# A Traceable Spectral Radiation Model of Radiation Thermometry
## 4 Results
### Effective Parameters as Instrumental Settings for the Scalar Inverse Model  
The equilibrium equation (\(\epsilon(\lambda)+\tau(\lambda)+\rho(\lambda)=1\)) is in practice usually extended to scalar operators (\(i\varepsilon+i\tau+i\rho=1\)). This simplification is correct only if the measured surface is an ideal (spectrally homogenous) grey body or if the spectral distributions of the signals involved are equal, i.e., in thermal equilibrium.  
Under real conditions, the effective scalar equivalent value of a spectral parameter is determined by the spectral distribution of the spectral parameters, the signal, and the spectral sensitivity of the instrument. Since the spectral parameters are represented according to the signals involved, the effective scalar parameters are independent and therefore the sum of the three instrumental parameters is not necessarily equal to 1 when measured under real conditions. The effective and mean scalars of the emissivity and reflectivity of the object are shown in Figure 15, where each of the parameters is expressed as a function of the influenced value, in this case the temperature of the emitting object and the temperature of the environment, assuming thermal equilibrium of the environment, which is necessary to satisfy the assumption that the radiance is distributed according to [PERSON]'s law.  
Figure 14: Theoretical error of grey body (GBRT) and real body (RBRT) radiation thermometry models, using inverse model (**a**) with constant parameters, spectrally averaged from 8 to 14 μm at various environmental temperatures; and (**b**) effective parameters, weighted to the signal and the instrument in accordance with the hybrid inverse model. The model error of the hybrid model does not change with environmental temperature and can be easily further decreased by improving the resolution of the inverse characteristic function, obtained as function fit in Figure 11.  
Unlike in the theory of radiation thermometry for grey bodies, the sum of effective parameters in the radiation thermometry of real objects is not necessarily equal to 1. This is evident in Figure 16, where the sum of the two effective parameters is plotted as a function of the object and environmental temperature.  
Figure 16: Sum of effective scalar emissivity and reflectivity with (**a**) object and (**b**) environmental temperature. Multiple plots show variation of (**a**) environmental and (**b**) object temperature.  
Figure 15: Shown are (**a**) the spectral emissivity and spectral reflectivity of the measured surface; (**b**) the spectrally averaged (8–14 μm) and effective emissivity as a function of the temperature of the measured object; and (**c**) the spectrally averaged (8–14 μm) reflectivity and effective reflectivity as a function of the environmental temperature, assuming pure emission according to [PERSON]’s law, e.g., when the environment has (a single) homogeneous temperature. Thermally non-homogenous environments are expected under real conditions, in which case, spectral radiance of the environment can be either measured or theoretically derived by summation of emission for this calculation method.  
The transmissivity was set to 0 and omitted for opaque bodies. Considering that the model error, obtained from the plotted effective parameters in Figure 14b was negligible, the effective parameters are considered valid.  
The sum of the effective atmospheric emissivity and the transmissivity at fixed atmospheric temperature (Figure 17) also deviates from 1, but the effect is less pronounced.