coefficients are evaluated for the molar mass of the natural gas. The molar mass of the mixture is given by [30]

$$M_{\mathrm{mix}}=\sum _{i=1}^N{x}_i{M}_i,$$

(23)

where $x_i$ denotes the (normalized) amount fraction of component i, $M_i$ its molar mass and $M_{\mathrm{mix}}$ the molar mass of the mixture. The outcomes of the numerical method given in Equations (20) and (21) are compared with the values obtained using the analytic formula, viz., [33],

$${\displaystyle \frac{{\partial }^{*}\stackrel{-}{M}}{\partial x_i}}=M_i-{\displaystyle \frac{1}{N}}\sum _{j=1}^N{M}_j.$$

The numerically evaluated sensitivity coefficients are obtained using Equation (18). In Table 1, the sensitivity coefficients are given using the numerical method ($c_{\mathrm{num}}$) versus analytic differentiation, including the constraint on the $x_i$ ($c_{\mathrm{an}}$).

The sensitivity coefficients in the ultimate two columns of Table 1 are identical up to four decimal figures, which is sufficient for the propagation of measurement uncertainty using the LPU. In the numerical evaluation of the sensitivity coefficients, $\epsilon$ was chosen to be 1% of the smallest amount fraction in the composition. This choice ensures that $\mathit{x}\pm \delta \mathit{x}$ are valid compositions. According to ISO/IEC Guide 98-3 [17], clause 5.1.3 NOTE 2, a perturbation in the order of a standard uncertainty is adequate for the purpose of propagating measurement uncertainty. This choice of $\epsilon$ just does that. The only concern can be that for compositions involving components at the μmol ${\mathrm{mol}}^{-1}$-level, this choice may lead to unacceptable cancellation in the calculation of the differences.

Using a constant value of $\epsilon$ and the first order Formula (18), there are in total N calls to the function f needed. For equations like Equation (23), this feature is not important, but the more involved the evaluation of f is, the more important this deliberation becomes.

The differences between the sensitivity coefficients using Equation (18) and the ones obtained using analytic differentiation are in the order of 10⁻¹² $\mathrm{g}{\mathrm{mol}}^{-1}$. Using Equation (19) instead, and thus obtaining the symmetric partial derivatives, the errors are slightly smaller in absolute value but still in the order of 10⁻¹² $\mathrm{g}{\mathrm{mol}}^{-1}$. This computation requires $2N$ calls to the function f.

Table 2. Amount fractions, molar masses and sensitivity coefficient with respect to the amount fractions for an eleven-component natural gas composition.

Component	x	M	${\mathit{c}}_{\mathbf{num}}$	${\mathit{c}}_{\mathbf{ana}}$
	mol ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$	g ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$	g ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$	g ${\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$
Nitrogen	0.130841	28.0134	−24.8135	−24.8135
Carbon dioxide	0.025217	44.0095	−8.8174	−8.8174
Methane	0.807295	16.04246	−36.7845	−36.7845
Ethane	0.030572	30.06904	−22.7579	−22.7579
Propane	0.004048	44.09562	−8.7313	−8.7313
i-Butane	0.000845	58.1222	5.2953	5.2953
n-Butane	0.000845	58.1222	5.2953	5.2953
neo-Pentane	0.000025	72.14878	19.3219	19.3219
i-Pentane	0.000150	72.14878	19.3219	19.3219
n-Pentane	0.000112	72.14878	19.3219	19.3219
n-Hexane	0.000048	86.17536	33.3484	33.3484

shows the same comparison, but now for an 11-component natural gas mixture. The differences between the numerically evaluated sensitivity coefficients and the ones obtained through analytic differentiation are in the order of 10⁻⁹ $\mathrm{g}{\mathrm{mol}}^{-1}$. This performance is also sufficient for propagating the measurement uncertainty when computing the molar calorific value H or the compressibility factor Z.

The calculations can be readily performed in any kind of software, from mainstream spreadsheet software to environments like R [53] or Python [54]. The arguments of a function f that do not belong to the composition can be handled as usual. They can be evaluated numerically directly using the approach described in GUM:1995 [17] (clause 5.1.3 NOTE 2).

4.2. Compressibility Factor from the GERG-2008 Equation of State

To illustrate the method, the computation of the compressibility factor Z using the GERG-2008 equation of state [11,34] is revisited. Apart from the model uncertainty stated in ISO 20765-2 [34], the uncertainty associated with the measured composition $\mathit{x}$, pressure p and temperature T is propagated using the LPU for dependent input quantities from ISO/IEC Guide 98-3 [17].

The normalized natural gas composition is given in

Table 3. Normalized composition of a natural gas containing five components, expressed in amount fractions ($\mathrm{cmol} {\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$).

