Optics Communications 366 (2016) 154–162
Contents lists available at ScienceDirect
Optics Communications
journal homepage: www.elsevier.com/locate/optcom
Application of the LSQR algorithm in non-parametric estimation of
aerosol size distribution
Zhenzong He a, Hong Qi a,n, Zhongyuan Lew a, Liming Ruan a, Heping Tan a, Kun Luo b
a
b
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
State Key Laboratory of Clean Energy Utilization, Department of Energy Engineering, Zhejiang University, Hangzhou 310027, PR China
art ic l e i nf o
a b s t r a c t
Article history:
Received 19 July 2015
Received in revised form
14 December 2015
Accepted 17 December 2015
Based on the Least Squares QR decomposition (LSQR) algorithm, the aerosol size distribution (ASD) is
retrieved in non-parametric approach. The direct problem is solved by the Anomalous Diffraction Approximation (ADA) and the Lambert–Beer Law. An optimal wavelength selection method is developed to
improve the retrieval accuracy of the ASD. The proposed optimal wavelength set is selected by the
method which can make the measurement signals sensitive to wavelength and decrease the degree of
the ill-condition of coefficient matrix of linear systems effectively to enhance the anti-interference ability
of retrieval results. Two common kinds of monomodal and bimodal ASDs, log-normal (L-N) and Gamma
distributions, are estimated, respectively. Numerical tests show that the LSQR algorithm can be successfully applied to retrieve the ASD with high stability in the presence of random noise and low susceptibility to the shape of distributions. Finally, the experimental measurement ASD over Harbin in China
is recovered reasonably. All the results confirm that the LSQR algorithm combined with the optimal
wavelength selection method is an effective and reliable technique in non-parametric estimation of ASD.
& 2015 Elsevier B.V. All rights reserved.
Keywords:
The least squares QR decomposition algorithm
Aerosol size distribution
Optimal wavelength set
Non-parametric estimation
1. Introduction
Atmospheric aerosols usually play a crucial role in influencing
the Earth's radiation balance and reducing the visibility [1,2]. The
interest in the field of atmospheric aerosol properties has been
sustained owing to the great number of engineering applications
such as atmospheric science, astrophysics, and remote sensing, etc.
For example, modeling the optical properties of aerosols is important for remote sensing applications as well as for estimating
the direct climate forcing effect of particulate matter in the atmosphere [3]. Moreover, the aerosols cause significant negative
influence on the human health and environment. Especially on the
Earth's surface, where people live and the highest concentrations
of aerosols are found, the aerosols present a serious health hazard.
Nowadays, the particle matter (PM), regarded as a criteria pollutant, has been a hot research topic for numerous studies in the
context of air pollutions [4,5]. A major obstacle to the reliable
determination of the particle size distribution of PM is the lack of
accurate measuring and modeling of the complex optical properties of various aerosols. In addition, in-situ measuring the properties of aerosols presented in the atmosphere is of great relevance
to further understanding the influence of aerosols and their optical
n
Corresponding author.
E-mail address: qihong@hit.edu.cn (H. Qi).
http://dx.doi.org/10.1016/j.optcom.2015.12.040
0030-4018/& 2015 Elsevier B.V. All rights reserved.
properties on human health and atmospheric environment [6,7].
Generally, there are three important properties of aerosols, i.e.
aerosol size distributions (ASDs), aerosol optical depths (AODs),
and Ångström exponents. Especially, accurate understanding and
modeling of the optical properties of aerosols depend heavily on
the knowledge of the particle size distribution. The ASD has a
significant influence on radiative transfer and meteorological
phenomena and plays an important role in determining the climatic trends [8,9]. The ASD is also regarded as a vital evaluation
criterion of environmental quality, and the influence of atmospheric aerosols on human health depends heavily on the
knowledge of aerosol size. Moreover, any uncertainty in the ASD
can lead to an uncertainty in estimating the AOD, and then to an
uncertainty in radiative forcing by the aerosols [8,10,11]. Thus,
without accurate measurement of the ASD, AOD, and Ångström
exponent, especially the ASD, the effects of the aerosols on the
climate, meteorology, human health, and air quality, would remain
highly uncertain. To date, although several global ground-based
aerosol observation networks have been established to study the
properties of the atmospheric aerosols, e.g. AERONET, MODIS [12],
accurate determination of the ASD is still regarded as an unsolved
problem and needs further research.
