# 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
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., \(\tau(\lambda)\to\tau_{eff}\) in  
Figure 12: Hybrid inverse model utilizes effective parameter calculation according to spectral radiances in the system of measurement. Equation (21)). Independent contributions (e.g., emission of atmosphere) are equal on both sides of equation and can therefore be subtracted from expression. \[ \begin{array}{l}\int\left(M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda \Big{)}\,\varepsilon(\lambda)+M_{\text{env}}(\lambda)\rho(\lambda)\Big{)}\tau( \lambda)S(\lambda)d\lambda\\ =\tau_{\text{eff}}\int\left(M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda \Big{)}\,\varepsilon(\lambda)+M_{\text{env}}(\lambda)\rho(\lambda)\right)S( \lambda)d\lambda\end{array}  \tag{21}\]  
Expressing from Equation (21), effective transmissivity \(\tau_{\text{eff}}\) can be expressed. \[ \begin{array}{l}\tau_{\text{eff}}\Big{(}M_{obj},\tau(\lambda) \Big{)}=\frac{\int\left(M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda\Big{)} \,\varepsilon(\lambda)+M_{\text{env}}(\lambda)\rho(\lambda)\Big{)}\tau( \lambda)S(\lambda)d\lambda}{\int\left(M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda\Big{)}\,\varepsilon(\lambda)+M_{\text{env}}(\lambda)\rho(\lambda) \right)S(\lambda)d\lambda}\end{array}  \tag{22}\]  
For advanced use, effective transmissivity \(\tau_{\text{eff}}\) can be separated into two entities \(\tau_{\text{eff,obj}}\) (23) and \(\tau_{\text{eff,env}}\) (24), according to the direct model, each with individual contributions (15) corresponding to its own contributing emitter and spectral distribution of the affected radiance. \[\tau_{\text{eff,obj}}\Big{(}T_{\text{obj}}\Big{)}=\frac{\int M_{\text{BB, obj}}\Big{(}T_{\text{obj}},\lambda\Big{)}\,\varepsilon(\lambda)\tau(\lambda)S( \lambda)d\lambda}{\int M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda\Big{)} \,\varepsilon(\lambda)S(\lambda)d\lambda} \tag{23}\]  
\[\tau_{\text{eff,env}}\left(M_{\text{env}}(\lambda)\right)=\frac{\int M_{ \text{env}}(\lambda)\rho(\lambda)\tau(\lambda)S(\lambda)d\lambda}{\int M_{ \text{env}}(\lambda)\rho(\lambda)S(\lambda)d\lambda} \tag{24}\]  
Similarly, the effective atmospheric emissivity (25) can be calculated. \[\varepsilon_{\text{eff,atm}}\big{(}T_{\text{atm}}\big{)}=\frac{\int M_{\text{ BB,atm}}\big{(}T_{\text{atm}},\lambda\big{)}\,\varepsilon_{\text{atm}}( \lambda)S(\lambda)d\lambda}{\int M_{\text{BB,atm}}\big{(}T_{\text{atm}}, \lambda)S(\lambda)d\lambda} \tag{25}\]  
Continuing substitutions in Equation (20), the effective emissivity (26) and reflectivity (27) can be expressed. As effective transmissivity \(\tau_{\text{eff}}\) 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. \[\varepsilon_{\text{eff}}\Big{(}T_{\text{obj}}\Big{)}=\frac{\int M_{\text{BB, obj}}\Big{(}T_{\text{obj}},\lambda\Big{)}\,\varepsilon(\lambda)S(\lambda)d \lambda}{\int M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda\Big{)}S(\lambda)d\lambda} \tag{26}\]  
\[\rho_{\text{eff}}\big{(}M_{\text{env}}(\lambda)\big{)}=\frac{\int M_{\text{env }}(\lambda)\rho(\lambda)S(\lambda)d\lambda}{\int M_{\text{env}}(\lambda)S( \lambda)d\lambda} \tag{27}\]  
A rule can be derived for converting any spectral influence parameter to effective scalar value \(p(\lambda)\to p_{\text{eff}}\) (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 \(1\).