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METROLOGIA

Metrologia 44 (2007) 152–160

doi:10.1088/0026-1394/44/2/006

Two procedures for the estimation of the uncertainty of spectral irradiance measurement for UV source calibration A-F Obaton, J Lebenberg, N Fischer, S Guimier and J Dubard Laboratoire National de M´etrologie et d’Essais (LNE), 29 Avenue Roger Hennequin, Trappes Cedex 78197, France E-mail: [email protected]

Received 27 November 2006 Published 20 March 2007 Online at stacks.iop.org/Met/44/152 Abstract The measurement uncertainty of the spectral irradiance of an UV lamp is computed by using the law of propagation of uncertainty (LPU) as described in the ‘Guide to the Expression of Uncertainty in Measurement’ (GUM), considering only a first-order Taylor series approximation. Since the spectral irradiance model displays a non-linear feature and since an asymmetric probability density function (PDF) is assigned to some input quantities, the usage of another process was required to validate the LPU method. The propagation of distributions using Monte Carlo (MC) simulations, as depicted in the supplement of the GUM (GUM-S1), was found to be a relevant alternative solution. The validation of the LPU method by the MC method is discussed with regard to PDF choices, and the benefit of the MC method over the LPU method is illustrated.

1. Introduction Over the last 20 years, the need for accurate ultraviolet (UV) measurements of source spectral irradiance has increased enormously. Calibration of solar simulators used for testing solar products and calibration of UV sources used for water purification are two examples among others. In order to measure the spectral irradiance with the lowest uncertainty, the LNE has developed a new set-up to calibrate UV sources in the spectral range 240 nm to 350 nm by comparison with a standard lamp. The set-up has been characterized and the uncertainty on the spectral irradiance of a tungsten–halogen lamp compared with a deuterium standard lamp has been evaluated using the classical method: ‘law of propagation of uncertainty’ (LPU) described in the ‘Guide to the Expression of Uncertainty in Measurement’ (GUM) [1]. However, the mathematical model describing the spectral irradiance is not linear. Consequently, it would be recommended to carry out a second-order Taylor expansion on using the LPU method. Moreover, some probability density functions (PDFs) associated with uncertainties are not symmetric, which implied corrections had to be made by the LPU method. A solution to overcome these difficulties is to apply an alternative method of evaluation of measurement. This method, called the propagation of distributions using Monte Carlo (MC) simulations, is fully described in the GUM-S1 [2]. This 0026-1394/07/020152+09$30.00

method is based on the principle of random sampling by the MC method (MCM) using software called ‘Crystal Ball’, including a pseudo-random number generator. It requires a definition of the PDF for each input quantity associated with the measurand. Both methods are developed to estimate the uncertainty on the spectral irradiance of a tungsten–halogen lamp. The significance of such a comparison is to validate the LPU method by the MCM. In this paper, we first provide a detailed description of the measurement set-up and the UV source calibration method. Next the uncertainty components are identified and their evaluation methods are described. Then the calibration uncertainty is calculated using the two methods. Finally, the results are discussed regarding the validation procedure.

2. Calibration method and experimental set-up The method used to calibrate the spectral irradiance of UV sources is by comparison with a standard lamp (figure 1). In this paper, the calibration of a tungsten–halogen lamp has been performed by comparison with a deuterium standard lamp. The radiation from the lamp under calibration (fixed current supply) or the standard lamp is collected using an integrating sphere with Spectralon internal coating. A shutter in front of the integrating sphere enables one to measure the dark signal level of our set-up. The lens between the integrating

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Two procedures for the estimation of the uncertainty of spectral irradiance measurement

Standard source High voltage

Photo multiplier Translation stage

Integrating sphere

() shutter

Double monochromator (grating+prism)

Electrometer

lens

PC

Source under test

Figure 1. Schematic description of the set-up used to measure the spectral irradiance of an UV source.

sphere and the entrance slit of the monochromator is positioned in a 2f configuration. The monochromator is a double subtractive monochromator composed of a 600 grooves/mm diffraction grating (wavelength selector) and a prism (order selector). After the beam goes through the monochromator, it is detected by a photomultiplier tube (PMT). The PMT is supplied with a high voltage (1000 V) and the delivered current is measured with an electrometer. The entire system is computer-controlled using Labview software. The two lamps are positioned on a translation stage which enables one to measure them alternately. From such a setting, it is the current delivered by the PMT for the tested lamp I and the standard lamp Istd which are measured directly, rather than the spectral irradiance of the tested lamp E. The dark current of the PMT drifts with time, so the following four-step measurement cycle for each wavelength has been chosen to take this behaviour into account: (1) shutter closed, the dark signal Z¯ 1 in ampere is the average of 10 successive acquisitions, (2) shutter open, the standard lamp being in front of the integrating sphere, the standard lamp signal I¯std in ampere is the average of 10 successive acquisitions, (3) shutter open, the lamp under calibration being in front of the integrating sphere, the lamp signal I¯ in ampere is the average of 10 successive acquisitions, and (4) shutter closed, the dark signal Z¯ 2 in ampere is the average of 10 successive acquisitions. During a run on the 240 nm to 350 nm spectral range, this cycle is repeated for each wavelength with a 5 nm step. The electrometer and the monochromator are initialized after each run. We performed 10 runs and we calculated the average of the signals over all runs I¯¯, Z¯¯ 1 , Z¯¯ 2 , I¯¯std and for each wavelength. Since the input quantities (I¯, I¯std , Z¯ 1 and Z¯ 2 ) are repeatedly observed, the choice of the estimator of the measurand should be discussed. Let us consider a functional relationship between the measurand Y and a set of input variables (X1 , . . . , Xp ): Y = f (X1 , . . . , Xp ).

