## Abstract

A precise measurement for the aerosol particle size distribution and fog droplet size distribution simultaneously in the open atmosphere is proposed. The extinction coefficient and small-angle forward scattering measurement are integrated into the detection for particle size distribution in the open atmosphere, and can achieve the fine detection of the particles in the atmosphere with radius between 0.1 to 30 μm. The key technology including optimal scattering angle in small-angle forward scattering measurement and optimal wavelengths selection are discussed and solved in detail. The fourteen different particle size distributions including aerosol size distributions and fog droplet size distributions are used for the determination of optimal forward scattering angle and wavelengths. The optimal forward scattering angle is calculated to be 1.1°. Seven wavelengths for extinction coefficients and five wavelengths for forward scattering coefficients are chosen for the retrieval of particle size distribution in the measurement. The regularization inversion of optical parameters for the retrieval of particle size distribution is described. The aerosol particle size distributions measured by particle spectrometer and actual fog particle size distributions are used for the method test and the reconstructions of particle size distributions. The inversion results show that the method can achieve the precise measurements of aerosol particle size distribution and fog droplet size distribution. The error influence on the inversion results of distributions is discussed. Based on the sensitivity analysis of inversion results, the feasibility of measurement in the real atmosphere is analyzed and discussed, and the scheme of detection system is provided.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Aerosols are solid or liquid particles with the radius of 0.001-50 μm suspended in the atmosphere, and most particle radius range from 0.01 to 5 μm. Aerosols play an important role in many atmospheric processes. Fog is an aerosol system consisting of a large number of tiny droplets of water or ice crystals suspended in the atmosphere near the ground [1], and the radius of fog droplet is mainly distributed between 2 and 15 μm, with a maximum of 50 μm. Due to the strong scattering or absorption of fog droplets or aerosol particles, the horizontal visibility is usually less than several kilometers when fog or haze/smog occurs [2]. In North China, fog and haze weathers often occur, and the microphysical properties of particles in the atmosphere are prone to change with atmospheric environment (temperature and relative humidity). Although fog and haze are different weather phenomena, while they often occur simultaneously, and change each other. Because fog and haze weather have an important impact on the atmospheric environment, climate change and residents' health, it is necessary to study fog droplet size distribution (FDSD) and aerosol particle size distribution (APSD). In this paper, particle size distribution (PSD) is used to represent the total particle size distribution of fog droplet and aerosol. The simultaneous detection of FDSD and APSD cannot be achieved using the current instruments.

At present, the measurements of PSD mainly include in situ measurements and remote sensing measurement for profiles. FDSD is usually detected using the three-purpose droplet spectrometer and the droplet spectrometer [3]. The principle of the three-purpose drop spectrometer is to use inertia to drop the droplets on the glass coated with the oil, and then the droplet size and number can be directly counted by using a microscope or count on the photomicrograph. The shortcoming of this method is the collection of droplet samples takes a long time, and continuous real-time observation cannot be performed. At the same time, the mechanical counting is an error-prone process [4]. The droplet spectrometer is capable of continuous observation and producing one FDSD per minute, but the air is sampled into the instrument by an air pump, the real atmospheric condition such as temperature and humidity is changed during this process. There are many apparatus for APSD such as wide-range particle spectrometer (WPS) [5] and optical particle counters [6]. WPS is a recently introduced commercial instrument with the unique capability to measure APSD from 0.005 to 5 µm of radius. The instrument includes a scanning mobility spectrometer (SMS) comprised of a differential mobility analyzer (DMA), a condensation particle counter (CPC) for particle measurement from 0.005 to 0.25 µm and a laser particle spectrometer (LPS) for measurement in the range from 0.2 to 5 µm. Air is pumped into the instrument by a sampling pump. As for the optical particle counter, the sampled air is drawn into the scattering cavity, the particle is illuminated by the incident radiation, and the detector quantifies the intensity of the scattered light in an angular range [7]. Then the particle size is determined by the amplitude of the electric pulse, and the number by counting the electric pulse [8].

Besides WPS and optical particle counters, some equipment or technology based on the principle of forward scattering is used for the measurement of aerosol or cloud [9–13]. M. Bouvier, et al. provided a novel approach for in situ soot size distribution measurement based on spectrally resolved light scattering. The two-angle (90° and 7°) optical technique was proposed to enable the measurement of the two parameters of a given size distribution [9].The size distribution of semitransparent irregularly shaped mineral dust aerosol samples was determined using a commonly used laser particle-sizing technique in the Ref [10]. The size distribution was derived from intensity measurements of singly scattered light at various scattering angles close to the forward-scattering direction at a wavelength of 632.8 nm.

The multi-wavelength lidar [14] and sun- photometer [15] can also detect APSD in the atmosphere with high spatial or temporal resolution. Lidar receives the backscattering signal of particles at the multiple wavelengths for the retrieval of APSD. The solar photometer is also a widely used instrument for the aerosol remote sensing. The aerosol optical thickness at different wavelengths can be obtained by measuring the direct Sun radiation. The APSD in the total atmosphere can be obtained from the solar photometer.

There are various deficiencies for the above measurements. The disadvantage of the three-purpose droplet spectrometer is that the APSD cannot be continuously observed. For the droplet spectrometer, the wide-range particle spectrometer and the optical particle counter, the disadvantages are the actual atmospheric environment such as temperature and humidity is changed by the air pump. And all of them cannot detect APSD and FDSD simultaneously. As for lidar and solar photometer, the optical parameters for the retrieval of APSD are limited, and the information about the large particle (radius is larger than 5 μm) is not so much [16]. When fog weather occurs, these two instruments can hardly work. And, it is difficult to achieve fine detection and retrieval of PSD with a radius of 0.1μm to several microns (including the aerosol particle and fog droplet) [16,17].

