A Traceable Spectral Radiation Model of Radiation Thermometry

Featured Application

Radiation thermometry of real objects under real conditions.

Abstract

Despite great technical capabilities, the theory of non-contact temperature measurement is usually not fully applicable to the use of measuring instruments in practice. While black body calibrations and black body radiation thermometry (BBRT) are in practice well established and easy to accomplish, this calibration protocol is never fully applicable to measurements of real objects under real conditions. Currently, the best approximation to real-world radiation thermometry is grey body radiation thermometry (GBRT), which is supported by most measuring instruments to date. Nevertheless, the metrological requirements necessitate traceability; therefore, real body radiation thermometry (RBRT) method is required for temperature measurements of real bodies. This article documents the current state of temperature calculation algorithms for radiation thermometers and the creation of a traceable model for radiation thermometry of real bodies that uses an inverse model of the system of measurement to compensate for the loss of data caused by spectral integration, which occurs when thermal radiation is absorbed on the active surface of the sensor. To solve this problem, a hybrid model is proposed in which the spectral input parameters are converted to scalar inputs of a traditional scalar inverse model for GBRT. The method for calculating effective parameters, which corresponds to a system of measurement, is proposed and verified with the theoretical simulation model of non-contact thermometry. The sum of effective instrumental parameters is presented for different temperatures to show that the rule of GBRT, according to which the sum of instrumental emissivity and instrumental reflectivity is equal to 1, does not apply to RBRT. Using the derived models of radiation thermometry, the uncertainty of radiation thermometry due to the uncertainty of spectral emissivity was analysed by simulated worst-case measurements through temperature ranges of various radiation thermometers. This newly developed model for RBRT with known uncertainty of measurement enables traceable measurements using radiation thermometry under any conditions.

1. Introduction and Literature Review

Since the SARS outbreak in China in 2002, influenza A (H1N1) in 2009, Ebola virus disease (EVD) in 2014 and coronavirus disease in 2019 (COVID-19), when non-contact thermometry was mainly recognized as a practically convenient method for measurement (screening) of human body temperature [1,2,3,4,5], radiation thermometry has continuously expanded to new industries and applications, including the consumer market. Thanks to the progress in the production of thermal radiation sensors in recent decades, the instrumentation in the field of radiation thermometry is now technologically advanced. However, there are also studies available which demonstrate the limitations of the non-contact thermometry [6,7,8,9,10,11].

Despite practical advances, the theory of radiation thermometry with spectral-band sensors has not advanced in recent years. While non-contact temperature measurements are in practice often conducted on simplified grey bodies, no methodology has yet been developed for measuring the temperature of real bodies. The authors of this article strive to introduce traceability into the practice of radiation thermometry of real bodies. To achieve traceable measurements under real conditions, a mathematical model of measurement must be derived and measurement uncertainty evaluated.

1.1. Black Body Radiation Thermometry (BBRT)

A black body (BB) is an ideal radiation source that has a maximum (or ideal, perfect) emissivity value (of 1) at all wavelengths ($\epsilon \left(\lambda \right)=1$) and in all directions, and does not reflect incident radiation. Therefore, the result of a non-contact temperature measurement on a black body always corresponds only to the temperature of the object and is independent of other environmental influences (neglecting the influence of the atmosphere).

While traceability is easily obtained with such measurements, the black body radiation thermometry (BBRT) method is only applicable to measurements on black bodies, which is an idealization of real properties and only suitable (with negligible effects) when laboratory black bodies are used. In practice, laboratory black bodies are the best approximation of the black body with the highest achievable emissivity, which ideally approaches the emissivity of the theoretical black body (${\epsilon }_{LBB}\left(\lambda \right)\approx {\epsilon }_{BB}\left(\lambda \right)=1$). Despite the ease of measurement, black bodies are difficult to achieve in practice and are usually not feasible.

Established protocols for calibration of non-contact temperature measuring instruments, and consequently most scientific publications in this field, mostly refer to calibrations of radiation thermometers using black bodies. Such calibration protocols are described in [12,13,14,15,16,17,18,19]. BBRT is considered traceable and is therefore used by calibration laboratories to transfer radiation references. This means that black body calibrations are used to transfer reference measurements from the highest levels of traceability to lower levels.

The BBRT model is used by the simplest measuring instruments that operate with a fixed instrumental emissivity value of 1 (e.g., transfer reference thermometers and many industrial sensors), and is also available in the majority of radiation thermometers, which have adjustable instrumental parameters that can be set to a value of 1.

1.2. Grey Body Radiation Thermometry (GBRT)

Grey body radiation thermometry (GBRT) is a simplification used for measuring grey bodies. It assumes that all influential quantities in the system of measurement are spectrally homogeneous and therefore independent of the observed wavelength. Based on this simplification, the spectral behaviour can be summarized with scalar values (constants), and their effect can be considered spectrally independent.

