# Machine Learning to Predict the Global Distribution of Aerosol Mixing State Metrics

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Mixing State Metric $\chi $

#### 2.2. Particle-Resolved Aerosol Modeling

#### 2.3. GEOS-Chem-TOMAS Dataset

#### 2.4. Design of the Training and the Testing Scenarios

#### 2.5. Machine Learning as Applied to PartMC

_{2}] < 30 ppb)?”, and if this is false then it outputs $\chi =0.8$. A depth-n tree allows up to n-way interactions between feature variables.

## 3. Results

#### 3.1. Predicting $\chi $ for the Bulk Aerosol Population

#### 3.2. Predicting $\chi $ for Individual Size Bins

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of aerosol mixing states for four different aerosol populations that have the same bulk composition. The blue and red color represent aerosol species with different hygroscopicity: (

**a**) fully external mixture; (

**b**,

**c**) intermediate mixing states; and (

**d**) internal mixture. The mixing state metric $\chi $ measures the degree of internal mixing, ranging from 0% to 100%.

**Figure 2.**Global distribution of fraction of hygroscopic species as simulated by GEOS-Chem-TOMAS for the month of January for particles of $\sim 358$ nm.

**Figure 3.**Relative error in CCN concentration when neglecting aerosol mixing state as a function of aerosol mixing state index $\chi $. Each dot represents an aerosol population from Ching et al. [14]. CCN concentration was evaluated at a supersaturation of 0.6%.

**Figure 4.**Schematic of the learning architecture used to train, test, and use the machine-learning model.

**Figure 5.**(

**Left**) Mean error in the predicted $\chi $ values from the testing data set as a function of tree depth for the gradient boosted regression tree model; and (

**Right**) true $\chi $ values versus model-predicted values for our final model (corresponding to depth 8 in the left panel).

**Figure 6.**Global distribution of $\chi $ from the machine-learning model, at 06:00 UTC on January 1 (

**left**), and June 1 (

**right**), 2010. The model used to predict $\chi $ here is trained on the PartMC output that includes the entire particle population.

**Figure 7.**Global distribution of size-resolved $\chi $ values from the machine-learning model based GEOS-Chem-TOMAS inputs for the months of: January (

**top**); and July (

**bottom**). (

**Left**) $\chi $ for size bin 8, bin median diameter is 89.4 nm. (

**Right**) $\chi $ for size bin 14, bin median diameter is 2024 nm. The colored boxes show the regions over which data were averaged for display in Figure 9.

**Figure 8.**Global distribution of fraction of hygroscopic species as simulated by GEOS-Chem-TOMAS for the months of: January (

**top**); and July (

**bottom**). (

**Left**) $\chi $ for size bin 8, bin median diameter is 89.4 nm. (

**Right**) $\chi $ for size bin 14, bin median diameter is 2024 nm.

**Figure 9.**Size-resolved values of $\chi $ forselected regions in: January (

**left**); and July (

**right**). See Figure 7 for the location of these regions.

Quantity | Meaning |
---|---|

$\mu}_{i}^{a$ | Mass of species a in particle i |

${\mu}_{i}={\displaystyle \sum _{a=1}^{A}}{\mu}_{i}^{a}$ | Total mass of particle i |

${\mu}^{a}={\displaystyle \sum _{i=1}^{N}}{\mu}_{i}^{a}$ | Total mass of species a in population |

$\mu ={\displaystyle \sum _{i=1}^{N}}{\mu}_{i}$ | Total mass of population |

${p}_{i}^{a}=\frac{{\mu}_{i}^{a}}{{\mu}_{i}}$ | Mass fraction of species a in particle i |

${p}_{i}=\frac{{\mu}_{i}}{\mu}$ | Mass fraction of particle i in population |

${p}^{a}=\frac{{\mu}^{a}}{\mu}$ | Mass fraction of species a in population |

**Table 2.**Definitions of aerosol mixing entropies, particle diversities, and mixing state index. In these definitions, we take $0\mathrm{ln}0=0$ and ${0}^{0}=1$. This table is taken from Riemer and West [13].