During the last two decades, various methods have been developed to determine the ASDs, i.e. the aerodynamic method, the
optical measurement method, electrical mobility and condensation method, electrical sensing zone method, ultrasonic
Z. He et al. / Optics Communications 366 (2016) 154–162
Nomenclature
Cond [A] the condition number of matrix A
the particle diameter, μm
D
f (D )
the volume frequency distribution
the intensity of the laser, W/(m2 sr)
I
the total intensity of the laser, W/(m2 sr)
I0
the geometric thickness of the particle system, m
L
the complex refractive index
m
the total number concentration of the suspended
ND
particle system
the number of subintervals
N
the extinction efficiency
Q ext
the number of the incident wavelengths
S
Greeks symbols
measurement method, and the electron microscopy method
[4,13]. For the aerosol particle properties can be derived by measuring a variety of scattering properties, such as extinction or
scattering information at multiple wavelengths, scattering information at multiple angles, or multiple-scattering information,
the optical measurement method coupled with inverse techniques
has drawn much attention in the field of estimating ASD [14]. In
addition, the optical measurement method also provides a sufficient size-measuring resolution over a broad range of sizes (from
nanometer to millimeter), which covers the main size range of
aerosols (between 0.001 and 10 μm) [5]. The optical measurement
method usually contains the spectral extinction measurement,
dynamic light scattering measurement, combined scattering, and
extinction measurement, angular scattering measurement etc.
[13]. Among these techniques, the spectral extinction technique is
most viable, for it only requires a simple optical layout and can be
realized by using the commercial spectrophotometer [15]. Usually,
the spectral extinction technique is based on the Lambert–Beer
law, and the general description of light absorption and scattering
properties of particles is described by the Mie theory. Unfortunately, the calculation of the Mie theory can only be used to
predict radiative properties of spherical particles, and the calculation is time-consuming [1,15]. To remedy this problem, the
Anomalous Diffraction Approximation (ADA) was introduced by
Van de Hulst [16] to study the optical properties of non-spherical
particles. Lots of numerical simulation and experimental results
showed that the ADA could be successfully applied to calculating
the radiative properties of spherical and non-spherical particles
[17,18]. Thus, the ADA is used to solve the direct problem in the
present research.
Generally speaking, the methods for determining ASD can be
classified into three different categories, the analytic inversion
model, the dependent model, and the independent model [18].
Under the dependent model, the ASD is known beforehand to
satisfy certain distribution and retrieved by inverse algorithms,
which is widely used in various science and engineering applications recently [5,19,20]. However, under the independent model,
the prior distribution information of ASD cannot be known in
practice. To determine the ASD under the independent model,
there are parametric and non-parametric estimation approaches.
In the parametric estimation under the independent model, an
assumed distribution function, e.g. Johnson's SB (J-SB) function and
the modified beta (M-β) function, is employed to approximately
estimate the ASD [18,19]. Due to the assumed distribution functions can only estimate the ASD approximately, the deviation between the actual distribution and estimated results is inevitable
155
the incident wavelength of the laser, μm
the relative deviation of the particle size distribution
the sensitivity coefficient
the transmittance of the particle system
the size parameter of particles
λ
δ
η
τ
χ
Subscripts
est
ext
L-N
Gamma
max
mea
min
true
the estimated value
the extinction efficiency
the log-normal distribution
the Gamma distribution
the maximum value
the measurement value
the minimum value
the true value
[21,22]. Different from the parametric estimation approach, there
is no assumed distribution function in the non-parametric estimation approach, and the ASD is divided into many subintervals
and recovered by measuring the spectral extinction data of multiwavelengths. In other words, this approach is independent of any
given a priori information of ASD. Consequently, the non-parametric estimation approach can avoid the inevitable discrepancy
of the parametric estimation approach due to the deviation of
assumed ASD function from the true distribution. The only obstacle in the non-parametric estimation approach is to solve the
Fredholm integral equation of the first kind, a well-known illposed problem which leads to highly unstable solutions because
even small noise components in the measured quantities can
cause extremely large spurious oscillations in the solutions [21].
Fortunately, the Fredholm integral equation of the first kind could
be discretized and solved by many inverse algorithms, e.g. the
generalized eikonal approximation (GEA) method [23,24], the
conjugate gradient algorithm (CGA) [25], and the generalized
cross-validation (GCV) [26]. The Least Squares QR decomposition
(LSQR) algorithm, one of the iterative regularization methods
based on Lanczos bidiagonalization and QR factorization, was first
developed by Paige and Saunders [27] in 1982 to solve the discrete
optimization problems. This algorithm is demonstrated to be well
suited for the simultaneous estimation of unknown functions or
parameters with high numerical reliability as well as removing the
numerical difficulty associated with the singularity of the adjoint
equation in various circumstances [27,28]. However, to the best of
our knowledge, few studies have investigated the application of
LSQR algorithm to retrieve the ASD. The objective of present work
is to apply the LSQR algorithm to estimate the spherical ASD
theoretically and experimentally. The remainder of this research is
organized as follows. First, an optimal wavelengths selection algorithm is studied. Then, the common monomodal and bimodal
ASDs, i.e. the log-normal (L-N) and Gamma distributions are estimated by the LSQR. In the sequel, the effect of measurement
errors on the accuracy of estimation is investigated. Meanwhile,
the actual measurement ASD in Harbin of China is reconstructed
experimentally. The main conclusions and prospects for further
research are provided finally.