While GBRT is theoretically similar to BBRT in terms of scalar calculation algorithm, it extends the application of radiation thermometry to idealized real bodies—grey bodies.

GBRT requires scalar input parameters, generally referred to as instrumental parameters, which enable compensation of inevitable environmental influences. Input parameters should correspond to the entire system of measurement, which consists of the measured object (and its temperature), the environmental radiation, the transfer path and the measuring instrument. In GBRT, only the influence of actual grey bodies, which are spectrally independent, can be correctly compensated.

The GBRT model and its variations are currently used by most radiation thermometers on the market with an adjustable emissivity parameter. This approach has been documented in numerous non-scientific (e.g., Fluke 4181 calibrator manual [20]) and scientific [21] sources and thoroughly by Saunders [22], who modelled the response of a fixed instrumental emissivity radiation thermometer in calibration with a black body calibrator. Although the article focuses on calibration, the mathematical model is also applicable to measurements. The grey body model by Saunders utilized radiances (total radiant powers throughout the spectrum, denoted $M$), obtained by the Sakuma–Hattori approximation of Planck’s law. Spectral interactions are not considered later in this model, limiting model performance to spectrally homogenous influence properties of GBRT.

1.3. Real Body Radiation Thermometry (RBRT)

Some authors previously attempted to incorporate the emissivity characterisation within correction of thermal calibration by expressing common parameters [23]; however, such solutions are not applicable upon a change of environmental conditions and require re-calibration.

Currently, best practice in radiation thermometry of real bodies (RBRT) is to adjust the input parameters of the GBRT model while matching the measurement and reference temperature of the surface, obtained with a contact thermometer.

Repeated evaluations of required instrumental parameters are required even in radiation thermometry of grey bodies because of the thermal dependence of spectral influence parameters. While thermal dependence of influence parameters is documented for many materials and often expected in practice, spectral interactions of spectral influence parameters in the system of measurement are always neglected in the GBRT model; therefore, GBRT is not suitable for RBRT, where spectral parameters are non-homogenously distributed.

In order to correctly perform RBRT, instrumental parameters of GBRT should take into account the spectral distribution of influence parameters and the spectral radiances that are emitted and absorbed at each stage of the physical process of RBRT.

The objective of this study was to define a mathematical model for the process of radiation thermometry (including BBRT, GBRT and RBRT) and to define inverse models that serve to compensate for the inevitable influences within the physical model, namely emission, reflection and atmospheric transmission.

2. The Direct (Physical) Model of Radiation Thermometry

2.1. The Full System of Measurement

In radiation thermometry, thermal radiation from objects is transferred to the radiation thermometer according to the physical laws of radiation. In measurement systems, influences of the environment are usually inevitable; therefore, these influences are taken into account and compensated for in the measuring instrument. These influences include reflection, transmission through the air (and possibly optics and filter) and absorption to the sensor surface (Figure 1).

Radiation thermometer sensors measure the object’s radiant heat flux to its surface. Conventional thermometers use thermal sensors that absorb radiation on an exposed, absorbing surface, and convert it to thermal radiation. This is the case with the most commonly used types of thermopile: MEMS and pyroelectric sensors.

The temperature difference between the exposed and the unexposed side of the absorbing surface is measured and calculated to the incident heat flux, using characteristic functions of the sensor that are beyond the scope of this research and are considered properly calibrated and traceable. The self-emission of the sensor is considered mathematically correctly compensated within the scope of this research, as it is purely dependent on the temperature of the sensor.

The output data of the sensor and the complementary input–output characteristic of the sensor constitute the incident radiance, which is assumed as compensated for natural heat dissipation, both through the case and through the optics. These influences are considered mathematically easy to compensate for, and are beyond the scope of this research.

The incident radiance data of the sensor are later converted by another mathematical model (within the instrument) into the temperature of the radiation source, taking into account the inevitable influences of the environment and reflective properties of the object.

The operation and behaviour of the latter model, which is the inverse of the physical model, is the subject of this article. It represents the opposite of the physical model of radiation transfer, described in Figure 1.

2.2. Thermal Radiation Emission, Transfer and Absorption

Every black body with a thermodynamic temperature $T$ emits thermal radiation in the form of spectral radiance $M_{\mathrm{B}\mathrm{B}}(T,\lambda )$ (Figure 2) in accordance with Planck’s law (1),

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

(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.

The actual emission of any real surface is the emission of the black body (2), spectrally reduced by the spectral emissivity of the surface.

$$M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{obj},\lambda \right)=M_{\mathrm{B}\mathrm{B}}(T_{obj},\lambda )$$

(2)

A general form for radiation of any real object $M_{obj}\left(\lambda \right)$ is described by Equation (3), where $M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)$ represents object’s raw black body emission (2) under an assumption of homogenous temperature; $M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)$ represents incident environmental radiation which is reflected from the surface of a medium; $M_{\mathrm{b}\mathrm{g}}\left(\lambda \right)$ represents spectral radiance behind the object in the case of transmission through a translucent material; and $\epsilon \left(\lambda \right)$, $\tau \left(\lambda \right)$ and $\rho \left(\lambda \right)$ represent spectral transmissivity, spectral emissivity and spectral reflectivity of the object, respectively.

