## 3 Results and Discussion
### Uncertainty Propagation Analysis  
The microAeth\({}^{\circledR}\) AE51 assumes a constant attenuation cross section, so the first term of inside square root of Equation (5) goes to zero. The spot area also ought to be constant but it could be changed due to leakage from failed sealing on the contact between the filter and the conduit. The spot size can be measured using a magnifier or digital image analysis, but the uncertainty of the spot size was assumed to be 2% [7]. Uncertainty of the flow rate may ensue not only from the stability of a flow controller but also that of a pump. The uncertainty of a flow controller could be 1.5%, as assumed by [PERSON] et al. [7] and the uncertainty of a pump may be 5% [20]. We liberally assume the uncertainty of the flow rate as 5%, which includes only the uncertainty of a pump. The uncertainty of the measurement time should be zero. However, we noticed that a time lag occurred after the 10-day-long measurement. At the beginning of the test, the clock of the device was synchronized with the clock in the laptop  
Figure 11: Long term zero test using filter-cleaned particle free air. Black carbon (BC) concentration, attenuation (ATN) and Flow rate are shown together with battery status.  
computer used for data collection. At the end of the test, however, the clock of the device lagged behind 30 s. Thus, we can assume that the uncertainty of time is non zero, though it is a very small number. The time lag divided by the entire measurement duration (10 days) is calculated to be \(3.47\times 10^{-3}\%\). The last term of the inside square root of Equation (5) may dominate overall uncertainty.  
Here, we can assume \(\left(\frac{\delta I_{0}}{I_{0}}\right)_{t+\Delta t}\approx\left(\frac{\delta I _{0}}{I_{0}}\right)_{t}\) and \(\left(\frac{\delta I}{I}\right)_{t+\Delta t}\approx\left(\frac{\delta I}{I} \right)_{t}\) because the reference signal and sensing signal do not change much during the time interval especially when the sampling time is not long. Thus, the last term inside square root of Formula (5) can be written as below.  
\[2\left(\left(\frac{\delta I_{0}}{I_{0}}\right)_{t}\right)^{2}+2\left(\left( \frac{\delta I}{I}\right)_{t}\right)^{2} \tag{6}\]  
Incorporating Equation (5) and formula (6) gives us the following Equation.  
\[\frac{\delta BC}{BC}=\sqrt{\left(\frac{\delta\sigma_{ATN}}{\sigma_{ATN}} \right)^{2}+\left(\frac{\delta A}{A}\right)^{2}+\left(\frac{\delta Q}{Q} \right)^{2}+\left(\frac{\delta\Delta t}{\Delta t}\right)^{2}+2\left(\left( \frac{\delta I_{0}}{I_{0}}\right)_{t}\right)^{2}+2\left(\left(\frac{\delta I }{I}\right)_{t}\right)^{2}} \tag{7}\]  
The error of reference signal (\(\delta I_{0}\)/\(I_{0}\)) and that of sensing signal (\(\delta\)/\(I\)) can be estimated from the collected raw data. The errors of reference signal and sensing signal were calculated by dividing the standard deviations (1-\(\sigma\)) by the averages of the reference signal and sensing signal during entire measurement, respectively. The error of reference signal (\(\delta I_{0}\)/\(I_{0}\)) was estimated to be 0.3% and the error of sensing signal (\(\delta\)/\(I\)) was estimated to be 0.3% or 4.3% for the filtered clean air and the indoor BC, respectively. Then, the overall uncertainty for indoor BC is calculated to be 8%. The overall uncertainty for the filtered clean air must be smaller than 8% because the error of sensing signal for the filtered clean air is smaller than that for the indoor BC.