2. Direct problem
2.1. The principle of the spectral extinction method
The fundamental principle of the spectral extinction method is
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Z. He et al. / Optics Communications 366 (2016) 154–162
based on the ADA model and the Lambert–Beer Law. When a
collimated light beam of intensity I0 impinges on a suspension
particle system with a refractive index which is different from that
of the dispersant medium, the transmitted light is scattered and
absorbed by the particles, which causes the attenuation of the
light [29]. If the optical thickness is thin and the independent
scattering dominates, the transmitted light intensity I at different
incident wavelengths λ i can be calculated as follows [22]:
⎧
I (λ1)
3
= −
× L × ND ×
⎪ ln
I0 (λ1)
2
⎪
⎪
dD
⎪
⎪
⋯⋯
⎪
⎪
I (λ i )
3
⎪ ln
= −
× L × ND ×
⎨
I0 (λ i )
2
⎪
dD
⎪
⎪
⋯⋯
⎪
⎪
I (λ S )
3
⎪ ln
=−
× L × ND ×
I0 (λ S )
2
⎪
⎪
⎩
dD
∫D
Dmax
min
Q ext (λ1, m1, D)
f (D)
D
∫D
Dmax
min
Q ext (λ i , mi , D)
f (D)
D
⎡ a1,1
⎢a
⎢ 2,1
A=⎢ ⋮
⎢
⎢
⎢⎣ a S,1
...
a1, N ⎤
⎥
⋯
⋯
ai − 1, j
⎥
a i, j − 1 a i, j
ai, j + 1 ⋮⎥
⎥
⋮
ai + 1, j ⋮
⎥
...
a S, N ⎥⎦
(8)
(9)
T
f = ⎡⎣ f (D1), …, f (Dj ), …, f (DN ) ⎤⎦
(10)
where ai, j = − 3/2 × L × ND Q ext (λ i , mi , Dj ) /Dj , bi = ln [I (λ i ) /I0 (λ i )].
∫D
Dmax
min
Q ext (λ S , mS , D)
f (D)
D
2.2. The methodology of the LSQR
(1)
⎡1
exp ( − iρi )
1 − exp ( − iρi ) ⎤
⎥
Q ext (λ i , mi , D) = 4 × Re ⎢ − i ×
+
⎢⎣ 2
⎥⎦
ρi
ρi2
(2)
where ρi = πD (mi − 1) /λ i , i = −1 .
The edge effective term of the spherical particles is depicted as
[17]:
2c 0
χ 2/3
(3)
where c0 is a function of the complex refractive index in general
and is approximately 0.99163 for optically soft particles; χ = πD/λ
is the size parameter of the particle.
Considering the edge effects, the fixed expression for the extinction efficiency of large particles is written as [17]:
Q ext = Q ad (1 + Q edge/Q ad ) .
⎧
⎫
1
⎬,
Q ext = Q ad ⎨ 1 +
Q
c
Q
2/
+
1/
[
(
+
1
)]
⎩
⎭
edge
1
ad
Compatible linear systems: A⋅f = B
(11)
Least−squares problem: min A⋅f − B
(12)
A ∈ RS × N
S
where
and B ∈ R . The LSQR is based on the bidiagonalization procedure of Golub and Kahan [33] listed as follows:
⎧ β u = B, α v = AT u ,
1 1
1
⎪ 1 1
⎪ βi + 1ui + 1 = Avi − αi ui ,
⎨
⎪ αi + 1vi + 1 = AT ui + 1 − βi + 1vi ,
⎪
⎩ i = 1, 2, ...
(13)
The scalars
αi ≥ 0 and
βi ≥ 0 are chosen to make
‖ui ‖2 = ‖vi ‖2 = 1. It is easy to verify that ui ∈ κi (AAT , u1 ) and
vi ∈ κi (AAT , v1 ). For the bidiagonalization procedure, there exist the
following properties: (i) suppose that k steps of the procedure Eq.
