A Monte Carlo Simulation of Measurement Uncertainty in Radiation Thermometry Due to the Influence of Spectral Parameters

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Authors have defined a model of radiation thermometry, which is validated and demonstrated to be used for uncertainty contribution analysis and assessment in any non-contact thermometry measurement application under any conditions.

Abstract

While radiation thermometry is well-developed for laboratory calibrations using high-emissivity sources, the effect of spectral emissivity in real-world conditions, where emissivity ranges from 0 to 1, is usually not considered. Spectral parameters that influence non-contact temperature measurements are often neglected even in laboratory conditions. These parameters become more important with decreasing emissivity and at lower temperatures, leading to increased uncertainty contributions to the measurement result. In this manuscript, we analyze the impact of various influential spectral parameters using the constructed spectral Monte Carlo simulation of radiation thermometry. The investigation covers the influence of spectral and related parameters, namely spectral emissivity, reflection temperature, spectral sensitivity and atmospheric parameters of temperature, relative humidity and distance of the path in the atmosphere. Simulation results are compared to experimental results, overestimating sensitivity to humidity by 23–27% and sensitivity to emissivity and reflected temperature within 10% at given conditions. Multiple cases of radiation thermometer (RT) use are simulated for measurement uncertainty: high temperature RT use as the reference in calibration by comparison, the use of a flat plate calibrator for RT calibration, measurements with a RT using emissivity input data from literature with relatively high uncertainty and temperature measurements with a RT using emissivity data, obtained with FTIR spectroscopy with relatively low uncertainty. Findings suggest that spectral uncertainty contributions are often unjustifiably underestimated and neglected, nearing extended uncertainty contribution of 1.94 °C in calibration practices using flat plate calibrators with emissivity within 0.93 and 0.97 and 1.72 °C when radiation thermometers with spectral ranges, susceptible to atmospheric humidity, are used on black bodies.

1. Introduction

1.1. Research Topic

Currently, uncertainty of radiation thermometry is mostly considered in calibration laboratories, where it is calculated from the individual contributions and presented on a calibration certificate. While the radiation sources in the laboratory exhibit negligible reflective uncertainties, the increasing number of flat-plate calibration devices with decreased emissivity presents a challenge for calibration laboratories, because it requires an evaluation of the uncertainty of emissivity. Meanwhile, based on our experience, the calculation of the measurement uncertainty in real conditions is rare. In our laboratory, which serves as a national laboratory for thermodynamic temperature and humidity, we maintain a reference value and perform calibrations of radiation thermometers for various industries and institutions. While calibrations are performed on laboratory equipment and the calibration certificate only states the best-case calibration uncertainty for laboratory conditions, this uncertainty does not account for actual real-world conditions, where influence parameters are non-ideal and often poorly estimated or known by the end users. As the uncertainty calculation is complex and users are often not familiar with radiation thermometry, the calibration uncertainty from a calibration certificate is often presumed, even in a real-world use, thus often underestimating the uncertainty of measurement by an order of magnitude.

Having identified the problem in the perception of radiation thermometers, the authors of this paper strive to investigate the actual uncertainty contributions in a real-world setting. This research focuses on uncertainty contributions, caused by unavoidable influential spectral parameters and immediately related parameters in the process of radiation thermometry. Specifically, the spectral parameters are emissivity and reflectivity, transmissivity of the atmospheric transfer path and spectral sensitivity of the instrument. These parameters are physically further influenced by reflected temperature (or rather hemispherical combination), as well as the distance in atmospheric transfer path, its relative humidity, temperature and atmospheric pressure.

1.2. Existing Works

The majority of existing publications on the topic of radiation thermometry uncertainty are based on analytical calculations in idealized (spectrally uniform) spectral sensitivity ranges or central wavelengths of the spectral range [1]. Many authors [2,3,4] solve the integration of Planck’s law using the Sakuma–Hattori approximation [5], which is an approximation of Planck’s law or, more precisely, its narrow-band spectral integral, without any influence of the spectral parameters, namely spectral sensitivity, spectral emissivity and spectral reflectivity. Such a type of solution is only appropriate for monochromatic pyrometers and can inherently operate under the principle of the grey body system, where influence parameters are not considered as spectrally variable and are instead considered as constant at a single wavelength.

Meanwhile, the field of radiation thermometry is dominated by pyrometers, which operate in relatively wide spectral ranges (e.g., 8–14 µm) [6,7,8], and by spectral parameters, which are numerically obtained and cannot be simplified to an analytical function for analytical integration. Furthermore, the spectral integral of Planck’s law, which represents radiant emission, cannot be calculated analytically, especially after multiplication with numerically obtained spectral parameters. Numeric integration is therefore the only appropriate method for simulation of broad-spectrum radiation thermometry, which was concluded as well by [9,10].

1.3. Study Design

The intention of this paper is to use a numeric spectral model of radiation thermometry, as closely resembling the physical process of non-contact temperature measurement and the inverse calculation within the radiation thermometer, to simulate the input–output relationship of uncertainty contributions under any conditions. Such a model can provide insight into the actual states and behavior of the system. While this model is sufficient for analytical uncertainty analysis, which calculates partial uncertainty contributions of influence parameters and then combines them to the theoretical total uncertainty [1] under assumed covariances (in practice usually set to 1), we desire to confirm these assumptions through probability distribution simulations, which can account for these covariances within a multiparametric system and provide representative and case-specific results, which can serve as a reference for comparison with other methods.

In order to obtain the most representative results, the Monte Carlo simulation method was chosen. A numeric model of radiation thermometry is used in a Monte Carlo simulation to compare how uncertainty distributions of various parameters, corresponding to real-world conditions, are propagated to the uncertainty distribution of the temperature measurement in radiation thermometry.

Despite possible statistical noise in the result and high computational demand, it assures the correct propagation of the probability distribution throughout the simulated numeric system. The Monte Carlo simulation method is also an entirely numeric method, permitting numeric input parameters (such as emissivity spectrum measurements) and discrete boundary conditions, such as a spectral emissivity value limit of 1.

While the Monte Carlo simulation uses random sampling, a probability mapping method such as Latin hypercube sampling or even orthogonal sampling could be used to further improve the performance of this simulation model. Despite possible benefits, the Monte Carlo method was chosen for generating reference data as the most reliable method, providing reference for possible future models with improved performance.

The model performance is finally compared with experimental results and results from other literature, while case-specific results of four general use cases are presented.

2. Formation of the Model of Radiation Thermometry

Real objects with a thermodynamic temperature emit thermal emission, part of which is transmitted to the measuring instrument and detected by the detector.

A model of the physical process, named the direct model, is constructed to correspond to the physical laws in the process of radiation thermometry (depicted in Figure 1), governing emission, travel and absorption of thermal radiation. As the goal of radiation thermometry is to obtain a true temperature reading, the direct model will be inverted to obtain the inverse model, corresponding to the mathematical inverse of the physical process. The resulting function converts from radiance to true temperature. A model of radiation thermometry is a combination of both, the direct and the inverse model, permitting simulation of non-contact temperature detection.