In order to measure the precise APSD and FDSD simultaneously in the open atmosphere, a photoelectric detection measurement based on forward scattering of a wide-spectrum light is proposed in this paper. Based on the principles of lidar and solar photometer for APSD, extinction method and small-angle forward scattering method are integrated into the detection of PSD. Different particle information can be obtained from the extinction and small-angle forward scattering signal, extinction for small particles, and forward signal for large particles such as fog droplets or sand dust. The key technology of this method including the determination of scattering angle and the wavelengths are discussed and optimized. The regularization inversion of photoelectric detection data for retrieval of PSD is described. The known APSDs and FDSDs are used for the method test and the reconstructions of PSD.

## 2. APSD and FDSD

There are distinctly different distribution models in the atmosphere. Usually the aerosol number size distribution is used to represent the different sizes of the particle group [18]. APSD can generally be described by several lognormal distributions. It can be described by the following Eqs [16,19]:

Here, *n*(*r*) denotes the number concentration distribution, *k* is the number of modes, *N _{i}* is the total particle number of the

*i*th mode,

*r*describes the mode radius of APSD, and ln

_{mi}*σ*is the mode width of the

_{i}*i*th mode. Usually four main types of aerosol in the troposphere can be distinguished: urban industrial aerosol, biomass burning aerosol, desert dust, and marine origin aerosol. The same distribution can be written for volume concentration distribution

*v*(

*r*), which is preferred, because in volume concentration representation both fine and coarse modes are relatively easy to distinguish [16]. Both

*n*(

*r*) and

*v*(

*r*) have the same standard deviation

*σ*and the relationships between radius and concentration for each mode are [20]:

The parameters for these typical aerosols are shown in the Table 1 [16]. Besides *v*(*r*), the volume concentration distribution is often expressed in terms of d*V*/dln(*r*), and d*V*/dln(*r*) = r × *v*(*r*).

FDSD is usually the generalized gamma distribution proposed by Deirmendjian [21]. It can be written as:

Here*a*,

*b*,

*c*and

*d*are the parameters describing the FDSD.

## 3. The measurement of APSD and FDSD simultaneously

In the multi-wavelength lidar, the combined use of extinction and backscatter coefficients is important for the retrieval of APDS and aerosol microphysical propertie [16,22,23]. The extinction and backscatter efficiencies at different wavelength are sensitive to the particles of different diameters. The wavelength of laser is limited and the most commonly used wavelengths are 0.355 μm, 0.532 μm and 1.064 μm, so the corresponding kernel functions are not sensitive to size variations of big particles with radius greater than 5μm and small particles with radius less than 0.1 μm [16]. According to the Mie theory, the first peak of the extinction efficiency corresponds to the diameter of the particle [24] and it is generally applicable to the range of particle radius from 0.01 to 4 μm when light is visible beam. However, the small-angle forward scattering coefficient, which is sensitive to large particles in the air, can reflect particle radius of approximately 1 μm to several tens of microns [17]. Therefore, combined with the method of remote sensing for APSDs, it is proposed to use multi-wavelengths’ photoelectric detection methods to realize the precise detection of PSD simultaneously. If the extinction and forward scattering coefficients at the wide wavelength range are obtained, PSD can be retrieved by using the inversion technology. The air pumping and drying in this method are not required, and the PSD in the real environment can be obtained. The schematic diagram of PSD measurement in open environment is shown in Fig. 1.

The basic principle of this detection system is as follows: a wide-spectrum light beam (wavelength is in the range of 0.2 to 3 μm) collimated by a lens with long focal length emits a beam into the atmosphere. The light is scattered by the aerosol particles or fog droplets in the atmosphere. After light travels a distance *L* in the atmosphere, a lens collects the attenuated scattered light at different directions to the focal plane. The optical fibers placed on the focal plane transmit the forward scattered light into the spectrometer. The measured air can be considered to be uniform in a short distance. So the energy received at the wavelength of *λ* by the spectrometer are

*J*(

_{α}*λ*) is the power received by the spectrometer at the direction of light transmission,

*J*

_{0}(

*λ*) is the emitted power of wavelength λ at the exit of light source,

*T*(

*λ*) is the optical transmittance of detection system,

*α*

_{a}(

*λ*) is the extinction coefficient of the aerosol particles or fog droplets at the wavelength of

*λ*,

*α*

_{m}(

*λ*) is the extinction coefficient of air molecule which can be obtained by the standard model [25],

*J*(

_{β}*θ*,

*λ*) is the power at the angle of

*θ*,

*β*

_{a}(

*θ*,

*λ*) is the forward scattering coefficient of particles at the angle of

*θ*and at the wavelength of

*λ, β*

_{m}(

*θ*,

*λ*) is the forward scattering coefficient of air molecule,

*L*is the optical path length in the atmosphere. The extinction coefficient and the forward scattering coefficient at the angle of

*θ*can be expressed from the above Eqs. as follows:

The optical data *α*_{a}(*λ*) and *β*_{a}(*θ*, *λ*) are related to the physical quantities of PSD by the Fredholm integral Eqs. of the first kind, and can be expressed as [16,26]:

