Method for estimating effects of unknown correlations in spectral irradiance data on uncertainties of spectrally integrated colorimetric quantities

Abstract

Spectral irradiance data are often used to calculate colorimetric properties, such as color coordinates and color temperatures of light sources by integration. The spectral data may contain unknown correlations that should be accounted for in the uncertainty estimation. We propose a new method for estimating uncertainties in such cases. The method goes through all possible scenarios of deviations using Monte Carlo analysis. Varying spectral error functions are produced by combining spectral base functions, and the distorted spectra are used to calculate the colorimetric quantities. Standard deviations of the colorimetric quantities at different scenarios give uncertainties assuming no correlations, uncertainties assuming full correlation, and uncertainties for an unfavorable case of unknown correlations, which turn out to be a significant source of uncertainty. With 1% standard uncertainty in spectral irradiance, the expanded uncertainty of the correlated color temperature of a source corresponding to the CIE Standard Illuminant A may reach as high as 37.2 K in unfavorable conditions, when calculations assuming full correlation give zero uncertainty, and calculations assuming no correlations yield the expanded uncertainties of 5.6 K and 12.1 K, with wavelength steps of 1 nm and 5 nm used in spectral integrations, respectively. We also show that there is an absolute limit of 60.2 K in the error of the correlated color temperature for Standard Illuminant A when assuming 1% standard uncertainty in the spectral irradiance. A comparison of our uncorrelated uncertainties with those obtained using analytical methods by other research groups shows good agreement. We reestimated the uncertainties for the colorimetric properties of our 1 kW photometric standard lamps using the new method. The revised uncertainty of color temperature is a factor of 2.5 higher than the uncertainty assuming no correlations.

Monte Carlo method for determining uncertainty of total ozone derived from direct solar irradiance spectra: Application to Izaña results

“…The wavelength steps of the spectral data affect the magnitudes of the uncertainties in TOC created by spectrally random components. The number of data points N which is smaller with a larger wavelength step interval, affects uncertainties with a factor of √ N as 5 demonstrated in (Kärhä et al (2017b)). …”

“…where index i depicts the order of complexity of the deviation (Kärhä et al (2017b)), λ 1 = 300 nm or 305 nm depending on the spectroradiometer and λ 2 = 340 nm limit the wavelength range of the analysis, and the phase φ i of the base function is an equally distributed MC variable between 0 and 2π. In addition, f 0 (λ) = 1 is used to account for constant offset.…”

“…We have developed a Monte Carlo (MC) based model to estimate the uncertainties of the derived TOC values. One frequently overlooked problem with uncertainty evaluation is that the spectral data may hide systematic wavelength dependent errors due to correlations (Kärhä et al (2017b(Kärhä et al ( , 2018; Gardiner et al (1993)). The reason often is that the correlations are not known.…”

Monte Carlo analysis of uncertainty of total atmospheric ozone derived from measured spectra

“…Kärhä et al have recently proposed a method to study measurement errors that unknown correlations may introduce in derived quantities and applied this to study uncertainties of the correlated color temperature [2]. Spectral error functions with varying order of complexity are formed and used to find maximum errors that the uncertainties permit.…”

“…Spectral error functions with varying order of complexity are formed and used to find maximum errors that the uncertainties permit. In this paper, we test the method [2] to study uncertainties of the total atmospheric ozone determined from spectral measurements of direct solar UV irradiance [3].…”