# Multi-spectral infrared thermography using finite element emissivity modelling for high-temperature measurements on metallic surfaces

Kamal Ennass, Benjamin Remy, Vincent Schick, Johann Meulemans

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Kamal Ennass, Benjamin Remy, Vincent Schick, Johann Meulemans. Multi-spectral infrared thermography using finite element emissivity modelling for high-temperature measurements on metallic surfaces. Infrared Physics and Technology, 2025, 150, pp.105999. 10.1016/j.infrared.2025.105999. hal-05195885

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Research paper

# Multi-spectral infrared thermography using finite element emissivity modelling for high-temperature measurements on metallic surfaces

Kamal Ennass<sup>a,b,c,*</sup>, Benjamin Remy<sup>a,b</sup>, Vincent Schick<sup>a,b</sup>, Johann Meulemans<sup>b,c</sup>

## ARTICLE INFO

**Keywords:**

High temperature

Inverse methods

Infrared thermography

Multi-spectral thermometry

Opaque materials

## ABSTRACT

Accurate high-temperature radiation thermometry of opaque surfaces (from 800 to 1200 K) is critical for controlling industrial processes and ensuring product quality. However, its accuracy is limited by inadequate knowledge of the material's optical properties and their spectral and thermal dependencies. Multi-spectral thermography addresses this challenge by identifying the surface temperature while considering its emissivity spectra. Here, we propose a method combining infrared radiometric measurements with deterministic inverse modelling, where emissivity is interpolated using a P1-1D finite element polynomial basis to attenuate the systematic and stochastic components of the identification error. Monte Carlo simulations and experimental validations using an induction heater and a multi-spectral infrared camera calibrated at four wavelengths demonstrate transient-state temperature uncertainties below 5%. Numerical hyperspectral assessments with 20 wavelengths further demonstrate significant bias reduction with increased spectral resolution. These findings establish an advancement towards reliable, dynamic, non-contact temperature measurement in industrial metallurgical environments.

## 1. Introduction

Material manufacturing typically relies on precise transient temperature regulation under extreme conditions to achieve the desired physical properties. The associated temperature accuracy dictates performance characteristics and reduces production costs for various industries (infrastructure, automotive, aerospace, electronics, etc.). Notable manufacturing processes include vacuum induction melting of metal, continuous casting of molten metal, thermal tempering of flat glass, forming and laminating of windshields, production of ceramics by electrofusion, or concrete in rotary kilns.

Long-established temperature monitoring techniques such as thermocouples rely on direct contact and are thus regarded as intrusive to the heat propagating inside the material. Moreover, these sensors present a risk of damage at high temperatures and are not adapted to moving targets. These limitations can be bypassed by employing non-invasive approaches, which rely on physical quantities that do not necessitate contact with the target object.

Radiation-based instruments containing infrared sensors have showcased significant potential for high-temperature identification (here defined as superior to 800 K) due to highly exploitable data signals, as

opposed to other spectral bands. Initial implementations of non-contact thermometry consisted in mono- and bi-spectral measurements [1–9] coupled with inverse techniques that incorporate blackbody and grey-body assumptions, respectively. Despite their extensive use in various industrial applications, these methods can result in substantial temperature errors due to their inadequate behavioural assumptions of the target's optical properties [10].

In recent years, multi-spectral thermometry has emerged as an improvement over past techniques by identifying the emissivity spectra simultaneously with the surface temperature of opaque targets. For this method, the challenge resides in identifying the adequate emissivity model that accurately describes its spectral dependency. This resulted in past works demonstrating irregular outcomes depending on the material, the operating wavelengths, and the heating conditions (temperature, surface conditions, environment). For instance, Duvaut [11] tested mathematical (polynomial, exponential, and fractional) and physical (Maxwell, Hagen-Rubens, and Edwards) emissivity models on austenitic stainless steel alloys and titanium for temperatures between T = 500 K and the melting point in the visible and infrared range. The author reported identification errors under 3% for the most

<footnote><sup>a</sup> Laboratoire Énergies et Mécanique Théorique et Appliquée, Université de Lorraine, CNRS, 2 Av. de la Forêt de Haye, 54500, Vandœuvre-lés-Nancy, France</footnote>

<footnote><sup>b</sup> Laboratoire commun Canopée, CNRS, Université de Lorraine, Saint-Gobain, France</footnote>

<footnote><sup>c</sup> Saint-Gobain Research Paris, 39 Quai Lucien Lefranc, 93303, Aubervilliers, France</footnote>

<footnote>* Corresponding author at: Laboratoire Énergies et Mécanique Théorique et Appliquée, Université de Lorraine, CNRS, 2 Av. de la Forêt de Haye, 54500, Vandœuvre-lés-Nancy, France.</footnote>

<footnote>E-mail address: kamalennass@gmail.com (K. Ennass).</footnote>

<footnote>https://doi.org/10.1016/j.infrared.2025.105999</footnote>

<footnote>Received 25 March 2025; Received in revised form 29 May 2025; Accepted 3 July 2025</footnote>

<footnote>Available online 17 July 2025</footnote>

<footnote>1350-4495/© 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
suitable model, which varies based on the choice of metallic surface. Pierre [12] worked on the simultaneous identification of temperature and emissivity of molten metals (around 2750 K and 1697 K for niobium and steel, respectively) using deterministic and Bayesian techniques with linear and non-linear models. The first-order polynomial formulation provided the most accurate estimations. Rodiet [13] determined optimal operating wavelengths in the infrared region that improve the signal-to-noise ratio (SNR) of radiometric measurements performed at T = 623 K. These wavelengths minimise the temperature standard deviation to limit the propagation of experimental measurement uncertainty on the estimated values. The emissivity behaviour of an oxidised cast iron sample was then fitted using a second-order polynomial, which led to a mean identification error of 0.3% with a standard deviation of 0.6%. For a detailed state of the art of multi-spectral thermometry, we invite the reader to refer to [10,14].

This paper aims to combine multi-spectral infrared thermography with deterministic inverse techniques for metallurgical applications at high temperatures. The main objective is to develop an emissivity model that describes the spectral behaviour of various metals with sufficient accuracy to improve surface temperature identification. The first section of this study presents sensitivity and correlational analyses of the proposed models and identifies their optimal wavelengths. The second section describes the experimental apparatus of multi-spectral infrared thermography and its principle of operation. The final section reports on Monte Carlo numerical simulations and experimental assessments of mono-, bi-, and multi-spectral methods applied to various metallic surfaces subjected to transient high temperatures.

## 2. Methodology

In thermal imaging systems, the signal provided by each sensor is proportional to the incoming flux of the thermal scene. The latter is referred to as irradiance and encompasses the radiant flux emitted by the target object as well as numerous perturbations stemming from its surroundings (refer to Section 3).

The spectral radiant flux emitted by a real opaque surface at a given temperature T, per unit area and wavelength λ, is quantified by the formula below:

$$M_{\lambda}(T) = \epsilon_{\lambda,T} M_{\lambda}^{0}(T) \quad (1)$$

With $M_{\lambda}^{0}(T)$ being the blackbody exittance given by Planck's law:

$$M_{\lambda}^{0}(T) = \frac{C_{1}\lambda^{-5}}{\exp\left(\frac{C_{2}}{\lambda T}\right) - 1} \quad (2)$$

Where $\epsilon_{\lambda,T}$ is the spectral emissivity of the non-black surface, corresponding to its ability to emit thermal radiation compared to that of an ideal emitter (a blackbody), and $C_1 = 3.74 \times 10^{-16}$ W m$^2$ and $C_2 = 1.43 \times 10^{-2}$ m K are the first and second radiation constants, respectively.

### 2.1. Multi-spectral predictive model and fitting technique

As the lack of knowledge of the emitting surface's optical properties constitutes a major obstacle to an accurate temperature identification, multi-spectral thermometry assumes that its emissivity can be modelled as a wavelength-dependent analytical function. The approach presented in this paper consists in using P1-1D finite element polynomials to model the spectral behaviour of non-ideal emitters. The resulting monochromatic radiant flux is expressed as follows:

$$\varphi_{\lambda}(T, \epsilon_{\lambda_1}, \epsilon_{\lambda_2}, \dots) = \left( \sum_{j=1}^{N-1} \epsilon_{\lambda_j} H_j(\lambda) \right) M_{\lambda}^{0}(T) \quad (3)$$

With:

$$\forall j \in [1, N-1] : \quad H_j(\lambda) = \begin{cases} \frac{(\lambda - \lambda_{j-1})}{(\lambda_j - \lambda_{j-1})}, & \lambda \in [\lambda_{j-1}, \lambda_j] \\ \frac{(\lambda_{j+1} - \lambda_j)}{(\lambda_{j+1} - \lambda_j)}, & \lambda \in [\lambda_j, \lambda_{j+1}] \end{cases} \quad (4)$$

The emissivity parameter $\epsilon_{\lambda_j}$ corresponds to its spectral value at user-defined $\lambda_j$ of $\lambda_{model} = \{\lambda_1, \lambda_2, \dots, \lambda_{N-1}\}$ and $H_j(\lambda)$ the P1-1D hat function that peaks at $\lambda_j$ (see Eq. (4)). The $N-1$ emissivity parameters are simultaneously estimated with the surface temperature and require at least $N$ spectral measurements as input to the inverse model. The finite element emissivity model introduced in Eq. (3) does not take into account the emissivity's temperature dependence as the estimation occurs at a constant temperature level.

