# Determination of Aerosol Size Distribution from Angular Light-Scattering Signals by Using a SPSO-DE Hybrid Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The principle of the ALSM

#### 2.2. The Principle of the SPSO

#### 2.3. The Principle of the SPSO-DE Hybrid Algorithm

**Step****1.**- Initialize the system control variables of SPSO-DE hybrid algorithm, such as ${N}_{\mathrm{s}}$, ${N}_{\mathrm{c}}$, ${N}_{\mathrm{p}}$, $\epsilon $, ${C}_{1}$, ${C}_{2}$, $F$ and ${C}_{R}$.
**Step****2.**- Obtain the initial particles’ position and velocity for all the swarm randomly; calculate and evaluate the corresponding objective function value ${F}_{\mathrm{obj},}{}_{i}$ of each particle; record the initial personal best position ${\mathbf{P}}_{i}$ and objective function value $pbes{t}_{i}$ of each particle; record the global best position ${\mathbf{P}}_{\mathrm{g}}$ and objective function value $gbest$ of the swarm.
**Step****3.**- Finish the iteration, if the calculation and iteration satisfy the stop criteria:
- (i)
- $gbest$ is less than the tolerance ε;
- (ii)
- iteration number is not less than the user-defined iteration limit ${N}_{\mathrm{c}}$.

**Step****4.**- Update and obtain the new particle velocity and position according to Equations (3) and (4).
**Step****5.**- Calculate new ${F}_{\mathrm{obj},}{}_{i}$ and evaluate the new positions for each particle.
**Step****6.**- Compare the old ${F}_{\mathrm{obj},}{}_{i}$ with the new one for each particle. Update the corresponding particle information if the new ${F}_{\mathrm{obj},}{}_{i}$ is superior
**Step****7.**- Use the DE algorithm to modify the personal best food source positions according to Equations (5) and (6).
**Step****8.**- Determine if the current personal best position and global best position are superior than the old ones, if so, update them. Loop to Step 3.

## 3. The Aerosol Size Distribution and Optical Constants

## 4. Sensitivity Analysis of Optical Measurement Signals to Characteristic Parameters in ASDs

## 5. Numerical Simulation

#### 5.1. Comparison of SPSO and Hybrid SPSO-DE Algorithms

^{−16}or the iteration number is larger than 1000, finish the calculation and output the results. The investigation shows that the convergence speed and the value of the objective function of the SPSO-DE hybrid algorithm are superior to the SPSO algorithm, which means the local optima and low convergence accuracy exiting in the SPSO can be avoided in the hybrid SPSO-DE algorithm.

#### 5.2. Retrieval of the Aerosol Size Distribution

## 6. Conclusions

- (1)
- To obtain more accurate results, the measurement wavelengths should be better selected from the short wavelength region for SEM (e.g., $\lambda \in \left[0.2,\text{\hspace{0.17em}}1.4\right]\text{\hspace{0.17em}}\mathsf{\mu}\mathrm{m}$ in this study), and the measurement angles should better be selected from the forward angle region for ALSM (e.g., $\theta \in \left[0,\text{\hspace{0.17em}}30\right]\text{\hspace{0.17em}}\mathrm{deg}$ in this study).
- (2)
- The SPSO-DE hybrid algorithm can converge much faster and obtain a lower fitness function value within a smaller number of generations than the SPSO algorithm in retrieving the ASDs.
- (3)
- The retrieval results by using ALSM show better convergence accuracy and robustness than those by using SEM, which attributes to the different distributions of objective function value for these methods. Moreover, considering only single spectral information of aerosol is required, the ALSM provides a more effective and reliable technique to obtain spherical ASDs. Future research will focus on the application of ALSM in studying spheroidal ASDs.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

${C}_{R}$ | crossover constant |

$D$ | particle diameter, μm |

$f(D)$ | unknown volume frequency distribution |

${F}_{\mathrm{obj}}$ | objective function value |

$F$ | mutant factor in the DE algorithm |

$I(\theta )$ | angular light-scattering intensity at the angle $\theta $, $\mathrm{W}/({\mathrm{m}}^{2}\cdot \mathrm{sr})$ |