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

(3)

Interaction of incident radiation with a medium at any wavelength is described by the equilibrium Equation (4).

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

(4)

According to Kirchhoff’s law [24], emissivity $\epsilon \left(\lambda \right)$ is equal to absorptivity $\alpha \left(\lambda \right)$ at individual wavelengths (5).

$$\epsilon \left(\lambda \right)=\alpha \left(\lambda \right)$$

(5)

2.3. Input Data of the Physical Model

2.3.1. Object’s Emission and Environmental Radiation Reflection

Equation (6) represents the emission of the surface in the direct model as the spectral radiance of a black body $M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(\lambda ,T_{obj}\right)$ from Equation (2), reduced by the spectral emissivity factor $\epsilon \left(\lambda \right)$.

$$M_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}\left(\lambda \right)=M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{obj},\lambda \right)\epsilon \left(\lambda \right)$$

(6)

According to Equation (4), the only complementary process of emission for real opaque surfaces is reflection. The radiation (7) reflected from the object is considered as being the raw environmental spectral radiance $M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)$, reduced by the spectral reflectivity factor $\rho \left(\mathsf{\lambda }\right)$ (8).

$$M_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\left(\lambda \right)=M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\rho \left(\lambda \right)$$

(7)

$$\rho \left(\lambda \right)=1-\epsilon \left(\lambda \right)$$

(8)

The directional dependence of any influence parameter is omitted from this (simplified) model and is considered accounted for in the raw environmental radiation $M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)$ data. This simplification does not affect the performance of the model, as long as the environmental radiation data are obtained in agreement with the spatial reflectivity characteristic of the surface in the given direction of observation.

The directional spectral emissivity of any material can be measured with IR spectroscopy. Databases exist that cover the directional spectral emissivities of various materials, such as ECOSTRESS [25] (formerly ASTER [26]) and USGS Spectral Library Version 7 [27]. The normal (perpendicular) spectral emissivity of Topaz from the USGS Spectral Library is used as an example in this research (Figure 3).

2.3.2. Atmospheric Transmission and Self-Emission

The emitted and reflected radiation of the object $M_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(\lambda \right)$ travels through the atmosphere toward the radiation thermometer. During travel, the radiation is partially absorbed according to the spectral absorptivity of the air, while most of the radiation is transmitted according to the spectral transmissivity of the air $\tau \left(\lambda \right)$. According to Kirchhoff’s law (5), spectral absorptivity equals spectral emissivity, therefore atmospheric self-emission is present and can be considered as black body emission $M_{\mathrm{B}\mathrm{B},\mathrm{a}\mathrm{t}\mathrm{m}}\left(T,\lambda \right)$ (10) reduced by the spectral emissivity of the atmosphere ${\epsilon }_{\mathrm{a}\mathrm{t}\mathrm{m}}\left(\lambda \right)$ (11).

$$M_{\mathrm{s}\mathrm{e}\mathrm{n}}\left(\lambda \right)=M_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(\lambda \right)\tau \left(\lambda \right)+M_{\mathrm{B}\mathrm{B},\mathrm{a}\mathrm{t}\mathrm{m}}\left(\lambda \right){\epsilon }_{\mathrm{a}\mathrm{t}\mathrm{m}}\left(\lambda \right)$$

(9)

$$M_{\mathrm{B}\mathrm{B},\mathrm{a}\mathrm{t}\mathrm{m}}\left(T,\lambda \right)=M_{\mathrm{B}\mathrm{B}}(T,\lambda )$$

(10)

$${\epsilon }_{\mathrm{a}\mathrm{t}\mathrm{m}}\left(\lambda \right)=1-\tau \left(\lambda \right)$$

(11)

The spectral transmissivity of the atmosphere is relatively high under usual indoor conditions, as already determined by the authors for the purpose of this study [28]. The spectral transmissivity of air is shown in Figure 4 for a 10 m atmospheric path; however, a 1 m measuring distance—along with 45% relative humidity, 1013.25 hPa atmospheric pressure and 25 °C atmospheric temperature—is used for calculations in this article.

2.3.3. Optical System of the Measuring Instrument

The transmission path may also include possible transparent windows and the optical system of the thermometer. Publicly available spectral data of the optics of pyrometers and thermal imagers are limited; however, the properties of the optics are usually included in the spectral sensitivity characteristic of the measuring instrument.

Remaining radiances, detected by the sensor and originating within the measuring instrument, including the optical system, are considered device-compensated and outside the scope of this research.