Through the finite element formulation, each estimated emissivity $\epsilon_{\lambda_j}$ is linked together by piecewise linear interpolation. The advantage of using P1 Lagrange functions is that they ensure the continuity of the model and its derivative almost everywhere (excluding $\lambda_j$ for the derivative). The method can be generalised with other P2 or P3 polynomials (cubic splines) to model smooth functions. Furthermore, the offset between each $H_j(\lambda)$ centred at $\lambda_j$, ensures that their sum is equal to 1, while preventing any one-on-one correlation.

The radiant flux model in Eq. (3) is non-linear with respect to the surface temperature, which entails a non-linear optimisation process for inferring the optimal solution. Consequently, we employ non-linear least squares which, unlike ordinary least squares (OLS), requires an iterative process to reach the solution that minimises the cost function. The latter is defined as the weighted sum of residuals between measured and predicted monochromatic radiant flux, as given by Eq. (5). The scaling factor $W_i$ is equal to the reciprocal of the measurement noise's standard deviation $\sigma_{noise,i}$ at the operating wavelength $\lambda_{exp,i}$.

$$J(T, \epsilon_{\lambda_1}, \epsilon_{\lambda_2}, \dots) = \sum_{j=1}^{N} W_j^2 \left( \varphi_{\lambda_{exp,j}}^{\text{meas}} - \varphi_{\lambda_{exp,j}}(T, \epsilon_{\lambda_1}, \epsilon_{\lambda_2}, \dots) \right)^2 \quad (5)$$

In this paper, the iterative minimisation is conducted with the deterministic Levenberg-Marquardt algorithm, a hybrid method combining the gradient descent algorithm and the Gauss-Newton algorithm by using a damping parameter $\mu$, as dictated by Eq. (6).

$$\hat{\beta}^{(n+1)} = \hat{\beta}^{(n)} + \frac{X^{(n)^T}(\varphi^{\text{meas}} - \varphi^{(n)})}{\left(\left(X^{(n)^T}X^{(n)}\right) + \mu^n \text{diag}\left(X^{(n)^T}X^{(n)}\right)\right)} \quad (6)$$

With $\hat{\beta}^{(n)}$ being the nth approximation of unknown parameters vector $\beta$, and $X$ the Jacobian matrix containing the direct model's sensitivities with respect to it.

In addition to the finite element model (FEM), the polynomial formulation, often found in literature [7,10,12,13], will also be employed for comparison (see Eq. (7)). Contrary to the previous model, only the emissivity parameter $a_0$ contains a physical significance parallelling the described variable $\epsilon_{\lambda,T}$.

$$\varphi_{\lambda}(T, a_0, a_1, \dots) = (a_0 + a_1 \lambda + \dots + a_{N-2} \lambda^{N-2}) M_{\lambda}^{0}(T) \quad (7)$$

### 2.2. Parameter sensitivity analysis

Since the determination of $\beta$ heavily relies on computing the inverse of $X^T X$ in Eq. (6), and the systematic bias is proportional to the stochastic error introduced by measurement noise [15], the matrix must be well-conditioned to ensure great accuracy and robustness across iterations. This can be guaranteed if the parameters are not correlated and provide non-zero sensitivity coefficients, indicating influential parameters. In the presence of correlations, sensitivities showcase a similar evolution and are proportional, which results in high variances in correlated parameters and, consequently, several local minima in the cost function. For an extensive dive into the necessary tools for solving inverse problems and identifying the most prominent parameters, we recommend referring to [16–20].

A practical quantitative representation of parameter identifiability is the Vcor matrix that contains relative standard deviations $\sigma_{\hat{\beta}_i}$ in the diagonal and correlation coefficients $\rho_{i,j} = \frac{\text{cov}(\hat{\beta}_i, \hat{\beta}_j)}{\sigma_{\hat{\beta}_i} \sigma_{\hat{\beta}_j}}$ as off-diagonal entries, with $\text{cov}(\hat{\beta}_i, \hat{\beta}_j)$ being the covariance. The variance-covariance
matrix provides us with these quantities according to the formula below:

$$ cov(\hat{\beta}) = \begin{bmatrix} \sigma_{\hat{\beta}_1}^2 & cov(\hat{\beta}_1, \hat{\beta}_2) & \dots & cov(\hat{\beta}_1, \hat{\beta}_M) \\ cov(\hat{\beta}_1, \hat{\beta}_2) & \sigma_{\hat{\beta}_2}^2 & \dots & cov(\hat{\beta}_2, \hat{\beta}_M) \\ \vdots & \vdots & \ddots & \vdots \\ cov(\hat{\beta}_1, \hat{\beta}_M) & cov(\hat{\beta}_2, \hat{\beta}_M) & \dots & \sigma_{\hat{\beta}_M}^2 \end{bmatrix} \quad (8) $$

With:

$$ cov(\hat{\beta}) = (X^T X)^{-1} X^T E(ee^T) X (X^T X)^{-1} \quad (9) $$

Where e is the Gaussian additive noise corrupting the radiometric measurements and displaying a zero mean and a wavelength-dependent variance $\sigma_{noise}^2$. The latter is contained in the diagonal entries of $E(ee^T)$. For a given vector $\beta$ containing M unknown parameters:

$$ X = \begin{bmatrix} \frac{\partial \varphi_{\lambda_{\text{exp},1},T}}{\partial \hat{\beta}_1} & \frac{\partial \varphi_{\lambda_{\text{exp},1},T}}{\partial \hat{\beta}_2} & \dots & \frac{\partial \varphi_{\lambda_{\text{exp},1},T}}{\partial \hat{\beta}_M} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial \varphi_{\lambda_{\text{exp},N,T}}}{\partial \hat{\beta}_1} & \frac{\partial \varphi_{\lambda_{\text{exp},N,T}}}{\partial \hat{\beta}_2} & \dots & \frac{\partial \varphi_{\lambda_{\text{exp},N,T}}}{\partial \hat{\beta}_M} \end{bmatrix} \quad (10) $$

To align with the constraints of the metrological tool described in Section 3, which is restricted to four monochromatic filters (N = 4), the sensitivity analysis will be conducted for a matching number M of unknown parameters (surface temperature and three emissivity-related parameters) under both finite element and polynomial formulations. It is worth noting that increasing parameter dimensionality through refinement could offer an additional advantage to the finite element approach as it allows a more precise piecewise fitting of the emissivity spectra. The implementation of hyper-spectral radiometry is further explored in Section 4.1.3.

### 2.2.1. Finite element model

Since the parameter vector $\beta = (T, \varepsilon_{\lambda_1}, \varepsilon_{\lambda_2}, \varepsilon_{\lambda_3})$ exhibits dissimilar units, the following scaled sensitivities are computed instead:

$$ X_T^* = T \frac{\partial \varphi(\lambda, T, \varepsilon_{\lambda_1}, \varepsilon_{\lambda_2}, \varepsilon_{\lambda_3})}{\partial T} = \varphi(\lambda, T, \varepsilon_{\lambda_1}, \varepsilon_{\lambda_2}, \varepsilon_{\lambda_3}) \frac{C_2}{\lambda T} \frac{\exp\left(\frac{C_2}{\lambda T}\right)}{\exp\left(\frac{C_2}{\lambda T}\right) - 1} \quad (11) $$

$$ \begin{aligned} \forall j \in [1, 3] : & X_{\varepsilon_{\lambda_j}}^* = \varepsilon_{\lambda_j} \frac{\partial \varphi(\lambda, T, \varepsilon_{\lambda_1}, \varepsilon_{\lambda_2}, \varepsilon_{\lambda_3})}{\partial \varepsilon_{\lambda_j}} \\ & = \varepsilon_{\lambda_j} H_j(\lambda) \frac{C_1 \lambda^{-5}}{\exp\left(\frac{C_2}{\lambda T}\right) - 1} \end{aligned} \quad (12) $$

While the scaled sensitivity coefficients above do not imply any one-on-one correlation, a linear dependence involving more than two parameters may occur. The underlying correlation can be uncovered by defining a linear regression problem involving the scaled sensitivities, as dictated by Eq. (13). The fitting coefficients contained in $\alpha$ are obtained through ordinary least squares.