(13) have been taken, then the vectors v1, v2, ... , vk + 1 and
u1, u2 , ... , uk + 1 are the orthonormal basis of the Krylov subspaces
κi (AAT , u1 ) and κi (AAT , v1 ), respectively; (ii) the procedure Eq. (13)
will stop at step m′ if and only if min {μ, η} is m′, where μ is the
grade of v1 with respect to AAT and η is the grade of u1 with respect to AAT [32]. Further details of the LSQR algorithm are
available in Ref. [27] and not repeated for the sake of brevity.
3. Retrieval of aerosol size distribution
3.1. The models of aerosol size distribution
c1 = m − 1 .
(5)
To solve Eq. (1) numerically, the formula should be discretized
as follows:
N
Q ext (λ i , mi , Dj )
3
I (λ i )
f (Dj ) dDj
= −
× L × ND × ∑
2
Dj
I0 (λ i )
j =1
The basic idea of the LSQR algorithm is to convert the arbitrary
coefficient matrix into a square matrix firstly, and then solve the
least square solution of the equations by the Lanczos process [30].
The core of the LSQR can be described as follows [31,32]:
(4)
For small particles, the extinction efficiency is expressed as
[17]:
ln
(7)
B = A⋅f
T
B = ⎡⎣ b1, …, bi , …, bS ⎤⎦
where I (λ i ) /I0 (λ i ) is the transmittance τ (λ i ) at wavelength λ i , which
can be measured by the spectrograph or actinometer;
Q ext (λ i , mi , D) is the extinction efficiency factor of a single particle
which is a complex function of the diameter of particle D , the
wavelength λ i incident on the medium and the complex refractive
index mi ; ND is the total number of concentration depending on
the total amount of the suspended particle system; Dmax and Dmin
denote the upper and lower integration limits, respectively; f (D) is
the unknown volume frequency distribution needed to be determined; L is the geometric thickness of the particle system; S
denotes the number of measurement wavelengths. According to
the ADA model, the extinction efficiency factor Q ext (λ i , mi , D) for
aerosol particles can be described as [17]:
Q edge =
described as a discrete version by matrix equation as follows:
(i = 1, 2, ... , S )
(6)
where N denotes the number of subintervals which the particle
size range [ Dmin , Dmax ] is divided into. Therefore, Eq. (1) can be
Different models are used to study the aerosol size distribution,
e.g. Junge power law model, Gamma model, exponential model,
and L-N model. The Junge model was widely used to study the
aerosol size at the near-ground atmospheric layer in the original
researches for its simplicity, but it is not often used any more [34].
The Gamma model is usually employed to investigate the cloud
drops by many of the cloud microphysics models in the Weather
Research and Forecasting (WRF) model [35]. The exponential
model is a kind of special case of the Gamma model and was
Z. He et al. / Optics Communications 366 (2016) 154–162
applied to model the distributions of rain drops and snow aggregates by Marshall [36,37]. The L-N model, one of the most
popular aerosol size distribution models, has been used for a long
time to describe size distribution of particle properties in atmospheric aerosols and is often applied in the modeling of aerosols,
clouds, and precipitation as well as in the comparison of experimental data [38]. In the present work, two common monomodal
ASDs, i.e. the monomodal L-N and Gamma distributions, and two
common bimodal ASDs, i.e. the bimodal L-N and Gamma distributions, are estimated. The mathematical representations of
their volume frequency distributions are as follows [13,19]:
( )=
f Lmono
−N D
2⎤
⎡
1
1 ⎛ ln D − ln D¯ ⎞ ⎥
× exp ⎢ − ⎜
⎟
⎢⎣ 2 ⎝
⎠ ⎥⎦
ln σ
2π D ln σ
mono
f Gamma
(D) = Dα × exp ( −βD γ )
(14)
Table 1
The true values of characteristic parameters of ASD.