${I}_{0}$ | total intensity of the laser, $\mathrm{W}/({\mathrm{m}}^{2}\cdot \mathrm{sr})$ |

$i,{i}_{1},{i}_{2}$ | Mie scattering functions |

$k$ | absorption index of optical constant |

${m}_{\lambda}$ | optical constant of particle at wavelength $\lambda $ |

$M$ | number of the measurement wavelengths or angles |

$n$ | refractive index of optical constant |

${N}_{0}$ | total number concentration of the suspended particle system |

${N}_{\mathrm{c}}$ | maximum number of iterations |

${N}_{\mathrm{s}}$ | total number of the particles in the swarm |

${N}_{\mathrm{p}}$ | number of the inversion parameters (dimensions) |

${P}_{i},\text{\hspace{0.17em}}{P}_{\mathrm{g}},{P}_{\mathrm{m}}$ | the individual, global and mean best positions |

${Q}_{\mathrm{ext}}$ | extinction efficiency |

${V}_{i}$ | velocity of the $i\mathrm{th}$ particle |

${X}_{i}^{}$ | position of the $i\mathrm{th}$ particle |

## Greeks Symbols

$\lambda $ | incident wavelength of the laser, μm |

$\chi $ | sensitivity coefficient |

$\theta $ | scattering light measurement angle |

$\delta $ | relative deviation of the aerosol size distribution |

$\tau $ | transmittance of the particle system |

## Abbreviations

est | estimated value |

L-N | Log-Normal distribution |

Gamma | Gamma distribution |

max | maximum value |

min | minimum value |

true | true value |

## References

- Chen, Q.X.; Yuan, Y.; Shuai, Y.; Tan, H.P. Graphical aerosol classification method using aerosol relative optical depth. Atmos. Environ.
**2016**, 135, 84–91. [Google Scholar] [CrossRef] - Gong, W.; Zhang, S.; Ma, Y. Aerosol Optical Properties and Determination of Aerosol Size Distribution in Wuhan, China. Atmosphere
**2014**, 5, 81–91. [Google Scholar] [CrossRef] [Green Version] - Esparza, A.E.; Fitzgerald, R.M.; Gill, T.E.; Polanco, J. Use of light-extinction method and inverse modeling to study aerosols in the Paso del Norte Airshed. Atmos. Environ.
**2011**, 45, 7360–7369. [Google Scholar] [CrossRef] - Ladji, R.; Yassaa, N.; Balducci, C.; Cecinato, A. Particle size distribution of n-alkanes and polycyclic aromatic hydrocarbons (PAHS) in urban and industrial aerosol of Algiers, Algeria. Environ. Sci. Pollut. Res.
**2014**, 21, 1819–1832. [Google Scholar] [CrossRef] [PubMed] - Kokhanovsky, A.A.; Leeuw, G.H. Satellite Aerosol Remote Sensing over Land; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Chen, Q.X.; Yuan, Y.; Huang, X.; Jiang, Y.Q.; Tan, H.P. Estimation of surface-level PM 2.5 concentration using aerosol optical thickness through aerosol type analysis method. Atmos. Environ.
**2017**, 159, 26–33. [Google Scholar] [CrossRef] - Aerosol Robotic Network (AERONET). Available online: http://aeronet.gsfc.nasa.gov/ (accessed on 6 July 2018).
- Moderate Resolution Imaging Spectroradiometer (MODIS). Available online: https://modis.gsfc.nasa.gov/ (accessed on 6 July 2018).
- Wang-Li, L.; Cao, Z.; Buser, M.; Whitelock, D.; Parnell, C.B.; Zhang, Y. Techniques for measuring particle size distribution of particulate matter emitted from animal feeding operations. Atmos. Environ.
**2013**, 66, 25–32. [Google Scholar] [CrossRef] - He, Z.Z.; Mao, J.K.; Han, X.S. Non-parametric estimation of particle size distribution from spectral extinction data with PCA approach. Powder Technol.
**2018**, 325, 510–518. [Google Scholar] [CrossRef] - He, Z.Z.; Qi, H.; Yao, Y.C.; Ruan, L.M. Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm. Appl. Therm. Eng.
**2015**, 88, 306–314. [Google Scholar] [CrossRef] - Ye, M.; Wang, S.; Lu, Y.; Hu, T.; Zhu, Z.; Xu, Y. Inversion of particle-size distribution from angular light-scattering data with genetic algorithms. Appl. Opt.
**1999**, 38, 2677–2685. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, Y.; Liang, G.; Pan, Z. Inversion of particle size distribution from light-scattering data using a modified regularization algorithm. Particuology
**2010**, 8, 365–371. [Google Scholar] [CrossRef] - He, Z.Z.; Hong, Q.; Chen, Q.; Ruan, L.M. Retrieval of aerosol size distribution using Improved Quantum-behaved Particle Swarm Optimization from spectral extinction measurement. Particuology
**2014**, 133, 245–252. [Google Scholar] - Dombrovsky, L.A.; Baillis, D. Thermal Radiation in Disperse Systems: An Engineering Approach; Begell House: New York, NY, USA, 2010. [Google Scholar]
- Vargas-Ubera, J.; Aguilar, J.F.; Gale, D.M. Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods. Appl. Opt.
**2007**, 46, 124–132. [Google Scholar] [CrossRef] [PubMed] - Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia, 27 November–1 December 1995; pp. 1942–1948. [Google Scholar]
- Qi, H.; Ruan, L.M.; Zhang, H.; Wang, Y.; Tan, H.P. Inverse radiation analysis of a one-dimensional participating slab by stochastic particle swarm optimizer algorithm. Int. J. Therm. Sci.
**2007**, 46, 649–661. [Google Scholar] [CrossRef] - Yuan, Y.; Yi, H.L.; Shuai, Y.; Liu, B.; Tan, H.P. Inverse problem for aerosol particle size distribution using SPSO associated with multi-lognormal distribution model. Atmos. Environ.
**2011**, 45, 4892–4897. [Google Scholar] [CrossRef] - Wei, L.Y.; Qi, H.; Ren, Y.T.; Ruan, L.M. Application of stochastic particle swarm optimization algorithm to determine the graded refractive index distribution in participating media. Infrared Phys. Technol.
**2016**, 79, 74–84. [Google Scholar] [CrossRef] - Wu, Y.; Lu, J.; Sun, Y. An improved differential evolution for optimization of chemical process. Chin. J. Chem. Eng.
**2008**, 16, 228–234. [Google Scholar] [CrossRef] - Lu, S.; Sun, C.; Lu, Z. An improved quantum-behaved particle swarm optimization method for short-term combined economic emission hydrothermal scheduling. Energy Convers. Manag.
**2010**, 51, 561–571. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Yuan, Y.; Yi, H.L.; Shuai, Y.; Wang, F.Q.; Tan, H.P. Inverse problem for particle size distributions of atmospheric aerosols using stochastic particle swarm optimization. J. Quant. Spectrosc. Radiat. Transf.
**2010**, 111, 2106–2114. [Google Scholar] [CrossRef] - Deepak, A.; Gerber, H.E. Report of the Experts Meeting on Aerosols and Their Climatic Effects, Williamsburg, VA, 28–30 March 1983; World Meteorological Organization: Geneva, Switzerland, 1983. [Google Scholar]
- D’Almeida, G.A.; Koepke, P.; Shettle, E.P. Atmospheric Aerosols: Global Climatology and Radiative Characteristics; A Deepak Pub.: Hampton, VA, USA, 1991. [Google Scholar]
- Kamrunnahar, M.; Braatz, R.D.; Alkire, R.C. Parameter Sensitivity Analysis of Pit Initiation at Single Sulfide Inclusions in Stainless Steel. J. Electrochem. Soc.
**2004**, 151, 90–97. [Google Scholar] [CrossRef] - Renard, J.B.; Dulac, F.; Berthet, G.; Lurton, T.; Vignelles, D.; Jégou, F.; Tonnelier, T.; Jeannot, M.; Couté, B.; Akiki, R. LOAC: A small aerosol optical counter/sizer for ground-based and balloon measurements of the size distribution and nature of atmospheric particles—Part 1: Principle of measurements and instrument evaluation. Atmos. Meas. Tech.
**2016**, 9, 1721–1742. [Google Scholar] [CrossRef] [Green Version] - Lurton, T.; Renard, J.B.; Vignelles, D.; Jeannot, M.; Akiki, R.; Mineau, J.L.; Tonnelier, T. Light scattering at small angles by atmospheric irregular particles: Modelling and laboratory measurements. Atmos. Meas. Tech.
**2014**, 7, 931–939. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Sensitivity analysis of spectral transmittance signals to characteristic parameters in ASDs: (