2.1. The Direct Model of Radiation Thermometry

A direct (physical) model of radiation thermometry, based on a real process and its hypothetical inverse, is depicted in Figure 2. This model would be ideal for the construction of a mathematical inverse; however, the spectral radiance measurement is practically not achievable.

An object with thermodynamic temperature $T$ emits spectral radiation $M_{\mathrm{B}\mathrm{B}}(T,\lambda )$ according to Planck’s law (1), where $h$ is the Planck constant, $k_B$ is the Boltzmann constant, $c$ is the speed of light in the medium and $\lambda$ is the wavelength.

$$M_{\mathrm{B}\mathrm{B}}(T,\lambda )={\displaystyle \frac{2h{c}^2}{{\lambda }^5}}{\left[exp\left({\displaystyle \frac{hc}{\lambda k_{B}T}}\right)-1\right]}^{-1}$$

(1)

Cylindrical operators in Figure 2 correspond to individual emission according to Planck’s law. In the case of environmental radiation, this emission is simplified to a homogeneous emission of a single temperature, assuming a homogenously heated environment.

Square and triangular operators correspond to multiplication with (spectral) parameters, functions or summation, accounting for inevitable influences in the practice of radiation temperature measurement. According to the equilibrium Equation (2), the spectral parameters in rectangular operators are traditionally bound to sum to 1 at all wavelengths.

$$\epsilon \left(\lambda \right)+\tau \left(\lambda \right)+\rho \left(\lambda \right)=1$$

(2)

Influence parameters are spectral emissivity $\epsilon \left(\lambda \right)$, spectral reflectivity $\rho \left(\lambda \right)$ and spectral transmissivity $\tau \left(\lambda \right)$. In case of absorption, Kirchhoff’s law dictates that absorptivity equals emissivity. Instead of absorptivity, this research uses relative spectral sensitivity $S\left(\lambda \right)$.

Upon interaction with the medium, spectral radiation is scaled according to the corresponding spectral influence parameter (3), e.g., the object’s black body emission $M_{obj}$ is scaled by the spectral emissivity, the spectral radiance of the object’s background $M_{\mathrm{b}\mathrm{g}}$ is transmitted according to spectral transmissivity and environmental spectral radiation $M_{\mathrm{e}\mathrm{n}\mathrm{v}}$ is reflected according to spectral reflectivity.

$$M_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(\lambda \right)=\underset{\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}{\underbrace{M_{obj}\left(T_{obj},\lambda \right)\epsilon \left(\lambda \right)}}+\underset{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}{\underbrace{M_{\mathrm{b}\mathrm{g}}\left(\lambda \right) \tau \left(\lambda \right)}}+\underset{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\underbrace{M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right) \rho \left(\lambda \right)}}$$

(3)

Object’s actual emission is the black body radiation, scaled by the spectral emissivity $\epsilon \left(\lambda \right)$ of an object. Meanwhile, the reflection of the environment is environmental radiation, scaled by spectral reflectivity $\rho \left(\lambda \right)$ of an object. Considering the opaque surface of measurement, transmission of background radiation is set as zero and omitted.

Similarly, for translucent materials, the combined radiation further travels through the air and is influenced according to the spectral transmissivity $\tau \left(\lambda \right)$, the spectral radiance $M_{atm}$ and the spectral emissivity ${\epsilon }_{air}\left(\lambda \right)$ of the air in a radiation transfer path. Approaching the sensor, the spectral sensitivity $S\left(\lambda \right)$ of a detector determines the scale of absorption of the constructed spectral radiation signal (4).

$$M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=\int \left(\left(M_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\mathsf{\lambda }\right)+M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\rho \left(\mathsf{\lambda }\right)\right)\tau \left(\mathsf{\lambda }\right)+{M_{\mathrm{a}\mathrm{t}\mathrm{m}}\left(T_{\mathrm{a}\mathrm{t}\mathrm{m}},\lambda \right)\mathsf{\epsilon }}_{\mathrm{a}\mathrm{t}\mathrm{m}}\left(\mathsf{\lambda }\right)\right)S\left(\lambda \right)d\lambda$$

(4)

While the sensor casing’s own radiation emission contributes to sensor measurement, it can be compensated and does not contribute any spectral influence. Therefore, it is omitted from this model under the assumption of correct compensation.

The combined spectral radiation at this point is absorbed into the instrument and in an ideal case, it would be measured as the spectral radiation. The role of the spectral inverse model is to mathematically reverse the behavior of the physical model and to obtain the object temperature.

For wide-band radiation thermometers, this model is practically not achievable due to the absence of a spectral component in the measurement.

2.2. Scalar Model of Radiation Thermometry

Considering this spectral limitation of real sensors, a scalar model, depicted in Figure 3, was derived.

This model operates under grey body principles, meaning it is only correct when spectrally uniform influence parameters are present. As the spectral radiance is spectrally integrated during measurement, only scalar values can be obtained for further use in the model. By operating with scalar values and spectral means of influence parameters, spectral interactions are omitted from the inverse model, resulting in small errors, which are usually neglected for the convenience of this method.

The black body sensor characteristic function of a sensor (5) represents conversion between the temperature of a black body source $T_{BB}$ and the black body to sensor radiance (BBS) $M_{BBS}$, which is practically measured as the black body radiance $M_{\mathrm{B}\mathrm{B}}$, accounting for the spectral sensitivity $S\left(\lambda \right)$ of the sensor.

$$M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T_{BB}\right)=\int M_{\mathrm{B}\mathrm{B}}(T_{BB},\lambda )S\left(\lambda \right)d\lambda$$

(5)

In its inverse form, the characteristic function (Figure 4) converts black-body to sensor radiance measurement $M_{BBS}$ to black body temperature $T$, accounting for the spectral sensitivity of the sensor when measuring emission of a black body according to Planck’s law. This characteristic can be derived experimentally or theoretically from spectral sensitivity $S\left(\lambda \right)$, using Planck’s law.

The operation of the scalar inverse model can be summarized with Equations (5) and (6), where iε, iρ, iε_atm and iτ are input settings of the radiation thermometer, often referred to as instrumental parameters, M_meas is measured radiance on the sensor, $M_{\mathrm{B}\mathrm{B}\mathrm{S}}$ is black body-sensor characteristic radiance and T_BBS is its inverse function, black body sensor radiance to temperature characteristic.

$$T_{meas}=T_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(\left(\left(M_{meas}-{M_{BBS}\left(T_{atm}\right)·i\epsilon }_{atm}\right)·{i\tau }^{-1}-M_{BBS}\left(T_{env}\right)·i\rho \right)·i\epsilon \right)$$

(6)

Although spectrally uniform influence parameters never occur in the real world, such a calculation method has long been used to simplify the calculation of temperature in radiation thermometry. For the purpose of spectral emissivity uncertainty contribution analysis, this model can be sufficient, when emissivity is expressed as grey (with spectrally independent, constant emissivity) with an associated value and uncertainty.

2.3. The Hybrid Model of Radiation Thermometry

Authors have further developed the hybrid model of radiation thermometry, permitting mathematically correct use of spectral input parameters in the inverse model.