Function types
Monomodal
L-N
Bimodal
Gamma
L-N
Gamma
δ=
f Lbi− N (D) = n ×
+ (1 − n) ×
2⎤
⎡
1
1 ⎛ ln D − ln D¯ 2 ⎞ ⎥
× exp ⎢ − ⎜
⎟
⎢⎣ 2 ⎝
⎠ ⎥⎦ (16)
ln σ 2
2π D ln σ 2
bi
f Gamma
(D) = n × Dα1 × exp ( −β1D γ1 )
+ (1 − n) × Dα 2 × exp ( −β2 D γ 2 )
Parameters
True values
¯ σ)
(D,
( α, β )
(D¯ 1, σ1, D¯ 2, σ 2, n)
(α1, β1, α 2, β2, n)
(2.0, 2.0)
(4.0, 2.0)
(1.5, 2.0, 8.0, 1.2, 0.7)
(6.0, 2.2, 0.8, 2.0, 0.15)
Table 1. For the purpose of investigating the reliability and feasibility of the algorithm, the relative deviation of ASDs δ, which
means the sum of the deviation error between the probability
distribution estimated from the inverse calculation and the true
ASD in every subinterval, is studied to evaluate the quality of estimated results, and its mathematic expression is described as:
(15)
2⎤
⎡
1
1 ⎛ ln D − ln D¯ 1 ⎞ ⎥
× exp ⎢ − ⎜
⎟
⎢⎣ 2 ⎝
ln σ1
⎠ ⎥⎦
2π D ln σ1
157
{
2
N
∑i = 1 ⎡⎣ Nest (D˜ i ) − Ntrue (D˜ i ) ⎤⎦
{
2
N
∑i = 1 ⎡⎣ Ntrue (D˜ i ) ⎤⎦
1/2
}
1/2
}
(18)
where N denotes the number of subintervals which the particle
size range [ Dmin , Dmax ] is divided into; D̃i is the midpoint of the ith
subinterval [ Di , Di + 1]; Ntrue (D˜ i ) is the true number concentration
distribution in the ith subinterval; Nest (D˜ i ) is the estimated number
concentration distribution in the ith subinterval.
3.2. Selection of an optimal wavelength set
(17)
where D̄ , D̄1 and D̄2 are the characteristic diameters of these
distribution functions; σ , σ1 and σ 2 are the narrowness indices of
the distribution; n is the weight coefficient between the two peaks
in bimodal distribution and is limited as 0 ≤ n ≤ 1; α , α1, α2, β , β1,
β2 , γ , γ1 and γ2 are the characteristic parameters of the Gamma
distribution functions. Usually, in the modified form, γ , γ1, γ2 = 1,
so only parameters α , β and α1, α2, β1, β2 , n in the monomodal and
bimodal Gamma distributions need to be investigated,
respectively.
Most aerosols are polydisperse, with a range from a few molecules (10 3 μm) to a few tens of micrometers, and different
aerosol types favor different size ranges [34]. For example, aerosols
deriving from combustion sources gas-to-particle conversion such
as sulfates and most organics, tend to be an order magnitude
smaller (0.05–0.40 μm) than those aerosols originating from
wind-driven processes such as sea salt ( 1 μm) and dust (0.40–
10 μm). However, as a whole, most of the aerosols are included in
the size range 0.001 to 10 μm, which also corresponds to the
aerosol major impact on radiation [5,34]. Thus, in the present
work, the size range of aerosols is set from 0.001 to 10 μm.
Roughly, for most dry aerosols, the real part of the complex
refractive index is about 1.53, with a small imaginary part of
0.5–10 10 2. If the aerosols absorb water vapor and become
hygroscopic particles, the value of the complex refractive index
will be diluted and decrease towards the value of pure water with
complex refractive index of 1.33þ0.00 i. For instance, the oceanic
aerosols, mainly composed of hygroscopic sea salt, almost have no
absorption, and the real part of the refractive index is around 1.38,
which is just slightly larger than that of pure water. Soot or black
carbon has a very high absorption with a representative complex
refractive index of 1.75 þ0.45 i, although black carbon is most often found internally mixed with other diluent particles, especially
for other organic materials [19,34]. In this study, the complex refractive index of the aerosols is assumed to be fixed as 1.53 þ0.02 i
at different wavelengths for the sake of simplicity.
The true values of the ASDs studied in the study are listed in
Since the spectral extinction data contain some important information about the aerosol particle system, it is necessary to put
insight into the influence of the spectrum on the retrieval accuracy
of ASD. Different wavelengths lead to different influences in retrieving the ASD. Usually, the selected wavelengths are expected to
carry more useful information, which means the spectral transmittance are different from each other as much as possible. In the
case of retrieving the ASD, the spectral transmittance, a measurement signal, carries important information of the particle
system. So, the optimal spectral transmittance at different wavelengths should be irrelevant to each other and sensitive to the
wavelength. With this idea, the optimal wavelength region should
be found. In the present study, the sensitivity analysis of transmittance to the wavelength is employed to find the optimal wavelength region. The sensitivity coefficient is one of the most important characteristic parameters in the sensitivity analysis, which
is the first derivative of the transmittance τ (λ ) to a certain measurement wavelength λ . The sensitivity coefficient at every wavelength η (λ ) is defined as:
η (λ ) =
∂τ (λ )
τ (λ + Δλ ) − τ (λ − Δλ )
=
∂λ
2Δλ
(19)
where Δ represents a tiny change, Δλ > 0. It is obvious that if the
sensitivity coefficients are either small or correlated with one
another, the estimation problem will be very sensitive to measurement errors and difficult to be solved. Therefore, the optimal
wavelength region for inverse problems is the region with larger
absolute value of sensitivity coefficient generally. Using the measured value in this range can increase the amount of information
required by inverse parameters to improve the retrieval accuracy.
