## 2 Formation of the Model of Radiation Thermometry
### Scalar Model of Radiation Thermometry  
Considering this spectral limitation of real sensors, a scalar model, depicted in Figure 3, was derived.  
This model operates under grey body principles, meaning it is only correct when spectrally uniform influence parameters are present. As the spectral radiance is spectrally integrated during measurement, only scalar values can be obtained for further use in the model. By operating with scalar values and spectral means of influence parameters, spectral interactions are omitted from the inverse model, resulting in small errors, which are usually neglected for the convenience of this method.  
The black body sensor characteristic function of a sensor (5) represents conversion between the temperature of a black body source \(T_{BB}\) and the black body to sensor radiance (BBS) \(M_{BBS}\), which is practically measured as the black body radiance \(M_{\text{BB}}\), accounting for the spectral sensitivity \(S(\lambda)\) of the sensor.  
\[M_{\text{BBS}}(T_{BB})=\int M_{\text{BB}}(T_{BB},\lambda)S(\lambda)d\lambda \tag{5}\]  
In its inverse form, the characteristic function (Figure 4) converts black-body to sensor radiance measurement \(M_{BBS}\) to black body temperature \(T\), accounting for the spectral sensitivity of the sensor when measuring emission of a black body according to [PERSON]'s law. This characteristic can be derived experimentally or theoretically from spectral sensitivity \(S(\lambda)\), using [PERSON]'s law.  
The operation of the scalar inverse model can be summarized with Equations (5) and (6), where \(i\epsilon\), \(ip\), \(i\epsilon_{\rm atm}\) and \(i\tau\) are input settings of the radiation thermometer, often referred to as instrumental parameters, \(M_{\rm meas}\) is measured radiance on the sensor, \(M_{\rm BBS}\) is black body-sensor characteristic radiance and \(T_{\rm BBS}\) is its inverse function, black body sensor radiance to temperature characteristic.  
\[T_{meas}=T_{\rm BBS}\bigg{(}\bigg{(}(M_{meas}-M_{BBS}(T_{atm})\cdot i\epsilon_ {atm})\cdot i\tau^{-1}-M_{BBS}(T_{env})\cdot i\rho\bigg{)}\cdot i\epsilon\bigg{)} \tag{6}\]  
Figure 4: Characteristic function of VOx-based microbolometer sensor.  
Figure 3: Scalar model of radiation thermometry. Note that rather than inverting, \(S(\lambda)\) is mathematically distributed through the inverse model to the black body to sensor (BBS) characteristic. 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.  
Although spectrally uniform influence parameters never occur in the real world, such a calculation method has long been used to simplify the calculation of temperature in radiation thermometry. For the purpose of spectral emissivity uncertainty contribution analysis, this model can be sufficient, when emissivity is expressed as grey (with spectrally independent, constant emissivity) with an associated value and uncertainty.