Quantity | Name | Units | Range | Meaning |
---|---|---|---|---|

${H}_{i}={\displaystyle \sum _{a=1}^{A}}-{p}_{i}^{a}\mathrm{ln}{p}_{i}^{a}$ | Mixing entropy of particle i | — | 0 to $\mathrm{ln}A$ | Shannon entropy of species distribution within particle i |

${H}_{\alpha}={\displaystyle \sum _{i=1}^{N}}{p}_{i}{H}_{i}$ | Average particle mixing entropy | — | 0 to $\mathrm{ln}A$ | average Shannon entropy per particle |

${H}_{\gamma}={\displaystyle \sum _{a=1}^{A}}-{p}^{a}\mathrm{ln}{p}^{a}$ | Population bulk mixing entropy | — | 0 to $\mathrm{ln}A$ | Shannon entropy of species distribution within population |

${D}_{i}={e}^{{H}_{i}}={\displaystyle \prod _{a=1}^{A}}{\left({p}_{i}^{a}\right)}^{-{p}_{i}^{a}}$ | Particle diversity of particle i | Effective species | 1 to A | Effective number of species in particle i |

${D}_{\alpha}={e}^{{H}_{\alpha}}={\displaystyle \prod _{i=1}^{N}}{\left({D}_{i}\right)}^{{p}_{i}}$ | Average particle (alpha) species diversity | Effective species | 1 to A | Average effective number of species in each particle |

${D}_{\gamma}={e}^{{H}_{\gamma}}={\displaystyle \prod _{a=1}^{A}}{\left({p}^{a}\right)}^{-{p}^{a}}$ | Bulk population (gamma) species diversity | Effective species | 1 to A | Effective number of species in the bulk |

$\chi =\frac{{D}_{\alpha}-1}{{D}_{\gamma}-1}$ | Mixing state index | — | 0% to 100% | Degree to which population is externally mixed ( $\chi =0$%) versus internally mixed ($\chi =100$%) |

**Table 3.**Number concentration, ${N}_{\mathrm{a}}$, of the initial aerosol population. The aerosol size distributions are assumed to be lognormal and defined by the geometric mean diameter, ${D}_{\mathrm{g}}$, and the geometric standard deviation, ${\sigma}_{\mathrm{g}}$.

Initial/Background | ${\mathit{N}}_{\mathbf{a}}/{\mathbf{cm}}^{-3}$ | ${\mathit{D}}_{\mathbf{g}}$/$\mathsf{\mu}$m | ${\mathit{\sigma}}_{\mathbf{g}}$ | Composition by Mass |
---|---|---|---|---|

Aitken mode | 1800 | 0.02 | 1.45 | 49.64% ${\left({\mathrm{NH}}_{4}\right)}_{2}{\mathrm{SO}}_{4}$ + 49.64% SOA + 0.72% BC |

Accumulation mode | 1500 | 0.116 | 1.65 | 49.64% ${\left({\mathrm{NH}}_{4}\right)}_{2}{\mathrm{SO}}_{4}$ + 49.64% SOA + 0.72% BC |

**Table 4.**List of input parameters and their sampling ranges and procedures to construct the scenario library. See the main text for details.