Figs. 1 and 2 depict the transmittance and the sensitivity
coefficient for different ASDs listed in Table 1 at different wavelengths. From Fig. 1, it can be found that when the wavelength is
no more than 9 μm, the transmittances at different wavelengths
show obvious difference. While λ is larger than 9 μm, τ (λ ) almost
does not change much between different wavelengths. The similar
conclusion can be drawn clearly in Fig. 2. From Fig. 2, it can be seen
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Z. He et al. / Optics Communications 366 (2016) 154–162
follow:
1.00
Transmittance, τ (λ )
Cond [A] = ‖A‖ ∙ ‖A−1‖
Monomodal L-N
Monomodal Gamma
Bimodal L-N
Bimodal Gamma
0.96
0.92
0.84
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Wavelengths, λ (µm)
Fig. 1. The transmittances
wavelengths.
for
different
ASDs
at
different
measurement
Transmittance, ( )
1.00
Monomodal L-N
Monomodal Gamma
Bimodal L-N
Bimodal Gamma
0.96
0.92
0.88
0.84
where ‖A‖ is the norm of matrix A ;
matrix A ; ‖A−1‖ denotes the norm of matrix A−1. Moreover, the
relationship between Cond [A] and the perturbation error can be
described as:
‖δf‖
≤ Cond [A]
‖f‖
0.88
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Wavelengths, ( m)
Fig. 2. The sensitivity coefficients for different ASDs at different measurement
wavelengths.
that when λ is no more than 6 μm, the τ (λ ) is sensitive to the
change of wavelengths. While λ is larger than 6 μm, the value of
η (λ ) is almost equal to zero, which means that different wavelengths in that region show no obvious effect on retrieving ASD
more accurately. Considering the actual laser spectrums, the optimal wavelength region in this study is set from 0.5 μm to 6 μm.
When the optimal wavelength region is fixed, the number of
the measurement wavelengths should be considered. Usually, to
improve the retrieval results, more measurement wavelengths are
employed and the number of the wavelength always exceeds the
number of subintervals N, which the aerosol particle size range
[ Dmax , Dmin ] is divided into. Actually, to ensure the uniqueness of
the solutions for Eq. (7), only N wavelengths is enough, for too
many wavelengths will not only result in matrix A to be more illconditioned, but also lead the retrieval to be time-consuming. So,
in the present study, the particle size range [Dmax, Dmin]is divided
into 50 subintervals, and only 50 wavelengths are employed.
When the total number of the measurement wavelengths is fixed,
how to select the appropriate wavelengths to improve the inverse
accuracy is also very important. To make this point, Zuo and
coworkers [39] proposed a wavelength-selecting method to improve the accuracy in retrieving the ASD. In their study, the condition number of matrix A is considered, which can be derived as
(20)
A−1 is the inverse matrix of
∙
‖δB‖
‖B‖
(21)
where δf denotes error of vector f due to the perturbation error of
vector B . For the problem of retrieving the ASD, these mean the
error of retrieval results due to the measurement noise added to
the transmittance. From Eq. (21), it can be found that smaller value
of Cond [A] can make the retrieval results to be more unjammable
to some extent. Therefore, the selected optimal wavelength set
should be able to minimize the value of Cond [A].
To select the optimal wavelength set in the range of [0.5, 6] μm,
the Particle Swarm Optimization (PSO) algorithm is employed. The
PSO algorithm, first introduced in 1995 by Kennedy and Eberhart
[40], is characterized to be simple in concept, easy to implement,
and computationally efficient. Unlike other heuristic techniques,
e.g. the Genetic Algorithms (GA), the General Regression Neural
Networks (GRNN), the PSO has a flexible and well-balanced mechanism to enhance the global and local exploration abilities,
which has been studied extensively to solve various kinds of inverse problems in recent years [41–43]. The details of the PSO are
available in our previous work [44] and not repeated here for
brevity. The social cognitive confidence coefficients in the PSO are
set as 1.0 and the inertia weight is set as 0.5. The fitness function
value is Cond [A], the swarm size of particles is 50, and the search
space of wavelengths is set in the range of [0.5, 6] μm. The termination criteria are set as: (1) when the iteration accuracy is
below 10 8 and (2) when the maximum generation number of
3000 is reached.