(1)

An estimate y of the measurand is given by y = f (x1 , . . . , xp ),

(2)

where (x1 , . . . , xp ) is the set of estimates for the p input quantities. Metrologia, 44 (2007) 152–160

If we assume that these input variables are repeatedly observed through a sample of n independent sets (x1k , . . . , xpk ), the best estimate x1 (for example) of the quantity X1 is the sample average x¯1 of the x1k values. This leads one to consider two alternative estimation methods, detailed in the GUM H2–H4 [1] and by Bich et al in [3]. The first way to estimate the measurand Y consists of the function evaluation of the sample average. In terms of random variables, this first method, referring to method 1, yields the estimate Yˆ1 = f (X¯ 1 , . . . , X¯ p ). (3) The second approach consists of calculating the average of the function evaluated from each element of the sample. This second way of averaging, referring to method 2, leads to a different estimate: 1
Yˆ2 = f (X1k , . . . , Xpk ). n k=1 n

(4)

Approximate expressions for expectations of both estimates can be found by using power series development [3]:  p 2 
1 ∂ f  × u2 (xj ), (5) E(Yˆ1 ) = f (x1 , . . . , xp ) +  2 ∂ X¯ j2  ¯ j =1

Xj =xj

where u2 (xj ) = E(X¯ j − xj )2 and

 p 2 
∂ f 1  E(Yˆ2 ) = f (x1 , . . . , xp ) +  2 j =1 ∂Xj2 

× σ 2 (Xj ), (6)

Xj =xj

where σ 2 (Xj ) = E(Xj − xj )2 . In the case of a non-linear model, the expectations of the estimates are different. In [1, H.2 and H.4], the two methods are compared, leading to the suggestion that the second method is more appropriate. However, since these examples provide similar results by using either method, they do not allow one to draw any conclusion on this matter. As far as we are concerned, the mathematical model that gives the spectral irradiance E has a non-linear feature. It is therefore necessary to address the following question: which method is more relevant with regard to our measurement process? This question is particularly relevant since the two methods so far provided different numerical estimations. Indeed, the difference between the two expectations is actually of the same order of magnitude as the standard uncertainty to certain wavelengths. Firstly we assumed that the ten sets of random variables (I¯, I¯std , Z¯ 1 , 153

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Z¯ 2 ) which are associated with the ten series of acquisitions could be considered as independent and identically distributed. Secondly, we made the hypothesis that the input quantities, repeatedly measured (I¯, I¯std , Z¯ 1 , Z¯ 2 ), are thought to display an essentially unique value so that the observed noise is considered as purely experimental. This assumption is clearly stated in [1, note 4.1.1] where ‘it is assumed that the physical quantity itself can be characterized by a unique value’. Under this specific assumption, according to [3], method 1 should be preferred. Furthermore, the authors conclude ‘that the approach proposed in [2] is compliant with method 1, which is consistent with the GUM uncertainty framework’. Considering the aim of this work, which is to compare results obtained by the LPU and MC methods, we will thus estimate the measurand according to method 1.

3. List of the contributory uncertainties All the uncertainties which can affect the evaluation of the spectral irradiance E of the lamp under calibration are defined below: • uncertainty due to repositioning of the translation stage, • uncertainty due to the alignment of the lamp in front of the calibration set-up, • uncertainty due to stray light inherent to the set-up, • uncertainty due to the wavelength calibration of the monochromator u(cλmono ), • uncertainty due to the spectral bandwidth of the monochromator u(cλ ), • uncertainty due to linearity of the PMT u(cPM ), • uncertainty due to the stability of the high voltage power supply of the PMT, • uncertainty due to the current measured by the electrometer u(celectro ), • uncertainty due to the short-term instability of the set-up, • uncertainty due to the calibration of the standard lamp u(cstd ), • uncertainty due to the ageing of the standard lamp between std ), two consecutive calibrations u(cdrift • uncertainty due to the interpolation of the data points of the standard lamp calibration certificate u(cinterpol ), • uncertainty due to the distance between the lamps and the integrating sphere u(cdist ) • repeatability uncertainty σ .

4. Mathematical relationship between the measurand and the input quantities The mathematical expression between the measurand E and the input quantities is given by the following expression:

 Z¯¯ +Z¯¯ +2C z I¯¯ + Cλmono + Cλ + Celectro − 1 2 2 electro  

E=   Z¯¯ +Z¯¯ +2C z std std + Celectro − 1 2 2 electro I¯¯std + Cλstdmono + Cλ 



  std std  + CPM   (Estd + Cstd + Cdrift + Cinterpol + Cdist ) + Cdist , (7) 154

where Estd is the spectral irradiance of the standard lamp given by the calibration certificate and C denotes corrections which will be considered as zero average correction except Celectro and Cdist in one particular case that we will describe below. There is a linear correlation between Cstd and I¯¯std with r(Cstd , I¯¯std ) = 1. Though Z¯¯ 1 and Z¯¯ 2 represent the same physical quantity at different times, they are assigned to white noise, so they are not correlated. The influence of a correlation between the Celectro has been evaluated and found to be negligible. Other corrections are not correlated.

