# A Traceable Spectral Radiation Model of Radiation Thermometry
## 3 The Inverse Model of Radiation Thermometry
### Derivation of the Inverse Model of Radiation Thermometry
#### 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). \[ \begin{split} i\varepsilon=\varepsilon_{\text{eff}}\big{(}M_{ \text{BBS}}\big{(}T_{\text{obj}}\big{)}\big{)},\,\rho=\rho_{\text{eff}}(M_{ \text{S,env}})\\ i\tau=\tau_{\text{eff}}\big{(}M_{\text{obj}}(\lambda)\big{)},\, i\varepsilon_{\text{atm}}=\varepsilon_{\text{eff,atm}}(M_{\text{BBS}}(T_{\text{atm}}))\end{split}  \tag{19}\]  
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  
Figure 11: The inverse characteristic function of a VOx sensor. The characteristic is obtained by inverting the characteristic function of the sensor from Equation (13) and Figure 6.  
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 \(S(\lambda)\). 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 [PERSON]'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. \[M_{meas}=\int\Big{(}\Big{(}M_{\text{BB,obj}}(\lambda,T)\varepsilon(\lambda)+ M_{\text{env}}(\lambda)\rho(\lambda)\Big{)}\tau_{\text{eff}}+M_{\text{BB, atm}}(\lambda,T)\varepsilon_{\text{eff,atm}}\Big{)}S(\lambda)d\lambda \tag{20}\]  
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.