## 2 Formation of the Model of Radiation Thermometry
### The Direct Model of Radiation Thermometry  
A direct (physical) model of radiation thermometry, based on a real process and its hypothetical inverse, is depicted in Figure 2. This model would be ideal for the construction of a mathematical inverse; however, the spectral radiance measurement is practically not achievable. An object with thermodynamic temperature \(T\) emits spectral radiation \(M_{\text{BB}}(T,\lambda)\) according to [PERSON]'s law (1), where \(h\) is the [PERSON] constant, \(k_{B}\) is the [PERSON] constant, \(c\) is the speed of light in the medium and \(\lambda\) is the wavelength. \[M_{\text{BB}}(T,\lambda)=\frac{2 hc^{2}}{\lambda^{5}}\bigg{[}exp\bigg{(}\frac{hc }{\lambda k_{B}T}\bigg{)}-1\bigg{]}^{-1} \tag{1}\]  
Cylindrical operators in Figure 2 correspond to individual emission according to [PERSON]'s law. In the case of environmental radiation, this emission is simplified to a homogeneous emission of a single temperature, assuming a homogenously heated environment. Square and triangular operators correspond to multiplication with (spectral) parameters, functions or summation, accounting for inevitable influences in the practice of radiation temperature measurement. According to the equilibrium Equation (2), the spectral parameters in rectangular operators are traditionally bound to sum to 1 at all wavelengths. \[\varepsilon(\lambda)+\tau(\lambda)+\rho(\lambda)=1 \tag{2}\]  
Influence parameters are spectral emissivity \(\varepsilon(\lambda)\), spectral reflectivity \(\rho(\lambda)\) and spectral transmissivity \(\tau(\lambda)\). In case of absorption, [PERSON]'s law dictates that absorptivity equals emissivity. Instead of absorptivity, this research uses relative spectral sensitivity \(S(\lambda)\). Upon interaction with the medium, spectral radiation is scaled according to the corresponding spectral influence parameter (3), e.g., the object's black body emission \(M_{obj}\) is scaled by the spectral emissivity, the spectral radiance of the object's background  
Figure 2: The direct model of radiation thermometry, converting object temperature to radiance, detected by the sensor. Cylinders represent black body emissions, square factors multiplication, triangles summation and orange paper sheets represent the probability distribution function of influence parameters. Double-trailed lines represent spectral radiances, single-trailed lines represent scalar radiance values, whereas the interrupted lines specify parameter or temperature. is transmitted according to spectral transmissivity and environmental spectral radiation \(M_{\text{env}}\) is reflected according to spectral reflectivity. \[M_{\text{obj}}(\lambda)=\underbrace{M_{\text{obj}}\Big{(}T_{\text{obj}},\lambda \Big{)}\varepsilon(\lambda)}_{\text{object emission}}+\underbrace{M_{\text{bg}}(\lambda)\ \tau(\lambda)}_{\text{transmission}}+\underbrace{M_{\text{env}}(\lambda)\ \rho(\lambda)}_{\text{reflection}} \tag{3}\]  
Object's actual emission is the black body radiation, scaled by the spectral emissivity \(\varepsilon(\lambda)\) of an object. Meanwhile, the reflection of the environment is environmental radiation, scaled by spectral reflectivity \(\rho(\lambda)\) of an object. Considering the opaque surface of measurement, transmission of background radiation is set as zero and omitted. Similarly, for translucent materials, the combined radiation further travels through the air and is influenced according to the spectral transmissivity \(\tau(\lambda)\), the spectral radiance \(M_{atm}\) and the spectral emissivity \(\varepsilon_{air}(\lambda)\) of the air in a radiation transfer path. Approaching the sensor, the spectral sensitivity \(S(\lambda)\) of a detector determines the scale of absorption of the constructed spectral radiation signal (4).