$$ X_T^* = \alpha_1 X_{\varepsilon_{\lambda_1}}^* + \alpha_2 X_{\varepsilon_{\lambda_2}}^* + \alpha_3 X_{\varepsilon_{\lambda_3}}^* \quad (13) $$

$$ \Rightarrow \hat{\alpha} \approx (\hat{\alpha}_1, \hat{\alpha}_2, \hat{\alpha}_3) = \begin{bmatrix} X_{\varepsilon_{\lambda_1}}^* & X_{\varepsilon_{\lambda_2}}^* & X_{\varepsilon_{\lambda_3}}^* \\ X_{\varepsilon_{\lambda_1}}^* & X_{\varepsilon_{\lambda_2}}^* & X_{\varepsilon_{\lambda_3}}^* \end{bmatrix}^{-1} \begin{bmatrix} X_{\varepsilon_{\lambda_1}}^* \\ X_{\varepsilon_{\lambda_2}}^* \\ X_{\varepsilon_{\lambda_3}}^* \end{bmatrix} X_T^* \quad (14) $$

In the numerical and experimental results section, various wavelength-specific FEM configurations $\lambda_{model} = \{\lambda_1, \lambda_2, \lambda_3\}$ are implemented with $\lambda_1$ and $\lambda_3$ corresponding to the first and last measurement wavelength, respectively. The impact of the central wavelength $\lambda_2$ should then be investigated at the second and third measurement wavelengths ($\lambda_2 = \lambda_{exp,2}$ or $\lambda_2 = \lambda_{exp,3}$) and their mean value $\lambda_2 = \frac{\lambda_{exp,2}+\lambda_{exp,3}}{2}$ as well. Therefore, the sensitivity analysis is performed at the metrological tool's operating wavelengths $\lambda_{exp} = \{2.00, 2.35, 2.85, 4.00 \, \mu\text{m}\}$ and the noise levels $e = \{1.59, 0.30, 0.26, 0.14\} \times 10^7 \, \text{W. m}^{-2} \, \text{m}^{-1}$ obtained experimentally through blackbody measurements at 1000 K.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Combination of emissivity scaled sensitivities $X_{\varepsilon_{\lambda_j}}^*$ as a function of $X_T^*$, in log scale, for various FEM configurations. The associated linear regression coefficients are: $\hat{\alpha} \approx (7.20, 5.98, 3.61)$ for $\lambda_2 = 2.35 \, \mu\text{m}$, $\hat{\alpha} \approx (7.20, 5.38, 3.70)$ for $\lambda_2 = 2.60 \, \mu\text{m}$, and $\hat{\alpha} \approx (7.09, 5.04, 3.70)$ for $\lambda_2 = 2.85 \, \mu\text{m}$.</figcaption></figure>

Fig. 1 showcases the fitting results of Eq. (13) at nominal values $\beta = (T, \varepsilon_{\lambda_1}, \varepsilon_{\lambda_2}, \varepsilon_{\lambda_3}) = (1000, 1.00, 1.00, 1.00)$ given that the sensitivity coefficients are functions of the unknown parameters. One can see that the linear dependency between the sensitivity coefficients varies with $\lambda_2$, with the FEM configuration centred at the mean wavelength ($\lambda_2 = 2.60 \, \mu\text{m}$) yielding an optimal linear combination of $X_{\varepsilon_{\lambda_j}}^*$ that overlaps the identity line of $X_T^*$ over the experimental wavelength range.

Our goal for the sensitivity analysis is to identify the wavelength configuration that produces the least correlations possible between unknown parameters. Hence, the configuration $\lambda_{model} = \{2.00, 2.35, 4.00 \, \mu\text{m}\}$ appears to be the best candidate due to its higher residuals compared with other formulations, though its parameters remain nearly dependent ($\sum_{i=1}^N \alpha_i \tilde{\beta}_i^* \approx \tilde{0}$). In fact, while the rank of the matrix $X^T X$ remains at 4 since the linear combination of all scaled sensitivities is not exactly zero, the outcome of such quasi-dependence reduces the degree of freedom and leads to multiple local minima tainted with erroneous estimations in the optimisation problem. The effect of an underlying correlation can also be perceived by computing the Vcor matrix in Eqs. (15) to (17) for matching nominal values of $\beta$.

Higher correlation coefficients are perceived between the parameters by employing the FEM centred at $\lambda_2 = 2.60 \, \mu\text{m}$, which is in good agreement with the fitting results in Fig. 1. The resulting relative standard deviation of temperature is around 27%, meaning that the simultaneous identification of the model's parameters cannot be achieved accurately. The stochastic uncertainty on the temperature can be reduced by positioning $\lambda_2$ at the second or third measurement wavelength. Once again, the configuration $\lambda_{model} = \{2.00, 2.35, 4.00 \, \mu\text{m}\}$ displays more favourable conditions for obtaining an invertible matrix $X^T X$ and an effective estimation due to lower correlation coefficients and relative temperature uncertainty (0.08% compared with 0.58% for $\lambda_2 = 2.85 \, \mu\text{m}$).

$$ \text{For } \lambda_2 = 2.35 \, \mu\text{m} : \text{Vcor}(\beta) = \begin{bmatrix} 0.0008 & -0.9609 & -0.9993 & -0.9991 \\ -0.9609 & 0.0065 & 0.9602 & 0.9600 \\ -0.9993 & 0.9602 & 0.0052 & 0.9981 \\ -0.9991 & 0.9600 & 0.9981 & 0.0032 \end{bmatrix} \quad (15) $$

$$ \text{For } \lambda_2 = 2.60 \, \mu\text{m} : \text{Vcor}(\beta) = \begin{bmatrix} 0.2687 & -1.0000 & -1.0000 & -1.0000 \\ -1.0000 & 1.9326 & 1.0000 & 1.0000 \\ -1.0000 & 1.0000 & 1.4459 & 1.0000 \\ -1.0000 & 1.0000 & 1.0000 & 0.9937 \end{bmatrix} \quad (16) $$
<figure><img src="image_4.png" /><figcaption>Fig. 2. Combination of emissivity scaled sensitivity coefficients $X_{\epsilon_{ij}}^*$ as a function of $X_T^*$, in log scale. The associated linear regression coefficients are: $\hat{\alpha}=(19.60, -19.64, 11.14)$ for $\lambda_{ref} = 4.00 \mu m$.</figcaption></figure>

$$ \text{For } \lambda_2 = 2.85 \, \mu m: \quad V_{cor}(\beta) = \begin{bmatrix} 0.0058 & -0.9999 & -1.0000 & -1.0000 \\ -0.9999 & 0.0397 & 0.9999 & 0.9999 \\ -1.0000 & 0.9999 & 0.0293 & 1.0000 \\ -1.0000 & 0.9999 & 1.0000 & 0.0213 \end{bmatrix} \quad (17) $$

### 2.2.2. Polynomial model

The scaled sensitivity coefficients of the polynomial model (Eq. (7)) for $\beta = (T, a_0, a_1, a_2)$ are:

$$ X_T^* = T \frac{\partial \varphi(\lambda, T, a_0, a_1, a_2)}{\partial T} = \varphi(\lambda, T, a_0, a_1, a_2) \frac{C_2}{\lambda T} \frac{\exp\left(\frac{C_2}{\lambda T}\right)}{\exp\left(\frac{C_2}{\lambda T}\right) - 1} \quad (18) $$

$$ \forall j \in [0, 2] : X_{aj}^* = a_j \frac{\partial \varphi(\lambda, T, a_0, a_1, a_2)}{\partial a_j} = a_j \lambda^j \frac{C_1 \lambda^{-5}}{\exp\left(\frac{C_2}{\lambda T}\right) - 1} \quad (19) $$

The terms multiplying the blackbody exitance in Eq. (19) showcase dissimilar units depending on the emissivity parameter. To ensure consistency in both their significance and order of magnitude, the following normalised formulation is utilised instead:

$$ \varepsilon(\lambda) = a_0 + a_1 \left( \frac{\lambda}{\lambda_{ref}} \right) + \dots + a_{N-2} \left( \frac{\lambda}{\lambda_{ref}} \right)^{N-2} \quad (20) $$

Once more, linear dependence analysis is conducted to prove whether $\sum_{i=1}^N \alpha_i \vec{X}_j^* = \vec{0}$ is only true if $\alpha_i = 0 \forall i \in [1, N]$, implying that the sensitivities are linearly independent. For $\lambda_{ref} = 4.00 \mu m$, the acquired fitting results are plotted in Fig. 2 in the experimental spectral range for $\beta = (1000, 1.00, 1.00, 1.00)$. The plot exhibits near-perfect agreement between the fitted sensitivities and the spectral dependence of the scaled temperature sensitivity. The strong correlations involving the formulation's unknown parameters are also perceived in the off-diagonal terms of the Vcor matrix (see Eq. (21)). The polynomial model thus yields a relatively high standard deviation on the temperature (1.22% of the nominal value $T = 1000$ K) compared with the FEM for $\lambda_2 = 2.35 \mu m$ and $\lambda_2 = 2.85 \mu m$.

$$ V_{cor}(\beta) = \begin{bmatrix} 0.0122 & -0.9998 & 0.9984 & -0.9981 \\ -0.9998 & 0.2289 & -0.9994 & 0.9992 \\ 0.9984 & -0.9994 & 0.2137 & -1.0000 \\ -0.9981 & 0.9992 & -1.0000 & 0.1202 \end{bmatrix} \quad (21) $$

<figure><img src="image_5.png" /><figcaption>Fig. 3. Relative temperature standard deviation, in semi-log scale, as a function of surface temperature and various proposed methods (emissivity parameters are assigned nominal values of 1.00).</figcaption></figure>

Fig. 3 plots the relative standard deviation levels of proposed multi-spectral formulations for dynamic temperatures ranging between 800 and 1200 K. The standard deviations are inversely proportional to the surface temperature due to an increasing signal-to-noise ratio as the emissivity is assumed to be temperature-independent. It is concluded that the FEM centred at $\lambda_2 = 2.35 \mu m$ generates the least stochastic errors throughout the temperature range (reaching between 0.04 and 0.28%), which is attributed to lower correlation levels and a better conditioning of the matrix $X^T X$.

