# Particle Size Distributions and Extinction Coefficients of Aerosol Particles in Land Battlefield Environments

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, and selected the optimal PSD function to evaluate their extinction coefficients in the laser wavelength range of 0.249~12 μm. The results showed that smoke screens, ground dust, and soil explosion dust exhibited particle size ranges of 0.7~50 µm, 1~400 µm, and 1.7~800 μm, respectively. The lognormal distribution had the best goodness of fit for fitting the PSDs of these aerosol particles in the six unimodal PSD functions, followed by the gamma and Rosin–Rammler distributions. For the bimodal aerosol particles with a lower span, the bimodal lognormal PSD functions exhibited the best goodness of fit. The graphite smoke screen exhibited the highest extinction coefficient, followed by the copper and iron powder smoke screens. In contrast, the ground dust and soil explosion dust exhibited the lowest extinction coefficients, reaching their minimum values at a wavelength of approximately 8.2 μm. This study provides a basis for analyzing and improving the detection and recognition performance of lasers in land battlefield environments.

## 1. Introduction

## 2. Methods

#### 2.1. Aerosol Particle Sampling

#### 2.2. PSD Measurement

#### 2.3. Fitting and Evaluation of PSD Functions

^{2}are frequently used to compare the goodness of fit of two or more functions. The RMSE and R

^{2}values can be expressed as follows [35]:

^{2}does not necessarily imply a better goodness of fit, as the degree of freedom also affects the R

^{2}value, which will increase if more parameters are introduced, but this does not imply a better goodness of fit [36]. Thus, we used the adjusted R

^{2}as a modified version of R

^{2}accounting for the degrees of freedom to compare the goodness of fit of the different PSD functions in this study. The adjusted R

^{2}can be expressed as follows [36]:

#### 2.4. Extinction Coefficient Calculation

^{3}∙µm

^{−3}can be given by [38,39]:

## 3. Results and Discussion

#### 3.1. PSDs of Aerosol Particles

#### 3.2. Goodness of Fit of PSD Functions

^{2}values of the PSD functions listed in Table 2 used for fitting the PSDs of the graphite smoke screen, copper powder smoke screen, iron powder smoke screen, ground dust, and soil explosion dust. The RMSE values of the exponential and Rayleigh distributions were higher than those of the lognormal, generalized gamma, gamma, and Rosin–Rammler distributions (Figure 6a). The RMSE values of the Rosin–Rammler distribution were higher than those of the gamma distribution. The RMSE values of the lognormal distribution were very close to those of the generalized gamma distribution, and their RMSE values were essentially the minimum among the six PSD functions. Because the RMSE is the average distance between the values predicted by the fitting curve and the measured values, taking values in the range [0, +∞), the lower the RMSE, the better the goodness of fit of the fitting function [44]. Thus, a comparison of the RMSE values indicated that the unimodal PSD functions with the optimal goodness of fit for fitting these aerosol particles were the lognormal and generalized gamma distributions, followed by the gamma and Rosin–Rammler distributions; the worst were the exponential and Rayleigh distributions.

^{2}values of the exponential and Rayleigh distributions were partially negative (Figure 6b), indicating that their goodness of fit was worse than the average line or that their PSD functions omitted a constant [45]. Their adjusted R

^{2}values were chiefly less than those of the lognormal, generalized gamma, gamma, and Rosin–Rammler distributions. The adjusted R

^{2}values of the Rosin–Rammler distribution were lower than those of the gamma distribution. The adjusted R

^{2}values of the lognormal distribution were very close to those of the generalized gamma distribution, and their adjusted R

^{2}values ranged from 0.775 to 0.997, which mainly represented the maximum among the six PSD functions. The fitting curves of the lognormal and generalized gamma distributions essentially coincided (Figure 7). The positive adjusted R

^{2}is a percentage of the response variable variation explained by the fitting function, and the closer R

^{2}is to 1, the better the goodness of fit of the fitting function. The results obtained by comparing the adjusted R

^{2}values were consistent with those obtained by comparing the RMSE values. Furthermore, the adjusted R

^{2}values of the lognormal and gamma distributions, when used to fit the aerosol particles exhibiting a unimodal distribution and a bimodal distribution with a high span value of 11.35, ranged from 0.956 to 0.997, which was remarkably close to 1. However, for aerosol particles with a bimodal distribution and low span values of no more than 4.0, the adjusted R

^{2}values of the lognormal and gamma distributions ranged from 0.775 to 0.852.