*α*

_{a}(

*λ*) denote the extinction coefficient of particles at the wavelength of

_{n}*λ*,

_{n}*β*

_{a}(

*θ*,

*λ*) denote the forward scattering coefficient at the angle of

_{n}*θ*and at the wavelength of

*λ*,

_{n}*v*(

*r*) describes the volume concentration distribution, the lower integration limit is defined by

*r*

_{min}, and the upper limit is determined by

*r*

_{max},

*K*(

_{α}*r*,

*m*,

*λ*) is the extinction kernel function,

_{n}*K*(

_{β}*r*,

*θ*,

*m*,

*λ*) is the forward scattering kernel at the angle of

_{n}*θ*and the refractive index of

*m*, and

*P*(

*r*,

*θ*,

*m*,

*λ*) is the phase function of a single particle. The scattering coefficients change with wavelength and the scatter angles. The key problem in this method for PSD is the determination of the forward scattering angle and the selection of the optimal wavelength for extinction and forward scatter.

_{n}The PSD can be retrieved from the extinction coefficient and forward scattering coefficient of the spectrometers using the regularization algorithm or the principal component analysis technique [27,28]. In this method, no air pumps are needed, so the actual state of the air detected is not changed. Besides, its’ detection speed is fast, and real-time non-interference detection can be realized.

Figure 2 shows the extinction efficiencies [29] and the kernels as the function of particle diameter at the wavelengths of 0.2 μm, 1.06 μm and 3 μm when the refractive index *m* is 1.50-0.01*i*. The kernels are calculated from the extinction efficiencies and the geometric cross section π*r*^{2} as

From the Fig. 2, the extinction efficiency strongly depends on particle size, and it gradually converges to 2 with the increase of particle size. The maximum peak diameters of extinction efficiencies change with the wavelength, and balance it [29]. Therefore, the extinction coefficient can better reflect the particle size characteristics of this region [17]. The detection of large particles can be realized by obtaining the extinction at longer wavelength. However, it is difficult to obtain and detect the light at mid-infrared wavelength or longer-band beams. Therefore, we propose to use small-angle forward scattering coefficients combined with the extinction coefficients to achieve the detection of large particles in the air. Figure 3 shows the variation of the scattering phase function of the particle with the diameter of particles at 0.3 μm, 0.55 μm and 1.06 μm at the angle of 2° and the refractive index is 1.50-0.01*i*. From Fig. 3, the maximum peak of the scattering phase function is greater than the extinction efficiency, and it can reflect the characteristics of large particles.

## 4. The selection of the scattering angle and the wavelengths

#### 4.1 The determination of the forward scattering angle

According to the Mie theory, the intensity distribution of the scattered light is the function of the angle [29]. The angle, at which the scattering coefficient is the most sensitive to PSDs difference, should be chosen in the measurement for improving the detection accuracy and sensitivity.

The scattering phase function *Pʹ*(*λ*, *θ*, *m*) of PSD is a weighted superposition of the scattering phase function of a single particle on the particle swarm:

*Q*

_{sca}(

*r*) is the aerosol scattering efficiency at the radius of

*r*.

The scattering phase functions of 14 different PSDs, which represent most aerosol distributions in the atmosphere, are chosen to calculate the optimal scattering angle. These 14 distributions include 10 APSDs in different atmospheric humidity conditions and 4 FDSDs. According to the ref [30], the parameters of 5 typical APSDs are chosen and shown in Table 2 and Fig. 4(a). These distributions are basically obtained by optical particle counter, representing the APSD at low relative humidity.

When the relative humidity increases in the atmosphere, the hygroscopic growth of aerosols would occur. Because of the lack of measured data, the APSDs under high relative humidity are obtained by the theoretical calculation. The radius of aerosol particles under high relative humility (RH) can be calculated by the formula (13) and the distributions of wet haze after moisture absorption can be estimated [31]:

*r*

_{0}denotes the radius of aerosol particles,

*r*(

*RH*) denotes the radius after hygroscopic growth,

*RH*denotes the relative humidity, and

*d*is 3.5 generally [31]. 5 new APSDs are obtained from the Table 2 and the Eq. (12). They are shown in Fig. 4(b). Four FDSDs are selected from the Ref [3,32]. for our analysis, and their parameters are shown in Table 3. The four FDSDs are shown in Fig. 4(c).

The scattering phase functions of above 14 PSDs with the scattering angle are calculated and shown in Fig. 5. We can see that the scattering phase functions change with the PSDs and angles and there are large variations among the scattering phase functions when the angles are between 0~2°.

In order to select the optimal scattering angle with the highest sensitivity to the PSDs (i.e., the difference in the scattering phase function of different PSDs is the largest), the standard deviation coefficient *CV* is calculated for the dispersion degree of the particle phase function at different angles, which is expressed as:

*σ*is the standard deviation of 14 curves in Fig. 5,

*μ*is their average value. The curve of

*CV*with respect to

*θ*is shown in Fig. 6. From the Fig. 6, the value of

*CV*change greatly with the angle

*θ*, the large value can be found in the range of 0.1°~1.2° and 157°~178°. The angle where the standard deviation coefficient

*CV*is greater than 1(i.e., 0.1 to 1.1 degrees) is selected in our design.

In order to find the optimal angle for the PSD measurement, the overlapping area of the scattering phase function and PSDs are also calculated. Define the overlapping area as the value of the normalized particle spectrum and the phase function intersecting in the same coordinate system. Its’ schematic diagram is shown in Fig. 7. The value reflects the amount of information about PSD.