### 2.3. Optimal wavelength selection

In an attempt to attenuate the propagation of stochastic uncertainty from observations (measured radiant flux) to unknown parameters, Rodiet [13] determined optimal operating wavelengths for surface temperature identification. These wavelengths can be computed using an analogous law to Wien's displacement law (which identifies the wavelength corresponding to the peak of blackbody radiation solely from the reciprocal of the temperature) for mono- and bi-spectral measurements. Regarding multi-spectral pyrometry, optimal wavelengths were derived by minimising the temperature standard deviation, given by Eqs. (8) and (9), through an optimisation problem. The experimental operating wavelengths $\lambda_{exp}$ employed in this paper are adjacent to the optimal values procured by Rodiet under a second-order polynomial model for steady-state temperature measurements at $T = 623$ K. The aim of employing this set of wavelengths for higher surface temperatures is to assess the identification accuracy in a non-optimal setting, as industrial applications typically involve an extensive range of temperature levels.

In this section, the minimisation of $\sigma_T$ is performed with the interior-point algorithm [21,22] embedded in MATLAB through the fmincon toolbox [23], which allows a constrained estimation of $\lambda_{opt}$ within the metrological tool's operating spectral range [1.50, 5.10 $\mu m$] (see Section 3). The trust-region algorithm was also implemented for comparison but demonstrated less stability as a function of the surface temperature.

#### 2.3.1. Finite element model

The first results concern the finite element formulation with an evenly spaced initial guess $\lambda_{ini} = \{1.50, 2.70, 3.90, 5.10 \mu m\}$ and a measurement noise $e$ assumed to be identical to the experimental values.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Constrained selection of optimal multi-spectral wavelengths as a function of simulated surface temperature (FEM).</figcaption></figure>

The optimal wavelengths plotted in Fig. 4 were estimated at the un-
known parameters' nominal values $\beta = (T, 1.00, 1.00, 1.00)$, with $T$ rang-
ing from 800 to 1200 K, and for $\lambda_{model} = \{\lambda_{opt,1}, \frac{\lambda_{opt,1}+\lambda_{opt,4}}{2}, \lambda_{opt,4}\}$. Least-
squares results incorporating a central wavelength $\lambda_2$ set to the sec-
ond or third optimal wavelength produced two identical wavelengths
$\lambda_{opt,2} = \lambda_{opt,3}$ throughout the temperature range and are therefore
excluded from this section.

Optimising wavelengths under the blackbody assumption, as pro-
posed in [13], introduces a systematic bias by ignoring the mate-
rial's true emissivity behaviour. The resulting wavelengths are thus
only pseudo-optimal. However, incorporating the target's real optimal
properties revealed only minor deviations from the blackbody-derived
values, due to the inherent spectral range constraints applied during the
estimation. This observation justifies the blackbody assumption, as it
facilitates the practical implementation of multi-spectral thermography
across diverse metallurgical applications where emissivity is variable or
unknown.

While the procured wavelengths are inversely proportional to the
surface temperature, thus exhibiting a similar trend to mono- and bi-
spectral optimal values, the derivation of a displacement law $\propto T^{-1}$ was
unattainable. This is due to the quasi-constant trend resulting from the
imposed spectral constraints. Furthermore, the fluctuations observed in
the figure above are attributed to the significant noise levels and the
difficulty in identifying a consistent global minimum at each examined
temperature level.

Fig. 5 plots the relative temperature standard deviation computed at the temperature-dependent optimal wavelengths, at the mean temperature's optimal wavelengths $\lambda_{opt}^{\text{FEM}}(T = 1000 \text{ K}) = \{1.80, 2.98, 3.37, 4.95 \text{ µm}\}$, and at $\lambda_{exp} = \{2.00, 2.35, 2.85, 4.00 \text{ µm}\}$. The trends suggest that utilising the mean temperature's optimal wavelengths provides near-minimal temperature uncertainties over the dynamic range of interest. This exempts us from acquiring an extensive range of monochromatic filters or high-cost hyper-spectral cameras. Moreover, the procured trends highlight the deterioration of the temperature's stochastic error due to non-optimal measurements, as $\sigma_T$ computed at $\lambda_{exp}$ results in a 187 to 225% increase from values computed at $\lambda_{opt}^{\text{FEM}}(T = 1000 \text{ K})$.

2.3.2. Polynomial model

The constrained minimisation of $\sigma_T$, derived from Eq. (9) by incor-
porating the sensitivity coefficients of Eq. (7), reveals a higher sensitiv-
ity towards the initial guess $\lambda_{ini}$. Fig. 6 plots the resulting optimal wave-
lengths for $\beta = (T, 1.00, 1.00, 1.00)$ and $\lambda_{ini} = \{2.00, 3.00, 4.00, 5.00 \text{ µm}\}$,
which proved to generate fewer fluctuations compared with other
guesses tested beforehand.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Relative standard deviation of temperature as a function of simulated surface temperature (FEM).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Constrained selection of optimal multi-spectral wavelengths as a function of simulated surface temperature (Polynomial model).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Relative standard deviation of temperature as a function of simulated surface temperature (Polynomial model).</figcaption></figure>
Fig. 7 reveals a significant increase in both optimal and experimental standard deviations, with $\lambda_{exp}$ generating 318% to 463% compared with $\lambda_{opt}^{Poly}(T = 1000 \text{ K}) = \{1.65, 3.09, 4.26, 5.07 \text{ µm}\}$. This is a direct consequence of the substantial noise amplification role that the numerical approximation of the ill-conditioned matrix $X^T X$ plays in the case of a second-order polynomial formulation.

### 3. Experimental apparatus

This section delves into the experimental components involved in implementing multi-spectral infrared thermography for surface temperature measurements on opaque materials. The key objective was to design an experimental setup capable of quantifying the temperature error arising from the proposed methods while limiting the influence of numerous disturbances that could affect the measurement of the radiant flux emitted by the target sample, and thereby, its temperature.

As illustrated in Fig. 8, the experimental bench consists primarily of an opaque target sample heated by induction using a CELES MP 6 heater, and a Mikron M330-EU cavity blackbody, which serves as a reference for calibrating a Cedip/FLIR® Titanium 560M infrared camera over a temperature range of 573 K to 1973 K. The blackbody utilises a resistive heater along a closed-end tube cavity, and a B-type thermocouple for measuring the emitter's temperature with an uncertainty of ±0.25% of the reading ±1 °C.

The infrared camera offers a maximum spatial resolution of 640 × 512 pixels and supports predefined and user-defined sub windows. While the image sampling frequency is set to 50 Hz, it could be increased up to 4980 Hz for a reduced resolution of 16 × 4 pixels [24]. The system is equipped with a 50 mm lens featuring an aperture of $f/2.0$.

The focal-plane array comprises InSb quantum detectors, providing sensitivity to wavelengths from 1.50 to 5.10 µm. The sensors exhibit a thermal sensitivity of 25 mK at 298 K, representing the noise-equivalent temperature difference (NETD) between two points. To mitigate the thermal noise, which originates from the thermal agitation of electrons in conductive materials, and enhance detector performance, the measuring device incorporates Stirling cycle cooling, replacing earlier models that typically relied on liquid nitrogen at 77 K.

The camera also incorporates a motorised filter wheel equipped with band-pass filters characterised by a central wavelength (CWL) and bandwidth (BW), thereby restricting measurements to narrow spectral bands. This allows for monochromatic imaging at selected wavelengths $\lambda_{exp} = \{2.00, 2.35, 2.85, 4.00 \text{ µm}\}$.

