Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 1, 2024



Article Number  21  
Number of page(s)  4  
DOI  https://doi.org/10.1051/jeos/2024019  
Published online  24 May 2024 
Short Communication
Uncertainty analysis of spectral flux measurement using Monte Carlo simulation
^{1}
TÜBİTAK UME, National Metrology Institute of Türkiye Barış Mah., Dr. Zeki Acar Cad., Gebze, Kocaeli 41470, Türkiye
^{2}
Gebze Technical University, 2254. Sok., No: 2, Gebze, Kocaeli 41400, Türkiye
^{3}
TechnoTeam Bildverarbeitung GmbH, WernervonSiemensStraße 5, DE98693 Ilmenau, Germany
^{4}
PTB, Bundesallee 100, 38116 Braunschweig, Germany
^{*} Corresponding author: senel.yaran@tubitak.gov.tr
Received:
5
September
2023
Accepted:
9
April
2024
In photometry, spectrally integrated quantities are commonly used, and their uncertainties are calculated using classical approaches. Since determining correlations at different wavelengths and effects on spectrally integrated quantities is rather complex, a noise modification of the spectral value using Monte Carlo simulation was applied to estimate unknown correlations in the uncertainty of total luminous flux based on integrated spectral luminous flux values. For this aim, an LED with 6500 K correlated colour temperature was measured in an integrated sphere flux measurement system. The correlations between the measurements at different wavelengths were analysed, and the uncertainty boundaries of the integrated quantity and total luminous flux were obtained.
Key words: Uncertainty / Monte Carlo / Base functions / Integrated quantities / Correlations between quantities at different wavelengths
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://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
Photometric parameters like total luminous flux, colour coordinates, photometer responsivity, spectral mismatch correction factor, etc., are attained by spectrally integrating quantities. The guide to the expression of uncertainty in measurement (GUM) uncertainty framework or Monte Carlo (MC) simulations [1–3] are commonly used to assign uncertainties of the parameters at different wavelengths in spectral measurements. However, fully uncorrelated contributions will average out while integrating the quantities over wavelength, and the fully correlated contributions can be assigned to the integrated value without any change. The question is: what happens with partly correlated quantities?
The correlation between the spectral values of photometric parameters at various wavelengths possesses considerable intricacy. To overcome the challenges involved in determining the correlation between values at different wavelengths, Kärhä, in 2017 [4], introduced an innovative MC based technique.
In this study, the novel MCbased method is utilised for analysing the uncertainty of the total luminous flux, one of the photometric integrated quantities.
2 Theoretical framework
The luminous flux is one of the main quantities of the light sources. This quantity value is determined using several methods and systems based on spatial and/or spectral measurements. The total luminous flux can be determined depending on the measurement system and the integration of spatial and/or spectral measurement values.
As the luminous flux measurement, spectral measurements are generally preferred for more accurate results. To obtain the total luminous flux value, the spectral measurement results are integrated over the visible spectrum (typically from approximately 380 nm to 780 nm).
The relation between the spectral measurements at different wavelength positions is found to be rather complex. The fully uncorrelated contributions will average out while integrating the quantities over wavelength, and the fully correlated contributions can be assigned to the integrated value without any change. But the calculation of the partial correlations is very difficult.
Since there is a great difficulty in calculating the correlation between values at different wavelengths, a noise modification of the spectral value using the novel MC simulation was applied and used for analysing and estimating the possible effects of partly correlated uncertainties in the measurement of spectral flux data to spectrally integrated total luminous flux quantity. A series of orthogonal base functions were used in this MCbased method to simulate potential systematic deviations.
The uncertainty of spectral radiant flux at each wavelength, expressed in equation (1) and the uncertainty of the total luminous flux, expressed in equation (2) were calculated using classical approaches GUM framework without any correlation contribution.
The uncertainty sources of the spectral radiant flux are given in Table 1.$${\varphi}_{\mathrm{eT}}\left(\lambda \right)={\varphi}_{\mathrm{eR}}\left(\lambda \right)\bullet \left(\frac{{y}_{\mathrm{T}}\left(\lambda \right)}{{y}_{\mathrm{R}}\left(\lambda \right)}\right)\bullet \left(\frac{{y}_{\mathrm{AR}}\left(\lambda \right)}{{y}_{\mathrm{AT}}\left(\lambda \right)}\right)$$(1)
Uncertainty budget of the radiant flux of the lamp at several wavelengths.