The hybrid model permits the use of spectral input parameters, improving traceability of the scalar inverse model. As radiation thermometer sensors operate on the principle of absorption, only the total radiance is measured, whereas the spectral distribution of this radiance is unknown. Despite spectral data loss in the sensor, traceable measurements are enabled in the hybrid model, which accounts for the missing data by means of predicting spectral components. However, the traceability is limited to systems with fully known influential spectral parameters and spectral radiances, including uncertainties.

The hybrid model operates the same way as the scalar model from Figure 3; however, here, the instrumental parameters account for spectral distributions in the inverse model using the effective influence parameters. The effective parameters (e.g., $i\epsilon$) are calculated as scalar equivalents of corresponding spectral parameters (e.g., $\epsilon \left(\lambda \right)$) according to the hybrid model method for effective parameter calculation (7)–(10) for a given temperature [11]. Their values correspond to the spectral parameter’s effective attenuation of the radiance component in magnitude, as exhibited at the sensor.

$${i\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T_{obj}\right)={\displaystyle \frac{\int M_{BB}\left(T_{obj},\lambda \right) \epsilon \left(\lambda \right) S\left(\lambda \right) d\lambda }{\int M_{BB}\left(T_{obj},\lambda \right) S\left(\lambda \right) d\lambda }}$$

(7)

$${i\rho }_{eff}\left(T_{env}\right)={\displaystyle \frac{\int M_{BB}\left(T_{env},\lambda \right) \rho \left(\lambda \right) S\left(\lambda \right) d\lambda }{\int M_{BB}\left(T_{env},\lambda \right) S\left(\lambda \right) d\lambda }}$$

(8)

$${i\tau }_{eff}\left(M_{obj},\tau \left(\lambda \right)\right)={\displaystyle \frac{\int \left(M_{BB}\left(T_{obj},\lambda \right) \epsilon \left(\lambda \right)+M_{BB}\left(T_{env},\lambda \right) \rho \left(\lambda \right)\right) \tau \left(\lambda \right) S\left(\lambda \right) d\lambda }{\int \left(M_{BB}\left(T_{obj},\lambda \right) \epsilon \left(\lambda \right)+M_{BB}\left(T_{env},\lambda \right) \rho \left(\lambda \right)\right) S\left(\lambda \right) d\lambda }}$$

(9)

$${\mathrm{i}\mathsf{\epsilon }}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{a}\mathrm{t}\mathrm{m}}\left(T_{atm}\right)={\displaystyle \frac{\int M_{BB}\left(T_{atm},\lambda \right) {\epsilon }_{atm}\left(\lambda \right) S\left(\lambda \right) d\lambda }{\int M_{BB}\left(T_{atm},\lambda \right) S\left(\lambda \right) d\lambda }}$$

(10)

Considering mathematically correct performance, the hybrid model is used for uncertainty simulations, permitting traceable calculation with spectral input parameters, as well as permitting scalar input values in case of grey body systems.

2.4. Final Constructed Model of Radiation Thermometry

A numeric simulation of the radiation thermometry process was constructed in the Matlab R2022a environment. Within the direct model, spectral radiant emission of a given object temperature $T_{obj}$ is calculated (11), propagated through the direct model $D\left(T\right)$ (Figure 2), accounting for influence parameters (summarized with $x$) and measured by the sensor as radiance $M_{meas}$ (4).

$$M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=D\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},x\right)$$

(11)

Scalar radiance value is then propagated through the hybrid inverse model I(M) (Figure 3) and compensated for various influential parameters (e.g., ε(λ)) to the black body radiance, characteristic to the spectral sensitivity of an instrument. The effective parameters are calculated as scalar equivalents of the spectral influence parameters according to the hybrid model method for effective parameter calculation [11] and correspond to the spectral parameter’s effective attenuation of the spectral radiance, as measured by the sensor.

The resulting radiance is finally converted through the sensor’s radiance to signal characteristic function to a simulated temperature measurement T_meas (12).

$$T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=I\left(M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(\lambda \right),x\right)$$

(12)

The numeric model of radiation thermometry process (13) is constructed based on the hybrid model of radiation thermometry (5)–(10), where ε(λ) is spectral emissivity of the surface, S(λ) spectral sensitivity of the detector, T_refl the temperature of reflected surroundings, T_air temperature of air in the transfer path and RH, d and p the relative humidity, distance and atmospheric pressure of the transfer path in air.

$$T_{meas}=I\left(D\left(T_{obj},\epsilon \left(\lambda \right),S\left(\lambda \right),T_{refl},T_{air},RH,d,p\right),\epsilon \left(\lambda \right),S\left(\lambda \right),T_{refl},T_{air},RH,d,p\right)$$

(13)

3. Uncertainty Simulation

3.1. Analytical Uncertainty Simulation Using the Model of Radiation Thermometry

Building on the model of radiation thermometry (13), uncertainty simulation consists of simulations of the radiation thermometry process with deviating input parameters $x$ in line with parameter uncertainty u_x (14).

$$T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=I\left(D\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},x\pm u_{\mathrm{x}}\right),x\right)$$

(14)

In line with the usual use of radiation thermometers, actual real-world parameters of the direct model are considered uncertain (x ± u_x), whereas instrument input parameters of the inverse model are of a numeric nature and therefore fixed at a certain value (x).

In this sense, when a model of radiation thermometry $RT\left(T,x\right)$ is stated with input uncertainties, uncertainty deviations apply to the direct model, whereas the inverse model operates with fixed parameters without uncertainty deviations, as fixed values are set on the radiation thermometer (15).

$$T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=RT\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},x\pm u_{\mathrm{x}}\right)=I\left(D\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},x\pm u_{\mathrm{x}}\right),x\right)$$

(15)

In the analytical calculation of uncertainty, individual uncertainty contributions are calculated by comparing the calculation of one positive and one negative standard deviation of the input parameter x ± u_x with the simulated object temperature. Considering both positive and negative deviations, the greater deviation is used for uncertainty results in this study (16).

$$u_{meas,x}=T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}-T_{\mathrm{o}\mathrm{b}\mathrm{j}}=RT\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},x\pm u_{\mathrm{x}}\right)-T_{\mathrm{o}\mathrm{b}\mathrm{j}}$$

(16)

3.2. The Monte Carlo Simulation Model of Radiation Thermometry for Uncertainty Simulation

In a Monte Carlo simulation, a high number of temperature measurements is simulated from deviated input parameters and statistically evaluated for the expected uncertainty of radiation thermometry. Built from the model of radiation thermometry (13), the result of uncertainty simulation, regardless of input parameter distribution, is the standard deviation of simulated temperature measurement errors (17), considering the guide to the expression of uncertainty in measurement—propagation of distributions using a Monte Carlo method [12].

$$u_{meas}=std\left(RT\right(T_{obj},x\pm u_x\left)\right)·k$$

(17)