^{2}value ranges of bimodal lognormal probability density functions were 0.093~0.209%∙µm

^{−1}(Figure 8a) and 0.938~0.980 (Figure 8b), respectively. Comparing the RMSE and adjusted R

^{2}values between lognormal and bimodal lognormal PSD functions indicated that the bimodal lognormal PSD functions exhibited better goodness of fit than lognormal PSD functions for fitting aerosol particles with a bimodal PSD, excepting for SED-A with a higher span value of 11.354 (Figure 8).

#### 3.3. Extinction Coefficients of Aerosol Particles

^{−1}, respectively (Figure 11). Notably, at the same laser wavelength and volume concentration, the graphite powder smoke screen exhibited the highest extinction coefficient, followed by the copper powder and iron powder smoke screens, whereas those of the ground dust and soil explosion dust were the lowest. In other words, the attenuation effects of the graphite, copper powder, and iron powder smoke screens on the laser beam transmission were more significant than those of ground dust and soil explosion dust. Additionally, the extinction coefficients of ground dust and soil explosion dust showed similar trends with respect to wavelength variation: (a) in the UV, VIS, and NIR bands (0.249~1.4 μm), their extinction coefficients roughly increased with wavelength; (b) in the SWIR, MWIR, and LWIR bands (1.4~12 μm), their extinction coefficients exhibited oscillatory variations with wavelength and showed trough values of approximately 3, 6.2, and 8.2 μm, respectively; and (c) their extinction coefficients reached a minimum at a wavelength of approximately 8.2 μm, followed by the region near a wavelength of 6.2 μm. Therefore, when prioritizing the anti-interference capabilities of laser detection and identification in ground dust or soil explosion dust environments, laser devices operating at wavelengths near 6.2 μm and 8.2 μm should be the optimal choices.

## 4. Conclusions

^{2}, the lognormal, gamma, and Rosin–Rammler distributions were better suited for describing the PSDs of the aerosol particles, with their goodness of fit decreasing in this order. For the aerosol particles exhibiting bimodal PSDs with a lower span value (≤4.0), bimodal lognormal distributions had the best goodness of fit.

^{3}∙µm

^{−3}), the graphite smoke screen exhibited the highest extinction coefficient (0.717~0.932 µm

^{−1}), followed by the copper powder smoke screen (0.439~0.508 µm

^{−1}) and iron powder smoke screen (0.357~0.444 µm

^{−1}). In contrast, the extinction coefficients were lowest for the ground dust (0.081~0.339 µm

^{−1}) and soil explosion dust (0.061~0.227 µm

^{−1}). Specifically, these extinction coefficients reached their minimum values at a wavelength of 8.2 μm, followed by the region near a wavelength of 6.2 μm.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Ground dust sampling: (

**a**) physical diagram of ground dust sampling device; (

**b**) real scene of ground dust sampling.

**Figure 3.**Physical images of aerosol particle samples. GrSS, CuSS, and FeSS, respectively, represent the graphite, copper powder, and iron powder smoke screen samples; GD-Bt, GD-Fs, and GD-Xz, respectively, represent the ground dust samples collected from Baotou City, Fangshan District, and Xuzhou City, with soil types of loam, silt, and silt loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam.

**Figure 4.**Scanning electron microscope images of aerosol particle samples. GrSS, CuSS, and FeSS, respectively, represent the graphite, copper powder, and iron powder smoke screen samples; GD-Bt, GD-Fs, and GD-Xz, respectively, represent the ground dust samples collected from Baotou City, Fangshan District, and Xuzhou City, with soil types of loam, silt, and silt loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam.