Φ_{eT}(λ): spectral radiant flux values of the measured lamp,
Φ_{eR}(λ): spectral radiant flux values of the reference lamp,
y_{T}(λ): spectral measured values when the measured lamp is on,
y_{R}(λ): spectral measured values when the reference lamp is on,
y_{AT}(λ): spectral measured values when the measured lamp is pleased in the integrating sphere and only the auxiliary lamp is on,
y_{AR}(λ): spectral measured values when the reference lamp is pleased in the integrating sphere and only the auxiliary lamp is on,$${\varphi}_{\mathrm{T}}={K}_{\mathrm{m}}\bullet {\int}_{380\mathrm{nm}}^{780\mathrm{nm}}V\left(\lambda \right)\bullet {\varphi}_{\mathrm{eT}}\left(\lambda \right)\bullet \mathrm{d}\lambda $$(2)
Φ_{T}: total luminous flux value of measured lamp,
Φ_{eT}(λ): spectral radiant flux values of the measured lamp,
V(λ): luminous efficiency function,
K_{m}: maximum spectral luminous efficacy for photopic vision.
Later, the novel MCbased method was applied to the spectral values. The novel MCbased method has been employed to account for the correlations at different wavelengths for the total luminous flux uncertainty analysis.
For this aim, the spectral radiant flux values, Φ_{eT}(λ), undergo a modification process in equation (3) by adding a component based on an uncertaintyweighted random error function, given in equation (4). The error function includes the number of base functions and the weight as given in equation (5) and equation (6), respectively.$${\varphi}_{\mathrm{eTn}}\left(\lambda \right)={\varphi}_{\mathrm{eT}}\left(\lambda \right)\bullet \left(1+\delta \left(\lambda \right)\bullet {u}_{\mathrm{c}}\left(\lambda \right)\right)$$(3) $$\delta \left(\lambda \right)=\sum _{i=0}^{N}{\gamma}_{i}\bullet {f}_{i}\left(\lambda \right)$$(4) $${f}_{i}\left(\lambda \right)=\sqrt{2}\bullet \mathrm{sin}\left[i\bullet \left(2\pi \bullet \frac{\lambda {\lambda}_{1}}{{\lambda}_{2}{\lambda}_{1}}\right)+{\varphi}_{i}\right]$$(5) $$\left\{{\gamma}_{0},{\gamma}_{1},\dots ,{\gamma}_{N}\right\}=\left\{\frac{{Y}_{0}}{\sqrt{{Y}_{0}^{2}+{Y}_{1}^{2}+\dots +{Y}_{N}^{2}}},\frac{{Y}_{1}}{\sqrt{{Y}_{0}^{2}+{Y}_{1}^{2}+\dots +{Y}_{N}^{2}}},\dots ,\frac{{Y}_{N}}{\sqrt{{Y}_{0}^{2}+{Y}_{1}^{2}+\dots +{Y}_{N}^{2}}}\right\}$$(6)
Φ_{eTn}(λ): spectral radiant flux with the random error function,
u_{c}(λ): combined uncertainty of spectral radiant flux,
δ(λ): random error function,
γ_{ i }: weight function,
f_{ i }(λ): base function,
Φ_{ i }: phase,
λ_{1} and λ_{2}: start and end wavelength of the range,
Y_{ i }: random variables,
N: number of weight and base functions
The Y_{ i } variables, which relate to weighting the base functions, and the Φ_{ i } phase terms, which correspond to base functions, are randomly generated with 20,000 iterations. Following each random generation of these variables, the resultant modified radiant flux values are computed for every wavelength in equation (3). The f_{0}(λ) function is assumed to be equal to one as a special case to have the full correlation. Several random error functions given in equation (4) are illustrated in Figure 1 to show their pattern.
Fig. 1 Error functions for some N values. 
3 Modification and results
An LED light source was measured using the integrating sphere flux measurement system with a spectrometer. Using the substitution method, the spectral radiant flux values of the LED light source were calculated in equation (1) for each wavelenth. Then integrating the spectral radiant flux values in equation (2), the total luminous flux was obtained. The uncertainty of total luminous flux, expressed in equation (2), was calculated using classical approaches GUM framework as 0.69%.