A partial uncertainty contribution of an individual parameter (e.g., u_meas,ε) can be simulated by deviating the individual parameter, e.g., emissivity ε(λ) ± u_ε (18),

$$u_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s},\mathsf{\epsilon }}=k\sqrt{{\displaystyle \frac{{\sum }_i^n{\left(RT\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},S\left(\lambda \right),\epsilon \left(\lambda \right)\pm u_{\epsilon },T_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}},T_{\mathrm{a}\mathrm{i}\mathrm{r}},RH,d,p\right)-T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)}^2}{n-1}}}$$

(18)

Whereas the combined measurement uncertainty simulation simulates the distribution of all parameters simultaneously (19).

$$u_{meas}=k\sqrt{{\displaystyle \frac{{\sum }_i^n{\left(RT\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},S\left(\lambda \right),\epsilon \left(\lambda \right)\pm u_{\epsilon },T_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}\pm u_{T_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}},T_{\mathrm{a}\mathrm{i}\mathrm{r}}\pm u_{T_{\mathrm{a}\mathrm{i}\mathrm{r}}},RH\pm u_{\mathrm{R}\mathrm{H}},d\pm u_{\mathrm{d}},p\pm u_{\mathrm{p}}\right)-T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)}^2}{n-1}}}$$

(19)

Numerous model iterations of Monte Carlo analysis are required to properly account for parameter statistical distributions. In this research, simulations were performed with n = 10,000 iterations at each temperature point, simulated in a 1–18 μm spectral range (further dependent on S(λ)) with 0.001 μm spectral resolution.

The Monte Carlo simulation process is depicted in Figure 5. Through statistical analysis of simulation results, uncertainty is calculated. Two main probability distribution functions are common for input parameters in the field of radiation thermometry, namely the normal (Gaussian, Figure 6A) and the uniform distribution (Figure 6B). The normal distribution is easily summarized with the standard deviation (or one σ). The uniform distribution can be further summarized with the uncertainty of uniform distribution or in the form of a boundary interval, which corresponds to a 100% coverage interval. The uncertainty can be calculated from the boundary interval by dividing the half-width by the square root of 3.

The Monte Carlo method of uncertainty simulation was chosen for its ability to correctly consider the correlated effects of multiple simultaneous input parameter deviations and to provide the output probability distributions, caused by case-specific input parameters with various probability distributions. The Monte Carlo simulation is therefore beneficial over analytical methods in terms of accounting for covariances in multivariate systems and providing a probability distribution of the output; however, it has the drawback of high computational demand and noisy results, as evident later in the validation of emissivity.

4. Simulation Model Input Parameters

4.1. Emissivity and Reflectivity

In practice, the uncertainty of emissivity varies and is very much case-specific. If a traceable emissivity measurement is required, spectral emissivity must be evaluated. Spectral emissivity of an object’s surface is a material property, specific to the surface finish, angle of observation, wavelength and temperature. The majority of emissivity uncertainties in this research assume a uniform distribution.

In the case of the spectral emissivity evaluation using a spectrometer, the uncertainty is reported relative, according to Kaplan et al. [13] in the order of 1% of the measured value, ranging to 10% at low emissivity values.

However, usual practices rely on nominal spectral parameter values with much higher uncertainties, usually sourced from general emissivity charts and/or tables from literature. Such practices simplify objects as (spectrally independent) grey bodies with increased emissivity uncertainty, accounting for the inhomogeneity of spectral emissivity.

The emissivity of human skin is an important topic in medical non-contact temperature measurements. While consensus has never been made, most sources agree on the skin emissivity value 0.98 ± 0.01, as evident from

Table 1. A comparison of the sources of human skin emissivity value and uncertainty.

Author [Source]	Year	Spectral Range	Skin Emissivity
Hardy [14], in [15]	1934	>2 µm	0.989 ± 0.01
Büttner [16]	1937	0.3–50 µm	0.954 ± 0.004
Eckoldt [17]	1960	0.8–12 µm	0.939 ± 0.003
Buchmüller [16]	1961	3–15 µm	within 0.99 and 1
Gärtner [16]	1964	unknown	0.976 ± 0.015
Mitchell et al. [17]	1967	<20 µm	within 0.995 and 1
Patil in Williams [16]	1969	4–6 µm	0.990 ± 0.045
Patil in Williams [16]	1969	6–18 µm	0.972 ± 0.041
Patil in Williams [16]	1969	4–18 µm	0.975 ± 0.043
Steketee [15]	1973	3–14 µm	0.98 ± 0.01
Togawa [16]	1989	8–14 µm	0.969 ± 0.004

. Uniform (boundary) distribution has been assumed for this agreement.

In specific cases, such as for laboratory calibrations, the spectral emissivity is geometrically enhanced by using cavities. In such cases, emissivity is less material dependent and generally more spectrally homogeneous. The scalar emissivity value is usually calculated analytically from cavity dimensions and the general surface emissivity value.

4.2. Atmospheric Conditions

Authors have previously developed a model for the calculation of spectral transmissivity of an atmospheric transfer path [18]. Using the HITRAN database, spectral absorption lines were previously calculated for the air of representative atmospheric composition at reference atmospheric conditions, permitting further conversion to any desired conditions. Four atmospheric parameters were recognized to have an influence on radiation thermometer measurement, namely air temperature, relative humidity, transfer path distance and atmospheric pressure. The effects of atmospheric temperature and relative humidity are simulated in Figure 7. As evident from the figure, the influence of unit temperature on transmissivity is more than twice greater of the influence of unit relative humidity.

The best obtainable measurement capabilities of temperature and relative humidity range below 1 mK and 1% RH; however, this uncertainty does not correspond to the instability and non-homogeneity of climatic parameters within specific environments. Multiple uncertainty cases are thus considered in

Table 2. Best achievable measurement uncertainties for influence parameters in expanded form (k = 2). Unless otherwise noted, normal statistical distribution applies.

Parameter	Best Achievable Uncertainty of Measurement	Assessed Practical Uncertainty	Laboratory Calibration Uncertainty Without Atmospheric Compensation
Air temperature	0.2 K	1–2 K	4 K (uniform)
Relative humidity	1% RH	5–10% RH	20% RH (uniform)
Atmospheric pressure	0.15 hPa	10–20 hPa	20 hPa (uniform)
Transfer path distance	0.05 m	0.05–1 m	0.1 m

in ascending uncertainty ranges.

The reported best achievable measurement uncertainties of influence parameters (listed in column 2) and their statistical distributions (listed in column 3) are believed to be the best-case uncertainties in standard practices of climate parameter measurements. The best achievable measurement uncertainties of these input parameters have been assessed by the World Meteorological Organization’s (WMO) Expert Team on Surface Technology and Measurement Techniques (2004) in Annex 1.A of the Guide to Instruments and Methods of Observation [19].

While these uncertainties are regarded as the best achievable uncertainties of measuring climatic conditions, we expect higher practical uncertainties (listed in column 4) when radiation thermometers (RTs) are being used in the field and when environmental parameters are measured with higher uncertainties and less frequently adjusted in RT compensation software.