5. Assessment of all uncertainty components and associated distribution In this section, we will strive to identify the different uncertainty components and evaluate their contribution. Uncertainty due to the alignment of the lamp in front of the calibration set-up. The alignment of the lamp in front of the calibration set-up is made with a He–Ne laser; consequently the uncertainty will be neglected. Uncertainty due to stray light inherent to the set-up. To evaluate this uncertainty we measured a tungsten–halogen lamp combined with WG320- or WG335-type filters. The measured signal for wavelengths shorter than the cut-off wavelengths is equal to the dark signal. Therefore, the stray light contribution can be neglected. Uncertainty due to the wavelength calibration of the monochromator u(cλmono ). The difference in wavelength between the monochromator display value and the true value is checked before each calibration using the 253 nm and 296 nm rays of a mercury lamp. This uncertainty is lower than 0.35 nm for a 2.5 mm monochromator slit width. The stability and the repeatability of the wavelength scale of the monochromator are included in this uncertainty. Therefore, considering I (λ), the current as the function of the wavelength, the interval in which this quantity is included is |I (λ − 0.35 nm) − I (λ + 0.35 nm)|. The distribution which can be associated with this quantity is of rectangular shape as the measurement can take any value in this interval with equal probability. I (λ − 0.35 nm) and I (λ + 0.35 nm) are estimated by linear interpolation around each value of I (λ). Uncertainty due to the spectral bandwidth of the monochromator u(cλ ). The spectral distribution E(λ) of the light exiting the monochromator is the product of the source spectrum and the apparatus function of the monochromator. The central wavelength of the beam is given by the following relation:   λ2  λ2 λc = E dλ. (8) Eλ dλ λ1

λ1

In the case of a source with a flat spectrum, λc is equal to the monochromator wavelength setting, λref . For any other source, λc differs from λref . This difference is larger as the source spectrum exhibits a stronger gradient with wavelength. So we have considered the most unfavourable conditions to estimate this uncertainty by simulation, i.e. using the Metrologia, 44 (2007) 152–160

Two procedures for the estimation of the uncertainty of spectral irradiance measurement

spectrum of a solar simulator, which presents the strongest gradient with regard to the other sources that will be calibrated. We have calculated the spectral distribution of the output beam of the monochromator set at λref = 305 nm from λ1 = 300 nm to λ2 = 310 nm with a 0.1 nm step. Considering that the entrance and the exit slit of the monochromator are equally opened, the apparatus function of the monochromator is triangular. Its full width at half maximum has been experimentally determined to be 2.9 nm for an entrance and exit slit width of 2.5 mm. In this case, the difference λc −λref = 0.36 nm.   So the interval in which  I (λ) is included is I (λ − 0.36 nm) − I (λ + 0.36 nm). The distribution which can be associated with this quantity is of rectangular shape as the measurement can take any value in this interval with equal probability. I (λ − 0.36 nm) and I (λ + 0.36 nm) are estimated by a linear fit around each value of I (λ). Uncertainty due to linearity of the photomultiplier tube u(cPM ). The calibration of the PMT linearity is performed using a set-up based on the flux addition method. The current range extends from 0.2 nA to 0.2 µA with a 2× flux step. For each octave, an uncertainty is calculated according to the uncertainty evaluation associated with this method. In our experiment, the uncertainty must be evaluated on the current difference between I and Istd . This uncertainty is calculated by summing quadratically the uncertainties corresponding to the octaves between I and Istd . A normal distribution is assigned to this quantity. Uncertainty due to the current measured by the electrometer u(celectro ). According to the calibration certificate of the electrometer, corrections have to be made for each current, and the uncertainty on these corrections for the range in between 200 nA and 200 µA is ±(1 × 10−4 I + ud) (k = 2) and for the range in between 2 nA and 200 nA it is ±(2 × 10−4 I + 0.1 pA + 1 ud) (k = 2) where ‘ud’ is the last displayed digit. Appropriate corrections have been made for each current level and the associated uncertainty considered. A normal distribution has been associated with this input quantity. Uncertainty due to the calibration of the standard lamp u(cstd ). Uncertainty on the spectral irradiance of the standard lamp is given by the calibration certificate as a function of the wavelength every 10 nm. A normal distribution is associated with this uncertainty. Uncertainty due to the ageing of the standard lamp between std two consecutive calibrations u(cdrift ). This uncertainty has been estimated from the ageing of a tungsten coiled filament lamp (FEL) from 290 nm to 350 nm as the deuterium lamp was calibrated for the first time recently. It is calculated from twice the widest gap between two successive calibrations considering a rectangular distribution. For wavelengths lower than 290 nm, uncertainty associated with this wavelength has been considered. The reason why we chose a FEL lamp rather than another lamp is that we have calibration certificates from 1995 every three years and that this lamp is known to age quickly in this UV range. However, the drift of the output of our standard deuterium lamp was characterized over 30 h and is evaluated to be less than 0.2% so that the uncertainty used Metrologia, 44 (2007) 152–160

has been overestimated. Subsequently, the calculation will be refined considering the deuterium lamp. Uncertainty due to the interpolation of the data points of the standard lamp calibration certificate u(cinterpol ). The standard lamp is calibrated with a 10 nm wavelength step. In order to calculate the spectral irradiance for other wavelengths, a cubic spline interpolation is performed from the data points given by the calibration certificate. Numerical simulations on the black body spectrum show that the uncertainty on the interpolation is less than 0.1% and its associated distribution is rectangular. Uncertainty due to the distance between the lamps and the integrating sphere u(cdist ). The spectral irradiance is inversely proportional to the square of the distance (d) between the lamp and the integrating sphere, so  2 u2 (cdist ) ∝ 2/d 3 u2 (d). (9) The uncertainty on the distance is evaluated to be d = ±2 × 10−3 m for the tested lamp and d = ±5 × 10−4 m for the standard lamp. Because of the distance measurement process, we systematically overestimate the distance of the lamp. Therefore, an asymmetric PDF is associated with this uncertainty. In addition, we assume that the PDF decreases in the range 0 to d. Consequently, a right-angle-triangular distribution should be assigned to this quantity. However, it is easier to manipulate a symmetric distribution with the LPU method. So we have considered the expanded symmetric interval associated with a triangular distribution with a mean equal to zero and a range of 2d, thus √ (10) u(d) = d/ 6. But in order to measure the impact of the choice of a PDF on the calculation of the uncertainty we have also made the calculation using a right-angle-triangular distribution. In this condition, a correction on the spectral irradiance of both lamps was needed, i.e.

and

Ecorr = E (1 − (2d)/3d)

(11)

√ u(d) = d/ 18.

