Comparison of Aerosol Size-Distributions Using Linear-Regression,Genetic Algorithm,and Annealing Genetic Algorithm

Introduction

Methods for precise particle-size-distribution retrieval using the extinction method have been widely studied (Bockmann, 2001; Deshpande and Kamra, 2002; Madhavi, 2004; Lekhtmakher and Shapiro, 2005; Kulkarni and Wang, 2006; Wang et al., 2007; Olfert et al., 2008; Wang, 2008). The general principle of these methods is as follows.

Suppose that the initial radiance at wavelength λ is I₀(λ), and after passing through the atmosphere it becomes I₁(λ), according to the Lambert-Beer law it has (Zuo and Yang, 2007). \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document} \begin{align*} I_1 (\lambda) = I_0 (\lambda) \exp [ - \tau (\lambda) - \tau_{oth} (\lambda) ] \tag{1} \end{align*} \end{document}

where τ(λ) is the atmospheric optical depth of the aerosol and τ_oth(λ) is the total optical depths of other extinction factors in the air. τ(λ) can be expressed as (Hulst, 1957). \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document} \begin{align*} \tau (\lambda) = \int_{r_{\min}}^{r_{\max}} \pi r^2 Q (r, \lambda) n (r) dr \tag{2} \end{align*} \end{document}

in which r is the radius of the aerosol; r_min and r_max are the lower and upper limits of the radius respectively; Q(r, λ), which can be calculated by using Mie scattering theory (Hulst, 1957), is the extinction efficiency of the particle whose radius is r; and n(r) is the undetermined aerosol size-distribution. Theoretically, if τ(λ) is known, n(r) can be retrieved by solving equation (2). So far, there are two models to retrieve n(r), one is the dependent-model and the other is the independent-model.

The independent-model is more objective and more universal than the dependent model (Kulkarni and Wang, 2006). According to the idea of the independent-model, the radius section [r_min, r_max] is divided into M sub-sections, and in every sub-section n(r) is supposed to be a constant n_i. Then, equation (2) can be rewritten as \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document} \begin{align*} \tau (\lambda) = \sum_{i = 1}^M \pi r_i^2 Q (r_i, \lambda) n_i \Delta r \tag{3} \end{align*} \end{document}

where r_i is the average radius of particles in the i th sub-section. In this way, the problem is converted into solving a matrix equation. It seems that by solving equation (3), n_i can be retrieved with the number of the wavelengths being greater than or equal to M. However, usually the Mie matrix Q(r_i, λ) is ill-conditioned, which would introduce large error and instability into the retrieval results. In order to solve this problem, many techniques have been attempted. Previously, the linear-regression (LR) method was used, which is quite simple. Afterward, some optimal algorithms were applied in the retrieval, which can get more stable and accurate results.

This article will compare the retrieval stability and accuracy of the LR and two optimal algorithms—the genetic algorithm (Kim et al., 2007) (GA) and annealing genetic algorithm (AGA)—in detail. First, the GA and AGA are briefly introduced; then, the retrieval stability and accuracy of LR, GA, and AGA are compared by numerical simulation; afterward, the accuracy is further compared by the actual measurement; at last, the main results are summarized.

The GA and the AGA

GA is a class of probabilistic optimization algorithms that are inspired by biological evolution process. GA is particularly useful for hard problems where little is known about the underlying search space. The process of GA includes initialization, selection, mutation, variation, and so on. Although GA is not guarantee that always gives the optimum solution, it usually gives an approximately good solution in a short time.

However, the GA, although it is called the “global optimal algorithm,” is easy to be trapped in local optimum (Kim et al., 2007; Ye et al., 1999), which would also influence the stability and accuracy of the retrieval results. In recent years, the AGA was proposed (Yu et al., 2000; Sun, 2010), which combined the advantages of GA with those of simulated annealing algorithm and was more easy to search for the global optimum. It is expected to get more stable and more accurate retrieval results by using AGA.

Numeric Simulation

In the simulation, four supposed size-distributions serve as the standard size-distributions. Substituting the standard size-distributions into equation (3), the values of the τ(λ) can be obtained. Based on the simulated value of τ(λ), the aerosol size-distribution can be retrieved by using the LR, GA, and AGA, respectively. Then, the stability and accuracy of the three algorithms can be reflected by the correlation coefficients between the retrieved size-distributions and the standard size-distributions.

Standard size-distributions and the related simulation conditions

The four supposed standard size-distributions are shown in Fig. 1. Fig. 1a is a Junge distribution with its index −1.5, Fig. 1b is a Gauss distribution, Fig. 1c is a bimodal distributions, and Fig. 1d is triple-peak distribution. The radius range is 0.1–5 μm, which is divided into 50 sub-sections (therefore, at least 50 wavelengths are needed in the retrieval of the size-distribution), the complex refractive index of the aerosol is supposed to be 1.306-0.001i, and the 50 wavelengths used to measure values of the τ(λ)are equidistantly selected in the spectrum range of 300–900 nm.