2.3.4. Sensor Absorption and Characteristic Function

The combined spectral radiance travelling towards the measuring instrument is absorbed at the sensor. The actual detected (measured) spectral radiance $M_{meas}$ to the sensor shown in Figure 1 prior to integration is the incident spectral radiance to the sensor $M_{\mathrm{s}\mathrm{e}\mathrm{n}}\left(\lambda \right)$, reduced by the sensor’s relative spectral sensitivity $S\left(\lambda \right)$. Relative spectral sensitivity is used exclusively in this research. Absolute scaling is considered as accounted for in sensor calibration and does not influence the operation of the models.

$$M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=\int M_{\mathrm{s}\mathrm{e}\mathrm{n}}\left(\lambda \right)S\left(\lambda \right)d\lambda$$

(12)

Figure 5 presents the relative spectral sensitivity $S\left(\lambda \right)$ of the Heitronics KT19.01 and two temperature ranges of the Heitronics Transfer Radiation Thermometer II (TRT II), digitized from Heitronics pyrometer documentation [29], and the generic spectral sensitivity of vanadium oxide (VOx)-based microbolometer detectors of FLIR thermal imagers with 45° field-of-view optics [30].

Analytical solutions for the integration of Planck’s law are complex, impractical, and moreover infeasible when the discretely measured spectral sensitivity of the sensor and other influence parameters are taken into account. For this reason, we have opted for entirely numerical calculations.

A sensor’s characteristic function converts from the input quantity to the output quantity. The characteristic function is usually obtained by calibration. Sensors for thermal radiation convert the radiance to an electrical response, generally obtained numerically as a sensor reading.

However, in radiation thermometry, radiance is measured by integration of spectral radiance, according to Equation (13). Calibrations are usually limited to black bodies and therefore the input quantity of a sensor characteristic function is usually extended from radiant power to the temperature of a black body radiator. This assumption utilizes Planck’s law (1) for the missing relationship between the spectral radiance and the temperature of a black body.

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

(13)

For the purpose of this research, the output quantity of the sensor is considered correctly calibrated as the radiance of a black body $M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T\right)$, detected at the sensor in W sr⁻¹ m⁻². The sensor characteristic function is presented in Figure 6.

2.4. Constructed Direct Model of Radiation Thermometry

In radiation thermometry, numerous parameters interfere with the signal of interest, namely the radiation emitted by the surface of the measured object. Considering the described properties of the physical laws, a direct model of radiation thermometry can be generated according to Figure 1. Equation (14) represents the direct model of radiation thermometry.

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

(14)

The process flow of the direct model of radiation thermometry is also presented in Figure 7. The spectral information is lost during the spectral integration during absorption on the active surface of the sensor.

Individual contributions can be expressed (15).

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

(15)

2.5. Final Spectral Data of the Physical Model

A theoretical example of spectral behaviour within the direct model (14) is presented in Figure 8. A spectral emissivity characteristic was randomly generated for this research, which is intentionally non-homogeneous across the entire sensitivity spectrum of the sensor to emphasize the effect of non-homogeneous spectral emissivity, as can occur with real objects.

Figure 8a presents the raw spectral radiances $M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{obj},\lambda \right)$, $M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)$ and $M_{\mathrm{B}\mathrm{B},\mathrm{a}\mathrm{t}\mathrm{m}}\left(T_{atm},\lambda \right)$ of the three contributing sources of radiation. Figure 8b presents the influence parameters $\mathsf{\epsilon }\left(\mathsf{\lambda }\right)$, $\tau \left(\mathsf{\lambda }\right)$ and $S\left(\lambda \right)$. Figure 8c presents the detected spectral radiance of the three contributing sources, as expressed in Equation (15) prior to integration, and the sum of the three, as expressed in Equation (14) prior to integration.

3. The Inverse Model of Radiation Thermometry

3.1. Derivation of the Inverse Model of Radiation Thermometry

The direct model is based on a real physical model of conversion of object temperature to radiant power as a sensor value, including the inevitable influences of the environment. The inverse model, i.e., the inverse of the physical model, is used for the calculation from the radiant power on the sensor to the temperature of the object.

In all inverse models, inevitable influences of the measured object, environment and air, must be calculated and compensated for from the measured radiance. After compensating for these influences, only the black body radiation of the measured object should be present in the resulting signal as raw surface emission $M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T,\lambda \right)$, used in Equation (6) in the direct model (of the real system of measurement). This pure black body emission from Equation (1) can be converted to the object temperature using a black body radiance characteristic function, which is theoretically the inverse of the spectrally integrated Planck’s law.

3.1.1. Theoretically Ideal Spectral Direct and Inverse Models

Since the physical model is a spectral model, the inversion of this model results in a spectral inverse model. The correct combination of the direct and inverse models is shown in Figure 9. Spectral radiance is extinct at the sensor; therefore, the ideal traceable measurements should be made in spectral radiance, not radiance. The ideal direct and inverse models should omit spectral integration; therefore, such a model is not practically feasible with radiance sensors.