Since the coil attached to the induction system is cooled with a water circuit, the radiant flux emitted by the surrounding environment and reflected by the target's surface mainly stems from sources at an ambient temperature, as dictated by the measured irradiance in Eq. (22). This facilitates our efforts in evaluating the compatibility of proposed emissivity formulations with negligible measurement bias and a non-unitary apparent surface emissivity (as opposed to a tube furnace, where the sample and the surrounding walls exhibit adjacent temperatures).

$$ 
\begin{aligned}
I_{meas,\lambda}(T) &= I_{emitted,\lambda}(T) + I_{reflected,\lambda}(T_{env}) \\
&= \varepsilon_{\lambda,T} M_{\lambda}^{0}(T) + \rho_{\lambda,T} M_{\lambda}^{0}(T_{env})
\end{aligned} 
\quad (22) $$

With $\rho_{\lambda,T} = 1 - \varepsilon_{\lambda,T}$ for an opaque surface (Kirchhoff's law of thermal radiation), and $T_{env} = 300 \text{ K}$.

In practice, the received signal is measured in the form of digital levels (DL) due to the analogue-to-digital converter embedded in the metrological tool's system. Additionally, the measurements contain attenuated values of both emitted and reflected radiations (in addition to internal emissions of the camera's housing) after they traverse the optical system, which includes the filter wheel. To convert the outputs into spectral radiant flux values (in W. m$^{-2}$. m$^{-1}$), the calibration law

<figure><img src="image_10.png" /><figcaption>Fig. 8. Simplified illustration of the experimental bench. Source: Adapted from [25].</figcaption></figure>

in Eq. (23) is employed [26].

$$ DL = \left[ \frac{A_{\lambda}}{\exp\left(\frac{C_{2}}{\lambda T}\right) - 1} + B_{\lambda} + C_{\lambda}(T_{cam} - T_{cam,ref}) \right] \times (IT) + \text{offset} \quad (23) $$

Where $A_{\lambda}$ contains the constant of proportionality (view factor, electronic gain, optical transfer function, and detector yield) along with the numerator of Eq. (2), $B_{\lambda}$ represents the radiant contributions of the surrounding environment, and $C_{\lambda}(T_{cam} - T_{cam,ref})$ corrects the signal according to the difference between the camera's current temperature and the reference value at which $A_{\lambda}$ and $B_{\lambda}$ were estimated. These sensitivity-dependent parameters are determined during the calibration process with the cavity blackbody.

Another use for the calibrating device is non-uniformity corrections (NUC), which reduce detector drifts originating from changes in the surrounding thermal scene. For each detector, the tool determines the correction's slope and y-intercept by measuring the radiant flux at two temperature levels for a given integration time or dissimilar integration times $IT \pm 5\%$ for a constant temperature. In this paper, the second technique was employed as it does not necessitate altering the cavity's temperature.

### 4. Results and discussion

#### 4.1. Numerical results

Numerical investigations were conducted on three opaque materials — copper, stainless steel 321, and oxidised nickel 201 — selected for their differing optical properties, ranging from weakly to highly reflective. The simulations relied on synthetic measured data computed with Eq. (1) between 800 and 1200 K using the emissivity levels plotted in Fig. 9. The spectral emissivity is assumed to remain constant within the temperature range of interest ($\frac{\partial \varepsilon_{\lambda,T}}{\partial T} \approx 0$), thereby attributing variations in the signal-to-noise ratio to the transient temperature and emissivity (or reflectivity) levels.

Since the Levenberg-Marquardt algorithm does not allow bound constraints on unknown parameters, a practical approach is to formulate the FEM's emissivity parameters as $\varepsilon_{\lambda,j} = \exp(-|\beta_{j+1}|) \forall j \in [1, N-1]$, thus limiting possible solutions to physical values between 0 and 1.

The accuracy of the inverse model's output is susceptible to numerous types of errors, necessitating simulations under both noisy and
<figure><img src="image_11.png" /><figcaption>Fig. 9. Normal spectral emissivity of various metals as a function of wavelength [27, 28].</figcaption></figure>

noise-free conditions. Simulations conducted without noise allow for quantifying the sum of two types of systematic error: (i) material-specific errors, which stem from discrepancies between the true emissivity behaviour of the target and the assumed trend (e.g., blackbody, greybody, polynomial, FEM), and (ii) model-induced errors, which arise from structural limitations of the chosen model. These issues include convergence behaviour, sensitivity to initial guesses, underlying parameter correlations, and the consequent ill-conditioning of the inverse problem that is highly sensitive to parameter dimensionality and the number of spectral observations. Monte Carlo simulations incorporating noise-corrupted data assess the conditioning under experimental noise disturbances through (iii) stochastic deviations from noise-free accuracies. In practice, such disturbances include thermal fluctuations, parasitic radiant contributions, atmospheric absorption, and calibration imperfections, which are further discussed in Section 4.2. Additionally, the influence of wavelength selection — whether experimental or optimal — and the number of observations will also be examined, as they can affect multiple error types simultaneously.

### 4.1.1. Radiometric measurements at experimental wavelengths

Table 1 summarises the relative temperature error obtained with various pyrometric techniques for $\lambda_{exp} = \{2.00, 2.35, 2.85, 4.00 \ \mu m\}$ under zero-noise conditions. It is worth noting that the mono-spectral method infers radiance temperatures under the blackbody assumption, whereas the bi-spectral method estimates both a surface temperature and a grey emissivity. Since these techniques contain fewer unknown parameters, their inputs require less data. The simulated measurements were thus acquired at wavelengths adjacent to the optimal values provided by the aforementioned analogous laws [13]: $\lambda_{exp}^{mono} = 2.35 \ \mu m$ and $\lambda_{exp}^{bi} = \{2.00, 4.00 \ \mu m\}$.

Additional simulations were conducted by incorporating a range of initial temperatures (400 to 2000 K) and emissivity guesses (0.10 to 1.00). Results revealed enhanced robustness when initial values were equal to or overestimated relative to the true values. Conversely, underestimating the temperature guess by at least 100 K led to a deterioration in the FEM's identification accuracy, with this divergence becoming more pronounced as the spectral configuration became increasingly ill-conditioned—when $\lambda_2$ was shifted from 2.60 µm to 2.85 µm and then to 2.35 µm. For the emissivity parameters, the variable transformation in the FEM rules out using an initial emissivity of 1.00, as it would produce a zero increment in the iterative minimisation process. However, using non-unit guesses does not appear to affect the accuracy of the inferred temperatures. The mono-, bi-spectral, and polynomial models remained insensitive to the initialisation choice of both parameters. Consequently, a standard initial guess of 1200 K for temperature and

1.00 for emissivity parameters was adopted across all proposed models, except for the FEM, where the emissivity was initialised at 0.95.

Least squares optimisation results showcase directly proportional near-linear trends across all techniques except the FEM centred at $\lambda_2 = 2.60 \ \mu m$. The latter yields amplified errors in the case of copper for lower temperatures, which is attributed to the configuration's sensitivity to the initial guess.

According to the procured errors, it appears that a universal emissivity formulation cannot be established for all samples. Furthermore, the compensation phenomenon between the temperature and emissivity parameters, which can simultaneously produce near-zero cost functions and erroneous estimations due to both underlying correlations and the emissivity models' limited domain of validity, renders the cost function value an unreliable criterion for identifying the appropriate emissivity assumption. However, the FEM configuration centred at $\lambda_2 = \lambda_{exp,2}$ or $\lambda_2 = \lambda_{exp,3}$ provides relative temperature errors under 9.07%, whereas the second-order polynomial reaches up to 21.36% across the temperature range.

Monte Carlo simulations were carried out by randomly sampling 100 simulated data, for a given wavelength, using the corresponding noise level contained in $e$. To mimic experimental conditions further, an additive heat flux bias was introduced to the data and taken into account in the analytical model. This bias corresponds to the radiant flux $I_{reflected,\lambda}(T_{env})$ emitted by sources at an ambient temperature of $T_{env} = 300$ K and reflected by the target surface. Fig. 10 displays the statistical mean and standard deviation of estimated temperatures for simulated values ranging between 800 and 1200 K.

Concerning the statistical mean, employing the FEM centred at $\lambda_2 = 2.35 \ \mu m$ leads to accuracies near-identical to noise-free simulations (similar to the mono- and bi-spectral approaches) except for copper at lower range temperatures due to insufficient signal-to-noise ratio. Shifting the central wavelength to $\lambda_2 = 2.85 \ \mu m$ results in higher discrepancies from the values in Table 1, with the drift attenuating as the surface temperature rises. As for the FEM centred at $\lambda_2 = 2.60 \ \mu m$ and the polynomial model, we perceive unpredictable deviations from zero-noise estimations throughout the temperature range for highly reflective materials (i.e. copper) that lead to low SNR, and diminishing drifts for weakly reflective surfaces (i.e. oxidised nickel) as a function of temperature.