(12)

Repeatability uncertainty σ . This uncertainty is evaluated from the 10 runs of 10 successive measurements of Z1 , Istd , I and Z2 . The standard deviation of these 100 data points is calculated. Since all these quantities √ are independent, the standard deviation was divided by 10. It includes the uncertainty due to the repositioning of the translation stage between each series of 10 measurements, the uncertainty due to the stability of the high voltage power supply of the PMT and the uncertainty due to the short-term instability of the setup. Regarding this last uncertainty, the observed fluctuations on the signal are known from the surveillance carried out on the signals (I − (Z1 + Z2 )/2) and (Istd − (Z1 + Z2 )/2). One thousand successive measurements have been performed for three different wavelengths in order to estimate the statistical distribution associated with this quantity. It has been found to be normal. The significant uncertainties and their associated distributions are listed in table 1. 155

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Table 1. List of all the non-negligible uncertainties, distribution associated, values and units. Uncertainty component

Description

Distribution

u(cλmono )

Difference in wavelength on the monochromator

Rectangular

u(cλ )

Spectral bandwidth of the monochromator

Rectangular

urel (cPM ) u(celectro ) urel (cstd ) std ) urel (cdrift

urel (cinterpol )

urel (cdist )

σ

Units

Linearity of the photomultiplier tube Current measured by electrometer Calibration of the standard lamp Drift of the standard lamp between two consecutive calibrations

Normal

|I (λ − 0.35 nm) − I (λ + 0.35 nm)| √ 2 3 |I (λ − 0.36 nm) − I (λ + 0.36 nm)| √ 2 3  (n + 1)u2rel (3 dB)

Normal

Calibration certificate (k = 2)

A

Normal

Calibration certificate (k = 1)

Relative %

Interpolation of the data points of the standard lamp calibration certificate Distance between the lamp and the integrating sphere

Rectangular

Repeatability over 10 series of 10 successive measurements

0.0114 from 240 nm to 290 nm √ 3 0.0087 from 295 nm to 350 nm √ 3 0.1

Rectangular

Triangular and right-angle-triangular

Normal

6. Uncertainty evaluation using the LPU Following the process, the GUM uncertainty framework involves the application of the LPU (only the easy-to-use truncation at the first order has been computed). Writing partial derivatives, the variance of the spectral irradiance of the tested lamp is given by

 ∂E 2  2 u2 (e) = σ (i) + u2(cλmono ) + u2 (cλ ) + u2 (celectro ) ∂I

   std  ∂E 2  2 + σ (istd ) + u2 cλstdmono + u2 cλ ∂Istd

 std   std  ∂E 2  2 + u (cstd ) + u2 cdrift + u2 celectro ∂Estd

   std  ∂E 2  2 + σ (z1 ) + u2 cinterpol + u2 cdist ∂Z1

   ∂E 2 2 2 z 2 2 + 2u celectro + σ (z2 ) + u (cdist ) + u (cPM ) ∂CPM



∂E ∂E u(cstd )σ (istd ) . (13) +2 ∂Cstd ∂Istd The result is shown in table 2. The expanded uncertainty is given by U (e) = 2u(e).

7. Uncertainty evaluation using the propagation of distributions involving MCM We have just seen the manner of evaluating the measurement standard uncertainty u(e) of the measurand and the expanded 156

Value

2d 2d √ or √ with d = 0.3 m, d 18 d 6 d = 0.5 × 10−3 m for the standard lamp and d = 0.2 m, d = 2 × 10−3 m for the lamp  under calibration n ¯ 2 k=1 (xk − x) A(xi ) = n−1

A A Relative

Relative

Relative %

Relative

A

uncertainty U (e), according to the standard procedure proposed by [1]. However, the GUM uncertainty framework suffers from some important limitations which consist of the following. • The combined uncertainty u(e) is obtained at step 6 by combining the standard uncertainties of the input quantities using the LPU. This law originates from the truncation of Taylor’s series expansion at first-order terms. Because our model displays a non-linear feature, the linear approximation could thus lead to erroneous results. • A non-symmetric distribution can be assigned to an input quantity (type B evaluation of the uncertainty), which weighs down the LPU method if we do not want the asymmetric feature of the uncertainty to be lost. • A coverage factor needs to be calculated (by the Welch–Satterthwaite formula or chosen arbitrarily) in order to obtain a coverage interval. The alternative approach to the LPU method is the wellknown method of propagation of distributions [4–6]. The principle of this method is no longer to propagate through the model the uncertainty but rather the PDF for the input quantities in order to obtain the PDF for the measurand. Given the PDF for each input quantity, the PDF for the measurand Y can be analytically obtained through the Markov formula [2, 6]. In accordance with the notations in [2], the PDF for Y, gY (η), is given by  +∞  +∞ ... gX (ξ ) × δ(η − f (ξ )) dξ1 ... dξp , gY (η) = −∞

