## 3 Results
### Experiments  
We conduct an infrared radiation measuring experiment for the blackbody source in different temperatures. The blackbody source is settled horizontally 30 m away from the infrared thermal imager, with temperature increasing from 65 to 105 degrees C at intervals of 10 degrees C, as shown in Figure 6.  
Figure 5: Fitted straight line with the least square method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Temperature/\({}^{\circ}\)C** & **Radiance/W\(\cdot\)m\({}^{-2}\cdot\)str\({}^{-1}\)** & **DN Value** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Radiance and DN value collected in the radiometric calibration. The radiance calculated according to [PERSON]' law of radiation and corresponding DN value are listed in Table 4.  
According to Equations (17) and (18) the atmospheric transmittance \(\tau_{a\lambda}\) and atmospheric radiation \(\varepsilon_{a\lambda}f(T_{a})\) can be calculated through least square fitting in order to reduce error, values of which are approximately 88.8% and 0.0239 W/m\({}^{2}\)/sr. In comparison with MODTRAN, the atmospheric parameters used for the blackbody test are listed in Table 5.  
Input these parameters listed in Table 5 into MODTRAN, the atmospheric transmittance and atmospheric radiation are 83.9% and 0.0352 W/m\({}^{2}\)/sr, respectively. With all the parameters given above, the infrared radiation of targets can be calculated by Equation (12). The comparison of inversion results with the derivation from the analytical expression and with MODTRAN are given in Table 6. It is obvious that the inversion results derived by the expression in each row are closer to the actual radiance of the blackbody than that by MODTRAN, with the maximal error of 2.56% and 5.7% by the derivation from the analytical expression and by MODTRAN, respectively. To extend the verification to other tests, we also carried out a field test for a fan-shaped infrared target. Due to the relatively small size of the blackbody, we cannot acquire enough pixels over a long distance in some cases. The target with a bigger fan-shaped infrared target is used in the experiment, which consists of 10 pieces of heating plates, each with an 18-degree central angle, the radius of 2.25 m, and the surface emissivity of 0.52. The fan-shaped target is shown in Figure 7, with the aluminum box of the temperature controller. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Temperature/\({}^{\circ}\)C** & **Radiance/W\(\cdot\)m\({}^{-2}\)\(\cdot\)str\({}^{-1}\)** & **DN Value** \\ \hline
65 & 6.4034 & 4072 \\
75 & 8.4950 & 5298 \\
85 & 11.1051 & 6764 \\
95 & 14.3216 & 8605 \\
105 & 18.2395 & 11,207 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Radiance and DN value for the blackbody test. Figure 6: Blackbody source in field test with different temperatures in degree C: (**a**) 65; (**b**) 75; (**c**) 85; (**d**) 95; (**e**) 105. Two pieces of plates are chosen in the experiment, with the temperature increasing from 35 to 50 degrees C at intervals of 3 degrees C and with the ambient temperature of 28 degrees C, shown in Figure 8.  
The atmospheric parameters for the fan-shaped target test are listed in Table 7.  
The transmittance and atmospheric radiation are calculated by Equation (12) with values of 73.3% and 1.17 W/m\({}^{2}\)/sr, while by MODTRAN with values of 72.3% and 1.34 W/m\({}^{2}\)/sr, respectively. Table 8 shows the comparison of inversion results for the two hearting plates, indicating that for each row, the radiance derived from the analytical expression is also closer to the actual radiance than that from MODTRAN over a relatively long distance, with the maximal error of 10.2% and 16.7% by the derivation from the analytical expression and by MODTRAN, respectively.