Component	x	$\mathit{u}\left(\mathit{x}\right)$	${\mathit{u}}_{\mathbf{rel}}\left(\mathit{x}\right)$
	$\mathbf{cmol} {\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$	$\mathbf{cmol} {\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$
Nitrogen	3.280	0.022	0.67%
Carbon dioxide	2.421	0.019	0.77%
Methane	84.335	0.111	0.13%
Ethane	6.587	0.044	0.67%
Propane	3.378	0.110	3.27%

[41]. The correlation matrix associated with the composition is

$$R_{\mathit{x}}=\left[\begin{array}{ccccc}1\hfill & 0.0635\hfill & -0.0703\hfill & 0.0367\hfill & -0.1543\hfill \\ 0.0635\hfill & 1\hfill & -0.0605\hfill & 0.0320\hfill & -0.1341\hfill \\ -0.0703\hfill & -0.0605\hfill & 1\hfill & -0.2531\hfill & -0.8782\hfill \\ 0.0367\hfill & 0.0320\hfill & -0.2531\hfill & 1\hfill & -0.1609\hfill \\ -0.1543\hfill & \hfill -0.1341& -0.8782\hfill & -0.1609\hfill & 1\hfill \end{array}\right]$$

(24)

The actual conditions are $T=300.00 \mathrm{K}$ and $p=6.20 \mathrm{MPa}$. The standard uncertainty associated with the temperature is $u\left(T\right)=0.20 \mathrm{K}$ and that associated with the pressure is $u\left(p\right)=0.05 \mathrm{MPa}$.

The normalized composition $\mathit{x}$, expressed in amount fractions, is given in Table 3. The associated correlation matrix ${\mathit{R}}_{\mathit{x}}$ is shown in Equation (24). The covariance matrix ${\mathit{V}}_{\mathit{x}}$ can be obtained as [19] (definition 3.21)

$$V_x=D{R}_{x}D,$$

(25)

where $D=\mathrm{diag}\{u\left(x_1\right),\dots ,u\left(x_N\right)\}$.

For the sensitivity coefficients of the temperature and pressure, the process from ISO/IEC Guide 98-3 [17] (clause 5.1.3 NOTE 2) can be used. So, evaluating the equation of state for $\mathit{x},T+\delta T,p$ and a second time for $\mathit{x},T,p+\delta p$, enables evaluating these two partial derivatives, which are approximated by

$${\displaystyle \frac{\partial f}{\partial T}}={\displaystyle \frac{f(\mathit{x},T+\delta T,p)-f(\mathit{x},T,p)}{\delta T}},$$

(26)

and

$${\displaystyle \frac{\partial f}{\partial p}}={\displaystyle \frac{f(\mathit{x},T,p+\delta p)-f(\mathit{x},T,p)}{\delta p}}.$$

(27)

In these expressions, $\mathit{x}$ denotes the composition in Table 3. If desired, they can also be obtained exactly at the value of the estimate as shown in Equations (18) and (19) for differentiating f with respect to the amount fractions.

To obtain the sensitivity coefficients with respect to the composition using the GERG-2008 equation of state, Equations (18) and (20) were used. For the composition in Table 3, $\epsilon$ is set equal to 0.01 times the smallest amount fraction in $\mathit{x}$ ($=0.0002421\mathrm{mol}{\mathrm{mol}}^{-1}$), and using the directions as defined in Equation (21), the following compositions were obtained (see

Table 4. Compositions used to calculate the partial derivatives with respect to the compressibility factor calculated using the GERG-2008 equation of state (mol ${\mathrm{mol}}^{-1}$).

Component	${\mathit{x}}_0$	${\mathit{x}}_0+\mathbf{\delta }{\mathit{x}}_1$	${\mathit{x}}_0+\mathbf{\delta }{\mathit{x}}_2$	${\mathit{x}}_0+\mathbf{\delta }{\mathit{x}}_3$	${\mathit{x}}_0+\mathbf{\delta }{\mathit{x}}_4$
Nitrogen	0.03280	0.03263	0.03270	0.03273	0.03275
Carbon dioxide	0.02421	0.02438	0.02411	0.02414	0.02416
Methane	0.84334	0.84334	0.84354	0.84327	0.84329
Ethane	0.06587	0.06587	0.06587	0.06608	0.06582
Propane	0.03378	0.03378	0.03378	0.03378	0.03400

). From these, the compressibility factors at 300.0 $\mathrm{K}$ and 6.20 $\mathrm{M}$$\mathrm{Pa}$ were calculated (see

Table 5. Compressibility factors from the GERG-2008 equation of state and the values of the directional derivatives $b_j$ with respect to the measured composition $\mathit{x}$.

Component	x	$\mathit{x}+{\mathbf{\delta }\mathit{x}}_1$	${\mathit{x}}_2+\mathbf{\delta }\mathit{x}$	${\mathit{x}}_0+\mathbf{\delta }{\mathit{x}}_3$	${\mathit{x}}_0+\mathbf{\delta }{\mathit{x}}_4$
Z	0.869670	0.869622	0.869671	0.869609	0.869574
$b_j$		−0.20849	−0.00794	−0.26203	−0.40880

).