Fig. 3 shows the optimal wavelength set and the equal interval
wavelength set for monomodal Gamma distribution, which are all
selected from 0.5 μm to 6 μm. The Cond [A] for the optimal wavelength set is 7373.4, while that for equal wavelength set becomes 174447.2, which means that the anti-interference ability of
the transmittance under this wavelength set is so weak that every
minor measurement error added to the transmittance may introduce great error into the estimated results. Fig. 4 depicts the
Optimal wavelength set
Equal interval wavelength set
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Wavelengths (µm)
Fig. 3. Wavelengths
distribution.
selection
used
to retrieve
the
monomodal
Gamma
Z. He et al. / Optics Communications 366 (2016) 154–162
0.4
Monomodal Gamma distribution
True value
Optimal wavelength set
Equal interval wavelength set
0.3
Volume frequency distribution, fv(D)
Volume frequency distribution, fv(D)
0.4
0.2
0.1
0.0
159
Monomodal Gamma distribution
True value
LSQR, err=0%
LSQR, err=5%
0.3
0.2
0.1
0.0
0
2
4
6
8
10
0
2
4
10
0.4
Monomodal L-N distribution
True value
LSQR, err=0%
LSQR, err=5%
0.2
0.4
Volume frequency distribution, f(D)
estimated results by LSQR algorithm using different wavelength
sets. The relative deviations δ for the optimal wavelength set and
the equal interval wavelength set are 0.023 and 0.146, respectively.
It is obvious that the estimated curve using the optimal wavelength set shows satisfactory inverse accuracy, while that using
the equal wavelength set is unsatisfactory. In addition, by the same
method, the optimal wavelength sets for monomodal L-N distribution, bimodal Gamma and L-N distributions are also found.
The similar conclusions are also drawn. The optimal wavelength
sets are omitted here and interested readers can contact us for the
details.
With the help of the LSQR algorithm, the monomodal and bimodal ASDs are estimated using the optimal wavelength set, and
the random measurement errors added to the spectral transmittance I (λ i ) /I0 (λ i ) are also considered. The retrieval curves are depicted in Figs. 5–8, and the corresponding retrieval results are
listed in Table 2. When estimating the monomodal ASDs, it can be
found that the satisfactory results can be obtained by the LSQR
algorithm without random measurement errors. When the random errors are considered, the accuracy of retrieval results will be
deteriorated. However, as a whole, the estimated results are reasonable even in the presence of 5% random measurement errors.
The similar conclusions can be drawn in determining the bimodal
ASDs by the LSQR. Moreover, it is worth mentioning that when
Volume frequency distribution, fv(D)
8
Fig. 6. Retrieval results of the monomodal Gamma distribution of aerosols by LSQR.
Bimodal L-N distribution
True value
LSQR, err=0%
LSQR, err=5%
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10
Diameter ( m)
Fig. 7. Retrieval results of the bimodal L-N distribution of aerosols by LSQR.
0.25
Volume frequency distribution, f(D)
Fig. 4. Retrieval results of the monomodal Gamma distribution under different
wavelengths selection.
0.3
6
Diameter ( m)
Diameter ( m)
Bimodal Gamma distribution
True value
LSQR, err=0%
LSQR, err=5%
0.20
0.15
0.10
0.05
0.00
0
0.1
1
2
3
4
5
6
7
8
9
10
Diameter ( m)
Fig. 8. Retrieval results of the bimodal Gamma distribution of aerosols by LSQR.
0.0
0
2
4
6
8
10
Diameter ( m)
Fig. 5. Retrieval results of the monomodal L-N distribution of aerosols by LSQR.
there is 5% random error, the estimated curves of major peak agree
well with the true distribution and the other peak shows somewhat serious deviation. The results confirm the practical potential
of the proposed LSQR approach and show its effectiveness and
Z. He et al. / Optics Communications 366 (2016) 154–162
2.5
Aerosol size distributions
Monomodal L-N (2.0, 2.0)
Gamma (4.0, 2.0)
Bimodal
L-N (1.2, 2.0, 8.0, 1.2, 0.7)
Gamma
(6.0, 2.2, 0.8, 2.0, 0.15)
ASD in Harbin, China
Random errors (%)
Retrieval error δ
0
5
0
5
0
3
0
3
0
0.024006
0.066996
0.022987
0.082461
0.025781
0.095404
0.041215
0.111166
0.141428
Real part of the refractive index
Table 2
The retrieval results of ASDs by LSQR.
