# The Use of Gaussian Mixture Models with Atmospheric Lagrangian Particle Dispersion Models for Density Estimation and Feature Identification

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Tracer Data

#### 2.2. Lagrangian Atmospheric Transport and Dispersion Model

#### 2.3. Simulations of CAPTEX

#### 2.4. Histogram Method

#### 2.5. Simulations of Volcanic Eruption

#### 2.6. Gaussian Mixture Model

## 3. Results

#### 3.1. Density Reconstruction for a Tracer Experiment

#### 3.2. Density Reconstruction for a Volcanic Eruption

#### 3.3. Feature Identification

#### 3.3.1. Object Based Statistics

#### 3.3.2. Identify Where and When Simulations Diverge

#### 3.3.3. Feature Tracking

## 4. Discussion

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ARL | Air Resources Laboratory |

BGMM | Bayesian Gaussian mixture model |

CAPTEX | Cross Appalachian Tracer Experiment |

GMM | Gaussian mixture model |

HYSPLIT | Hybrid Single-Particle Langrangian Integrated Trajectory model |

KDE | Kernel density estimator |

LPDM | Lagrangian Particle Dispersion Model |

probability density function | |

n | number of Gaussians used in a fit |

N | number of computational particles found in a defined volume |

VAAC | Volcanic Ash Advisory Center |

## Appendix A. Shot Noise

**Figure A1.**Examples of shot noise in the histogram method. The top row (

**a**–

**c**) shows normalized histograms (blue bars) of the number of points found in a volume. The black line shows the Poisson distribution with the same mean. The position of computational particles in the volume are shown in the second and third rows. Positions from three of the simulations are plotted with the different colors indicating the computational particles from different simulations. The first and second columns (

**a**,

**b**,

**d**,

**e**,

**g**,

**h**) are from 50 simulations of the runKB with output every 10 min from 9 August, 12:10 UTC to 13:00 UTC. The first column (

**a**,

**d**,

**g**) looks at a smaller volume (${0.1}^{\circ}\times {0.1}^{\circ}\times $ 500 m) and only one particle size while the second column uses a volume of ${0.25}^{\circ}\times {0.25}^{\circ}\times $ 1 km and all the particle sizes. The last column (

**c**,

**f**,

**i**) is from 100 simulations of runB with output every 5 min from 25 September 18:00 UTC to 21:00 UTC and a volume of ${0.25}^{\circ}\times {0.25}^{\circ}\times $ 100 m. Although the histogram in (

**c**) looks somewhat flatter than the Poisson, this may be simply because 100 points is not enough to represent the distribution well.

## Appendix B. Setting the Number of Gaussians

**Figure A2.**Score, ${S}_{ij}$ as a function of number of Gaussians used in the GMM fit. Black and green lines indicate score for $i=j$ and red and blue lines indicate scores for $i\ne j$. (

**a**) RunC and RunD for CAPTEX1 at 09/20/1983 0 UTC. (

**b**) RunC and RunD for CAPTEX1 at 09/20/1983 03 UTC. (

**c**) RunKC and runKD at 08/09/2008 at 04 UTC. (

**d**) RunKC and RunKD at 08/10/2008 12 UTC.

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**Figure 1.**The top panels, (

**a**–

**c**) show particle positions at (

**a**,

**b**) 09/18/1983 21:00 UTC and (

**c**) 09/18/1983 from 21 to 24 UTC with particle positions output every 5 min. The Gaussian fits are shown in light blue and the colors of the particles indicate which cluster the particles were assigned to by the algorithm. The bottom panels, (

**d**–

**f**) show 3 h averaged concentrations calculated from the fits on a ${0.05}^{\circ}\times {0.05}^{\circ}$ 25 m grid from the ground to 25 m. A threshold of $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{pg}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$ has been applied and station measurements for the CAPTEX1 experiment are shown by the circles. (