−∞

(14) Metrologia, 44 (2007) 152–160

Two procedures for the estimation of the uncertainty of spectral irradiance measurement

Table 2. Spectral irradiance and standard uncertainties using the MC and LPU methods and assigning different symmetric or asymmetric PDFs to the uncertainty due to the distance between the lamps and the integrating sphere u(cdist ). LPU method MC method λ/nm 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350

−2

e/W m 5.28 × 10−5 6.67 × 10−5 8.13 × 10−5 1.087 × 10−4 1.286 × 10−4 1.680 × 10−4 2.190 × 10−4 2.792 × 10−4 3.421 × 10−4 4.291 × 10−4 5.269 × 10−4 6.521 × 10−4 7.942 × 10−4 9.539 × 10−4 1.1438 × 10−3 1.3668 × 10−3 1.5951 × 10−3 1.8533 × 10−3 2.1599 × 10−3 2.5110 × 10−3 2.9029 × 10−3 3.3709 × 10−3 3.8364 × 10−3

Cdist : symmetric PDF −2

u(e)/W m 8.5 × 10−6 6.4 × 10−6 6.4 × 10−6 7.4 × 10−6 7.7 × 10−6 9.6 × 10−6 1.13 × 10−5 1.39 × 10−5 1.62 × 10−5 2.03 × 10−5 2.44 × 10−5 2.89 × 10−5 3.44 × 10−5 4.12 × 10−5 3.99 × 10−5 4.52 × 10−5 4.37 × 10−5 5.16 × 10−5 5.87 × 10−5 6.95 × 10−5 7.68 × 10−5 8.97 × 10−5 7.66 × 10−5

−2

e/W m 5.32 × 10−5 6.72 × 10−5 8.19 × 10−5 1.096 × 10−4 1.296 × 10−4 1.693 × 10−4 2.207 × 10−4 2.814 × 10−4 3.448 × 10−4 4.325 × 10−4 5.311 × 10−4 6.574 × 10−4 8.005 × 10−4 9.616 × 10−4 1.1528 × 10−3 1.3776 × 10−3 1.6078 × 10−3 1.8680 × 10−3 2.1771 × 10−3 2.5306 × 10−3 2.9260 × 10−3 3.3975 × 10−3 3.8666 × 10−3

where X is the vector of input quantities, gX (ξ ) the PDF for X and δ the Dirac delta function [7]. An analytical determination of the PDF for Y does not suffer from any approximation and seems so far to be the ideal solution. However, apart from fairly simple examples [5], the multiple integral cannot be evaluated analytically. Thus, Supplement 1 to the GUM [2] provides a numerical method that implements the propagation of distributions using a MCM. This alternative approach, described in [3], follows a step-by-step procedure. • Specify the measurand, the input quantities and the measurement process, as well as the functional relationship relating Y to X = (X1 , .., Xp ). This step is common to both approaches implemented in this work. • Assign a PDF (normal, uniform, . . . ) to each input quantity according to available knowledge or a joint PDF to correlated quantities (for example, in our context Istd and Cstd are correlated). • Generate M sets of realizations of the p input quantities Xj by sampling from the PDFs. • Obtain an empirical PDF for the output by computing the model equation for each of the M sets generated in the preceding step. • Provide summarized information on the output quantity including: – the expectation of that quantity, – the standard deviation of that quantity and – the shortest coverage interval associated with a specified coverage probability. Metrologia, 44 (2007) 152–160

Cdist : asymmetric PDF −2

u(e)/W m 8.5 × 10−6 6.4 × 10−6 6.5 × 10−6 7.4 × 10−6 7.8 × 10−6 9.7 × 10−6 1.14 × 10−5 1.41 × 10−5 1.64 × 10−5 2.05 × 10−5 2.47 × 10−5 2.92 × 10−5 3.49 × 10−5 4.17 × 10−5 4.08 × 10−5 4.61 × 10−5 4.51 × 10−5 5.31 × 10−5 6.06 × 10−5 7.15 × 10−5 7.94 × 10−5 9.27 × 10−5 8.11 × 10−5

e/W m−2 5.28 × 10−5 6.67 × 10−5 8.13 × 10−5 1.088 × 10−4 1.286 × 10−4 1.680 × 10−4 2.190 × 10−4 2.792 × 10−4 3.421 × 10−4 4.292 × 10−4 5.270 × 10−4 6.523 × 10−4 7.943 × 10−4 9.541 × 10−4 1.144 × 10−3 1.3669 × 10−3 1.5953 × 10−3 1.8535 × 10−3 2.1602 × 10−3 2.5109 × 10−3 2.9032 × 10−3 3.3711 × 10−3 3.8366 × 10−3

u(e)/W m−2 8.5 × 10−6 6.4 × 10−6 6.4 × 10−6 7.4 × 10−6 7.7 × 10−6 9.6 × 10−6 1.13 × 10−5 1.39 × 10−5 1.62 × 10−5 2.02 × 10−5 2.44 × 10−5 2.88 × 10−5 3.44 × 10−5 4.11 × 10−5 4.00 × 10−5 4.51 × 10−5 4.37 × 105 5.15 × 10−5 5.87 × 10−5 6.93 × 10−5 7.68 × 10−5 8.97 × 10−5 7.67 × 10−5

From our experience, we choose the Crystal Ball software [8,9], among numerous applications, to perform the MCM. The choice of the number M of trials also appears to be important since a sufficient number of simulations are needed to ensure the convergence between the observed PDF for Y and the theoretical one. In [2] and [5], the authors recommend one to perform M = 106 trials. We decided to implement an adaptive MC approach involving the use of an increasing number of MC trials till the results (expectation, standard deviation, coverage interval) have stabilized. Depending on the wavelength and the numerical tolerance associated with the result, we carried out 2 × 105 to 106 trials.