3.1.2. Scalar Inverse Model

Due to spectral data loss at the sensor, most existing radiation thermometers and thermal imagers currently operate with radiances instead of spectral radiances and use the simplified scalar inverse model (Figure 10) and its derivatives. A scalar model is a generalization, used because the sensor reading is obtained as radiance (radiant power), which is a scalar number, rather than spectral radiance, which is defined as a spectrally distributed radiant power.

The scalar inverse model can be obtained from the scalar direct model (16); that is, the direct model (14) with scalar influence parameters ($\mathsf{\epsilon }\left(\mathsf{\lambda }\right)=\mathsf{\epsilon }$, $\tau \left(\mathsf{\lambda }\right)=\tau$, etc.). This scalar replacement can only be correctly implemented in the direct model under simplification of constant (spectrally independent) influence parameters. The spectrally homogenous (grey) properties of the model, which are required for correct operation, limit the applicability of this model to GBRT.

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

(16)

Assuming an opaque surface, the raw emission of the object $M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)$ can be considered as the emission of a black body of temperature $T_{\mathrm{o}\mathrm{b}\mathrm{j}}$ (2). Simplifying for single (homogenous) atmospheric temperature $T_{\mathrm{a}\mathrm{t}\mathrm{m}}$, the raw radiant contribution of the atmosphere can similarly be considered as the emission of a black body (10).

Raw environmental reflection can be simplified as the single temperature emission of a black body $M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(T_{env},\lambda \right)$ as well; however, under real conditions, the incident environmental spectral radiance $M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)$ is a sum of many incident radiances around the measured surface, determined according to the directional reflectivity characteristic of the observed object.

The black body sensor temperature to radiance characteristic of a sensor $M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T\right)$, obtained in Equation (13), can be expressed from the scalar direct model. Similarly, sensor sensitivity-weighted environmental radiance $M_{\mathrm{S},\mathrm{e}\mathrm{n}\mathrm{v}}$ can be expressed.

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

(17)

By expressing $M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)$, the scalar inverse model of GBRT can be obtained (18). Parameters marked with the prefix $i$- (e.g., $i\epsilon$) are scalar input parameters of the inverse model, called instrumental parameters. These parameters—instrumental emissivity, instrumental reflectivity, instrumental (atmospheric) transmissivity and instrumental atmospheric emissivity—represent the scalar input settings of an instrument’s mathematical model.

$$M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)=\frac{\frac{M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}-{i\epsilon }_{\mathrm{a}\mathrm{t}\mathrm{m}}{M}_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T_{\mathrm{a}\mathrm{t}\mathrm{m}}\right)}{i\tau }-i\rho M_{\mathrm{S},\mathrm{e}\mathrm{n}\mathrm{v}}}{i\mathsf{\epsilon }}$$

(18)

The process flow chart of the scalar inverse model is presented in Figure 10.

The temperature measurement can be obtained by inverting the sensor characteristic function $M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)$, which is obtained from Equation (13) and displayed in Figure 6. The inverse sensor characteristic function is presented in Figure 11.

As a scalar simplification, this model is appropriate for grey GBRT and inherently for BBRT, where interactions are not spectrally dependent. The vast majority of instruments in the field of radiation thermometry currently operate according to this model, as can be seen from the scalar input values.

Since the settings of the measuring instruments are often unknown, these input parameters are usually estimated, taken from literature or determined experimentally, usually without following the rules of traceability for the particular system of measurement. It is also common procedure to use constant instrumental parameters for all measurements of a given surface, regardless of the temperatures and spectral interactions involved.

Due to the omission of processing of the spectral information of the signal, this model and method are considered inaccurate. An improved version of this model is needed to take full advantage of the available accuracy and traceability of radiation thermometry. To do this, the spectral interactions must be accounted for in the inverse model.

3.1.3. Hybrid Model

While an ideal inverse for measurements under real conditions is not possible due to the loss of spectral data in sensor measurement, correct integration of the scalar inverse model for radiation thermometry of real bodies in real conditions (RBRT) can be achieved by correctly accounting for the influence of spectral parameters in instrumental parameters, according to Figure 12.

Since the spectral distribution data of the signal are lost in the direct model, it should be accounted for by the effective scalar factor to compensate for the discrepancy between the results of the scalar and the spectral inverse model (19).

$$\begin{gathered} i\mathsf{\epsilon }={\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)\right),\hspace{1em}\rho ={\rho }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(M_{\mathrm{S},\mathrm{e}\mathrm{n}\mathrm{v}}\right) \\ i\tau ={\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(M_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(\lambda \right)\right),\hspace{1em}{\mathrm{i}\mathsf{\epsilon }}_{\mathrm{a}\mathrm{t}\mathrm{m}}={\mathsf{\epsilon }}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{a}\mathrm{t}\mathrm{m}}\left(M_{\mathrm{B}\mathrm{B}\mathrm{S}}\left(T_{\mathrm{a}\mathrm{t}\mathrm{m}}\right)\right) \end{gathered}$$

(19)

These effective scalar parameters correspond to the actual effect of spectral influence parameters on spectral radiances in the direct model (14), as manifested in the measuring instrument.