**d**,

**e**) were calculated from separate fits to each time period and the panels above show the first such fit. (

**f**) was calculated from one fit to all time periods which is shown in (

**c**).

**Figure 2.**Comparison of density estimations. Color shows concentrations in pg m${}^{-3}$. Shown is a cross section of the simulated plume from the CAPTEX 1 experiment with station measurements shown by the circles. Simulated concentrations are 3 h averages on 09/18/1983 from 21 to 24 UTC, 3 h after the release. all figures have horizontal resolution of ${0.05}^{\circ}\times {0.05}^{\circ}\phantom{\rule{3.33333pt}{0ex}}\times $ 25 m except for the control runE, (

**d**) has resolution of ${0.25}^{\circ}\times {0.25}^{\circ}\times $100 m. Number of particles released per hour is shown at the top of each column. (

**a**–

**e**) are calculated using histogram method. (

**f**–

**k**) are calculated using a GMM with number of components annotated on the graph. Particles from the ground to 500 m were used in the fit. A threshold of 10 pg m${}^{-3}$ was applied to (

**f**–

**k**). (

**i**,

**j**) were calculated from fits to each time step in the averaging period, while (

**f**,

**g**,

**h**,

**k**) were calculated from a fit to all particles in the averaging period. (

**h**) and (

**k**) were calculated from Run C.

**Figure 3.**Statitics for CAPTEX experiments using a variety of density reconstructions. The rank, described by Equation (1) is shown in panel (

**a**). The four statistics which go into calculating rank are shown in panels (

**b**–

**e**) and the root mean square error, RMSE, is shown in (

**f**). Values for the standard simulation, runE, with standard concentration grid are shown by the black square. Values for concentrations calculated with the histogram and a high resolution grid are plotted slightly to the left of the black square. Values for concentrations calculated with the mixture model are plotted to the right of the black square. The high resolution grid of ${0.05}^{\circ}\times {0.05}^{\circ}\times $ 25 m was used unless otherwise noted in the legend. For the GMM, the time averaging was performed with a separate fit to each output time unless tmave = aggregate is noted in the legend.

**Figure 4.**Mass loading from simulations of the 2008 eruption of Kasatochi. The first and third rows (

**a**–

**c**,

**f**,

**h**,

**i**) display plots of mass loading with the simulation used and method of density reconstruction noted in the bottom right corner. The GMM used 50 Gaussians and output displayed at the same resolution as the histogram plots (${0.25}^{\circ}\times {0.25}^{\circ}$ grid). Directly below each mass loading plot is a histogram of the mass loading values with a threshold of 0.01 g m${}^{-2}$ applied. The histogram for (

**a**) is displayed as the red shaded region in each histogram plot for easy comparison.

**Figure 5.**Concentrations at longitude-151. Top row (

**a**–

**c**) concentrations calculated using histogram method resolution ${0.25}^{\circ}\times {0.25}^{\circ}\times 1000\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Second row (

**d**–

**f**) and third (

**g**–

**i**) row concentrations calculated using GMM with 50 Gaussians and plotted at resolution of ${0.1}^{\circ}\times {0.1}^{\circ}\times 200\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$. For the second row concentrations below below 0.01 g m${}^{-3}$ are not shown. The third row shows contour levels of output shown in second row and the black dots indicate position of computational particles within ${0.125}^{\circ}$ of the −151 line of longitude. The dots are shown smaller in (

**g**) for clarity. First column (

**a**,

**d**,

**g**) Run KA, second column (

**b**,

**e**,

**h**) Run KB and third column (

**c**,

**f**,

**i**) run KD.

**Figure 6.**Concentrations at longitude −145. Top row (

**a**–

**c**) concentrations calculated using histogram method. Second row (

**d**–

**f**) and third (

**g**–

**i**) row concentrations calculated using GMM with 50 Gaussians and plotted at the same resolution as the histogram method in the second row and the contours in the third row are for with a higher vertical resolution (${0.25}^{\circ}\times {0.25}^{\circ}\times $ 100 m). First column (

**a**,

**d**,

**g**) Run KA, second column (

**b**,

**e**,

**h**) Run KB and third column (

**c**,

**f**,

**i**) run KD. The black dots in the third row indicate position of particles within ${0.125}^{\circ}$ of the −145 line of longitude. The red dots indicate position of particles within ${0.5}^{\circ}$ of the −145 line of longitude.