The analysis of resulting standard deviations reveals a predominantly inverse proportional relationship with the surface temperature, which aligns with the findings in Section 2. Additionally, reducing the number of parameters, as is the case for the mono- and the bi-spectral methods, contributes to decreasing the stochastic error. For most temperature levels and each simulated sample, $\sigma_T$ is minimised — vis-à-vis other multi-spectral formulations — by positioning the emissivity parameters $\epsilon_{\lambda_j}$ at $\lambda_{model} = \{2.00, 2.35, 4.00 \ \mu m\}$, thus interpolating it only at the third measurement wavelength 2.85 µm. This confirms that the configuration favours a better conditioning of the matrix $X^T X$ and that the choice of wavelength parametrisation in Eq. (4), for a given set of operating wavelengths, plays a crucial role in attenuating the propagation of uncertainty, as established during the sensitivity analysis.

Both under zero-noise and noise-corrupted conditions, the FEM configuration centred at $\lambda_2 = \lambda_{exp,2}$, as well as the mono- and bi-spectral methods, exhibit stable convergence behaviour, with iteration counts largely unaffected by surface temperature and emissivity levels. In contrast, the FEM centred at $\lambda_2 = \lambda_{exp,3}$ shows increased sensitivity to noise, while the configuration at $\lambda_2 = 2.60 \ \mu m$ shows unstable convergence behaviour. Although the polynomial model benefits from reduced iteration counts at higher temperatures and emissivity, it consistently requires more iterations than other proposed methods.

Although the results are not contained in this paper for brevity, inverse temperature identifications involving unoxidised nickel and
<caption>**Table 1**</caption>

Temperature identification results for copper, stainless steel 321, and oxidised nickel 201 under zero-noise conditions and for $\lambda_{exp} = \{2.00, 2.35, 2.85, 4.00\mu m\}$. The highlighted value indicates the minimal error for a given temperature level.

<table><thead><tr><th rowspan="2">Sample</th><th rowspan="2">T<sub>real</sub>(K)</th><th colspan="3">Relative temperature estimation error (%)</th><th rowspan="2">2<sup>nd</sup>-order polynomial</th></tr><tr><th>Mono-spectral</th><th>Bi-spectral</th><th>FEM</th></tr><tr><td></td><td></td><td colspan="3">λ<sub>2</sub> = 2.35 µm</td><td></td></tr></thead><tbody><tr><td rowspan="3">Copper</td><td>800</td><td>28.60</td><td>6.85</td><td>5.87</td><td>5.44</td><td>5.03</td></tr><tr><td>1000</td><td>33.35</td><td>8.88</td><td>7.40</td><td>6.83</td><td>6.29</td></tr><tr><td>1200</td><td>37.48</td><td>11.16</td><td>9.07</td><td>8.27</td><td>7.53</td></tr><tr><td rowspan="3">Stainless steel 321</td><td>800</td><td>15.11</td><td>6.00</td><td>0.73</td><td>0.95</td><td>4.35</td></tr><tr><td>1000</td><td>18.19</td><td>7.76</td><td>0.91</td><td>1.76</td><td>5.43</td></tr><tr><td>1200</td><td>21.02</td><td>9.73</td><td>1.10</td><td>2.90</td><td>6.54</td></tr><tr><td rowspan="3">Oxidised nickel 201</td><td>800</td><td>3.22</td><td>2.72</td><td>3.93</td><td>4.64</td><td>5.19</td></tr><tr><td>1000</td><td>3.98</td><td>3.48</td><td>4.92</td><td>6.11</td><td>6.49</td></tr><tr><td>1200</td><td>4.73</td><td>4.31</td><td>5.97</td><td>7.85</td><td>7.80</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Mean relative surface temperature error and standard deviation at $\lambda_{exp}$ in the case of copper ((a) and (b)), stainless steel 321 ((c) and (d)), and oxidised nickel 201 ((e) and (f)) in semi-log scale.</figcaption></figure>

oxidised stainless steel 304 (see Fig. 9 for the corresponding spectra) established an emissivity threshold $\epsilon_{\lambda} = 0.7$ below which the multi-spectral FEM formulation demonstrates higher adequacy compared to the emissivity assumptions that make up one- and two-colour pyrometry.

### 4.1.2. Radiometric measurements at optimal wavelengths

For this subsection, Monte Carlo simulations were performed on copper and oxidised nickel 201 at the mean temperature's optimal wavelengths, $\lambda_{opt}^{FEM} = \{1.80, 2.98, 3.37, 4.95\mu m\}$ and $\lambda_{opt}^{Poly} = \{1.65, 3.09, 4.26, 5.07\mu m\}$, computed in Section 2.3. Wien's analogous laws yield
<figure><img src="image_13.png" /><figcaption>Fig. 11. Mean relative surface temperature error and standard deviation at $\lambda_{opt}$ in the case of copper (a) and (b), and oxidised nickel 201 (c) and (d) in semi-log scale.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Relative temperature standard deviation computed with the Vcor matrix for various target samples, at optimal wavelengths and $\lambda_{model} = \{1.80, 3.37, 4.95 \mu m\}$, in semi-log scale.</figcaption></figure>

$\lambda_{opt}^{Mono} = 2.41 \mu m$ and $\lambda_{opt}^{Bi} = \{1.83, 4.47 \mu m\}$. Here, the wavelength-dependent noise level tainting the simulated data is assumed to remain unaltered.

The results presented in Fig. 11 entail that $\lambda_{model} = \{2.00, 2.35, 4.00 \mu m\}$ contributes to lower standard deviations on the temperature compared to $\lambda_{model} = \{1.80, 3.37, 4.95 \mu m\}$. This is attributed to the diminished effectiveness of the optimal wavelengths for highly reflective materials when derived under constrained optimisation conditions. The amplification rate of $\sigma_T$ caused by the latter showcases a quasi-constant value of 16.75 for copper and 1.13 for oxidised nickel according to the trends in Fig. 12, computed with the Vcor matrix. As for the polynomial formulation, utilising optimal measurements significantly reduces the stochastic component for both

samples. Logically, their efficiency at minimising $\sigma_T$ increases with the emissivity.

It is important to recognise that performing measurements at different sets of wavelengths also alters the systematic errors, as the true emissivity trend could shift further from the model's fitting capacities. This accounts for the higher mean errors observed for copper in the configuration $\lambda_{model} = \{1.80, 3.37, 4.95 \mu m\}$ compared to $\lambda_{model} = \{2.00, 2.35, 4.00 \mu m\}$. While the same can be said for the polynomial formulation at lower temperatures, the inferred trend displays fewer fluctuations than with $\lambda_{exp}$.

Mono- and bi-spectral pyrometry achieve accuracies comparable to Table 1 due to the proximity of their optimal wavelengths to experimental values as well as the low variation rate of the emissivity spectra in Fig. 9. Both methods, as well as the finite element model, exhibit a similar iteration count compared to measurements performed at $\lambda_{exp}$. Conversely, the polynomial model shows improved convergence.

### 4.1.3. Hyper-spectral radiometric measurements

Hyper-spectral measurements are distinguished from previously cited approaches by the extensive number of observations acquired throughout the spectral range, forming an almost continuous spectrum [29]. For thermographic purposes, hyper-spectral techniques can be a powerful tool for optimising the measurement accuracy [30–32].

This subsection delves into a preliminary investigation into the feasibility of the hyper-spectral approach. To do so, N measurements are performed at a simulated temperature of $T_{real} = 1000$ K over the range [2.00, 4.00 µm] with $\Delta\lambda = cte$, except for N = 4, where $\lambda_{exp} = \{2.00, 2.35, 2.85, 4.00 \mu m\}$. In this study, the absence of simulated measurement noise allowed for quantifying the systematic fitting adequacy of various multi-spectral models. These models correspond to a 4-parameter (i.e. $\beta = (T, \epsilon_{\lambda_1}, \epsilon_{\lambda_2}, \epsilon_{\lambda_3})$) and a 10-parameter (i.e. $\beta = (T, \epsilon_{\lambda_1}, \dots, \epsilon_{\lambda_9})$) FEM formulation, in addition to the second-order polynomial. Results procured with FEM structures incorporating 5 to 9 parameters were omitted, as they did not demonstrate a progressive reduction in error. A substantial increase in parameter dimensionality is therefore needed for a noticeable improvement in model performance.
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Absolute surface temperature error in the case of copper (a), stainless steel 321 (b), and oxidised nickel 201 (c) and with $T_{real}$ = 1000 K and under zero-noise conditions, in semi-log scale. Null temperature errors appear as missing data in (a) and (b).</figcaption></figure>

The temperature errors in Fig. 13 demonstrate that by increasing the number of observations while retaining a limited number of parameters, the surface temperature error fluctuates as the measurement wavelengths take on various values to retain $\Delta\lambda = cte$. The trend persists until it reaches a non-zero plateau, indicating the presence of a residual systematic bias such as the limited number of linear subfunctions in the case of the FEM. Moreover, adding observations can result in additional drifts from the simulated temperature compared with a Stoltz system [33], where the number of parameters is equal to the number of observations. These findings suggest that the hyperspectral approach can aggravate the ill-posed nature of the proposed models by introducing additional emissivity values that do not fit the piecewise linear or polynomial trend, thus generating higher residuals.