The overlapping area of 14 PSDs and the scattering phase function at 0.1° to 1.1° are calculated, and shown in Fig. 7. The overlapping area increases with the scattering angle and reach the largest at 1.1°. The optimal forward scattering angle is chosen at 1.1° for the PSD measurement.

#### 4.2 The selection of wavelengths

The detection system designed in our method can theoretically obtain extinction coefficients and forward scattering coefficients of many wavelengths. It is not necessary to use the optical parameters of all wavelengths for the PSDs. Quite a number of wavelength signals are useless in this method. The optimal wavelengths need be singled out for the inversion. The retrieval of PSD can be realized by the selected multi-wavelengths extinction coefficient and forward scattering coefficient combined with the inversion algorithm. The APSDs varies greatly with time and location, which may be the superposition of single lognormal model, two lognormal models, three lognormal models or four lognormal models [22]. FDSD is usually the generalized gamma distribution. The single lognormal distribution contains three unknown and gamma distribution contains four unknowns, so three or four independent extinction coefficients and forward scatter coefficients are needed to accurately obtain the parameters of one mode. 12-16 optical parameters are needed to accurately reconstruct the PSD in the atmosphere.

### 4.2.1 The selection of wavelengths for extinction coefficients

To select the optimal wavelengths, the extinction coefficients of above 14 PSDs are calculated at the intervals of 50 nm in the range of 0.2 to 3 μm according to the Mie scattering theory. Figure 9 shows the extinction coefficients of these APSDs (Fig. 9(a)) and FDSDs (Fig. 9(b)) at the wavelengths of 0.2 to 3 μm.

According to Fig. 9, the extinction coefficients vary greatly with the PSDs. The extinction coefficient in foggy days is much larger than that in hazy days and it increases with the increase of relative humidity in air when haze occurs. Extinction coefficients of aerosol particle change greatly with the wavelength, especially in the range of 0.2 to 1.5 μm, while they hardly change with wavelength for fog droplets, as shown in Fig. 8, which indicates that the extinction coefficients from large particles are not sensitive to the variation of wavelength. According to the above section, the first peak of the extinction efficiency corresponds to the diameter of the particle. And APSDs can generally be described by lognormal distribution. The APSDs are more complex when the diameter of particle is less than 1.5 μm, which means that more information in this range needed to be obtained for the retrieval of PSD. Five wavelengths in this range are chosen for the retrieval of APSD, and the intervals between these five wavelengths are logarithmic equidistances. Since the extinction coefficients of 0.2 −1.5 μm mainly reflect the particle radius up to about 1 μm, the extinctions of two infrared wavelengths of 2 μm and 3 μm are also chosen for the retrieval of particles larger than 1 μm. The seven wavelengths are selected for extinction coefficient, and they are 0.2 μm, 0.337 μm, 0.525 μm, 0.88 μm, 1.45 μm, 2 μm and 3 μm.

### 4.2.2 The selection of the forward scattering wavelengths

Since it is necessary to simultaneously invert the size distribution of particles with radius of 0-30 μm, the forward small-angle scattering coefficient is added to achieve accurate inversion of large particles. Figure 10 shows the forward scattering coefficients of these APSDs (Fig. 10(a)) and FDSDs (Fig. 10(b)) at the wavelength of 0.2-3 μm. There are great differences between the extinction coefficients and forward scattering coefficients from Fig. 9 and Fig. 10. Similar to the Fig. 9, the forward scattering coefficients vary greatly with the PSDs, the values of fogs are greater than aerosols, and the forward scattering coefficients at wet haze day are also greater than the values at dry haze day. For APSD, the changing trends of forward scattering coefficients with wavelength are similar. As the wavelength increases, the forward scattering coefficients decrease rapidly. There are great changes for FDSD with wavelength, especially within 0.2 to 1 μm. These indicate that the forward scattering coefficients are sensitive to large particles. The method used in selecting the forward wavelength is the same as that of the extinction. For the forward kernel function, five wavelengths of equal logarithmic spacing between 0.3 and 1.06 μm are selected to be 0.3 μm, 0.41 μm, 0.56 μm, 0.78 μm, and 1.06 μm, respectively.

The extinction coefficients of 7 wavelengths and the forward scattering coefficients of 5 wavelengths are chosen for the measurement of PSDs. Supposed that the refractive index is 1.50-0.01*i*, these extinction and the forward scattering kernel functions at the angle of 1.1° are calculated and shown in Fig. 10. According to these kernels, their first peaks at those wavelengths cover the range of 0.1 to 30 μm, which means that the particles with the radius of 0.1 to 30 μm can be retrieved.

## 5. The retrieval algorithm and simulations for PSD

The possibility of using optical parameters to estimate the size distribution of stratospheric aerosols has been previously discussed by a number of researchers [27]. The regularization algorithm [23], the principal component analysis technique [27] and linear estimation algorithm [28] have been used for the aerosol bulk properties. The regularization algorithm is used most commonly for inverting multi-wavelength measurements, allowing the retrieval of particle size, concentration, and the main features of the particle size distribution [24]. Using the regularized inversion algorithm, the particle size distribution of aerosols can be obtained without assuming the initial complex refractive index and aerosol distribution. In previous research, the regularization algorithm is mainly used for the inversion of lidar data, and has been discussed elsewhere [16,24]. The inversion error of microphysical parameters from lidar data can reach 20% to 60% due to the limited optical parameters. In our method, the regularization algorithm is also selected for the retrieval of the PSD.