**Figure 8.**log probability of each particle position. The left column shows a projection onto the latitude, longitude plane and the right column shows a projection onto the height, longitude plane. The date is noted at the top of each figure. ${S}_{ij}$ indicates that the probabilities were calculated using the particle positions j and the fit to particle positions i. For example, ${S}_{41}$ shows positions of the smallest particle size with how probable it is that they could belong to a fit to the largest particle size. Note the differing color scales. Low values indicate poor fit and regions where the overlap between the location of the two particle sizes is low.

**Figure 9.**The top two rows show features identified by a BGMM with 10 Gaussians at three times. The black stars show the centers of the Gaussians. The colored dots indicate the position of the computational particles. The different colors belong to different groups. The width of each Gaussian is indicated by the blue shaded regions. The top row is a projection onto the height-longitude plane while the second row is a projection onto the latitude-longitude plane. The bottom row shows the position of the centers of the 10 Gaussians every hour from 10 August 12 UTC to 11 August 11 UTC. The plot in the left corner is a projection onto the latitude-longitude plane while the plot on the right is a three dimensional rendering of the positions.

**Table 1.**Summary of HYSPLIT simulations. RunC and RunD are identical except for the random seed which was used in the simulation. ${C}_{\ell}$ is given in Equation (2). For the CAPTEX simulations, the number of particles is the number of particles released over the 3 h emission period. For the Kasatochi simulations the number of particles is approximately the total number of particles released over 8 h. The number of particles changed due to deposition.

CAPTEX Simulations | |||||
---|---|---|---|---|---|

RunID | Number of | Horizontal | Vertical | Note | ${\mathit{C}}_{\ell}$ |

Particles | Resolution | Resolution | $\mathrm{pg}\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{-\mathbf{3}}$ | ||

A | 250,000 | ${0.05}^{\circ}\times {0.05}^{\circ}$ | 25 m | 39 | |

B | 50,000 | ${0.05}^{\circ}\times {0.05}^{\circ}$ | 25 m | 196 | |

C | 5000 | ${0.05}^{\circ}\times {0.05}^{\circ}$ | 25 m | 1958 | |

D | 5000 | ${0.05}^{\circ}\times {0.05}^{\circ}$ | 25 m | SEED = −4 | 1958 |

E | 50,000 | ${0.25}^{\circ}\times {0.25}^{\circ}$ | 100 m | Standard run | 2 |

Kasatochi Simulations | $\mathrm{mg}\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{-3}$ | ||||

KA | 500,000 | ${0.25}^{\circ}\times {0.25}^{\circ}$ | 1 km | Reference run | 0.08 |

KB | 53,000 | ${0.25}^{\circ}\times {0.25}^{\circ}$ | 1 km | 0.75 | |

KC | 26,000 | ${0.25}^{\circ}\times {0.25}^{\circ}$ | 1 km | SEED = −4 | 1.5 |

KD | 26,000 | ${0.25}^{\circ}\times {0.25}^{\circ}$ | 1 km | SEED = −6 | 1.5 |

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**MDPI and ACS Style**

Crawford, A.
The Use of Gaussian Mixture Models with Atmospheric Lagrangian Particle Dispersion Models for Density Estimation and Feature Identification. *Atmosphere* **2020**, *11*, 1369.
https://doi.org/10.3390/atmos11121369

**AMA Style**

Crawford A.
The Use of Gaussian Mixture Models with Atmospheric Lagrangian Particle Dispersion Models for Density Estimation and Feature Identification. *Atmosphere*. 2020; 11(12):1369.
https://doi.org/10.3390/atmos11121369

**Chicago/Turabian Style**

Crawford, Alice.
2020. "The Use of Gaussian Mixture Models with Atmospheric Lagrangian Particle Dispersion Models for Density Estimation and Feature Identification" *Atmosphere* 11, no. 12: 1369.
https://doi.org/10.3390/atmos11121369