The convergence characteristic of the FEM generally improves with increased spectral observations, demonstrating a decreasing iteration count followed by a plateau, mirroring the trend observed in the inferred temperature errors. As for the second-order polynomial, the convergence displays a slightly less stable behaviour, with the iteration count reached at $N=100$ occasionally surpassing that of the ill-conditioned Stoltz system.

By increasing the dimensionality of the finite element model to include nine emissivity parameters $\epsilon_{\lambda_j}$, the systematic hurdle is successfully extinguished. In fact, a Stoltz system accurately describes the spectral emissivity of simulated materials, with systematic temperature errors under $2 \times 10^{-7}$ K. Further refinement of the emissivity fitting (and thus the temperature identification) is achieved by incorporating more observations into the model, as the inferred trends show that utilising 20 equispaced wavelengths is sufficient for obtaining near-minimum errors below $10^{-10}$ K. While the iteration count is amplified for a Stoltz system with a higher number of parameters, the convergence is accelerated with additional spectral measurements.

## 4.2. Experimental results

Transient surface temperature measurements of three initially oxidised metallic target samples (copper, nickel, and cast iron) were conducted through multi-spectral thermography with the Cedip/FLIR® Titanium 560M infrared camera. Prior to measurements, each sample in Fig. 14 is polished with a machining centre and dry sanded on P800 grit sandpaper. Furthermore, the rear surface of the samples is drilled and equipped with a K-type thermocouple of diameter $D_{th}$ = $10^{-1}$ cm, providing a reference temperature for uncertainty quantification. The temperature gradient between the emitting surface and the thermocouple insertion depth ($d_{th}$ = 1.00 cm for copper and nickel, and $d_{th}$ = 0.50 cm for cast iron) is deemed negligible due to FlexPDE simulations showcasing low differences. Another potential source of error in thermocouple readings is the presence of alternating electromagnetic fields, which can heat the instrument's metallic components. To limit the penetration depth of the induced current to the sample's surface, we selected a high operating frequency of 230 kHz.

Temperature stability during open-air induction heating was quantified by measuring thermal fluctuations using the calibrated thermocouple prior to each radiometric measurement, to ensure consistency across spectral filters. During 30 min intervals, the samples demonstrated fluctuations within ±3 °C, which was deemed acceptable given the temperature scale.

The normal spectral emissivity levels of said samples are plotted in Fig. 15. The trends were procured at Saint-Gobain Research Paris through reflectivity measurements in two separate spectral bands. The first spectrometric quantitation was performed for wavelengths between 2.00 and 2.50 µm with the Lambda 900 UV/VIS/NIR spectrometer from PerkinElmer [34], containing a Peltier-cooled polycrystalline lead Sulphide (PbS) detector [34]. The second set of measurements concerned wavelengths ranging from 2.50 to 4.00 µm with the Mid-IR IntegratIR integrating sphere from PIKE [35], equipped with a deuterated triglycine sulphate (DTGS) detector. The spectrometric outputs of both devices are consistent at 2.50 µm and exhibit a similar extended
<figure><img src="image_16.png" /><figcaption>Fig. 14. From left to right: nickel (D = 2.5 cm, H = 1.9 cm), copper (D = 2.5 cm, H = 1.9 cm), and cast iron (D = 3 cm, H = 1 cm) samples after the heating procedure.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Normal spectral emissivity of oxidised target samples as a function wavelength, measured at room temperature after the heating procedure.</figcaption></figure>

trend in the case of nickel and cast iron. Nevertheless, the copper sample showcases a discontinuity at the said wavelength, followed by significant fluctuations. These outcomes are caused by non-uniform surface oxidation (see copper sample in Fig. 14) and the limited sample opening of both devices, which only allows us to illuminate a portion of the target's surface.

To quantify the reflected contribution $I_{reflected}$, two distinct approaches were tested beforehand on proposed methods. First, an inclusive technique in which the component explicitly appears in the analytical model, as given by Eq. (22). Second, a corrective approach [13] that estimates the offset between measured and target-emitted irradiances as a prior step to temperature identifications. This technique assumes that the parasitic irradiance remains constant between known initial and final surface temperatures $T_1$ and $T_2$, resulting in 8 unknown parameters (4 wavelength-dependent pairs of $(\epsilon_{\lambda_i,T}, K_i)$ in Eq. (24)) determined through weighted least-squares with the Levenberg-Marquardt algorithm. The estimated offset is subsequently subtracted from the measured data before utilising them as input for the inverse model.

$$ \forall i \in [1, 4] \quad \text{and} \quad \forall j \in [1, 2] : \phi_{\lambda_i}^{\text{meas}}(T_j) = \epsilon_i M_{\lambda_i}^0(T_j) + K_i \quad (24) $$

Based on the conducted comparative study, the inclusive technique consistently resulted in higher temperature accuracies across most pyrometric approaches. The exception was observed for the multi-spectral polynomial formulation, under which the corrective approach proved more effective.

Fig. 16 displays the initial and final thermal images provided by the metrological tool for $\lambda = 4.00$ µm during the induction heating process of the copper sample. The radiometric measurements were obtained by converting the digital levels using Eq. (23). Both images

showcase an increasingly pronounced fixed-pattern noise caused by non-uniformity corrections, which become less effective as the surface temperature deviates from the initial value. In addition to imaging artefacts, spatial deviations are also attributed to non-uniform surface conditions of the sample and various types of experimental noise, such as shot and flicker noise. The regions of interest (ROIs) presented in Fig. 16(a), along with others of varying sizes, were tested to identify pixels least affected by the fixed-pattern noise; however, no conclusive results were obtained under a non-uniform surface emissivity. Consequently, the findings of the experimental analysis will concern radiometric measurements averaged over the entire image due to the sheer amount of information per frame, monochromatic filter, and recording. The identified temperatures, along with the resulting temperature differences from the thermocouple reading, are presented in Fig. 17 for operating wavelengths identical to the numerical analysis. Given that the distance separating the object and the lens is sufficiently small ($d \approx 100$ cm, with $d$ being the optical path), we assume that the atmosphere is perfectly transparent. For higher object distances, the absorption spectrum of atmospheric gases in the infrared region has to be considered during the wavelength selection process.

Results procured with the FEM centred at $\lambda_2 = 2.60$ µm lead to unstable temperature errors that exceed $10^3$ K and are therefore excluded from this subsection.

The multi-spectral finite element model, primarily under the configuration $\lambda_{model} = \{2.00, 2.35, 4.00$ µm], yields the most accurate temperature estimations compared with other pyrometric techniques. It is noteworthy that when averaging the radiometric measurements across the focal-plane array, imaging artefacts introduce a bias that skews the mean heat flux, as though they represented additional non-zero measurement bias. The resulting deviations distort the identified temperature, an outcome that is further amplified by the FEM centred at $\lambda_2 = 2.85$ µm and the second-order polynomial, as predicted by the correlational analysis.

Experimentally, the temperature errors—notably in the case of the FEM centred at $\lambda_2 = 2.35$ µm—do not exhibit the near-linear trend perceived during numerical simulations. Assuming that the thermocouple readings are accurate, this mainly results from the temperature dependence of the fixed-pattern noise as well as the spectral emissivity, which affects the systematic bias and the reflected radiant contribution.

For each target sample, both the bi-spectral method and the FEM centred at $\lambda_2 = 2.85$ µm produce comparable temperatures to the mono-spectral technique as a result of estimated emissivity parameters adjacent to 1. As for the polynomial formulation, the inferred temperature differences exceed 15% for all samples and reach a maximum value of 45% in the case of nickel. Moreover, when the samples are at a near-constant temperature level towards the end of the measurement period, the deviation in the estimates attains higher values compared with the FEM. These considerable discrepancies are a result of high compensation rates between temperature and emissivity parameters, leading to spectral emissivity values that exceed unity while simultaneously minimising the cost function. This highlights another significant drawback to the polynomial model, as its parameters lack the physical significance that the finite element model possesses.

## 5. Conclusions and future perspectives

In this paper, non-contact infrared thermography was employed for high-temperature measurements on opaque surfaces using multispectral techniques to address the challenge of unknown optical properties. Two emissivity modelling approaches were evaluated: a piecewise linear finite element model (FEM) and a second-order polynomial. The FEM exhibited key advantages, including constrainable parameters reflecting the physical significance of the output emissivity, reduced underlying correlations, and greater flexibility in fitting spectral data through a finer discretisation and high-resolution spectral sampling.
<figure><img src="image_18.png" /><figcaption>Fig. 16. Measured irradiance (in W. m⁻². m⁻¹) with copper at initial (a) and final (b) frame and tested regions of interest, for λ = 4.00 µm and IT = {2000, 550, 200, 60 µs}.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 17. Experimental temperature identification results in the case of copper ((a) and (b)), cast iron ((c) and (d)), and nickel ((e) and (f)).</figcaption></figure>
Monte Carlo simulations incorporating spectrometric metallic emissivity spectra quantified the impact of a deliberately imperfect emissivity model on temperature accuracy. Numerical results demonstrated that the FEM consistently reduced both systematic and stochastic errors compared to the polynomial approach. Experimental validation on induction-heated metallic samples confirmed these findings, with the FEM yielding temperature uncertainties below 4.74%. In contrast, the polynomial model was more sensitive to uncorrected detector thermal drifts, leading to errors exceeding 15%.