0.5
Harbin, China, on 6th Mar. 2014 Dust like aerosol, Ref. Yi, (1993)
Real part
Real part
Imaginary part
Imaginary part
2.0
0.4
0.3
1.5
0.2
1.0
0.1
0.5
2.5
5.0
7.5
10.0
Wavelength, ( m)
Imaginary part of the refractive index
160
0.0
15.0
12.5
Fig. 11. The refractive indices of dust like aerosols and the actual aerosols in Harbin, China.
Fig. 9. The aerosol particle sampling system: (a) the aerosol particle collector,
(b) the flowmeter and (c) the vacuum pump.
robustness in estimating monomodal or bimodal ASD.
Volume frequency distribution, f(D)
4.0
3.5
3.0
Harbin, China, on 6th Mar. 2014
True distribution
Inverse results by LSQR
2.5
2.0
1.5
1.0
0.5
0.0
4. Experimental results
1
The reliability of the LSQR algorithm is also verified by retrieving the experimental measurement ASD in Harbin on 6th
March 2014. The experimental method consisted of several parts:
aerosols collection and the ASD measurement, spectral transmittance measurement, and reconstruction of the complex refractive
indices of aerosols. Fig. 9 shows the experimental installation of
the aerosol particle collection system. The aerosols were collected
by the aerosol particle collector with the help of the vacuum
pump. Then, one part of aerosols was measured by the AccuSizer
780 Syringe Injection Sample particle size analyzer to determine
the ASD. The other part of aerosols was made as a sample by tabletting machine, and the spectral transmittance was detected by
10
100
Diameter ( m)
Fig. 12. Retrieval results of the actual measurement ASD by LSQR.
the Fourier spectrum analyzer (See Fig. 10). Finally, combined with
the Mie theory and Kramers–Kronig (K–K) relation, the complex
refractive indices of aerosols in the sample were retrieved (see
Fig. 11). The details of the experiment are available in our previous
work [2]. From Fig. 11, it can be found that the measured refractive
indices of actual aerosols agree reasonably with those of the dust
like aerosol [45], especially for the imaginary part of the refractive
index. The optimal wavelength set for actual ASD is also studied,
Light
Source
Fixed
Mirror
Optical
Splitter
Sample
Detector
Movable
Mirror
PC
(a)
A/D
Convertor
Signal
Amplifier
(b)
Fig. 10. The experiment for measuring the complex refractive indices of aerosols: (a) Fourier spectrum analyzer and (b) the optical system of the Fourier spectrum analyser.
Z. He et al. / Optics Communications 366 (2016) 154–162
and interested readers can contact us for the details. The estimated
curve of the actual ASD is shown in Fig. 12, and the corresponding
results are also listed in Table 2. It is obvious that there is a reasonable agreement between the retrieval curve and the original
ASD. The deviation of estimated ASD may be attributed to the
uncertainty of complex refractive indices retrieved by the K–K
relation. Strictly speaking, the K–K relation requires the integral
over the entire wavelength range, i.e. from zero to infinity. However, the experiment measurement wavelength is finite, i.e. from
2.5 μm to 15 μm. So, using the K–K relation, the inevitable error
will be introduced into the results of the complex refractive index
which will finally influence the retrieval accuracy of the ASD [2].
5. Conclusions
Based on the ADA and the Lambert–Beer Law, the LSQR algorithm is applied in non-parametric estimation of ASD. This inverse
technique is independent of any given priori information about the
ASD. First, an optimal wavelengths selection method is developed
to improve the accuracy of retrieval results. The proposed optimal
wavelength set can make the measurement signals susceptible to
wavelengths and decrease the degree of the ill-condition of coefficient matrix of linear systems effectively to enhance the antiinterference ability of retrieval results. Then, the common monomodal and bimodal ASDs are estimated in the presence of random
measurement errors. Numerical tests show that this technique can
be successfully applied to retrieve ASD with high stability in the
presence of random noise and low susceptibility to the shape of
distributions. The results confirm that the LSQR can be employed
as an effective and robust technique to retrieve the ASD in the
non-parametric estimation approach. Then, the actual measurement ASD in Harbin China is studied, and the estimated results
also confirm the accuracy and reliability of the LSQR. As a whole,
the results obtained show that the proposed LSQR methodology is
a promising approach for non-parametric estimation of the ASD.
Further study will focus on performance improvement of the
LSQR-based methodology as well as the applications of LSQR in
the retrieval of the ASD for non-spherical particles.
Acknowledgments
The supports of this work by the National Natural Science
Foundation of China (Nos. 51476043 and 51576053) and the Major
National Scientific Instruments and Equipment Development
Special Foundation of China (No. 51327803) are gratefully acknowledged. A very special acknowledgment is made to the editors and referees who make important comments to improve this
paper.
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