Range | Sampling Method | |
---|---|---|

Environmental Variable | ||

RH | 10–100% | uniform within specified ranges ${}^{\left(1\right)}$ |

Latitude | 70${}^{\xb0}$ S–70${}^{\xb0}$ N | uniform |

Day of Year | 1–365 | uniform |

Temperature | based on latitude and day of year ${}^{\left(2\right)}$ | uniform |

Dilution rate | $1.5\times {10}^{-5}$ ${\mathrm{s}}^{-1}$ | constant |

Mixing height | 400 m | constant |

Gas phase emissions | ||

${\mathrm{SO}}_{2}$, ${\mathrm{NO}}_{\mathrm{x}}$, ${\mathrm{NH}}_{3}$, VOC | 0–100% of emissions in Riemer et al. [4] | non-uniform ${}^{\left(3\right)}$ |

Carbonaceous Aerosol Emissions (one mode) ${}^{\left(4\right)}$ | ||

${D}_{\mathrm{g}}$ | 25–250 nm | uniform |

${\sigma}_{\mathrm{g}}$ | 1.4–2.5 | uniform |

BC/OC mass ratio | 0–100% | non-uniform ${}^{\left(3\right)}$ |

${E}_{\mathrm{a}}$ | 0–1.6 × ${10}^{7}$ ${\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | non-uniform ${}^{\left(3\right)}$ |

Sea Salt Emissions (two modes) ${}^{\left(5\right)}$ | ||

${D}_{\mathrm{g},1}$ | 180–720 nm | uniform |

${\sigma}_{\mathrm{g},1}$ | 1.4–2.5 | uniform |

${E}_{\mathrm{a},1}$ | 0–1.69 × ${10}^{5}$ ${\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | non-uniform ${}^{\left(3\right)}$ |

${D}_{\mathrm{g},2}$ | 1–6 $\mathsf{\mu}$m | uniform |

${\sigma}_{\mathrm{g},2}$ | 1.4–2.5 | uniform |

${E}_{\mathrm{a},2}$ | 0–2380 ${\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | non-uniform ${}^{\left(3\right)}$ |

OC fraction | 0–20% | uniform |

Dust Emissions (two modes) ${}^{\left(6\right)}$ | ||

${D}_{\mathrm{g},1}$ | 80–320 nm | uniform |

${\sigma}_{\mathrm{g},1}$ | 1.4–2.5 | uniform |

${E}_{\mathrm{a},1}$ | 0–586,000 ${\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | non-uniform ${}^{\left(3\right)}$ |

${D}_{\mathrm{g},2}$ | 1–6 $\mathsf{\mu}$m | uniform |

${\sigma}_{\mathrm{g},2}$ | 1.4–2.5 | uniform |

${E}_{\mathrm{a},2}$ | 0–2380 ${\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ | non-uniform ${}^{\left(3\right)}$ |

hygroscopicity ($\kappa $) | 0.001–0.031 | uniform |

Bin Number | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|

Bin median diameter (nm) | 56.3 | 89.4 | 142 | 225.3 | 357.7 | 567.8 | 901.4 | 2024 | 6424 |

${R}^{2}$ | 19.68% | 65.31% | 79.68% | 87.63% | 90.87% | 89.45% | 81.87% | 70.94% | 36.42% |

Mean error | 12.55% | 9.13% | 7.21% | 5.99% | 5.16% | 5.51% | 6.91% | 8.64% | 11.86% |

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**MDPI and ACS Style**

Hughes, M.; Kodros, J.K.; Pierce, J.R.; West, M.; Riemer, N.
Machine Learning to Predict the Global Distribution of Aerosol Mixing State Metrics. *Atmosphere* **2018**, *9*, 15.
https://doi.org/10.3390/atmos9010015

**AMA Style**

Hughes M, Kodros JK, Pierce JR, West M, Riemer N.
Machine Learning to Predict the Global Distribution of Aerosol Mixing State Metrics. *Atmosphere*. 2018; 9(1):15.
https://doi.org/10.3390/atmos9010015

**Chicago/Turabian Style**

Hughes, Michael, John K. Kodros, Jeffrey R. Pierce, Matthew West, and Nicole Riemer.
2018. "Machine Learning to Predict the Global Distribution of Aerosol Mixing State Metrics" *Atmosphere* 9, no. 1: 15.
https://doi.org/10.3390/atmos9010015