**Figure 5.**Volume-based PSD graphics of reference aerosol particles: (

**a**) graphite smoke screen (GrSS); (

**b**) copper powder smoke screen (CuSS); (

**c**) iron powder smoke screen (FeSS); (

**d**) ground dust collected from Baotou City (GD-Bt, loam); (

**e**) ground dust collected from Fangshan District (GD-Fs, silt); (

**f**) ground dust collected from Xuzhou City (GD-Xz, silt loam); (

**g**) soil explosion dust collected from explosives blasted in silt (SED-A); (

**h**) soil explosion dust collected from explosives blasted in silt loam (SED-B); and (

**i**) soil explosion dust collected from explosives blasted in sandy loam (SED-C).

**Figure 6.**Heatmap of the RMSE and adjusted R

^{2}: (

**a**) RMSE; (

**b**) adjusted R

^{2}. LGN, GG, gamma, R-R, Exp, and Ra refer to the lognormal, generalized gamma, gamma, Rosin–Rammler, exponential, and Rayleigh distributions, respectively. GrSS, CuSS, and FeSS, respectively, represent the graphite, copper powder, and iron powder smoke screen samples; GD-Bt, GD-Fs, and GD-Xz, respectively, represent the ground dust samples collected from Baotou City, Fangshan District, and Xuzhou City, with soil types of loam, silt, and silt loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam.

**Figure 7.**Comparison between lognormal and generalized distributions for fitting reference aerosol particles: (

**a**) graphite smoke screen (GrSS); (

**b**) copper powder smoke screen (CuSS); (

**c**) iron powder smoke screen (FeSS); (

**d**) ground dust collected from Baotou City (GD-Bt, loam); (

**e**) ground dust collected from Fangshan District (GD-Fs, silt); (

**f**) ground dust collected from Xuzhou City (GD-Xz, silt loam); (

**g**) soil explosion dust collected from explosives blasted in silt (SED-A); (

**h**) soil explosion dust collected from explosives blasted in silt loam (SED-B); and (

**i**) soil explosion dust collected from explosives blasted in sandy loam (SED-C).

**Figure 8.**Comparisons of the RMSE and adjusted R

^{2}values between lognormal and bimodal lognormal PSD functions for fitting aerosol particles with a bimodal distribution: (

**a**) RMSE; (

**b**) adjusted R

^{2}. GD-Bt represents the ground dust samples collected from Baotou City with soil types of loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam.

**Figure 9.**Fitting curves of the lognormal, gamma, and Rosin–Rammler distributions for reference aerosol particles: (

**a**) graphite smoke screen (GrSS); (

**b**) copper powder smoke screen (CuSS); (

**c**) iron powder smoke screen (FeSS); (

**d**) ground dust collected from Baotou City (GD-Bt, loam); (

**e**) ground dust collected from Fangshan District (GD-Fs, silt); (

**f**) ground dust collected from Xuzhou City (GD-Xz, silt loam); (

**g**) soil explosion dust collected from explosives blasted in silt (SED-A); (

**h**) soil explosion dust collected from explosives blasted in silt loam (SED-B); and (

**i**) soil explosion dust collected from explosives blasted in sandy loam (SED-C).

**Figure 10.**Complex refractive indices of reference aerosol particles: (

**a**) real part depicted with a solid line; (

**b**) imaginary part depicted with a dashed line. UV, VIS, NIR, SWIR, MWIR, and LWIR, respectively, refer to ultraviolet, visible light, near-infrared, short-wave infrared, mid-wave infrared, and long-wave infrared. GrSS, CuSS, and FeSS represent the graphite, copper powder, and iron powder smoke screen samples, respectively.

**Figure 11.**Extinction coefficients of reference aerosol particles at the volume concentration of 1 µm

^{3}∙µm

^{−3}in the wavelength range of 0.249~12 µm. GrSS, CuSS, and FeSS, respectively, represent the graphite, copper powder, and iron powder smoke screen samples; GD-Bt, GD-Fs, and GD-Xz, respectively, represent the ground dust samples collected from Baotou City, Fangshan District, and Xuzhou City, with soil types of loam, silt, and silt loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam.