Since the effective emissivity depends on the spectral radiance of the raw (black body) emission and thus on the temperature of the object, which at the time of measurement is unknown, a recursive relationship is established between the spectral representation of the object’s emission and its temperature. The calculated object temperature can thus be improved iteratively. It is important to note that optimization is only required to better define the spectral distribution of the signal. Therefore, small deviations and fast regression to the optimal results are expected from the model throughout iterations; however, at least two iterations are required to vaguely approximate the spectral distribution of the object’s (temperature-dependent) raw emission.

By definition, spectral influence parameters are defined as the ratio between the real and hypothetically ideal radiant response (i.e., emission, reflection, transmission, absorption) at individual wavelengths. A scalar influence parameter corresponds to the ratio of the actual and theoretically ideal radiance. In terms of radiation thermometry, these radiant contributions correspond to the measured (spectral) radiance, as detected by the sensor with spectral sensitivity $S\left(\lambda \right)$.

A theoretically accurate implementation of spectral influence parameters in the scalar inverse model can be achieved by expressing with an effective (scalar) factor the overall effect of the spectral influence parameter on the measurement, as detected by the sensor. For this, the direct model of radiation thermometry (14) can be utilized. By taking into account Planck’s law for an object’s raw (black body) emission, spectral influence parameters and possibly other spectral radiances in the system of measurement (e.g., environmental reflection, atmospheric emission), spectral radiance contributions can be expressed for any stage of the direct (physical) model (14).

First, the effective transmissivity and emissivity of the atmosphere can be expressed. The effective transmissivity corresponds to the radiance of the joint emission and reflection of the atmosphere, whereas the effective emissivity of atmosphere corresponds to the spectral radiance of atmospheric emission.

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

(20)

The effective parameter (e.g., ${\tau }_{eff}$) is obtained through substitution of a spectral parameter with an effective parameter in the direct model (14) while retaining the result. This is achieved by equating a direct model with the spectral parameter and a direct model with the spectral parameter substituted for an effective scalar (e.g., $\tau \left(\mathsf{\lambda }\right)\to {\tau }_{eff}$ in Equation (21)). Independent contributions (e.g., emission of atmosphere) are equal on both sides of equation and can therefore be subtracted from expression.

$$\begin{array}{c}\int \left(M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\mathsf{\lambda }\right)+M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\rho \left(\mathsf{\lambda }\right)\right)\tau \left(\mathsf{\lambda }\right)S\left(\lambda \right)d\lambda \hspace{1em}\hspace{1em}\hfill \\ \hfill \hspace{1em}\hspace{1em}={\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}\int \left(M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\lambda \right)+M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\mathsf{\rho }\left(\lambda \right)\right)S\left(\lambda \right)d\lambda \end{array}$$

(21)

Expressing from Equation (21), effective transmissivity ${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be expressed.

$${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(M_{obj},\tau \left(\mathsf{\lambda }\right)\right)=\frac{\int \left(M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\mathsf{\lambda }\right)+M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\rho \left(\mathsf{\lambda }\right)\right)\tau \left(\mathsf{\lambda }\right)S\left(\lambda \right)d\lambda }{\int \left(M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\lambda \right)+M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\mathsf{\rho }\left(\lambda \right)\right)S\left(\lambda \right)d\lambda }$$

(22)

For advanced use, effective transmissivity ${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be separated into two entities ${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{o}\mathrm{b}\mathrm{j}}$ (23) and ${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{e}\mathrm{n}\mathrm{v}}$ (24), according to the direct model, each with individual contributions (15) corresponding to its own contributing emitter and spectral distribution of the affected radiance.

$${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)=\frac{\int M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\mathsf{\lambda }\right)\tau \left(\mathsf{\lambda }\right)S\left(\lambda \right)d\lambda }{\int M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\lambda \right)S\left(\lambda \right)d\lambda }$$

(23)

$${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{e}\mathrm{n}\mathrm{v}}\left(M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\right)=\frac{\int M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\mathsf{\rho }\left(\lambda \right)\tau \left(\mathsf{\lambda }\right)S\left(\lambda \right)d\lambda }{\int M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\mathsf{\rho }\left(\lambda \right)S\left(\lambda \right)d\lambda }$$

(24)

Similarly, the effective atmospheric emissivity (25) can be calculated.