**a**,

**c**) for L-N distribution, (

**b**,

**d**) for Gamma distribution.

**Figure 3.**Sensitivity analysis of angular light-scattering signals to characteristic parameters in ASDs: (

**a**,

**c**) for L-N distribution, (

**b**,

**d**) for Gamma distribution.

**Figure 4.**Comparison of objective function values of SPSO and hybrid SPSO-DE algorithms in retrieving monomodal Gamma distribution.

**Figure 5.**Reproducibility of different ASDs by using different optical measurement methods: (

**a**) monomodal L-N distribution; (

**b**) monomodal Gamma distribution; (

**c**) bimodal L-N distribution; (

**d**) bimodal Gamma distribution.

**Figure 6.**Distribution of the objective function values for different ASDs by using the SEM: (

**a**) L-N distribution; (

**b**) Gamma distribution.

**Figure 7.**Distribution of the objective function values for different ASDs by using the ALSM: (

**a**) L-N distribution; (

**b**) Gamma distribution.

**Figure 8.**Distribution of retrieval results of monomodal ASDs for 30 calculations by using the SEM and ALSM: (

**a**) and (

**b**) for SEM; (

**c**) and (

**d**) for ALSM.

Function Types | Parameters | True Values | |
---|---|---|---|

Monomodal | L-N | $(\overline{D},\text{\hspace{0.17em}}\sigma )$ | $(1.0,\text{\hspace{0.17em}}2.99)$ |

Gamma | $(\alpha ,\text{\hspace{0.17em}}\beta )$ | $(6.0,\text{\hspace{0.17em}}3.0)$ | |

Bimodal | L-N | $({\overline{D}}_{1},\text{\hspace{0.17em}}{\sigma}_{1},\text{\hspace{0.17em}}{\overline{D}}_{2},\text{\hspace{0.17em}}{\sigma}_{2},\text{\hspace{0.17em}}n)$ | $(2.5,\text{\hspace{0.17em}}1.1,\text{\hspace{0.17em}}6.0,1.2,0.2)$ |

Gamma | $({\alpha}_{1},\text{\hspace{0.17em}}{\beta}_{1},\text{\hspace{0.17em}}{\alpha}_{2},\text{\hspace{0.17em}}{\beta}_{2},\text{\hspace{0.17em}}n)$ | $(6.0,\text{\hspace{0.17em}}2.2,\text{\hspace{0.17em}}0.8,2.0,0.15)$ |

Function Types | Spectral Extinction Method, λ (μm) | Angular Light-Scattering Method, θ (deg) | |
---|---|---|---|

Monomodal | L-N | $(0.2,\text{\hspace{0.17em}}0.4,\text{\hspace{0.17em}}0.6,\text{\hspace{0.17em}}0.8)$ | $(5,\text{\hspace{0.17em}}10,\text{\hspace{0.17em}}15,\text{\hspace{0.17em}}20)$ |

