# A Traceable Spectral Radiation Model of Radiation Thermometry
## 3 The Inverse Model of Radiation Thermometry
### Derivation of the Inverse Model of Radiation Thermometry
#### 3.1.2 Scalar Inverse Model  
Due to spectral data loss at the sensor, most existing radiation thermometers and thermal imagers currently operate with radiances instead of spectral radiances and use the simplified scalar inverse model (Figure 10) and its derivatives. A scalar model is a generalization, used because the sensor reading is obtained as radiance (radiant power), which is a scalar number, rather than spectral radiance, which is defined as a spectrally distributed radiant power. The scalar inverse model can be obtained from the scalar direct model (16); that is, the direct model (14) with scalar influence parameters (\(\epsilon(\lambda)=\epsilon\), \(\tau(\lambda)=\tau\), etc.). This scalar replacement can only be correctly implemented in the direct model under simplification of constant (spectrally independent) influence parameters. The spectrally homogenous (grey) properties of the model, which are required for correct operation, limit the applicability of this model to GBRT. \[M_{meas}=\epsilon\tau\int M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda \Big{)}S(\lambda)d\lambda+\rho\tau\int M_{\text{env}}(\lambda)S(\lambda)d \lambda+\epsilon_{\text{atm}}\int M_{\text{BB,atm}}(T_{\text{atm}},\lambda)S( \lambda)d\lambda \tag{16}\]  
Figure 9: Theoretically ideal inverse of the direct model. Note that spectral integration is irreversible upon measurement. Therefore, measurement of spectral radiance is required for the ideal inverse model. Spectral integration was omitted from the direct model. Assuming an opaque surface, the raw emission of the object \(M_{\text{BB,obj}}\Big{(}T_{\text{obj}},\lambda\Big{)}\) can be considered as the emission of a black body of temperature \(T_{\text{obj}}\) (2). Simplifying for single (homogenous) atmospheric temperature \(T_{\text{atm}}\), the raw radiant contribution of the atmosphere can similarly be considered as the emission of a black body (10). Raw environmental reflection can be simplified as the single temperature emission of a black body \(M_{\text{env}}(T_{env},\lambda)\) as well; however, under real conditions, the incident environmental spectral radiance \(M_{\text{env}}(\lambda)\) is a sum of many incident radiances around the measured surface, determined according to the directional reflectivity characteristic of the observed object. The black body sensor temperature to radiance characteristic of a sensor \(M_{\text{BBS}}(T)\), obtained in Equation (13), can be expressed from the scalar direct model. Similarly, sensor sensitivity-weighted environmental radiance \(M_{\text{S,env}}\) can be expressed. \[M_{\text{meas}}=\varepsilon\tau M_{\text{BBS}}\Big{(}T_{\text{obj}}\Big{)}+ \rho\tau M_{\text{S,env}}+\varepsilon_{\text{atm}}M_{\text{BBS}}(T_{\text{atm}}) \tag{17}\]  
By expressing \(M_{\text{BBS}}\Big{(}T_{\text{obj}}\Big{)}\), the scalar inverse model of GBRT can be obtained (18). Parameters marked with the prefix \(i\) - (e.g., \(i\epsilon\)) are scalar input parameters of the inverse model, called instrumental parameters. These parameters--instrumental emissivity, instrumental reflectivity, instrumental (atmospheric) transmissivity and instrumental atmospheric emissivity--represent the scalar input settings of an instrument's mathematical model. \[M_{\text{BBS}}\Big{(}T_{\text{obj}}\Big{)}=\frac{M_{\text{meas}}-i\varepsilon_ {\text{atm}}M_{\text{BBS}}(T_{\text{atm}})}{i\varepsilon}-i\rho M_{\text{S,env }} \tag{18}\]  
The process flow chart of the scalar inverse model is presented in Figure 10.  
Figure 10: Model of radiation thermometry with the scalar inverse model. Parameters, beginning with \(i\) correspond to scalar instrumental parameters. Dotted lines represent transfer of non-radiation scalar data, such as temperature or calculated scalar equivalent of an influence parameter. The inverse black body radiance characteristic is a characteristic transformation between emitted radiance and temperature of a black body, which can be derived from [PERSON]’s law.