These results also highlight the limitations of grey- or blackbody assumptions for non-ideal emitters. When emissivity exceeds 0.7, one-colour pyrometry remains reliable; however, for low-emissivity surfaces, the multi-spectral FEM significantly reduces bias. Nevertheless, in industrial processes requiring high-precision thermal monitoring, contact-based methods still outperform optical thermometry unless emissivity behaviour is well-characterised.

Future work will focus on experimental radiometric assessments in vacuum conditions, targeting low-emissivity materials. This effort aims to further validate the potential of multi- and hyper-spectral thermometry under a finite element emissivity formulation, particularly in scenarios where the radiance temperature deviates significantly from the true surface temperature.

### CRediT authorship contribution statement

**Kamal Ennass:** Validation, Methodology, Formal analysis, Conceptualization, Software, Investigation, Writing – original draft, Data curation. **Benjamin Remy:** Project administration, Funding acquisition, Writing – review & editing, Supervision, Resources, Validation, Investigation. **Vincent Schick:** Supervision, Writing – review & editing, Project administration, Investigation, Validation, Resources, Funding acquisition. **Johann Meulemans:** Validation, Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition, Investigation, Conceptualization.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Kamal Ennass reports financial support was provided by Saint-Gobain Recherche. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This material is based upon work carried out at the University of Lorraine within the framework of a CIFRE agreement, supported by the National Association of Research and Technology (ANRT), and established between the Laboratory of Energetics, Theoretical and Applied Mechanics (LEMTA) and Saint-Gobain Research Paris.

### Data availability

Data will be made available on request.

### References

[1] Ph. Hervé, Ph. Masclet, A. Lefevre, I. Gobin, Ultra-violet pyrometry, in: Proc. of Fourth Symp. on Temp. and Thermal Meas. in Ind. and Sci., 1990, pp. 315–325.

[2] Ph. Hervé, A. Lefevre, E. Darcissac, Contrôle d'un processus métallurgique par mesure simultanée de la température et des propriétés thermooptiques, in: Proc. of Journées S.F.T., 1990.

[3] G. Sutter, L. Faure, A. Molinari, N. Ranc, V. Pina, An experimental technique for the measurement of temperature fields for the orthogonal cutting in high speed machining, Int. J. Mach. Tools Manuf. 43 (7) (2003) 671–678.

[4] R.R. Corwin, A. Rodenburgh, Temperature error in radiation thermometry caused by emissivity and reflectance measurement error, Appl. Opt. 33 (10) (1994) 1950–1957.

[5] P. Hervé, A. Sadou, Determination of the complex index of refractory metals at high temperatures: Application to the determination of thermo-optical properties, Infrared Phys. Techn. 51 (3) (2008) 249–255.

[6] Th. Pierre, B. Rémy, A. Degiovanni, Microscale temperature measurement by the multispectral and statistic method in the ultraviolet-visible wavelengths, J. Appl. Phys. 103 (3) (2008).

[7] P.B. Coates, Multi-wavelength pyrometry, Metrologia 17 (3) (1981) 103–109.

[8] P.B. Coates, The last square approach to multi-wavelength pyrometry, High Temp. - High Press. 20 (4) (1988) 433–441.

[9] P.B. Coates, Wavelength specification in multi-wavelength pyrometry, High Temp. - High Press. 20 (4) (1988) 443–448.

[10] A. Araújo, Multi-spectral pyrometry—a review, Meas. Sci. Tech. 28 (8) (2017) http://dx.doi.org/10.1088/1361-6501/aa7b4b.

[11] Th. Duvaut, Comparison between multiwavelength infrared and visible pyrometry: Application to metals, Infrared Phys. Techn. 51 (4) (2008) 292–299.

[12] Th. Pierre, J.-C. Krapez, H.R.B. Orlande, Ch. Rodiet, D. Le Maux, M. Courtois, Ph. Le Masson, B. Lamien, Simultaneous estimation of temperature and emissivity of metals around their melting points by deterministic and Bayesian techniques, Int. J. Heat Mass Transf. 183 (2022) http://dx.doi.org/10.1016/j.ijheatmasstransfer.2021.122077.

[13] Ch. Rodiet, B. Remy, A. Degiovanni, Optimal wavelengths obtained from laws analogous to the wien's law for monospectral and bispectral methods, and general methodology for multispectral temperature measurements taking into account global transfer function including non-uniform emissivity of surfaces, Infrared Phys. Techn. 76 (2016) 444–454, http://dx.doi.org/10.1016/j.infrared.2016.04.006.

[14] J.-C. Krapez, T. Pierre, Measurements without contact in heat transfer: multi-wavelength radiation thermometry. Principle, implementation and pitfalls, in: Proc. of Metti 8 Advanced School: Thermal Meas. and Inverse Tech., 2019.

[15] F. Uriz, B. Remy, A. Degiovanni, O. Verseux, Prise en compte du biais de modèle dans le calcul des écart-types des paramètres identifiés par méthode inverse, in: Proc. of Journées S.F.T., 2010.

[16] J.V. Beck, K.J. Arnold, Parameter Estimation in Engineering and Science, John Wiley & Sons, 1977.

[17] M.N. Özisik, H.R.B. Orlande, Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, 2000.

[18] H.R.B. Orlande, O. Fudym, D. Maillet, R.M. Cotta, Thermal Measurements and Inverse Techniques, CRC Press, 2011.

[19] D. Petit, D. Maillet, Techniques inverses et estimation de paramètres. partie 1, Techniques de l'ingénieur, 2008.

[20] D. Petit, D. Maillet, Techniques inverses et estimation de paramètres. partie 2, Techniques de l'ingénieur, 2008.

[21] R.H. Byrd, M.E. Hribar, J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim. 9 (4) (1999) 877–900.

[22] T.F. Coleman, Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optim. 6 (2) (1996) 418–445.

[23] Fmincon, MathWorks, inc., 2024, https://uk.mathworks.com/help/optim/ug/fmincon.html. (Accessed 16 October 2024).

[24] Titanium 560M Documentation, Cedip/FLIR®.

[25] A. Quattara, Caractérisation du refroidissement par jet liquide impactant une plaque métallique à haute température : l'influence de la composition du fluide sur le flux extrait (Ph.D. thesis), University of Henri Poincaré - Nancy 1, 2009, p. 60.

[26] Y. Souhar, Caractérisation thermique de matériaux anisotropes à hautes températures (Ph.D. thesis), Institut National Polytechnique de Lorraine, 2011.

[27] Y.S. Touloukian, D.P. DeWitt, Thermal Radiative Properties: Metallic Elements and Alloys, Thermophys. Properties of Matter 7, 1970.

[28] J. Hameury, A. Koenen, B. Hay, J. Wu, Ulf. Hammerschmidt, E. Rafeld, E. Pennewitz, E. Turzó-András, R. Strnad, A. Blahut, Identification and characterization of new materials for construction of heating plates for high-temperature guarded hot plates, Int. J. Thermophys. 39 (2018) 1–17.

[29] H.F. Grahn, P. Geladi, Techniques and Applications of Hyperspectral Image Analysis, John Wiley & Sons, 2007.

[30] W. Devesse, D. De Baere, P. Guillaume, High resolution temperature measurement of liquid stainless steel using hyperspectral imaging, Sensors 17 (1) (2017) 91.

[31] D.-X. Qu, J. Berry, N. Calta, M. Crumb, G. Guss, M. Matthews, Temperature measurement of laser-irradiated metals using hyperspectral imaging, Phys. Rev. Appl. 14 (1) (2020).

[32] A. Gorevoy, A. Machikhin, A. Bykov, A. Kren, Optimization of data acquisition algorithm for temperature and emissivity distribution measurement using snapshot hyperspectral imaging systems, Case Stud. Therm. Eng. 26 (2021).
[33] J.V. Beck, B. Blackwell, C.S. Clair, Inverse Heat Conduction: ill-Posed Problems, John Wiley & Sons, 1985.

[34] Lambda 800 UV/VIS and 900 UV/VIS/NIR, PerkinElmer, 2024, https://www.equipx.net/uploads/Perkin%20Elmer/PerkinElmerLambda800-900.pdf. (Accessed 12 June 2024).

[35] PIKE technologies, 2024, https://norlab.fi/file/3338/download?token=Mpar0RKe. (Accessed: 12 June 2024).