Gamma | $(0.2,\text{\hspace{0.17em}}0.4,\text{\hspace{0.17em}}0.6,\text{\hspace{0.17em}}0.8)$ | $(5,\text{\hspace{0.17em}}10,\text{\hspace{0.17em}}15,\text{\hspace{0.17em}}20)$ | |

Bimodal | L-N | $(0.2,\text{\hspace{0.17em}}0.4,\text{\hspace{0.17em}}0.6,\text{\hspace{0.17em}}0.8,\text{\hspace{0.17em}}1.0,\text{\hspace{0.17em}}1.2,\text{\hspace{0.17em}}1.4)$ | $(4,\text{\hspace{0.17em}}8,\text{\hspace{0.17em}}12,\text{\hspace{0.17em}}16,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}24,\text{\hspace{0.17em}}28)$ |

Gamma | $(0.2,\text{\hspace{0.17em}}0.4,\text{\hspace{0.17em}}0.6,\text{\hspace{0.17em}}0.8,\text{\hspace{0.17em}}1.0,\text{\hspace{0.17em}}1.2,\text{\hspace{0.17em}}1.4)$ | $(4,\text{\hspace{0.17em}}8,\text{\hspace{0.17em}}12,\text{\hspace{0.17em}}16,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}24,\text{\hspace{0.17em}}28)$ |

**Table 3.**System control parameters of the SPSO and hybrid SPSO-DE algorithms in retrieving monomodal Gamma distribution.

Parameters | N_{s} | N_{p} | N_{c} | $\mathit{\epsilon}$ | C_{1} | C_{2} | F | C_{R} |
---|---|---|---|---|---|---|---|---|

SPSO | 50 | 2 | 1000 | 10^{−16} | 1.0 | 1.0 | $\u2014$ | $\u2014$ |

SPSO-DE | 50 | 2 | 1000 | 10^{−16} | 1.0 | 1.0 | 0.5 | 0.4 |

Different Functions | Random Error | SEM | ASLM | |
---|---|---|---|---|

Retrieval Error, δ | Retrieval Error, δ | |||

Monomodal | L-N | 0% | 0.000125 | 0.000312 |

5% | 0.130401 | 0.064109 | ||

10% | 0.215168 | 0.162801 | ||

20% | 0.387621 | 0.265412 | ||

Gamma | 0% | 0.000322 | 0.000545 | |

5% | 0.162721 | 0.144108 | ||

10% | 0.187542 | 0.169336 | ||

20% | 0.365729 | 0.287532 | ||

Bimodal | L-N | 0% | 0.000156 | 0.003148 |

2% | 0.096542 | 0.085159 | ||

5% | 0.244494 | 0.232532 | ||

10% | 0.356279 | 0.328765 | ||

Gamma | 0% | 0.025242 | 0.090019 | |

2% | 0.145623 | 0.110330 | ||

5% | 0.181529 | 0.144053 | ||

10% | 0.319732 | 0.298621 |

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**MDPI and ACS Style**

He, Z.-Z.; Mao, J.-K.; Han, X.-S.
Determination of Aerosol Size Distribution from Angular Light-Scattering Signals by Using a SPSO-DE Hybrid Algorithm. *Computation* **2018**, *6*, 47.
https://doi.org/10.3390/computation6030047

**AMA Style**

He Z-Z, Mao J-K, Han X-S.
Determination of Aerosol Size Distribution from Angular Light-Scattering Signals by Using a SPSO-DE Hybrid Algorithm. *Computation*. 2018; 6(3):47.
https://doi.org/10.3390/computation6030047

**Chicago/Turabian Style**

He, Zhen-Zong, Jun-Kui Mao, and Xing-Si Han.
2018. "Determination of Aerosol Size Distribution from Angular Light-Scattering Signals by Using a SPSO-DE Hybrid Algorithm" *Computation* 6, no. 3: 47.
https://doi.org/10.3390/computation6030047