#### 5.1 The regularization algorithm

The solution *v*(*r*) of Eq. (9) and Eq. (10) can be approximated by the superposition of base functions *B _{j}*(

*r*):

*W*are weight coefficients.

_{j}*ε*

^{math}(

*r*) is the error in the solution. According to the Eqs. (7) (8) and (14), the optical coefficients can be rewritten as

*A*

_{(}

_{α}_{,}

_{β}_{)}

*(*

_{j}*m*) is the kernel function matrix, it is a 12 × 12 matrix in our method, and they can be calculated from the following Eqs.

Representing *W _{j}* and

*g*as vectors, the weight coefficients can be derived from the following relation:

_{p}*γ*is the Lagrange multiplier,

*H*is the smoothing matrix, and

*A*is the transpose of matrix

^{T}*A*. The value of

*γ*was decided by the minimum discrepancy principle [16]. And the logarithmic normal distribution function was selected as the base function for the inversion. According to our method, 12 optical parameters including the extinction and forward scattering coefficients were used for the retrieval of PSD. The extinction coefficients at the wide spectrum of 0.2-3 μm and forward scatter coefficients at 0.3-1.06 μm covering the ultraviolet to infrared range are combined with the regularized inversion algorithm to invert APSDs and FDSDs.

#### 5.2 Simulation

Four typical APSDs are selected to test our method and verify the rationality of wavelength and scattering angle selection. Figures 12(a) and 12(b) show the typical urban industrial APSDs from the Ref [11], Figs. 12(c) and 12(d) is the actual APSD measured by a Particle size spectrometer in haze day in Xi’an, China, and Fig. 12 (d) show the distribution after hygroscopic growth from Fig. 12(c). Their extinction coefficients at seven wavelengths (0.2 μm, 0.337 μm, 0.525 μm, 0.88 μm, 1.45 μm, 2 μm, 3 μm) and forward scattering coefficients at five wavelengths (0.3 μm, 0.41 μm, 0.56 μm, 0.78 μm, 1.06 μm) are calculated from the Eqs. (3) and (4). The selected complex refractive index of APSD1 and APSD2 in Fig. 11 is 1.50-0.01*i*, and the value of APSD3 and APSD4 is 1.40-0.03*i*.

Besides the APSDs, the complex refractive index is also the unknown quantity in the inversion. One of the most remarkable features of this regularization approach is that it does not need the a priori knowledge of the complex refractive index. The regularization algorithm is performed for the reconstructions of APSDs. All possible complex refractive indices are chosen for the inversion. The real parts of the refractive index varied from 1.3 to 1.7 in steps of 0.01, the imaginary parts varied from 0 to 0.03 in steps of 0.001. The inversion results of the complex refractive indices are 1.50-0.01*i* for APSD1, 1.52-0.011*i* for APSD2, 1.40-0.022*i* for APSD3 and 1.40-0.028*i* for APSD4. There is a small inversion error for complex refractive index. The four initial APSDs (black solid lines) and their retrievals (red dash lines) are illustrated in Fig. 12, and the corresponding error distributions (blue solid lines) with the particle radius are also calculated and shown in Fig. 12.

From the inversion results and their error curves, the APSDs in dry-haze day and wet-haze day with radius ranging from 0.1 μm to 13 μm can be well reconstructed using the data from the extinctions and forward scattering coefficients. The maximum point-to-point relative error is less than 10%, and their mean error is less than 5% for Figs. 12(a) and 12(b). The retrieval errors of measured distribution in Figs. 12(c) and 12(d) are larger than the models in Figs. 12(a) and 12(b). The measured distributions are not smooth curves, and the small peaks can be found. Great errors often occur at peak locations, while the basic distribution characteristics can be well retrieved. The maximum point-to-point error can reach 20% or worse, while the mean error is less than 15%. The microphysical parameters are the mean and integral properties of the particle ensemble, and they can be calculated from the APSD. The effective radius *r _{eff}*, volume concentration

*V*and number concentration

*N*are chosen for the error analysis [33]:

Their error is less than the above point-to-point error of distribution. The retrieval errors are all less than 8%. The errors of effective radius and volume concentration are less than 5%. These error values are far less than inversion error (20% to 60%) from multi-wavelength lidar data [16,24].

Two FDSDs [2,3] measured by the fog drop spectrometer are also chosen for the test. Fog drops are much larger than general aerosol particles, so the volume concentrations are much larger than them. The complex refractive index of fog is chosen as 1.33-0*i* [34]. The initial, retrieval FDSDs and their corresponding inversion error are shown in Fig. 13. From them, FDSDs can also be well reconstructed using the data in our method. The maximum point-to-point errors are less than 5% at the range of 0 to 30 μm. The large error often occurs at the range of low concentration. The retrieval errors of microphysical parameters are also calculated. The retrieval errors of microphysical parameters are less than 1%.

Besides the above APSDs and FDSDs, more APSDs and FDSDs are also simulated and reconstructed. The results show that they can be well retrieved using the selected optical parameters, and there are smaller inversion errors for FDSDs than the APSDs. From the above analysis, the method proposed can measure PSDs in the open atmosphere. Figure 14 shows a series of PSDs before and during a haze process (Fig. 14(a)) and their reconstruction results (Fig. 14(b)). The x-axis is the particle diameter, the y-axis is the time and the z-axis is the size distribution. The PSDs in Fig. 14(a) were measured in Xi’an in inland China from March 9 to 14, 2018, during which a haze episode occurred. Compared with the sunny day, the volume concentrations of fine and coarse particles increased simultaneously in haze day. The inversion can well reconstruct the initial APSDs from Fig. 14 no matter that it's sunny or hazy, which indicates that our inversion is stable.