Classification | Wavelength Range/µm | Typical Laser Wavelength/µm | |
---|---|---|---|

Ultraviolet (UV) | 0.249~0.38 | 0.249 | |

Visible light (VIS) | 0.38~0.75 | 0.532, 0.694 | |

Infrared (IR) | Near-infrared (NIR) | 0.75~1.4 | 0.905, 1.06 |

Short-wave infrared (SWIR) | 1.4~3 | 1.55, 2 | |

Mid-wave infrared (MWIR) | 3~6 | 4.6 | |

Long-wave infrared (LWIR) | 6~12 | 10.6 |

**Table 2.**Probability density functions [24] of various PSDs commonly employed for aerosol particles.

Function Name | Probability Density Function | Scale Parameter | Shape Parameter | Shape or Location Parameter |
---|---|---|---|---|

Generalized Gamma | $f(x;\alpha ,\beta ,\theta )=\frac{\beta {x}^{\alpha \beta -1}}{{\theta}^{\alpha \beta}\mathsf{\Gamma}(\alpha )}{\mathrm{e}}^{-{\left(x/\theta \right)}^{\beta}}$ | $\theta $ | $\alpha $ | $\beta $ |

Gamma | $f(x;\alpha ,\theta )=\frac{{x}^{\alpha -1}}{{\theta}^{\alpha}\mathsf{\Gamma}(\alpha )}{\mathrm{e}}^{-x/\theta}$ | $\theta $ | $\alpha $ | $\beta =1$ |

Rosin–Rammler | $f(x;\beta ,\theta )=\frac{\beta {x}^{\beta -1}}{{\theta}^{\beta}}{\mathrm{e}}^{-{(x/\theta )}^{\beta}}$ | $\theta $ | $\alpha =1$ | $\beta $ |

Exponential | $f(x;\theta )=\frac{1}{\theta}{\mathrm{e}}^{-x/\theta}$ | $\theta $ | $\alpha =1$ | $\beta =1$ |

Rayleigh | $f(x;\gamma )=\frac{x}{{\gamma}^{2}}\mathrm{exp}\left(-\frac{{x}^{2}}{2{\gamma}^{2}}\right)$ | $\theta =\sqrt{2}\gamma $ | $\alpha =1$ | $\beta =2$ |

Lognormal | $f(x;\mu ,\sigma )=\frac{1}{\sqrt{2\pi}\sigma x}\mathrm{exp}\left[-\frac{{(\mathrm{ln}x-\mu )}^{2}}{2{\sigma}^{2}}\right]$ | $\sigma $ | $\alpha \to \infty $ | $\mu $ |

**Table 3.**PSD characteristic parameters of reference aerosol particles. GrSS, CuSS, and FeSS, respectively, represent the graphite, copper powder, and iron powder smoke screen samples; GD-Bt, GD-Fs, and GD-Xz, respectively, represent the ground dust samples collected from Baotou City, Fangshan District, and Xuzhou City, with soil types of loam, silt, and silt loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam. D

_{peak}refers to the diameter corresponding to the peak value of PSD; D

_{4,3}refers to the volume-weighted mean diameter; D

_{10}, D

_{50,}and D

_{90}represent the particle diameters at which particles smaller than these sizes account for the 10%, 50%, and 90% of the total volume, respectively.

Sample | Characteristic | ${\mathit{D}}_{\mathbf{peak}}$/µm | ${\mathit{D}}_{4,3}$/µm | ${\mathit{D}}_{10}$/µm | ${\mathit{D}}_{50}$/µm | ${\mathit{D}}_{90}$/µm | Span |
---|---|---|---|---|---|---|---|

GrSS | Unimodal | 5.87 | 5.56 | 2.70 | 5.09 | 9.16 | 1.27 |

CuSS | Unimodal | 8.82 | 6.95 | 5.12 | 7.60 | 10.96 | 0.77 |

FeSS | Unimodal | 15.17 | 10.63 | 4.07 | 10.42 | 22.42 | 1.76 |

GD-Bt | Bimodal | 11.56, 88.58 | 69.71 | 6.74 | 49.84 | 168.81 | 3.25 |

GD-Fs | Unimodal | 13.25 | 26.36 | 5.71 | 14.85 | 57.43 | 3.48 |

GD-Xz | Unimodal | 44.94 | 32.82 | 8.69 | 26.26 | 66.54 | 2.20 |

SED-A | Bimodal | 26.11, 517.20 | 70.34 | 9.10 | 23.14 | 271.85 | 11.35 |

SED-B | Bimodal | 13.25, 200.00 | 106.57 | 7.20 | 63.60 | 261.35 | 4.00 |

SED-C | Bimodal | 34.25, 174.62 | 112.28 | 14.10 | 104.43 | 226.48 | 2.03 |

**Table 4.**Parameter values of PSD functions for fitting reference aerosol particles using the OriginPro 2021 software. The superscript of the parameter value represents its p-values for the t-test, and the p-value less than 0.0001 is omitted. GrSS, CuSS, and FeSS, respectively, represent the graphite, copper powder, and iron powder smoke screen samples; GD-Bt, GD-Fs, and GD-Xz, respectively, represent the ground dust samples collected from Baotou City, Fangshan District, and Xuzhou City, with soil types of loam, silt, and silt loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam.