8. Results, validation and discussion According to [2, section 8.1.1], ‘the domain validity for MCM is broader than that for the GUM uncertainty framework’. Therefore the results obtained by using the LPU method based on a first-order Taylor series approximation must be validated in comparison with those derived from the MCM. A statistical fitting procedure performed on the MC trials validated the approximation used to assign a normal distribution to the output quantity E. Therefore, the choice of 2 for the coverage factor is relevant. As the utilization of the validation process described in [2, section 8] is recommended, we implemented both methods and performed the validation procedure in two stages. • To define a numerical tolerance δ associated with the standard uncertainty u(e). 157

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6x10

-5

9x10

MC method LPU Cdist : symmetric PDF LPU Cdist : asymmetric PDF

-5

Contributions / (W/m )

2 2

8x10

-5

7x10

2

u(E) / (W/m )

Cstd CPM Cdist

-9

5x10

-5

6x10

-5

5x10

-5

-9

4x10

-9

3x10

-9

2x10

-9

4x10

1x10

-5

3x10

0 300

310

320

330

340

350

Wavelength / nm

Figure 2. Standard uncertainties using the MC and LPU methods and assigning different symmetric or asymmetric PDFs to the uncertainty due to the distance between the lamps and the integrating sphere u(cdist ).

To determine the appropriate tolerance δ, one should express the numerical result for uncertainty in the form c × 10l where c is the significant digit of the decimal (ndig ) and l is an integer. Thus, the tolerance is given by δ = (1/2) × 10l . For example, at λ = 300 nm, u(e) = 0.000 0349. As ndig = 1 can be assumed in the case of spectral irradiance measurements, this means that only the first significant digit is regarded as meaningful. Thus, u(e) is expressed as 3×10−5 , l = −5 and δ = 5×10−6 W m−1 . • To compare the coverage intervals obtained by both methods. In the case of an asymmetric PDF for the output quantity, the shortest MCM coverage interval must be provided. This is intended to determine whether there is any difference between the two intervals for all ndig digits regarded as meaningful. Therefore, the following absolute differences are computed:   dlow = y − Up − ylow  , (15)   dhigh = y + Up − yhigh  , where [ylow ; yhigh ] is the shortest MCM coverage interval and Up the expanded uncertainty given by the LPU method. Since this expanded uncertainty is associated with a p coverage probability, this validation applies for that specific coverage probability only. Then, if both differences, dlow and dhigh , are smaller than the tolerance δ, the results provided by the LPU method (involving a first-order Taylor series approximation) are validated. Table 2 gives the spectral irradiance of the tungsten– halogen lamp and its associated uncertainty for each wavelength from 240 nm to 350 nm. Figure 2 shows the standard uncertainties considering both choices of PDF for Cdist . Only the data on the spectral range 300 nm to 350 nm are displayed because they show significant differences between the two choices. The impact of the PDF choice is important because the uncertainty associated with the input quantity Cdist is one of the largest contributions. This is shown in figure 3. When the PDF is the same in both methods the 158

240

260

280

300

320

340

Wavelength / nm

Figure 3. Principal contributions of uncertainties.

results look similar. In order to confirm this observation we have calculated the numerical tolerance δ associated with the standard uncertainty u(e) and the coverage intervals obtained in both methods. The results are given in tables 3 and 4, respectively, for symmetric and asymmetric PDF associated with Cdist . In the case of a symmetric PDF choice, the large differences observed in the coverage intervals (table 3) confirm those observed in standard uncertainties (table 2). Namely, for many wavelengths the LPU method is not validated (table 3). This is firstly due to the difference observed in the expectations (table 2). Indeed, in the case of the triangular distribution no correction of distance has been carried out on the result. It is noticeable (table 4) that the fact of carrying out this correction (in the case of the right-angled triangle PDF) provides an expectation given by LPU close to that obtained with MCM. Secondly, one notices that this contributory uncertainty is far from being negligible for longer wavelengths (figure 3). Considering a symmetric triangular PDF, the width of the interval, assigned to the quantity Cdist , is twice as large, leading to a degradation of the uncertainty. Since the contribution of Cdist increases with the wavelength, the difference between the min(dlow , dhigh ) and the numerical tolerance increases in the same way with the wavelength. This means that although it is not simple to handle asymmetric PDF within the framework of LPU, it is nevertheless necessary. Considering only the first decimal digit regarded as meaningful in u(e), the LPU method is validated only for the asymmetric PDF except for one wavelength. However, for shorter wavelengths (the domain where the signal-to-noise ratio is small) one notices that the values of the difference dlow and dhigh have the same order of magnitude as δ. One can reasonably think that if the method is not validated at the wavelength 265 nm, it is rather due to an operational limit of the numerical tolerance than to a physical behaviour at this particular wavelength. Indeed, standard uncertainty does not particularly increase around 265 nm. Only the validation criterion is an absolute difference criterion. The δ(λ) function is a step function. Each step corresponds to an order of magnitude of uncertainty; that is, for a standard uncertainty of 6.4 × 10−6 and 9.7 × 10−6 , both numerical tolerances equal 10−7 . So as illustrated in table 5, the relative tolerance (δ/ylow ), Metrologia, 44 (2007) 152–160

Two procedures for the estimation of the uncertainty of spectral irradiance measurement

Table 3. Interval and numerical tolerance for a symmetric PDF associated with Cdist .