#### 5.3 Effects on the retrievals of errors in the optical data

The extinctions and forward scattering coefficients for the inversion of PSD can be obtained from the scattering signals *J _{α}*(

*λ*) and

*J*(

_{β}*θ*,

*λ*). The optical parameters may be affected by the systematic and random errors. From the Eqs. (5) and (6), systematic error in this measurement comes from different sources and need to be considered. Systematic errors can be due to nonlinearity of a photodetector, errors of calibration of the system constants and errors in the assumed atmospheric molecule density. Random errors arise naturally from the measurement process. In order to study the effect of optical parameter errors on the inversion results, two important microphysical parameters (effective radius and volume concentration) are selected to characterize the inversion results. The typical urban APSD in Fig. 12(a) is used for the analysis of inversion sensitivity. The initial refractive index is chosen as 1.5-0.001i. The optical parameters are generated from the Eqs. (7) and (8). We assume that the errors of optical parameters are −2% to 2%. The next step consists of applying a systematic bias to one optical datum at a time. For each of these induced biases, the inversion is performed and a new size distribution and set of microphysical parameters are then obtained. This procedure is applied to each of the fifteen optical data used in our measurement. Figure 15 and Fig. 16 present the sensitivity analysis for the retrieval of effective radius and volume concentration.

Figure 15 and Fig. 16 reveal that the retrievals of *V* are more sensitive to biases than *r _{eff}*. It is quite apparent that the retrievals are more sensitive to biases in the forward scattering coefficients. The low sensitivities are to biases in

*α*

_{a}(0.2 μm),

*α*

_{a}(1.06 μm),

*α*

_{a}(2 μm),

*α*

_{a}(3 μm),

*β*

_{a}(1.1, 1.06 μm), the high sensitivities are to biases in

*β*

_{a}(1.1, 0.41 μm),

*β*

_{a}(1.1, 0.56 μm). The positive slopes indicate higher values of

*reff*and

*V*when the optical data are affected by positive biases than when they are not affected by biases, while for negative slopes just the opposite occurs. According to Fig. 15 and Fig. 16, in order to obtain inversion error less than 20%, the error of optical parameters should be controlled within 2%.

## 6. The feasibility of the measurement in the real atmosphere

From the above simulations, the precise size distribution of aerosol particles and fog droplets in the open atmosphere can be obtained using the optical signals at the optimal forward scattering angle of 0° and 1.1° (~19 mrad) in the detection of a wide-spectrum light with the wavelength between 0.2 and 3 μm. The collimation of the light beam and the light propagation distance in the atmosphere are very important for our scattering measurement. From the section 3, the scattering phase functions of particles are sensitive to the angles. In order to make the change of *CV* less than 3%, the divergence angle of the light beam need be controlled within 1mrad. According to our system scheme in Fig. 1, Lens1 is a collimator, and the transmission lenses are not suitable due to the existence of chromatic aberration. Reflective lenses with long focal length can be used for the collimation of light beam with wide spectrum. The combination of the Cassegrain telescope with a focal length greater than 1 m and a fiber with a core diameter of 1 mm can ensure that the divergence angle is less than 1 mrad. Similarly, reflective lenses need be used for the convergence of beams. The light propagation distance in this method should not be too short in order to obtain sufficient detection sensitivity. Meanwhile, the optical path distance cannot be too long. Too long optical path distance not only increases the volume of the measurement instrument, but also affects the optical signals due to the atmospheric turbulence. The optimal distance would be different for aerosol and fog particles with different concentrations. It is considered that the optical path between Len1 and Len2 in Fig. 1 is folded by using two mirrors.

The designed optical system schematic diagram is shown in Fig. 17. Tungsten halide lamp with wide spectral range can be used as light source. The light beam is transmitted to the focal point of Cassegrain telescope1 (focal length = 2 m) through an optical fiber with 1mm core diameter. Fiber attenuator can be used to adjust the energy of output beam. The collimated light with a divergence angle of 0.5 mrad is emitted into the atmosphere after the Cassegrain telescope1. A variable diaphragm is used to control beam aperture. Two mirrors with high reflectivity are used to control the optical path length in the atmosphere. Adjust slightly the angle between the two mirrors, and the optical path length will change accordingly. By changing the aperture of the beam and the angle between the two mirrors, the path of the beam can be controlled to change from 1m to 1000 m. When the severe haze / fog occurs, short optical path length is needed. When the aerosol concentration in the air is low, long optical path length is needed. The Cassegrain telescope2 (focal length = 2m) is used to collect the attenuated and scattered light. Two fibers with 1mm core diameter placed on the focal plane of Cassegrain telescopes2 are used to receive the forward scattered light at angles of 0° and 1.1°. Adjust the optical axes of telescopes 1 and 2 so that they are parallel to each other. The fiber placed on the focal plane receives the attenuated light at the angle of 0°. Considering the divergence angle, the fiber receives all the transmitted beams with divergence angle of 0.5mrad. Another optical fiber is placed on the focal plane with a distance of 38.4 mm from the focal point, and it receives the 1.1°forward scattered light. Considering the receiving field of view, the second fiber receives the forward scattered light of 1.1° ± 0.0144°. These designs are feasible in real atmosphere.