Using Equation (20), the sensitivity coefficients can be computed from the quotients $\delta Z/\epsilon$ in the ultimate row of Table 5, see

Table 6. Values of the sensitivity coefficients for the compressibility factor calculated using the GERG-2008 equation of state using asymmetric (top) and symmetric directional derivatives.

Component	N2	CO2	CH4	C2H6	C3H8
Index  $\mathit{i}$	1	2	3	4	5
$\partial Z/\partial x_i$	0.31772	0.02287	0.160571	−0.13552	−0.36564
$\partial Z/\partial x_i$	0.31766	0.02285	0.160540	−0.13550	−0.36554

. The differences between the one- and two-sided constrained partial derivatives are irrelevant for the purpose of propagating the measurement uncertainty.

Using Equations (26) and (27), the sensitivity coefficients for temperature and pressure can be obtained. Setting $\delta T=0.2 \mathrm{K}$ and $\delta p=0.00372 \mathrm{MPa}$, the compressibility factor at a temperature $T+\delta T$ is 0.870025. The compressibility factor at a pressure $p+\delta p$ is 0.869601. From these and the compressibility factor at actual conditions (see Table 5, column labelled ${\mathit{x}}_{\mathbf{0}}$), the sensitivity coefficients can be obtained, whose values are −1.9326 × 10⁻⁸ ${\mathrm{Pa}}^{-1}$ and that with respect to temperature 1.7609 ${\mathrm{K}}^{-1}$.

The value of the compressibility factor Z is 0.869672 with a standard uncertainty $u\left(Z\right)$ of 0.000375. This standard uncertainty does not include the model uncertainty. According to ISO 20765-2 [34], the model uncertainty is 0.1% relative. Considering that most predicted compressibility factors will lie within $\pm 0.1$% of the predicted value, this relative uncertainty is interpreted as an expanded uncertainty with a coverage probability of 95%. Assuming a normal distribution, the relative standard uncertainty is 0.05%, so 0.000435. Combined with the uncertainty contribution from propagating the uncertainty associated with the composition, pressure and temperature provides a standard uncertainty $u\left(Z\right)$ of 0.000574. Expressed as a relative standard uncertainty, it is 0.066%. So, the magnitude of the model uncertainty is about the same as that arising due to the composition, pressure and temperature measurements.

5. Conclusions

In this work, it was shown how functions with constrained arguments can be numerically differentiated to obtain values for the sensitivity coefficients needed for propagating measurement uncertainty. The proposed procedure was demonstrated on the calculation of the compressibility factor of natural gas at metering conditions using the GERG-2008 equation of state.

The numerical differentiation method does not depend on a particular choice of directions for differentiation. The method provides a sensitivity matrix that yields the same covariance matrix for the output quantities of the measurement model as the Jacobian would if that matrix can be obtained. The method is applicable to all differentiable functions of a composition, so it can be applied to all models used in, e.g., physical chemistry. It is thereby a leap forward not only for custody transfer and billing, but also for engineering as a whole where safety margins need to be established.

The numerical method for obtaining the sensitivity coefficients extends the guidanceof the Guide to the Expression of Uncertainty in Measurement (GUM), which (silently) assumes that the partial derivatives can be formed by varying one by one the input quantities of the measurement model. This way of differentiating functions is neither possible in models having a composition as an argument, nor for any other measurement models where the arguments are subject to a mathematical constraint. The GUM should be revised replacedtoin the sense that either its scope should be narrowed to measurement models where the input quantities are unconstrained, or, better, it should duly deal with this kind of measurement model. The utility of the developed numerical method goes way beyond using it for models with chemical compositions only.

It was proven that the constrained and unconstrained partial derivatives with respect to the amount fractions of models can be used to propagate measurement uncertainty. With a properly formed covariance matrix, the outcome of the uncertainty propagation is identical. As many of the standards like ISO 6976 allow using an improper covariance matrix, it matters which partial derivatives are used with the law of propagation of uncertainty (LPU). The constrained partial derivatives provide uncertainties that are closer to the ones obtained with a properly formed covariance matrix, so these are to be preferred. The (unconstrained) partial derivatives in these standards do not live up to the explanations in ISO/IEC Guide 98-3 concerning sensitivity coefficients.

The numerical procedure is easy to implement on any platform that supports matrix multiplication, which includes mainstream spreadsheet software. The formulæ for the orthogonal matrix needed to perturb the composition can be readily implemented as well. It is founded on the same footing as the guidance provided in ISO/IEC Guide 98-3 [17], and can be used with the LPU from ISO/IEC Guide 98-3 and ISO/IEC Guide 98-3/Supplement 2 [19]. In the examples developed, the asymmetric partial derivatives were very close to the symmetric ones, so the former can be used as well. The GUM [17,19] presumes that the symmetric ones are used though.

In the example given for the compressibility factor from the GERG-2008 equation of state, the uncertainty contribution from propagating the uncertainty associated with the input quantities was of the same magnitude as the model uncertainty for the compression factor as given in ISO 20765-2. The example shows how to meet the requirement in clause 7.6 of this standard that requires that the uncertainty of the input quantities to the equation of state should be duly propagated.