MC method Shortest 95%/W m−2

LPU method Cdist : symmetric PDF Coverage interval/W m−2

λ/nm

ylow

yhigh

ylow

yhigh

dlow /W m−2

dhigh /W m−2

δ/W m−2

LPU validated

240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350

3.62 × 10−5 5.43 × 10−5 6.86 × 10−5 9.45 × 10−5 1.136 × 10−4 1.494 × 10−4 1.973 × 10−4 2.525 × 10−4 3.108 × 10−4 3.900 × 10−4 4.798 × 10−4 5.966 × 10−4 7.275 × 10−4 8.740 × 10−4 1.0663 × 10−3 1.2788 × 10−3 1.5099 × 10−3 1.7532 × 10−3 2.0457 × 10−3 2.3757 × 10−3 2.7532 × 10−3 3.1960 × 10−3 3.6866 × 10−3

6.94 × 10−5 7.94 × 10−5 9.38 × 10−5 1.234 × 10−4 1.437 × 10−4 1.868 × 10−4 2.416 × 10−4 3.068 × 10−4 3.741 × 10−4 4.694 × 10−4 5.756 × 10−4 7.096 × 10−4 8.622 × 10−4 1.0355 × 10−3 1.2224 × 10−3 1.4561 × 10−3 1.6810 × 10−3 1.9554 × 10−3 2.2753 × 10−3 2.6483 × 10−3 3.0547 × 10−3 3.5477 × 10−3 3.9872 × 10−3

3.62 × 10−5 5.44 × 10−5 6.90 × 10−5 9.47 × 10−5 1.141 × 10−4 1.500 × 10−4 1.979 × 10−4 2.533 × 10−4 3.120 × 10−4 3.916 × 10−4 4.817 × 10−4 5.990 × 10−4 7.307 × 10−4 8.782 × 10−4 1.0712 × 10−3 1.2853 × 10−3 1.5177 × 10−3 1.7619 × 10−3 2.0558 × 10−3 2.3877 × 10−3 2.7672 × 10−3 3.2121 × 10−3 3.7044 × 10−3

7.02 × 10−5 8.00 × 10−5 9.48 × 10−5 1.245 × 10−4 1.452 × 10−4 1.886 × 10−4 2.436 × 10−4 3.096 × 10−4 3.776 × 10−4 4.735 × 10−4 5.805 × 10−4 7.158 × 10−4 8.702 × 10−4 1.0449 × 10−3 1.2344 × 10−3 1.4698 × 10−3 1.6979 × 10−3 1.9742 × 10−3 2.2984 × 10−3 2.6735 × 10−3 3.0847 × 10−3 3.5829 × 10−3 4.0289 × 10−3

8 × 10−8 6 × 10−8 3.9 × 10−7 2.8 × 10−7 5.6 × 10−7 6.3 × 10−7 5 × 10−7 8 × 10−7 1.3 × 10−6 1.6 × 10−6 1.9 × 10−6 2.4 × 10−6 3.2 × 10−6 4.2 × 10−6 4.9 × 10−6 6.5 × 10−6 7.8 × 10−6 8.7 × 10−6 1.01 × 10−5 1.19 × 10−5 1.40 × 10−5 1.61 × 10−5 1.77 × 10−5

8.3 × 10−7 6.6 × 10−7 1.05 × 10−6 1.11 × 10−6 1.51 × 10−6 1.85 × 10−6 2.1 × 10−6 2.8 × 10−6 3.5 × 10−6 4.1 × 10−6 4.9 × 10−6 6.2 × 10−6 8.0 × 10−6 9.5 × 10−6 1.20 × 10−5 1.38 × 10−5 1.69 × 10−5 1.88 × 10−5 2.30 × 10−5 2.52 × 10−5 3.01 × 10−5 3.52 × 10−5 4.18 × 10−5

5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6

NO NO NO NO NO NO YES YES YES YES YES NO NO NO NO NO NO NO NO NO NO NO NO

Table 4. Interval and numerical tolerance for an asymmetric PDF associated with Cdist .

MC method Shortest 95%/W m−2

LPU method Cdist : asymmetric PDF Coverage interval/W m−2

λ/nm

ylow

yhigh

ylow

yhigh

dlow /W m−2

dhigh /W m−2

δ/W m−2

LPU validated

240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350

3.62 × 10−5 5.43 × 10−5 6.86 × 10−5 9.45 × 10−5 1.136 × 10−4 1.494 × 10−4 1.973 × 10−4 2.525 × 10−4 3.108 × 10−4 3.900 × 10−4 4.798 × 10−4 5.966 × 10−4 7.275 × 10−4 8.740 × 10−4 1.0663 × 10−3 1.2788 × 10−3 1.5099 × 10−3 1.7532 × 10−3 2.0457 × 10−3 2.3757 × 10−3 2.7532 × 10−3 3.1960 × 10−3 3.6866 × 10−3

6.94 × 10−5 7.94 × 10−5 9.38 × 10−5 1.234 × 10−4 1.437 × 10−4 1.868 × 10−4 2.416 × 10−4 3.068 × 10−4 3.741 × 10−4 4.694 × 10−4 5.756 × 10−4 7.096 × 10−4 8.622 × 10−4 1.0355 × 10−3 1.2224 × 10−3 1.4561 × 10−3 1.6810 × 10−3 1.9554 × 10−3 2.2753 × 10−3 2.6483 × 10−3 3.0547 × 10−3 3.5477 × 10−3 3.9872 × 10−3