In order to induce the inversion error of PSD, the optical parameters with high precision need be measured. According to the Eq. (5), the error of the extinction coefficient is determined by the error of optical path length *L*, the emitted power of light source *J*_{0}(λ), the detected power *J _{α}*(λ), the transmittance of optical system

*T*(λ) and the extinction coefficient of air molecule

*α*

_{m}

*(λ*).

*L*and

*T*(λ)are system constants, and they can be calibrated precisely. The fluctuation of the light intensity will not affect the result, because the term of intensity is a ratio. When haze or fog occurs in the atmosphere, the value of

*α*

_{m}

*(λ*) is far less than aerosol extinction. So the error of

*α*

_{m}

*(λ*) can be neglected. The error of the forward scattering coefficient comes from the error of the detected power at the angle of

*θ J*(

_{β}*θ*,

*λ*) and

*J*(λ). Similarly, the value is also a ratio, so the effect of the light intensity fluctuation can be neglected. According to the above discussion, the measurement accuracy of system will be high if the system constants can be accurately calibrated. And the calibration of system constants is not a difficult task.

_{α}## 7. Conclusion and discussion

A photoelectric detection method for APSDs and FDSDs of fog/haze/smog simultaneously in the open atmosphere is presented and described in this paper. Extinction measurement and small-angle forward scattering measurement are adopted for the retrieval of PSD in our method. The key technology including optimal scattering angle and optimal wavelengths selection is discussed and solved. The optimal forward scattering angle is calculated to be 1.1°, and seven wavelengths for extinction coefficients and five wavelengths for forward scattering coefficients are chosen for the measurement of PSD. The regularized inversion algorithm is used for the retrieval of APSDs and FDSDs. The six known APSDs and FDSDs were used for method test and the reconstructions of PSD. The simulation results show that the method can achieve the measurement of APSDs and FDSDs. The worst cases often occur at the radius less than 0.1μm for APSDs, the maximum relative error may reach 25%. The basic distribution characteristics can be well retrieved. For FDSDs, high precision results can be obtained. The optical parameters at several fixed wavelengths in the above method are chosen for the detection of PSD, and the optical parameters of these wavelengths can basically satisfy the detection of APSDs and FDSDs simultaneously. The urban industrial aerosol and fog are discussed in this paper, they can be thought of as spherical, so Mie theory are used for the calculation of optical coefficients. If there are non-spherical particles such as dust, the Mie theory will induce error. The kernels need to be generated in the range of size parameters $0.012\le x=\frac{2\pi r}{\lambda}\le 625$using the *T*-matrix method or DDA method. These contents will be discussed in future work.

## Funding

National Natural Science Foundation of China (NSFC) (41627807, 61575160 and 61875163).

## Acknowledgments

Thanks to the professor Mao Jietai from Peking University for giving the good suggestion about the aerosol size distribution measurement.

## References

**1. **D. Wu, “A discussion on difference between haze and fog and warning of Ash Haze Weather,” Meteorol. Monogr. **31**(4), 3–7 (2005).

**2. **R. N. Mahalati and J. M. Kahn, “Effect of fog on free-space optical links employing imaging receivers,” Opt. Express **20**(2), 1649–1661 (2012). [PubMed]

**3. **D. Liu, M. Pu, J. Yang, G. Zhang, W. Yan, and Z. Li, “Microphysical structure and evolution of four-day persistent fogs around Nanjing in December 2006,” Acta Meteorol. Sin. **67**(1), 147–157 (2009).

**4. **Z. Li and Z. Peng, “Physical and chemical characteristics of the Chongqing winter fog,” Acta Meteorol. Sin. **52**(4), 477–483 (1994).

**5. **B. Y. H. Liu, F. J. Romay, W. D. Dick, K.-S. Woo, and M. Chiruta, “A wide-range particle spectrometer for aerosol measurement from 0.010 µm to 10 µm,” Aerosol Air Qual. Res. **10**(2), 125–139 (2010). [CrossRef]

**6. **X. Li, Y. Gao, H. Wei, Y. Jiand, and H. Hu, “Development of optical particle counter with double scattering angles,” Opt. Precis. Eng. **17**(7), 1528–1534 (2009).

**7. **X. Li, Y. Gao, Y. Ji, and H. Hu, “Development optical particle counter of LED lamp-house,” J. Opt. Precision Eng. **14**(5), 802–806 (2006).

**8. **C. Chien, A. Theodore, C. Wu, Y. Hsu, and B. Birky, “Upon correlating diameters measured by optical particle counters and aerodynamic particle sizers,” J. Aerosol Sci. **101**, 77–85 (2016). [CrossRef]

**9. **M. Bouvier, J. Yon, G. Lefevre, and F. Grisch, “A novel approach for in-situ soot size distribution measurement based on spectrally resolved light scattering,” J. Quant. Spectrosc. Radiat. Transf. **225**, 58–68 (2019). [CrossRef]

**10. **B. Veihelmann, M. Konert, and W. J. van der Zande, “Size distribution of mineral aerosol: using light-scattering models in laser particle sizing,” Appl. Opt. **45**(23), 6022–6029 (2006). [CrossRef] [PubMed]

**11. **D. Baumgardner, H. Jonsson, W. Dawson, D. O’ Connor, and R. Newton, “The cloud, aerosol and precipitation spectrometer: a new instrument for cloud investigations,” Atmos. Res. **59–60**, 251–264 (2001). [CrossRef]