As many radiation thermometers do not compensate for the influence of atmospheric conditions, which is also the case for our Heitronics KT19.01 high temperature reference RT, the uncertainty of atmospheric conditions corresponds to change between calibration of the reference thermometer (at another institution) and subsequent use of this thermometer as the reference thermometer in calibrations at home institution. The laboratory calibration uncertainty (listed in column 5) is increased to account for the influence of the change in conditions.

4.3. Spectral Sensitivity

Spectral sensitivity characteristics of radiation thermometers are rarely known. Figure 8 displays characteristics of four RT detectors, obtained from the literature for simulation purposes.

• Heitronics TRT II with two detectors in two temperature ranges [20]:
• 8–14 µm sensor, measuring in a –50–300 °C range.
• 3.87 µm sensor, measuring in a 150–1000 °C range.
• Heitronics KT 19.01 II [20]:
• 2–2.7 µm sensor, measuring in the 350–2000 °C range.
• FLIR VOx-based microbolometer camera [21]:
• 8–14 µm sensor, measuring in a –40–2000 °C range.

4.4. General Simulation Parameters

All default simulation input parameters are listed in

Table 3. Input parameters of radiation thermometry uncertainty simulation and their default values.

Parameter	Value
Emissivity	(0.10–0.95)
Reflected temperature	23 °C
Air temperature	23 °C
Relative humidity	40% RH
Atmospheric pressure	1013.25 hPa
Transfer path distance	1 m

.

Spectral sensitivity function and its inverse characteristic are also applied in every simulation. Spectral resolution was 1 nm and 10,000 simulation iterations were conducted.

5. Validation of the Simulation Model

5.1. The Uncertainty of the Numeric Model of Radiation Thermometry

As long as no input deviations are fed through the model, spectral and instrumental input parameters of a radiation thermometer match and the actual temperature of the simulated object should be measured. The intrinsic numeric error of the model was found to range below 10⁻⁶ °C. This error is negligible and verifies the correct operation of the model.

5.2. Validation of the Humidity Sensitivity of the Model of Radiation Thermometry

An experimental analysis of the influence of humidity on temperature, as measured by the radiation thermometer Heitronics KT19.01 II, operating in the spectral range 2 µm to 2.7 µm, was performed at temperatures between 1000 °C and 1500 °C. Another radiation thermometer, the Sensortherm DS09, operating in the spectral range 0.7 µm to 1.1 µm, was used as a stable reference for temperature measurement. Theoretical spectral simulations of the influence of humidity on radiation thermometers assess high humidity dependency of measurements using Heitronics KT19.01 II and negligible influence for the DS09; therefore, the latter is considered a stable reference in this experiment. The DS09 has been calibrated at an external institution using a primary calibration method according to ITS-90 [22].

A high temperature furnace was measured at 1000 °C, 1250 °C and 1500 °C. Environmental conditions were modified by using air conditioning, a humidifier and a dehumidifier. Extremely dry conditions were obtained by running air conditioning in drying mode and with an additional dehumidifier for 24 h prior to measurements. Extremely humid conditions were created by increasing the temperature inside the laboratory while running an ultrasonic humidifier with a ventilator to aid with the dispersion and mixing of air within the laboratory. The humidifier was switched off 10 min prior to measurements to avoid light scattering on airborne water droplets. The atmospheric humidity within the laboratory was measured by a calibrated data logger.

Humidity was further calculated according to the Hardy ITS-90 [23] method and expressed as the partial pressure of water vapor, an independent parameter which corresponds to particle number density in the transfer path and, in turn, through optical density, to the transmissivity of the transfer path.

The difference between measurements of the two pyrometers was characterized as humidity-induced error and analyzed against the partial pressure of water vapor, revealing well-correlated dependence (Figure 9).

A planar function was fitted to the difference in measurements of the two pyrometers over the range of partial pressure of water vapor (x) and temperature of the black body (y). A polynomial fit function was chosen (20). As linear dependence on partial pressure of water vapor (x) was observed, a first-order polynomial was chosen, whereas the second-order polynomial was chosen for the temperature of the black body (y). Fit results are listed in

Table 4. Coefficients of the polynomial planar function, obtained by regression fitting.

Coefficients	Values
$p_{00}$	−22.93
$p_{10}$	0.2238
$p_{01}$	0.03724
$p_{11}$	−0.0007649
$p_{02}$	−1.334 × 10−5

. The expanded uncertainty of the repeatability of measurements of the fitted function was calculated as 0.57 °C.

$$fit\left(x,y\right)=p_{00}+p_{10}\ast x+p_{01}\ast y+p_{11}\ast x\ast y+p_{02}\ast y^2$$

(20)

Experimental results are compared with simulated results in Figure 10. The average discrepancy between simulated and experimentally measured humidity sensitivity at 1000 °C to 1500 °C ranges from 2.8 °C to 3.3 °C, respectively, whereas the correction of the KT19 was previously measured to range from 2.8 °C to 4.3 °C. After correcting the means of measurements with average discrepancy, corresponding to Heitronics KT 19.01 correction, the standard deviation of the difference between simulated and measured humidity sensitivity ranged from 0.64 °C to 1.27 °C.

The simulated effect of water vapor on measurement in range between 10 hPa and 24 hPa is summarized in

Table 5. Uncertainty and sensitivity coefficient results for the influence of atmospheric humidity (in water vapor partial pressure) on temperature measurement using KT19 in 2–2.7 μm spectral range.

Measured Temperature	Input Deviation	Simulated Deviation	Experimental Deviation	Experimental Sensitivity Mismatch
1000 °C	14 hPa	5.84 °C	7.57 °C	−22.9%
1200 °C	14 hPa	7.16 °C	9.72 °C	−26.3%
1300 °C	14 hPa	7.88 °C	10.79 °C	−26.9%
1500 °C	14 hPa	9.43 °C	12.93 °C	−27.1%

.

Discrepancy in sensitivity to humidity, as also evident from the difference in vertical separation between the graphs in Figure 10, can be attributed to discrepancy between simulation parameters and experimental conditions, namely mismatch in spectral sensitivity data (which was obtained by digitalization of manufacturer-supplied graph, which appears to have been smoothened), mismatch of other spectral model inputs, non-homogeneity of air temperature and humidity in front of the black body as well as the effect of convection of air within the black body cavity—the effect of which was neglected in simulation, but can be significant if the air within the cavity is not heated to the exact same temperature as the black body cavity.

5.3. Validation of the Emissivity Sensitivity of the Model of Radiation Thermometry

Simulation model results have been compared to experimental measurements at 250 °C according to case 2—measurement of the flat-plate calibrator FLUKE 4181 using Heitronics TRT II radiation thermometer (Figure 11).

The uncertainty due to emissivity was evaluated by comparing the reference measurements, obtained with nominal instrumental emissivity setting of 0.95, with subsequent deviated measurements, obtained with instrumental emissivity, which was deviated according to uniform distribution with boundary range of ±0.02 (emissivity uniformly distributed in range [0.93; 0.97]), to correspond to simulation settings. The effect of emissivity uncertainty was measured in a total of 200 cycles of reference and deviation measurements, presented in Figure 12. Upon every change of instrumental emissivity, a pause of 1 s was initiated before obtaining temperature measurements, in accordance with the response time of the radiation thermometer.