3.59 × 10−5 5.39 × 10−5 6.85 × 10−5 9.40 × 10−5 1.132 × 10−4 1.488 × 10−4 1.964 × 10−4 2.514 × 10−4 3.097 × 10−4 3.887 × 10−4 4.781 × 10−4 5.946 × 10−4 7.255 × 10−4 8.719 × 10−4 1.0638 × 10−3 1.2766 × 10−3 1.5080 × 10−3 1.7505 × 10−3 2.0427 × 10−3 2.3723 × 10−3 2.7497 × 10−3 3.1917 × 10−3 3.6832 × 10−3

6.97 × 10−5 7.94 × 10−5 9.41 × 10−5 1.236 × 10−4 1.440 × 10−4 1.871 × 10−4 2.417 × 10−4 3.071 × 10−4 3.745 × 10−4 4.696 × 10−4 5.758 × 10−4 7.100 × 10−4 8.631 × 10−4 1.0363 × 10−3 1.2238 × 10−3 1.4571 × 10−3 1.6827 × 10−3 1.9565 × 10−3 2.2777 × 10−3 2.6495 × 10−3 3.0568 × 10−3 3.5505 × 10−3 3.9900 × 10−3

3.0 × 10−7 4.2 × 10−7 1.8 × 10−7 4.8 × 10−7 3.3 × 10−7 5.3 × 10−7 9 × 10−7 1.1 × 10−6 1.0 × 10−6 1.3 × 10−6 1.7 × 10−6 2.0 × 10−6 2.1 × 10−6 2.1 × 10−6 2.5 × 10−6 2.2 × 10−6 1.9 × 10−6 2.7 × 10−6 3.0 × 10−6 3.4 × 10−6 3.5 × 10−6 4.3 × 10−6 3.4 × 10−6

3.9 × 10−7 1 × 10−7 3.5 × 10−7 1.7 × 10−7 3.9 × 10−7 3.7 × 10−7 1 × 10−7 3 × 10−7 5 × 10−7 2 × 10−7 2 × 10−7 4 × 10−7 9 × 10−7 9 × 10−7 1.4 × 10−6 1.1 × 10−6 1.7 × 10−6 1.1 × 10−6 2.3 × 10−6 1.2 × 10−6 2.1 × 10−6 2.8 × 10−6 2.8 × 10−6

5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−7 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6 5 × 10−6

YES YES YES YES YES NO YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES

Metrologia, 44 (2007) 152–160

159

A-F Obaton et al

Table 5. Comparison between relative and absolute tolerance. λ/nm 245 265

u(e)

ylow −6

6.4 × 10 9.6 × 10−6

dlow −5

5.43 × 10 1.494 × 10−4

4.2 × 10 5.3 × 10−7

for the same order of magnitude, is weaker for the greater value of u(e). At 245 nm, δ/ylow = 9.2 × 10−3 ; at 265 nm this quantity falls to 3.3×10−3 . Meanwhile, the relative differences between the two methods (dlow /ylow ) do not increase between 240 nm and 265 nm; they even tend to decrease. The method is thus not validated for 265 nm because the validation criterion (involving an absolute difference) is undoubtedly too severe for the greater values of uncertainty to a given order of magnitude. In the case where we considered two significant digits for the uncertainty the tolerance becomes ten times weaker and results of the LPU method are not validated for any wavelength. Various explanations can be advanced: mainly the non-linear model behaviour and the fact that an asymmetric PDF is associated with Cdist . However, the LPU method does not reflect the asymmetric feature of the PDF associated with the input quantity. This is why the MCM is more relevant to evaluate the uncertainty for this type of model.

9. Conclusions The calculation of the measurement uncertainty on the spectral irradiance of an UV lamp was performed using two methods: the LPU described in the GUM, considering only a firstorder Taylor series approximation, and the propagation of distributions using a MCM described in Supplement 1 to the GUM. The MCM validates the LPU method at one significant digit although the model giving the spectral irradiance is nonlinear. The reason is that the weight of the non-linearity is not too important. We have shown that the use of the MCM is more convenient, particularly in the case where asymmetric PDFs are associated with input quantities. Moreover, we pointed out an operational limit of the numerical tolerance.

160

dlow /ylow

δ −7

−7

5 × 10 5 × 10−7

δ/ylow −3

7.7 × 10 3.6 × 10−3

9.2 × 10−3 3.3 × 10−3

Acknowledgment The authors are grateful to Gilles Le Dortz from LNE who developed a program to realize the numerical analysis of the data wavelength by wavelength under Crystal Ball.

References [1] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP and OIML 1995 Guide to the Expression of Uncertainty in Measurement 2nd edn (Geneva: International Organization for Standardization) ISBN 92-67-10188-9 [2] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML 2006 Evaluation of measurement data — Supplement 1 to the Guide to the Expression of Uncertainty in Measurement—Propagation of distributions using a Monte Carlo method, draft [3] Bich W, Callegaro L and Pennecchi F 2006 Metrologia 43 S196–9 [4] Lira I 2002 Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance (Bristol: Institute of Physics Publishing) [5] Cox M G and Harris P M 2004 SSfM Best Practice Guide No 6: Uncertainty evaluation, Technical Report National Physical Laboratory, Teddington, UK [6] Cox M G and Siebert B R L 2006 Metrologia 43 S178–88 [7] Herrado M A, Asuero A G and Gonzalez A G 2005 Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte Carlo method: an overview Chemometr. Intell. Lab. Syst. 79 115–22 [8] Gonzalez A G, Herrador M A and Asuero A G 2005 Uncertainty evaluation from Monte Carlo simulations by using Crystal-Ball software Accreditation Qual. Assur. 10 149–54 [9] Crystal Ball 2002 Standard edition, Decisioneering Inc. http://www.crystalball.com

Metrologia, 44 (2007) 152–160