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

(25)

Continuing substitutions in Equation (20), the effective emissivity (26) and reflectivity (27) can be expressed. As effective transmissivity ${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ corresponds to influenced radiances, including emissivity and reflectivity, it can substitute for spectral transmissivity and can be, as a constant, expressed outside the integral and reduced from both sides of the division.

$${\mathsf{\epsilon }}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}}\right)=\frac{\int M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)\mathsf{\epsilon }\left(\mathsf{\lambda }\right)S\left(\lambda \right)d\lambda }{\int M_{\mathrm{B}\mathrm{B},\mathrm{o}\mathrm{b}\mathrm{j}}\left(T_{\mathrm{o}\mathrm{b}\mathrm{j}},\lambda \right)S\left(\lambda \right)d\lambda }$$

(26)

$${\rho }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\right)=\frac{\int M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)\rho \left(\mathsf{\lambda }\right)S\left(\lambda \right)d\lambda }{\int M_{\mathrm{e}\mathrm{n}\mathrm{v}}\left(\lambda \right)S\left(\lambda \right)d\lambda }$$

(27)

A rule can be derived for converting any spectral influence parameter to effective scalar value $p\left(\lambda \right)\to p_{\mathrm{e}\mathrm{f}\mathrm{f}}$ (28), which allows the calculation of any effective factor under any conditions and in any phase of the direct model. The effective scalar value is equal to the ratio between the actual measured radiance and the ideal radiance of the direct model (14), as it would be measured if the influence parameter was ideal, i.e., a value of 1.

$$p_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(\lambda \right)\right)=\frac{M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} p\left(\lambda \right)}\left(T\right)}{M_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t} p\left(\lambda \right)}\left(T\right)}$$

(28)

Spectral weights $W_p\left(\lambda \right)$ or $W_p\left(T,\lambda \right)$ (29) for averaging of spectral parameters can be expressed from Equations (22) and (25)–(27) and generated for different detectors and black body emissions, as shown in Figure 13. Spectral weights display spectral representation of the given parameter $p\left(\lambda \right)$ in the final measurement.

$${\mathrm{p}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\int W_p\left(\lambda \right)p\left(\lambda \right)d\lambda$$

(29)

Detectors with different spectral sensitivities differently respond to anomalies in spectral parameters due to temperature-induced shifts in radiance.

4. Results

4.1. Validation of the Hybrid Inverse Model of Radiation Thermometry

A measurement by radiation thermometry was simulated with a scalar and a hybrid inverse model. The model errors shown in Figure 14 are defined as measurement errors, calculated as the difference between the simulated theoretical measured temperature and the reference temperature of the object used for the simulation.

The model with constant instrumental values, obtained by the averaging of the spectral parameters, had large model errors. The model with effective scalar values, calculated for the signal and the instrument according to Equation (28), had negligible error.

4.2. Effective Parameters as Instrumental Settings for the Scalar Inverse Model

The equilibrium equation ($\epsilon \left(\lambda \right)+\tau \left(\lambda \right)+\rho \left(\lambda \right)=1$) is in practice usually extended to scalar operators ($i\epsilon +i\tau +i\rho =1$). This simplification is correct only if the measured surface is an ideal (spectrally homogenous) grey body or if the spectral distributions of the signals involved are equal, i.e., in thermal equilibrium.

Under real conditions, the effective scalar equivalent value of a spectral parameter is determined by the spectral distribution of the spectral parameters, the signal, and the spectral sensitivity of the instrument. Since the spectral parameters are represented according to the signals involved, the effective scalar parameters are independent and therefore the sum of the three instrumental parameters is not necessarily equal to 1 when measured under real conditions. The effective and mean scalars of the emissivity and reflectivity of the object are shown in Figure 15, where each of the parameters is expressed as a function of the influenced value, in this case the temperature of the emitting object and the temperature of the environment, assuming thermal equilibrium of the environment, which is necessary to satisfy the assumption that the radiance is distributed according to Planck’s law.

Unlike in the theory of radiation thermometry for grey bodies, the sum of effective parameters in the radiation thermometry of real objects is not necessarily equal to 1. This is evident in Figure 16, where the sum of the two effective parameters is plotted as a function of the object and environmental temperature.

The transmissivity was set to 0 and omitted for opaque bodies. Considering that the model error, obtained from the plotted effective parameters in Figure 14b was negligible, the effective parameters are considered valid.

The sum of the effective atmospheric emissivity and the transmissivity at fixed atmospheric temperature (Figure 17) also deviates from 1, but the effect is less pronounced.

4.3. Measurement Uncertainty Due to Uncertainty of Emissivity

The use of grey body measurement principles under real conditions is currently the standard practice in the field of radiation thermometry. Since the grey body simplification of spectral homogeneity never applies under real conditions, the uncertainty of spectral emissivity must be accounted for in the uncertainty of non-contact temperature measurement.