Aerosol Particle | Lognormal $\left(\mathit{\mu},\mathit{\sigma}\right)$ | Generalized Gamma $\left(\mathit{\alpha},\mathit{\beta},\mathit{\theta}\right)$ | Gamma $\left(\mathit{\alpha},\mathit{\theta}\right)$ | Rosin–Rammler $\left(\mathit{\beta},\mathit{\theta}\right)$ |
---|---|---|---|---|

GrSS | (1.63, 0.50) | (138.49, 0.17, 2.23 × 10^{−12}) | (4.91, 1.10) | (2.57, 5.78) |

CuSS | (2.03, 0.31) | (137.65, 0.28, 1.72 × 10^{−7}) | (11.47, 0.68) | (3.91, 8.22) |

FeSS | (2.36, 0.75) | (7.14^{0.4}, 0.55^{0.1}, 0.33^{0.8}) | (2.63, 4.47) | (1.88, 12.53) |

GD-Bt | (3.46, 1.22) | [139.63, 0.07, (2.90 × 10^{−29})^{0.01}] | (1.36, 29.22) | (1.20, 43.45) |

GD-Fs | (2.71, 0.77) | [136.58^{0.1}, 0.11, (1.99 × 10^{−18})^{0.8}] | (2.54, 6.49) | (1.83, 17.60) |

GD-Xz | (3.25, 0.86) | (139.66, 0.10, 1.40 × 10^{−20}) | (2.17, 13.53) | (1.66, 31.57) |

SED-A | (3.15, 0.74) | [17.46, 0.35, (6.36 × 10^{−3})^{0.6}] | (2.73, 9.35) | (1.92, 27.02) |

SED-B | (3.24, 1.02) | [138.26, 0.09, (1.73 × 10^{−24})^{0.6}] | (1.61, 19.06) | (1.29, 35.26) |

SED-C | (4.30, 1.30) | [141.03, 0.07, (1.68 × 10^{−31})^{0.7}] | (1.10, 89.38) | (1.05, 102.6) |

**Table 5.**Parameter values of bimodal lognormal probability density functions for fitting the aerosol particles with a bimodal distribution. GD-Bt represents the ground dust samples collected from Baotou City with soil types of loam; SED-A, SED-B, and SED-C, respectively, represent the soil explosion dust samples collected from explosives blasted in silt, silt loam, and sandy loam.

Aerosol Particle | $\mathit{w}$ | ${\mathit{\mu}}_{1}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{\mu}}_{2}$ | ${\mathit{\sigma}}_{2}$ |
---|---|---|---|---|---|

GD-Bt | 0.25 | 2.45 | 0.62 | 4.48 | 1.76 |

SED-A | 1.00 | 3.26 | 0.80 | 6.25 | 0.01 |

SED-B | 0.54 | 2.58 | 0.68 | 5.30 | 0.52 |

SED-C | 0.55 | 3.53 | 0.96 | 5.16 | 0.34 |

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**MDPI and ACS Style**

Gao, L.; Chen, H.; Chen, G.; Deng, J.
Particle Size Distributions and Extinction Coefficients of Aerosol Particles in Land Battlefield Environments. *Remote Sens.* **2023**, *15*, 5038.
https://doi.org/10.3390/rs15205038

**AMA Style**

Gao L, Chen H, Chen G, Deng J.
Particle Size Distributions and Extinction Coefficients of Aerosol Particles in Land Battlefield Environments. *Remote Sensing*. 2023; 15(20):5038.
https://doi.org/10.3390/rs15205038

**Chicago/Turabian Style**

Gao, Lijuan, Huimin Chen, Guang Chen, and Jiahao Deng.
2023. "Particle Size Distributions and Extinction Coefficients of Aerosol Particles in Land Battlefield Environments" *Remote Sensing* 15, no. 20: 5038.
https://doi.org/10.3390/rs15205038