Project supported by the Jiangsu Provincial Natural Science Foundation, China (Grant Nos. BK20170800 and BK20160794) and the National Natural Science Foundation of China (Grant No. 51606095).
Project supported by the Jiangsu Provincial Natural Science Foundation, China (Grant Nos. BK20170800 and BK20160794) and the National Natural Science Foundation of China (Grant No. 51606095).
† Corresponding author. E-mail:
Project supported by the Jiangsu Provincial Natural Science Foundation, China (Grant Nos. BK20170800 and BK20160794) and the National Natural Science Foundation of China (Grant No. 51606095).
The angular light-scattering measurement (ALSM) method combined with an improved artificial bee colony algorithm is introduced to determine aerosol optical constants and aerosol size distribution (ASD) simultaneously. Meanwhile, an optimized selection principle of the ALSM signals based on the sensitivity analysis and principle component analysis (PCA) is proposed to improve the accuracy of the retrieval results. The sensitivity analysis of the ALSM signals to the optical constants or characteristic parameters in the ASD is studied first to find the optimized selection region of measurement angles. Then, the PCA is adopted to select the optimized measurement angles within the optimized selection region obtained by sensitivity analysis. The investigation reveals that, compared with random selection measurement angles, the optimized selection measurement angles can provide more useful measurement information to ensure the retrieval accuracy. Finally, the aerosol optical constants and the ASDs are reconstructed simultaneously. The results show that the retrieval accuracy of refractive indices is better than that of absorption indices, while the characteristic parameters in ASDs have similar retrieval accuracy. Moreover, the retrieval accuracy in studying L-N distribution is a little better than that in studying Gamma distribution for the difference of corresponding correlation coefficient matrixes of the ALSM signals. All the results confirm that the proposed technique is an effective and reliable technique in estimating the aerosol optical constants and ASD simultaneously.
Aerosols play a crucial role in atmospheric radiative transfer by absorbing and scattering solar radiation, which results in climate degeneration, the greenhouse effect, etc. So, obtaining accurate and reliable data of the aerosol absorption and scattering properties is a prerequisite for studying the atmospheric radiative energy balance effectively.[1–5] Aerosol absorption and scattering properties depend on aerosol optical constants, aerosol size distribution (ASD), and morphology. Generally, the aerosol optical constant, a function of the chemical, phase composition and their mixing state, describes the optical properties and the interaction of the aerosol with radiation, while the ASD affects the meteorological phenomena and climatic trends.[6–10]
Generally, the aerosol optical constants and ASD cannot be obtained directly and often need to be retrieved with the help of some experimental data and corresponding inverse theories. Recently, coupled with some intelligent optimization algorithms, the optical measurement method, e.g., spectral extinction measurement, angular scattering measurement, optical depth measurement, or any combination of these, is regarded as a fast, nonintrusive, and convenient method to retrieve the ASD and aerosol optical constants.[5,11–18] However, accurate prediction of the optical constants and ASDs simultaneously is still regarded as a very challenging problem and needs further research.
In the present paper, the angular light-scattering measurement (ALSM) method, which has satisfactory performance in studying particle size distribution,[15,19,20] is proposed to estimate the aerosol optical constant and the ASD simultaneously. The inverse problem is solved by using an improved artificial bee colony (IABC) algorithm, which has been successful in solving the problem of retrieving the aerosol optical constants in our previous work.[11] Moreover, an optimized selection principle of measurement angles on the basis of the sensitivity analysis and principle component analysis (PCA) is proposed to improve the retrieval accuracy. The remainder of this paper is organized as follows. First, the principles of the ALSM method and the optimized selection principle of measurement angles are introduced. Then, the aerosol optical constants available on the AERONET and two common ASDs are estimated simultaneously under different random measurement errors. Finally, the main conclusions and prospects for further research are provided.
According to the principle of the ALSM method, when a non-polarized parallel incident light beam with intensity I0 impinges on a suspension particle system with the optical constants which are different from that of the medium (see Fig.
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In the present work, two common ASDs, i.e., the log-normal (L-N) and Gamma distributions, are studied. The mathematical representations of their volume frequency distributions are as follows:[24]
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![](cpb_27_5_059101/cpb_27_5_059101_ieqn1.gif)
![]() | Fig. 2. (color online) Aerosol optical constants available on AERONET.[27] |
![]() | Table 1. True values of characteristic parameters in the ASDs. . |
Since the ALSM signals contain some important information about the particle system, it is necessary to gain insight into the influence of the ALSM signals on the retrieval accuracy in simultaneous retrieval of the aerosol optical constants and ASDs. Usually, different measurement angles obtain different measurement signals, which will lead to different retrieval accuracy. So, selecting optimal measurement angles is very important. In the present study, the sensitivity analysis of the ALSM signals to the optical constants or characteristic parameters in the ASDs at different measurement angles is employed to find the optimized selection region of measurement angles. Then, the contribution rate of the ALSM signals based on the PCA approach is studied to select the optimal measurement angles within the optimized selection region obtained by the sensitivity analysis approach to improve the retrieval accuracy.