The highest measurement deviations are expected when the spectral parameter (emissivity) is at its highest deviation along the full (measurement) spectrum. Deviations of spectral emissivity in narrow spectral bands always produce a lesser or equal effect than deviations along the full spectrum (of measurement). Therefore, spectrally local uncertainties of the spectral emissivity are considered to be included in the uncertainty budget of the uncertainty interval of the total spectrum. Therefore, the calculation of the measurement of uncertainty interval is performed as the worst-case study, where the highest and the lowest expected deviation of the spectral emissivity (over the whole spectrum) define the uncertainty interval at the output of the measurement process.

Figure 18 shows the uncertainty intervals of GBRT due to the uncertainty of emissivity. Although a (spectrally homogeneous) grey body was used in this figure, any case-specific (spectral) emissivity and the corresponding (spectral) uncertainty can be used for the uncertainty calculation. In such a case, the highest possible uncertainty should be used.

Similarly, the ranges of relative uncertainty are shown in Figure 19. The relative uncertainty is expressed as the ratio between the total measurement uncertainty and the temperature difference between the object and the reflected environment.

5. Discussion

5.1. State of the Art in the Field of Radiation Thermometry

Currently the field of radiation thermometry appears to be technologically well supported. Radiation thermometers with state-of-the-art measuring capabilities are available for scientific, professional and consumer use.

Metrological conformity is in practice established for radiation thermometry of black bodies. Calibrations conducted on black body sources are well established and documented in the literature, often also accredited and traceable to national or international standards. Traceable use of radiation thermometers usually only includes BBRT; however, it is often also applied to GBRT while neglecting spectral interactions.

Through theory and scalar inverse models, the use of radiation thermometers can be extended to grey bodies while retaining traceability; however, this is generally not applicable to real conditions. The literature search for theoretically traceable radiation thermometry protocols for real bodies (RBRT) did not yield any existing results.

5.2. The Use of the Equilibrium Equation for Instrumental Settings in Radiation Thermometry under Real Conditions Is Theoretically Incorrect

As demonstrated by the simulation in Figure 15, the effective scalar equivalents of the spectral parameters do not necessarily sum to 1 (Figure 16) under real conditions. Therefore, the usual simplification of the equilibrium equation of the scalar inverse model ($i\epsilon +i\tau +i\rho =1$) is not valid under real conditions, when only a slight spectral non-homogeneity of the spectral factors exists.

The use of this equation under real conditions inevitably results in errors since the data of the spectral distribution of the signals are not taken into account, as shown in Figure 14.

5.3. Theoretically Derived Direct (Physical) Model and Hybrid Inverse, Validated by Simulation

Based on the results of literature review, a physical model of radiation thermometry was defined and scalar and hybrid inverse models were derived.

To correctly account for the interactions between the spectral distributions of the influence parameters and the involved signals, the hybrid inverse model should be used where the scalar influence parameters are determined as effective scalar parameters according to Equation (28).

Considering the results of the model error simulation in Figure 14, the hybrid inverse model, where the instrumental parameters are set to the effective values, was validated as a correct approach for radiation thermometry of real objects, resulting in negligible conversion error. This approach requires an individual calculation of the effective scalar equivalent for each spectral parameter and an iterative calculation of the object temperature.

5.4. Uncertainty of Radiation Thermometry Due to Uncertainty of Spectral Emissivity

Each surface can be classified as an idealized grey body with corresponding spectral homogeneity uncertainty. To assure traceability of measurements, input uncertainties must be evaluated and included in the total measurement uncertainty. A demonstration of the evaluation of uncertainty due to spectral emissivity is shown in Figure 18.

In general, the measurement uncertainty of any radiation thermometer is lowest near the apparent temperature of reflection and increases approximately linearly with the difference from the environment temperature. The linearity was explored in Figure 19, where the measurement uncertainty is expressed in relation to the temperature difference between the measured object and the 25 °C environment.

The results in Figure 18 indicate that different sensors with different spectral sensitivities respond differently to measurement uncertainty, presumably due to different emission/reflection ratios in different spectral sensitivity ranges and at different object temperatures.

As can be seen from the simulation results, the measurement uncertainty increases with decreasing emissivity of the surface (at constant input uncertainty of the emissivity).

As the emissivity uncertainty interval approaches an emissivity value of 0, the total measurement uncertainty will seemingly increase to infinity. This is also logical, since non-emissive surfaces cannot be measured due to the lack of emission.

Another error was found to occur frequently when the compensation algorithms overestimate the influence of influence parameters. In this case, the compensated signal may be calculated to a negative incident radiance of the environment. Overcompensation is very problematic when low temperatures (below 0 °C) are measured at higher environmental temperatures. Due to the steepness of the inverse sensor characteristic (Figure 11) at low temperatures, a small uncertainty in the inverse model translates to a high temperature measurement uncertainty. This is evident in Figure 19 for TRT II and FLIR detectors in 8–14 µm range.