The experimentally obtained standard uncertainty contribution due to the uncertainty of emissivity was assessed at 1.78 °C for the low temperature range and 0.87 °C for the high temperature range. The simulation model calculated these standard uncertainty contributions at 1.91 °C and 0.89 °C, which is only slightly higher. Simulation mismatch in comparison with experimental results is calculated in

Table 6. Input–output uncertainty and sensitivity coefficient results of 0.95 ± 0.02 (boundary interval) emissivity influence on uncertainty of temperature measurement.

Measured Temperature	Emissivity Deviation	Simulated Deviation	Experimental Deviation	Simulation Result Mismatch
250 °C, 8–14 μm	0.02/$\sqrt{3}$	1.90 °C	1.78 °C	+6.7%
250 °C, 3.87 μm	0.02/$\sqrt{3}$	0.89 °C	0.87 °C	+2.4%

. The difference can originate from spectral mismatch between simulation data and real experimental parameters (such as spectral sensitivity data, obtained by digitalization of the manufacturer-supplied graph, which appears to have been smoothened) as well as rounding of the instrumental emissivity setting in Heitronics TRT II, with a finite resolution of 0.001.

Table 6 presents a comparison of simulated and measured results of uncertainty contributions due to the influence of emissivity uncertainty.

5.4. Model Comparison to Other Results from Other Literature

While results for uncertainty evaluations are scarce and often insufficient data is available for comparison, a study by Saunders is available, which presents various uncertainty contribution plots [24]. Simulating results for these cases, the Monte Carlo simulation results are compared with reported values in

Table 7. Measurement uncertainty simulation results for examples from literature, compared to reported values for a radiation thermometer, operating in 8–14 μm spectral range.

Measured Temp.	Parameter	Input Deviation	Simulated Deviation	Reported Deviation	Simulation Mismatch
−50 °C	Emissivity	0.95 → 0.99	14 °C	13.3 °C	5.3%
500 °C	Emissivity	0.95 → 0.99	4.8 °C	5.1 °C	−5.9%
−50 °C	Reflection	20 ± 5 °C	0.133 °C	0.12 °C	10.8%
500 °C	Reflection	20 ± 5 °C	0.012 °C	0.012 °C	0%
−50 °C	Reflection	20 ± 5 °C	0.69 °C	0.63 °C	9.5%
500 °C	Reflection	20 ± 5 °C	0.06 °C	0.06 °C	0%

.

6. Results of Uncertainty Simulations for Multiple Cases of Use of Radiation Thermometry

Through years of activity as a calibration laboratory and in terms of academic activities, the measurement uncertainty of radiation thermometry was required (however unknown) in multiple cases of RT use. Four cases of radiation thermometry use were found of interest for future measurement activities and considered in this investigation. An appropriate instrument for each use is selected based on the operating temperature range and standard practice.

6.1. Case 1: Measurement Uncertainty of a Reference Radiation Thermometer with No Atmospheric Compensation in High Temperature Calibration Using a Laboratory Black Body Calibrator

Case 1 is the temperature measurement uncertainty of the laboratory black body calibration device. The usual emissivity of the laboratory cavity calibrators is 0.998 ± 0.002 (bounds). While laboratory calibrations are required for all radiation thermometers, our high temperature reference standard, Heitronics KT19.01 II, is simulated in this case as a reference thermometer, contributing to the uncertainty of the reference temperature data in calibrations of thermometers by comparison. While calibrations by comparison with a radiation thermometer reference are typically performed above 1000 °C, the full range of the thermometer is simulated in this case. The nominal focal distance of this thermometer is 733 mm and the best achievable uncertainty of distance is considered.

As no atmospheric correction is used by Heitronics KT19.01, the uncertainties of atmospheric conditions account for the change in conditions between the calibration of a reference thermometer and the use of this thermometer as a reference in laboratory calibrations. All influence parameters are listed in

Table 8. Model parameter values, standard uncertainties (k = 1) of the direct model parameters and standard uncertainty contributions (k = 1) of the simulated temperature measurement in case 1.

Parameter	Value with Standard Uncertainty	Probability Distribution	Standard Uncertainty Contributions at 1000 °C
Relative humidity	(40 ± 10)% RH	Uniform	1.50 °C
Air temperature	(23 ± 2) °C	Uniform	0.68 °C
Transfer path distance	(0.733 ± 0.05) m	Normal	0.42 °C
Emissivity	0.998 ± 0.00116 (0.002/$\sqrt{3}$)	Uniform	0.29 °C
Atmospheric pressure	(1013.25 ± 10) hPa	Uniform	0.063 °C
Reflected temperature	(23 ± 2) °C	Uniform	2.8 × 10−8 °C

with their partial contributions to the measurement uncertainty at 1000 °C. The Monte Carlo simulated combined standard measurement uncertainty at 1000 °C accounted for 1.72 °C.

Distributions of measurement parameters in partial contribution simulation are displayed in Figure 13. The histogram relating to the influence of relative humidity exhibits the highest influence of non-linearity in the system of radiation thermometry.

A histogram of partial contributions is presented in Figure 14.

Results of the Monte Carlo simulation in the form of standard uncertainty are plotted in Figure 15.

The histogram in Figure 16 displays the spread of combined measurement uncertainty simulation results at 1000 °C.

While the Monte Carlo simulation samples the model’s sensitivity to input uncertainties using a high number of samples, the analytical method conducts the sampling using only two simulations, one for the first positive standard deviation and one for the first negative standard deviation. While both methods can propagate uncertainty contributions through the model of radiation thermometry, the Monte Carlo method accounts for the output probability distribution, whereas the analytical calculation assumes a linear transformation.

Table 9. Standard uncertainty result comparison between the Monte Carlo and analytical simulation methods.

Parameter	Input Deviation	Monte Carlo Result	Analytical Result (Input Deviation ±σ)		Mismatch Between Analytical and Monte Carlo Result
		Standard Deviation	Highest	Mean	Highest	Mean
		(°C)	(°C)	(°C)	(%)	(%)
Relative humidity	10% RH	1.50	1.58	1.48	5.3%	−1.3%
Air temperature	2 °C	0.68	0.69	0.67	1.5%	−1.5%
Transfer path distance	0.05 m	0.42	0.43	0.42	2.4%	0.0%
Emissivity	0.00116 (0.002/$\sqrt{3}$)	0.29	0.29	0.29	0.0%	0.0%
Atmospheric pressure	10 hPa	0.063	0.063	0.062	0.0%	−1.6%
Reflected temperature	2 °C	2.8 × 10−8	3.7 × 10−8	2.5 × 10−8	32.1%	−10.7%
Measurement uncertainty (analytical sum)		1.73	1.80	1.71
Measurement uncertainty (Monte Carlo total)		1.72

presents the analytical simulation results of two methods (calculating the highest and the mean influence of a parameter’s one standard deviation in both directions and using an analytical formula to sum the total measurement uncertainty) and the Monte Carlo simulation results. The Monte Carlo result is used as a reference to compare analytical methods. Mismatch between the columns is a consequence of the non-linearity of the system of measurement.