Usually, sensitivity analysis studies how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be apportioned to different sources of uncertainty in its inputs. The sensitivity coefficient, which is the first derivative of the ALSM signals I(θ) to the optical constants or characteristic parameters in the ASDs at a certain measurement angle θ, is proposed to carry out the sensitivity analysis of the ALSM signals. The sensitivity coefficient χa(θ) is defined as[12,28]
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![](cpb_27_5_059101/cpb_27_5_059101_ieqn3.gif)
![]() | Fig. 3. (color online) Sensitivity coefficients of ALSM signals to optical constants or characteristic parameters in L-N distribution at different measurement angles. |
The PCA approach is a feature extraction method commonly used in pattern recognition. Its basic idea is to convert a set of observations of possibly correlated variables into a new set of values of linearly uncorrelated variables called principal components, each of which is a linear combination of the original variables. Each of the principal components contains different amount of information, which can be measured by their contribution rate. Therefore, the original variables can be replaced by several principal components with highest contribution rates, achieving the purpose of dimension reduction.[29] Moreover, the contribution rate of the original variables to these principal components can be calculated by using a formula suggested by Chiang et al.[30] The PCA approach was used by Tang to study the optimized selection method of measurement wavelengths to improve the retrieval accuracy of particle size distribution.[31]
To provide the optimized selection measurement angles and improve the accuracy in simultaneous retrieval of the aerosol optical constants and ASDs, the PCA approach is proposed to study the contribution rate of the ALSM signals at different measurement angles within the optimized selection region obtained by the sensitivity analysis. According to the PCA approach, various possible combinations of the optical constants and characteristic parameters are considered and form an ALSM signals matrix. In the present study, the measurement angles in studying the L-N distribution vary from [0○, 40○] ∪ [140○, 180○] in steps of 5○ (18 degrees), and those in studying the Gamma distribution vary from [0○, 60○] ∪ [160○, 180○] in steps of 5○ (18 degrees). The refractive indices n vary from 1.3 to 2.1 in steps of 0.05 (17 values), and the imaginary indices k vary from 0.00001 to 0.5 in steps of 0.05 (11 values), which means that there will be about 17 × 11 = 187 combinations of the optical constants. The characteristic parameters in the ASDs, e.g.,
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The eigenvectors and eigenvalues for the correlation coefficient matrix
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According to Tang’s work,[31] z is determined by letting Vz not less than 90%, which denotes that the original data can be represented well with the new z principal components. Therefore, the contribution rate of the ALSM signals at measurement angle θi to the overall z principal components is expressed as[32]
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The improved artificial bee colony algorithm is proposed as the inverse problem method, which has been used to study the optical constants successfully. The detail procedure of the IABC is not available in this paper and can be found in our previous work.[11]
For simultaneous estimation of the aerosol optical constants and ASD, there are four parameters to be determined, i.e., the refractive and absorption indices of optical constants (n and k) and the characteristic parameters in the ASDs (
Usually, the inverse problem is solved by minimizing the objective function Fobj, which is defined as the sum of the square residual between the simulated and the measured signal ratios. The mathematical expression of Fobj is derived as follows:
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![]() | Fig. 6. (color online) Comparison of objective function values under different scales of swarm size. |
![]() | Fig. 7. (color online) Comparison of objective function values under different numbers of measurement angles. |
![]() | Table 2. System control parameters of the IABC algorithm. . |
Considering that the inverse algorithm is a stochastic optimization method and the optimization process has certain randomness, all the calculations are repeated 30 times. The numerical simulation procedure is illustrated in Fig.
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![](cpb_27_5_059101/cpb_27_5_059101_ieqn7.gif)
![](cpb_27_5_059101/cpb_27_5_059101_ieqn8.gif)
![](cpb_27_5_059101/cpb_27_5_059101_ieqn9.gif)
To demonstrate that the optimized selection principle of measurement angles can improve the inverse accuracy effectively, we show in Table
![]() | Table 3. Retrieval results of different measurement angles selected according to different selection principles. . |
With the help of the IABC algorithm and the optimized selection principle of measurement angles, the optical constants and the ASDs mentioned above are reconstructed simultaneously. The searching ranges of the refractive and absorption indices are set as [1.3, 2.1] and [0.00001, 0.5], respectively, and the searching ranges of the characteristic parameters in ASDs are all set as [0.1, 10.1]. The average retrieval results of the optical constants are depicted in Figs.
![]() | Fig. 10. (color online) Retrieval results of aerosol optical constants under different random measurement errors in studying L-N distribution: (a) refractive index n, (b) absorption index k. |
![]() | Fig. 11. (color online) Retrieval results of aerosol optical constants under different random measurement errors in studying Gamma distribution: (a) refractive index n, (b) absorption index k. |
![]() | Fig. 12. (color online) Retrieval curves of different ASDs under different measurement random errors: (a) L-N distribution, (b) Gamma distribution. |
Figure
Combined with an improved artificial bee colony algorithm, the angular light-scattering measurement method is applied to estimate optical constants and aerosol size distribution simultaneously. Moreover, to improve the retrieval accuracy, an optimized selection principle of the ALSM signals based on the sensitivity analysis and principle component analysis is proposed. The following conclusions can be drawn. (i) To simultaneously estimate the aerosol optical constants and size distribution accurately, the number of measurement angles and the swarm scale in the IABC algorithm would be better set as 5 and 40, respectively. (ii) Comparing with random selection measurement angles, the optimized selection measurement angles can provide more useful information to ensure the accuracy of simultaneous estimation of the aerosol optical constants and size distribution. (iii) The retrieval accuracy of refractive indices of optical constants is better than that of absorption indices of optical constants, while the characteristic parameters in the ASDs have similar retrieval accuracy. (iv) The accuracy of the retrieval results in studying the L-N distribution is a little better than that in studying the Gamma distribution, which is attributed to the difference of the corresponding correlation coefficient matrixes of the ALSM signals.
Finally, as the aerosol particle is treated as a homogenic spherical particle in this study, the methodology presented here can also be used to study the simultaneous estimation of the optical constants and size distribution of other spherical particle dispersion medium, e.g., microalgae dispersion medium. Furthermore, as more measurement signals can be obtained by using the ALSM method, the methodology can also be proposed to simultaneously predict the optical constants and size distribution of non-spherical particles, which will be studied in our future work.
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