For the novel MCbased method for uncertainty analysis, a series of codes were developed in Python and are mainly based on the modules shared in empir19nrm02 GitHub repository [5, 6].
To determine the unpredictable uncertainty boundaries of the total luminous flux in equation (2), the spectral radiant flux values were disturbed using the MCbased method [4]. Total luminous flux was obtained by integrating the disturbed spectral radiant flux in equation (2).
The modification process was repeated for each of the N base function numbers. After each modification, the correlations between the values at different wavelengths were calculated, and the covariance and correlation matrices for N = 0, N = 1, N = 2, N = 5, and N = 200 are given in Figure 2.
Fig. 2 Correlation (on the left) and covariance (on the right) matrices for N given in ascending order. 
The case of full correlation, in a special case where N equals zero and the maximum effect of correlation where N equals one are seen in Figure 2.
The other matrices are displayed in Figure 2 for three different values of N, exemplary N = 2, N = 5, and N = 200, to facilitate a comparative analysis. The correlation matrices with N = 0 and N = 1 can be compared to the correlation matrices for these N values. Figure 2 illustrates a diminishing trend in correlations as the value of N increases.
The uncertainties associated with the total luminous flux were also depicted in Figure 3 versus N number after the modification.
Fig. 3 Uncertainty (k = 2) of the total luminous flux. 
The boundaries of the possible uncertainties, which cannot be unpredicted and determined, are discernible in Figure 3. The maximum uncertainty is observed when N attains unity. The convergence of total luminous flux uncertainty towards the calculated uncertainty value from the GUM framework is observed as N increases.
4 Conclusions
The total luminous flux of an LED light source with 6500 K CCT, a photometric integrating parameter, was attained by integrating the spectral radiant flux. The uncertainty calculation of the total flux using GUM framework [1] resulted in a value of 0.69%, which reflects the combined effects of various sources of uncertainty. However, the GUM analysis did not take into consideration any correlations that may exist between the different sources of uncertainty.
Since there is complexity in calculating the correlation between values at different wavelengths, a novel MCbased method was utilised for analysing and estimating the uncertainties of the spectrally integrated total luminous flux in this study. And the possible uncertainty distributions were attained. The maximum uncertainty is obtained 8.76% while GUM is 0.69%.
Funding
This project 19NRM02 RevStdLED has received funding from the EMPIR programme cofinanced by the Participating States and from the European Union’s Horizon 2020 research and innovation programme.
Conflicts of interest
The authors declare that they have no competing interests to report.
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors on request.
Author contribution statement
Şenel Yaran; writing – original draft preparation/writing – review and editing/investigation/methodology. Zühal Alpaslan Kösemen; writing – review and editing/software/investigation. Çağrı Kaan Akkan; investigation. Hilal Fatmagül Nişancı; software. Udo Krüger; supervision/software/conceptualization/methodology. Armin Sperlin; supervision/methodology.
References
 JCGM (2008) 100. Guide to the expression of uncertainty in measurement. [Google Scholar]
 JCGM (2008) 101. Supplement 1 to the “Guide to the expression of uncertainty in measurement” — Propagation of distributions using a Monte Carlo method. [Google Scholar]
 JCGM (2011) 102. Supplement 2 to the “Guide to the expression of uncertainty in measurement” – Extension to any number of output quantities. [Google Scholar]
 Kärhä P., Vaskuri A., Mäntynen H., Mikkonen N., Ikonen E. (2017) Method for estimating effects of unknown correlations in spectral irradiance data on uncertainties of spectrally integrated colorimetric quantities, Metrologia 54, 524–534. [CrossRef] [Google Scholar]
 19nrm02 E. (2023) Python package empir19nrm02. Available at https://github.com/empir19nrm02/empir19nrm02. [Google Scholar]
 Smet KAG (2020) Tutorial: the Luxpy Python toolbox for lighting and color science, LEUKOS 16, 3, 179–201. https://doi.org/10.1080/15502724.2018.1518717. [CrossRef] [Google Scholar]
All Tables
All Figures
Fig. 1 Error functions for some N values. 

In the text 
Fig. 2 Correlation (on the left) and covariance (on the right) matrices for N given in ascending order. 

In the text 
Fig. 3 Uncertainty (k = 2) of the total luminous flux. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.