6.2. Case 2: Measuring with a Low Temperature Reference Radiation Thermometer in Calibration of a Flat Plate Calibrator

Case 2 is the calibration uncertainty of a typical flat plate calibrator. While flat plate calibrators are simple and economically accessible, their design omits a black body cavity. Contrary to the cavity calibrators, flat plate calibrators only exhibit the spectral emissivity of a radiator material, without geometric enhancements. Radiator emissivity is therefore spectrally variable and prone to instability with both time and temperature, exhibiting greater emissivity uncertainty than cavity calibrators. Typical nominal emissivity of a flat plate calibrator is 0.95 ± 0.02 (bounds).

As the flat plate calibrators operate up to 500 °C, they are appropriate for calibrations of low temperature radiation thermometers, typically operating in the 8 μm–14 μm range. At LMK, evaluations of flat plate calibrators below 500 °C are performed by using the Heitronics TRT II reference pyrometer. As no atmospheric correction is used by the TRT II, atmospheric condition uncertainty corresponds to the change in conditions between instrument calibration and use. Similar input uncertainties as in case 1 are therefore considered and applied in this case. Partial uncertainty contributions in

Table 10. Model parameter values, standard uncertainties (k = 1, except for “boundary interval” with 100% coverage factor) of the direct model parameters and standard uncertainty contributions (k = 1) of the simulated temperature measurement in case 2.

Parameter	Value with Standard Uncertainty	Probability Distribution	Standard Uncertainty Contributions at 500 °C
Emissivity	0.95 ± 0.0116 (± 0.02/$\sqrt{3}$)	Uniform	1.94 °C
Reflected temperature	(23 ± 2) °C	Uniform	0.00030 °C
Transfer path distance	(0.38 ± 0.02) m	Normal	0.00023 °C
Relative humidity	(40 ± 10)% RH	Uniform	0.00023 °C
Atmospheric pressure	(1013.25 ± 10) hPa	Uniform	7.9 × 10−5 °C
Air temperature	(23 ± 2) °C	Uniform	7.8 × 10−5 °C

are obtained by simulating deviations of an individual input parameter. The combined standard measurement uncertainty at 500 °C is calculated separately and accounts for 1.94 °C.

Calculating individual uncertainty contribution as the standard deviation of the individually simulated measurement simulations, partial uncertainty contributions (in standard, k = 1 form) are listed and sorted by magnitude in Figure 17.

Combined measurement uncertainty is simulated with all input parameters simultaneously deviating according to the uncertainty distribution. A plot of combined measurement uncertainty simulation results is displayed in Figure 18.

Currently, the only established alternative method for the calculation of the influence of emissivity on measured temperature is an Excel spreadsheet by the Measurement and Standards Laboratory of New Zealand (MSL), accessible as an appendix to the Technical Guide 22 (TG22) [25]. Based on the spectral range of detectors, the uncertainty contribution due to emissivity was calculated and compared to both Monte Carlo (MC) simulation results and analytically (A) calculated results in Figure 19.

The mismatch in results of both Monte Carlo and analytical simulation compared to the Technical Guide 22 is plotted in Figure 20 and is within 0.073 °C in the range between −30 °C and 500 °C. While the Monte Carlo simulation results exhibit a mismatch mostly due to statistical noise, there is also an evident discrepancy between the two sensor types in the same spectral range (TRT LOW and FLIR VOx), behaving differently due to different spectral sensitivities within the same spectral range. The TG22 calculation method does not differ between these two detectors, as it only accounts for the spectral range and not for the spectral sensitivity of the instrument, within that range.

Table 11. Measurement uncertainty result comparison for emissivity uncertainty interval [0.93; 0.97] influence on uncertainty of temperature measurement.

Measured Temperature	Emissivity Deviation	Simulated Deviation	TG22 Deviation	Simulation Mismatch
250 °C, 8–14 μm	0.0116 (0.02/$\sqrt{3}$)	1.90 °C	1.92 °C	−0.84%
250 °C, 3.87 μm	0.0116 (0.02/$\sqrt{3}$)	0.89 °C	0.89 °C	0.073%

presents a comparison of simulated measurement uncertainty contributions due to the influence of emissivity uncertainty with those calculated using TG22 [25].

Uncertainty contribution due to emissivity was further simulated in Figure 21 as dependent on operating spectrum, as measured by a close-to-narrow spectral band radiation thermometer. For radiation thermometers, sensitive in a wider spectral range (wide-band), e.g., 8–14 μm, the highest uncertainty value within the operating spectral interval should be used.

6.3. Case 3: Measurement of Real Objects as Grey Bodies with Emissivity Uncertainty

Case 3 is the measurement of real objects. Any spectral emissivity $\epsilon \left(\lambda \right)$ of real objects can easily be summarized as a scalar value with the corresponding uncertainty interval ε(λ) = ε ± u_ε. A chart of expected measurement uncertainty of various spectral emissivity ranges would provide radiation thermometer users a suitable reference for assessing the uncertainty of their measurements in real-world conditions. The use of a general-purpose instrument is considered in this case, using the sensitivity characteristic of a thermal camera with a vanadium oxide (VOx) microbolometer detector in 8 μm–14 μm range.

In

Table 12. Model parameter values, standard uncertainties (k = 1) of the direct model parameters and standard uncertainty contributions (k = 1) of the simulated temperature measurement of human skin (0.97 ≤ ε(λ) ≤ 0.99) in case 3.

Parameter	Value and Standard Uncertainty	Probability Distribution	Standard Uncertainty Contributions for Skin at 35 °C
Emissivity	(0.10–0.95) ± 0.0058 (± 0.01/$\sqrt{3}$) (0.10–0.95) ± 0.058	Uniform	0.068 °C
Reflected temperature	(23 ± 1 °C)	Normal	0.019 °C
Air temperature	(23 ± 1) °C	Normal	0.0014 °C
Relative humidity	(40 ± 5)% RH	Normal	0.00094 °C
Transfer path distance	(1 ± 0.05) m	Normal	0.00096 °C
Atmospheric pressure	(1013.25 ± 10) hPa	Normal	0.00030 °C

, input parameter values and standard uncertainties are listed. The last column lists standard uncertainty contributions in the case of measuring the human skin temperature. Emissivity of the human skin is simulated as 0.98 ± 0.01 (boundary interval).

Partial uncertainty contributions in a case of measuring skin at 35 °C are compared in Figure 22. The combined standard measurement uncertainty of measuring the human skin accounted for 0.070 °C.

Figure 23 displays calculated measurement uncertainties of measuring grey body emissivities with ±0.10 uniform uncertainty interval boundaries.