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FIG. 1.

Standard size-distributions. (a) Junge distribution; (b) Gauss distribution; (c) bimodal distributions; (d) triple-peak distribution.

The results of the simulation

The correlation coefficients between the retrieved size-distributions and the standard size-distributions are shown in Figs. 2–4 (sorted in ascending order), which can reflect the stability and accuracy of the three algorithms.

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FIG. 2.

Correlation coefficients between the retrieval results from linear-regression and the standard size-distributions (sorted in ascending order) (a) of Junge distribution; (b) of Gauss distribution; (c) of bimodal distributions; (d) of triple-peak distribution. AOD, atmospheric optical depth.

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FIG. 3.

Correlation coefficients between the retrieval results from GA and the standard size-distributions (sorted in ascending order) (a) of Junge distribution; (b) of Gauss distribution; (c) of bimodal distributions; (d) of triple-peak distribution. GA, genetic algorithm.

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FIG. 4.

Correlation coefficients between the retrieval results from AGA and the standard size-distributions (sorted in ascending order) (a) of Junge distribution; (b) of Gauss distribution; (c) of bimodal distributions; (d) of triple-peak distribution. AGA, annealing genetic algorithm.

Fig. 2 shows the correlation coefficients between the retrieval results from LR and the standard size-distributions. It can be seen that to all the distributions, when errors of τ(λ) is 0, the retrieval results from the LR are accurate and stable completely, and the correlation coefficients are 1. When the upper limit of the errors of τ(λ) reach 1%, the stability of the retrieval results decrease sharply, and even the negative correlation coefficients appear. When the upper limit of the errors of τ(λ) reach 2% and 4%, the retrieval results become much more unstable.

Fig. 3 shows the correlation coefficients between the retrieval results from GA and the standard size-distributions. It can be seen that when the errors of the τ(λ) are 0, the results are quite stable and accurate, but the correlation coefficients are all slightly smaller than 1 and there are tiny differences among the correlation coefficients, which may be due to the differences of the concrete initial groups and the cross points of very retrieval. When the error is increased, the stability and accuracy are mildly decreased. However, generally speaking, even if the errors reach 4%, the retrieval results are still acceptable.

Fig. 4 shows the correlation coefficients between the retrieval results from AGA and the standard size-distributions. It can be seen that the results from AGA are very stable and quite accurate, even if the upper limit of the errors reach 4%, the minimum correlation coefficient exceeds 0.8.

Experimental Verification

The retrieval accuracy of the three algorithms is further verified by retrieving the size-distribution of the polystyrene particles in water. First, the size-distributions of two types of spherical polystyrene particles are measured precisely by using a microscope (JSM-5900LV) as the reference standards, and the results are shown by the histograms in Fig. 5. Then, the τ(λ) of polystyrene particles in the de-ionized water is measured by using a spectrograph. Finally, the size-distributions of the polystyrene particles are retrieved according to the τ(λ) by using the LR, GA, and AGA, respectively.

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FIG. 5.

The size-distributions of polystyrene particles. Histograms: measured by using microscope; the asterisks: retrieved by using linear-regression; the rings: retrieved by using GA; the diamonds: retrieved by using AGA. The correlation coefficient between the asterisks and the histograms is (a) 0.399, (b) 0.495. The correlation coefficient between the rings and the histograms is (a) 0.833, (b) 0.861. The correlation coefficient between the diamonds and the histograms is (a) 0.982, (b) 0.967.

The setup used to measure τ(λ) is shown in Fig. 6. An Xe lamp can emit polychromatic light stably. The output light from the cell can be guided into a spectrograph (300–900 nm) by a fiber. First, the cell is filled with pure de-ionized water, and the output light (I₀(λ)) is captured by using the spectrograph with 2000 times of sampling-average. Then, the particles are dispersed into the de-ionized water homogeneously, and output light (I₁(λ)) is captured again by using the same spectrograph. Finally, substituting I₀(λ) and I₁(λ) into equation (1), the values of the τ(λ) can be got.

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FIG. 6.

Setup used to measure the size-distribution.

Based on the τ(λ), the size-distributions are retrieved by using the LR, GA, and AGA respectively, with the results shown in Fig. 5. From Fig. 5, it can be seen that the results from the AGA are the best; the correlation coefficients between the retrieved size-distributions and the standard size-distributions exceed 0.9; the results from the GA, although they are worse than those from the AGA, are also acceptable; whereas the results from the LR are too bad to be accepted.

Conclusions

The stability and accuracy of retrieving the aerosol size-distribution by using LR, GA, and AGA are studied in detail. From the results of numeric simulation, it can be concluded that by using AGA the retrieval results are quite stable and accurate; by using the GA, the stability and accuracy are decreased mildly, but the results are still acceptable to some extent; by using the LR, the retrieval results are so unstable that the tiny errors of the τ(λ) could cause great errors of the retrieval results. This conclusion is further verified by the retrieval of the size distribution of the polystyrene particles in the water.