## 3 Analysis of Scale Effect of Simulation Data
### Effect of ACF Noise on Optimal IDS  
The noise level of ACF will be around 0.001, or even lower in the DLS measurement [46], so simulated experiments were performed at noise level of 0.0001, 0.005 and 0.001, respectively. Therefore, the simulated experiments were performed at noise level of 0.0001, 0.005, and 0.001, respectively. A large number of the simulation experiments show that, when IDS is bigger than 7 in WRIM, the inversion PSDs of the particles will be distorted. Thus, in the following experiments, we chose IDS 7 as the maximum. At three kinds of the noise levels, 0.0001, 0.005 and 0.001, respectively, the noisy ACFs of above two kinds of PSDs were inverted by WRIM. IDSs are used from 1 to 7 in the inversion. The inversion results are shown in Figure 5. Comparing _RE_ on various IDSs, the PSD recovered at the minimum _RE_ is optimal, and the corresponding scale is the optimal IDS. The inversion data of two kinds of PSDs are shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**IDS**} & \multicolumn{2}{c}{**Unimodal Distribution Particles**} & \multicolumn{2}{c}{**Bimodal Distribution Particles**} \\ \cline{2-5}  & **Peak Value (nm)** & **RE** & **Peak Value (nm)** & **RE** \\ \hline
1 & 434.11 & 0.6129 & —, — & 0.6174 \\
2 & 501.30 & 0.5261 & 197.94, 526.84 & 0.5485 \\
3 & 458.40 & 0.5207 & 196.72, 459.41 & 0.5511 \\
4 & 449.96 & 0.3471 & 205.79, 579.47 & 0.3334 \\
5 & 451.36 & 0.3598 & 193.96, 502.70 & 0.4668 \\
6 & 452.81 & 0.3257 & 188.89, 479.82 & 0.4998 \\
7 & 439.28 & 0.3963 & 222.10, 487.17 & 0.8255 \\
8 & 437.74 & 0.4051 & 217.95, 460.08 & 1.1538 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Inversion data of the unimodal and bimodal distribution particles on the different IDSs. Figure 5: _Cont_. From Figure 5, we can find that, with the increase of the noise level, the optimal IDS shows a downward trend. When the noise level is 0.0001, the optimal IDS of 300 nm\(\sim\)700 nm particles is 7. However, when the noise levels add to 0.0005 and 0.001, respectively, their optimal IDSs are reduced to 6 and 6. For 100 nm-700 nm particles, the noise levels are gradually added to 0.0001, 0.0005, and 0.001, respectively. Their corresponding optimal IDSs also show a downward trend, composed of 5, 4, and 4, respectively. The reason for above phenomenon can be interpreted from the following inversions of 100 nm-700 nm bimodal particles. In the inversion ranges of [120 nm, 1000 nm] and [100 nm, 700 nm], respectively, we inverted for ACFs with the noise level of 0.0001 and 0.001 in the space of scale 1. The inversion results are shown in Figure 6. From the results, we can see, at the low noise level 0.0001, when the inversion range is close to the true range of [100 nm, 700 nm], the inversion result is closer to the true PSD. On the contrary, when the inversion range is increased to [120 nm, 1000 nm], the error of its inversion PSD also increases. However, at the greater noise level 0.001, it is difficult to get the correct inversion result in the true inversion range of [100 nm, 700 nm]. Conversely, the result with greater inversion range [120 nm, 1000 nm] is more accurate. From the above inversion, we can draw a conclusion that the inversion of the greater noise data demands the greater inversion range. When IDS is large, it means that the inversion range of scale 1 becomes too small after multiple sub-inversions. Generally speaking, it is difficult to get the correct inversion result in the case of high noise. Therefore, ACF contains the higher noise level, so IDS should be smaller. In other words, the optimal scale of WRIM usually shows a downward trend with the increase of data noise.