6.4. Case 4: Precise and Traceable Measurement of Real Objects with Evaluated Spectral Emissivity

Case 4 is an attempt of traceable measurement of a real object, evaluated for emissivity by Fourier transform infrared spectroscopy (FTIRS). In this case, instead of the grey body emissivity, the actual spectral emissivity of non-oxidized copper, brass and stainless steel samples with a thin coating of chromium nitride (CrN) was measured with FTIRS (Figure 24) and used for uncertainty simulation. Effective temperature-dependent emissivity values, calculated using Equations (7)–(10) of the hybrid inverse model in the temperature range from –50 °C to 500 °C, ranged from 0.338 to 0.351 for copper, from 0.0325 to 0.0304 for brass and from 0.267 to 0.280 for stainless steel.

Table 13. Model parameter values, standard uncertainties (k = 1) of the direct model parameters and standard uncertainty contributions (k = 1) of the simulated temperature measurement in case 4.

Parameter	Value and Standard Uncertainty	Standard Uncertainty Contributions for CrN Coated Metals at 35 °C
		Stainless Steel	Brass
Reflected temperature	(23 ± 1) °C	2.4 °C	35 °C
Air temperature	(23 ± 1) °C	5.7 mK	51 mK
Emissivity	$\epsilon \left(\lambda \right)$ ± 0.005	0.57 mK	5.5 mK
Relative humidity	(40 ± 5)% RH	0.99 mK	0.96 mK
Transfer path distance	(1 ± 0.05) m	0.85 mK	2.1 mK
Atmospheric pressure	(1013.25 ± 10) hPa	0.26 mK	0.73 mK

presents the input uncertainties in this case. Uncertainty due to emissivity was simulated with scalar uncertainty deviations (ε(λ) ± u_ε). Kaplan et al. [13] calculated the best practically achievable relative spectral emissivity evaluation uncertainty in order of 1% of the evaluated emissivity value; however, as our FTIR spectrometer was not recently calibrated and optimized for low uncertainty, we chose to expand this standard relative uncertainty to a more realistic value of 5%. A fixed value for the general emissivity uncertainty of 0.005 and a uniform distribution was assumed in this simulation. As a general measurement case, the use of a thermal camera with a VOx microbolometer and practical uncertainties from Table 2 was assumed.

Figure 25 displays the calculated measurement uncertainties of measuring the three metal samples. As high measurement uncertainty was not expected when nearing the environmental temperature, we further analyzed the results. Emissivity compensation operates by subtracting measured radiance and (assumed) reflected environmental radiance, obtaining the emitted radiance of the surface as the difference. At low emissivity values and for temperatures close to the environmental temperature, the emitted radiance of the surface is calculated as very small in magnitude. However, the uncertainty of this near-zero difference is relatively high, as it is inherited from the reflected temperature. During division with very low emissivity, the uncertainty of the emitted radiance of the surface is greatly amplified. The uncertainty result is therefore exaggerated when measuring surface temperatures close to environmental temperature at very low emissivity values.

7. Conclusions

The constructed simulation model has been compared to experimentally obtained results. The simulated effect of water vapor on a measurement using KT19 underestimates the sensitivity to humidity between 23% and 27% throughout the expected atmospheric humidity range, which is significant. The simulated effect of emissivity uncertainty [0.93; 0.97] closely corresponds to experimental results. The experimentally obtained standard uncertainty contribution due to uncertainty of emissivity was over assessed by 6.7% for the low temperature range and 2.4% for the high temperature range in case 2. The effect of emissivity and reflection temperature was further compared to results from the literature, with the simulation corresponding to reported results within ±11%.

The uncertainty analysis was performed for multiple envisioned cases of radiation thermometry use. A model of radiation thermometry was constructed and analytical and Monte Carlo simulations were performed, the latter by using a high number (10,000) of input parameter deviation calculations.

Uncertainty simulations focused on spectral influence parameters (e.g., spectral sensitivity of the detector) and temperatures, which they are associated with. Five parameters were of key interest in uncertainty simulations: spectral emissivity, radiant temperature of reflection, relative humidity and temperature of the air as well as the atmospheric distance. No other uncertainty contributions have been accounted for in the obtained results, such as calibration uncertainty of the instrument, drift, etc. These uncertainty contributions were omitted in this study and should be added when stating the full measurement uncertainty of the instrument. Specified uncertainty results are expressed as standard uncertainties (k = 1) and can be expanded by the use of multiplication with a coverage factor, e.g., k = 2.

Case 1 considers using Heitronics KT19.01 as a reference thermometer in calibration by comparison. As no atmospheric correction was used, the uncertainty of atmospheric conditions has been increased to correspond to the change in conditions between the calibration of a reference instrument and subsequent calibrations with that reference instrument. As Heitronics KT19.01 is notorious for its humidity dependence due to the spectral range of measurement, the calculated expanded measurement uncertainty of about 8 °C is in line with expectations. Compensation of atmospheric influences has been proposed to improve the measurement uncertainty of this instrument.

Case 2 is a radiation thermometer calibration by using a flat plate calibrator. Despite manufacturer’s common claims of low calibration uncertainties when using flat plate calibration devices, nominal emissivity uncertainties of typical radiators (0.95 ± 0.02) contribute significantly to the uncertainty of calibration, in this case reaching the expanded measurement uncertainty nearing 4 °C. By choosing radiation thermometers which operate in shorter wavelength ranges, it is possible to obtain lower uncertainty, however the influence of spectral emissivity should be considered as well.

Case 3 is a general case of thermal imager use. While atmospheric compensation is present with thermal imagers, emissivity is usually unknown or poorly assessed. Uncertainties, calculated for rather high input uncertainties of emissivity, correspond to the use of radiation thermometers with real objects using general emissivity lookup tables. Obtained uncertainties are high; therefore, such radiation thermometry practice can be expected to yield questionable measurements. Measurement of skin temperature, on the other hand, considering that emissivity data is well known with a low uncertainty, can be conducted very precisely in controlled environments, with expanded measurement uncertainty of 0.14 °C, not accounting for the uncertainty of measurement instrument and uncertainty of subject acclimatization.

In Case 4, the measurement uncertainty was high near environmental temperatures. With low emissivity values, the inverse algorithm calculates the difference between measurement and theoretical reflection, which is very small—close to zero in magnitude—and relatively highly sensitive to temperature uncertainty contributions. This small difference is then divided by emissivity, which is also low, resulting in a high uncertainty contribution. Uncertainty of measurements with low emissivity is therefore very high, especially when the difference in temperature between the measured surface and reflection is very low.

The results of these simulations provide a realistic representation of uncertainties encountered in practical applications outside of the laboratory settings. In real-world scenarios, factors such as varying environmental conditions and material emissivity are less controlled, leading to greater measurement uncertainty. Our extensive experience in the field corroborates these findings, demonstrating that spectral and environmental parameters significantly influence practical measurement accuracy. The discrepancies observed in the simulated cases align with common challenges faced during field applications, underscoring the need for careful consideration of these factors to improve the reliability of non-contact temperature measurements in real-world settings. The four cases that were investigated all share similar experimental layouts; however, the resulting uncertainty contributions are very different in terms of dominant contributions by components and uncertainty budgets, proving that uncertainty budgets in radiation thermometry are multivariate spectral problem, which is very case-specific and nonlinear and should therefore be individually calculated for every application using a spectral uncertainty simulation model.