## 3 Uncertainty Simulation
### The Monte Carlo Simulation Model of Radiation Thermometry for Uncertainty Simulation  
In a Monte Carlo simulation, a high number of temperature measurements is simulated from deviated input parameters and statistically evaluated for the expected uncertainty of radiation thermometry. Built from the model of radiation thermometry (13), the result of uncertainty simulation, regardless of input parameter distribution, is the standard deviation of simulated temperature measurement errors (17), considering the guide to the expression of uncertainty in measurement--propagation of distributions using a Monte Carlo method [12].  
\[u_{meas}=std\big{(}RT\big{(}T_{obj},x\pm u_{\text{x}}\big{)}\big{)}\cdot k \tag{17}\]A partial uncertainty contribution of an individual parameter (e.g., \(u_{\text{meas},\epsilon}\)) can be simulated by deviating the individual parameter, e.g., emissivity \(\epsilon(\lambda)\pm u_{\epsilon}\) (18),  
\[u_{\text{meas},\epsilon}=k\sqrt{\frac{\sum_{i}^{n}\left(RT\left(T_{\text{obj} },S(\lambda),\epsilon(\lambda)\pm u_{\epsilon},T_{\text{refl}},T_{\text{air}}, RH,d,p\right)-T_{\text{obj}}\right)^{2}}{n-1}} \tag{18}\]  
Whereas the combined measurement uncertainty simulation simulates the distribution of all parameters simultaneously (19).  
\[u_{\text{meas}}=k\sqrt{\frac{\sum_{i}^{n}\left(RT\left(T_{\text{obj}},S( \lambda),\epsilon(\lambda)\pm u_{\epsilon},T_{\text{refl}}\pm u_{T_{\text{ rad}}},T_{\text{air}}\pm u_{T_{\text{far}}},RH\pm u_{\text{BL}},d\pm u_{ \epsilon},p\pm u_{p}\right)-T_{\text{obj}}\right)^{2}}{n-1}} \tag{19}\]  
Numerous model iterations of Monte Carlo analysis are required to properly account for parameter statistical distributions. In this research, simulations were performed with \(n\) = 10,000 iterations at each temperature point, simulated in a 1-18 \(\upmu\)m spectral range (further dependent on \(S(\lambda)\)) with 0.001 \(\upmu\)m spectral resolution.  
The Monte Carlo simulation process is depicted in Figure 5. Through statistical analysis of simulation results, uncertainty is calculated. Two main probability distribution functions are common for input parameters in the field of radiation thermometry, namely the normal (Gaussian, Figure 6A) and the uniform distribution (Figure 6B). The normal distribution is easily summarized with the standard deviation (or one \(\sigma\)). The uniform distribution can be further summarized with the uncertainty of uniform distribution or in the form of a boundary interval, which corresponds to a 100% coverage interval. The uncertainty can be calculated from the boundary interval by dividing the half-width by the square root of 3.  
Figure 5: A flow chart of Monte Carlo simulation of temperature measurement uncertainty due to input parameter uncertainty. Cylinders represent black body emissions, square factors multiplication, triangles summation and orange paper sheets represent the probability distribution function of influence parameters. Double-trailed lines represent spectral radiances, single-trailed lines represent scalar radiance values, whereas the interrupted lines specify parameter or temperature.  
The Monte Carlo method of uncertainty simulation was chosen for its ability to correctly consider the correlated effects of multiple simultaneous input parameter deviations and to provide the output probability distributions, caused by case-specific input parameters with various probability distributions. The Monte Carlo simulation is therefore beneficial over analytical methods in terms of accounting for covariances in multivariate systems and providing a probability distribution of the output; however, it has the drawback of high computational demand and noisy results, as evident later in the validation of emissivity.