**12. **K. M. Manfred, R. A. Washenfelder, N. L. Wagner, G. Adler, F. Erdesz, C. C. Womack, K. D. Lamb, J. P. Schwarz, A. Franchin, V. Selimovic, R. J. Yokelson, and D. M. Murphy, “Investigating biomass burning aerosol morphology using a laser imaging nephelometer,” Atmos. Chem. Phys. **18**(3), 1879–1894 (2018). [CrossRef]

**13. **D. B. Curtis, M. Aycibin, M. A. Young, V. H. Grassian, and P. D. Kleiber, “Simultaneous measurement of light-scattering properties and particle size distribution for aerosols: Application to ammonium sulfate and quartz aerosol particles,” Atmos. Environ. **41**(22), 4748–4758 (2007). [CrossRef]

**14. **H. Di, H. Hua, Y. Cui, D. Hua, T. He, Y. Wang, and Q. Yan, “Vertical distribution of optical and microphysical properties of smog aerosols measured by multi-wavelength polarization lidar in Xi’an, China,” J. Quant. Spectrosc. Radiat. **188**, 28–38 (2017). [CrossRef]

**15. **Y. Ma, W. Gong, L. Wang, M. Zhang, Z. Chen, J. Li, and J. Yang, “Inversion of the haze aerosol sky columnar AVSD in central China by combining multiple ground observation equipment,” Opt. Express **24**(8), 8170–8185 (2016). [CrossRef] [PubMed]

**16. **I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, K. Franke, and D. N. Whiteman, “Inversion of multiwavelength Raman lidar data for retrieval of bimodal aerosol size distribution,” Appl. Opt. **43**(5), 1180–1195 (2004). [CrossRef] [PubMed]

**17. **D. R. Lv, X. J. Zhou, and J. H. Qiu, “Principle and numerical experiment of integrated remote sensing for aerosol size distribution using extinction and small- angle scattering coefficients,” Sci. China **12**, 1516–1523 (1981).

**18. **H. M. Wei, W. Zhao, and X. C. Dai, “Influence of fog and aerosol particles forward-scattering on light extinction,” Opt. Precis. Eng. **26**(6), 1354–1361 (2018). [CrossRef]

**19. **A. K. Jagodnicka, T. Stacewicz, G. Karasiński, M. Posyniak, and S. P. Malinowski, “Particle size distribution retrieval from multiwavelength lidar signals for droplet aerosol,” Appl. Opt. **48**(4), B8–B16 (2009). [CrossRef] [PubMed]

**20. **H. Horvath, R. L. Gunter, and S. W. Wilkison, “Determination of the coarse mode of the atmospheric aerosol using data from a forward-scattering spectrometer probe,” Aerosol Sci. Technol. **12**(4), 964–980 (1990). [CrossRef]

**21. **D. Deirmendjian, “Scattering and polarization properties of water clouds and hazes in the invisible and near infrared,” Appl. Opt. **3**(2), 187–196 (1964). [CrossRef]

**22. **H. Di, J. Zhao, X. Zhao, Y. Zhang, Z. Wang, X. Wang, Y. Wang, H. Zhao, and D. Hua, “Parameterization of aerosol number concentration distributions from aircraft measurements in the lower troposphere over Northern China,” J. Quant. Spectrosc. Radiat. Transf. **218**, 46–53 (2018). [CrossRef]

**23. **D. Müller, U. Wandinger, and A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory,” Appl. Opt. **38**(12), 2346–2357 (1999). [CrossRef] [PubMed]

**24. **H. Di, Q. Wang, H. Hua, S. Li, Q. Yan, J. Liu, Y. Song, and D. Hua, “Aerosol microphysical particle parameter inversion and error analysis based on remote sensing data,” Remote Sens. **10**(11), 1753 (2018). [CrossRef]

**25. **D. N. Whiteman, “Examination of the traditional raman lidar technique. II. Evaluating the ratios for water vapor and aerosols,” Appl. Opt. **42**(15), 2593–2608 (2003). [CrossRef] [PubMed]

**26. **O. Dubovik and M. D. King, “A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements,” J. Geophys. Res. **105**(D16), 20673–20696 (2000). [CrossRef]

**27. **D. P. Donovan and A. I. Carswell, “Principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction measurements,” Appl. Opt. **36**(36), 9406–9424 (1997). [CrossRef] [PubMed]

**28. **I. Veselovskii, O. Dubovik, A. Kolgotin, M. Korenskiy, D. N. Whiteman, K. Allakhverdiev, and F. Huseyinoglu, “Linear estimation of particle bulk parameters from multi-wavelength lidar measurements,” Atmos. Meas. Tech. **5**(5), 1135–1145 (2012). [CrossRef]

**29. **C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (Wiley, New York, 1983).

**30. **X. Li, L. Qie, J. Li, K. Zhou, and Q. Xu, “Characteristics of Size Distribution of Haze Particles,” Daqi Yu Huanjing Guangxue Xuebao **6**(04), 274–279 (2011).

**31. **J. Yang, “Z. LiandS. Huang, “Influence of Relative Humidity on Shortwave Radiative Properties of Atmospheric Aerosol Particles,” Chin. J. Atmos. Sci. **23**(02), 239–247 (1999).

**32. **H. Wei, L. Shao, and T. Li, “Fog Field Scattering Analysis with Drop Size Distribution,” Las. Optoelect. Prog. **51**(12), 93–99 (2014).

**33. **D. Müller, U. Wandinger, and A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. **38**(12), 2358–2368 (1999). [CrossRef] [PubMed]

**34. **D. Segelstein, *The Complex Refractive Index of Water*, (University of Missouri, Kansas City, 1981).