Quantifying errors in the aerosol mixing-state index based on limited particle sample size

Figures & data

Figure 1. Frequency distribution of ${\chi }_{\text{samp}}^N$ for one example population (${\chi }_{\text{ref}}=46\%$) and different sample sizes. For each sample size, the sampling process was repeated 1000 times. The dashed colored lines correspond to the average ${\overline{\chi }}_{\text{samp}}^N$ for the specified sample size.

Figure 2. Distribution of sampled population mixing-state index ${\chi }_{\text{samp}}^N$ and reference mixing-state index χ_ref for increasing sample sizes based on the simulated scenario library described in Section 2.3. The solid black line is the 1:1 line.

Figure 3. Sampled ${\chi }_{\text{samp}}^N$ as a function of χ_ref from data obtained in Pittsburgh, PA. The one-to-one line is drawn for reference.

Figure 4. Distribution of average sampled ${\overline{\chi }}_{\text{samp}}^N$ (averaged over the 1000 repeats) and χ_ref for increasing sample sizes based on the simulated scenario library described in Section 2.3. The one-to-one line is drawn for reference.

Figure 5. Distribution of average sampled ${\overline{D}}_{\alpha ,\mathrm{s}\mathrm{amp}}^N$ (mean particle diversity, averaged over the 1000 repeats) and $D_{\alpha ,\text{ref}}$ (reference mean particle diversity) for increasing sample sizes based on the simulated scenario library described in Section 2.3. The one-to-one line is drawn for reference.

Figure 6. Distribution of average sampled ${\overline{D}}_{\gamma ,\mathrm{s}\mathrm{amp}}^N$ (total diversity, averaged over the 1000 repeats) and $D_{\gamma ,\text{ref}}$ (reference total diversity) for increasing sample sizes based on the simulated scenario library described in Section 2.3. The one-to-one line is drawn for reference.

Figure 7. 95%-Confidence intervals for sampled ${\chi }_{\text{samp}}^N$ values for sample sizes of N = 10, 100, 1000, and 10,000 particles based on the simulated scenario library described in Section 2.3.

Figure 8. Schematic to illustrate Theorem 3. Here we consider two sampled populations with α-diversities of $H_{\alpha ,1}$ and $H_{\alpha ,2},$ which we assume have average exactly equal to the reference value $H_{\alpha ,\mathrm{r}\mathrm{e}\mathrm{f}}$ (this is true on average, as we saw from Theorem 1). The exponential function maps entropies H to diversities D, and because it is convex the average sampled value, ${\overline{D}}_{\alpha ,\mathrm{s}\mathrm{amp}},$ will be greater than the reference value, $D_{\alpha ,\mathrm{r}\mathrm{e}\mathrm{f}}.$

Figure 9. Schematic to illustrate Theorem 4 in the case of a two-species aerosol population, where p is the mass fraction of the first species. We consider two sampled populations with first-species mass-fractions of p₁ and p₂, which we assume have average exactly equal to the first-species reference mass fraction of p_ref (this is exactly true on average by (43(43) $${\mathbb{E}}_{s\sim p_{\mathit{s},\text{tot}}}[\underset{X_{s,\text{tot}}}{\underbrace{{\mathbb{E}}_{i\sim p_{s,\mathit{i}}}[X_{s,i}]}}]={\mathbb{E}}_{I\sim p_{\mathit{I}}}\left[X_I\right].$$(43) )). The diversity function is concave (for 2 or 3 species) so the average sampled value, ${\overline{D}}_{\gamma ,\mathrm{s}\mathrm{amp}},$ will be less than the reference value, $D_{\gamma ,\mathrm{r}\mathrm{e}\mathrm{f}}.$

Data availability

The output of the particle-resolved modeling scenario library can be accessed at https://doi.org/10.13012/B2IDB-2774261_V1.