the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# A novel method for calculating ambient aerosol liquid water content based on measurements of a humidified nephelometer system

### Chun Sheng Zhao

### Gang Zhao

### Jiang Chuan Tao

### Wanyun Xu

### Nan Ma

### Yu Xuan Bian

Water condensed on ambient aerosol particles plays significant
roles in atmospheric environment, atmospheric chemistry and climate. Before
now, no instruments were available for real-time monitoring of ambient
aerosol liquid water contents (ALWCs). In this paper, a novel method is
proposed to calculate ambient ALWC based on measurements of a
three-wavelength humidified nephelometer system, which measures aerosol light
scattering coefficients and backscattering coefficients at three wavelengths
under dry state and different relative humidity (RH) conditions, providing
measurements of light scattering enhancement factor *f*(RH). The proposed ALWC calculation method includes two steps: the first
step is the estimation of the dry state total volume concentration of ambient
aerosol particles, *V*_{a}(dry), with a machine
learning method called random forest model based on measurements of the
“dry” nephelometer. The estimated *V*_{a}(dry) agrees
well with the measured one. The second step is the estimation of the volume
growth factor Vg(RH) of ambient aerosol
particles due to water uptake, using *f*(RH) and the
Ångström exponent. The ALWC is calculated from the
estimated *V*_{a}(dry) and Vg(RH). To validate the new method, the ambient ALWC calculated
from measurements of the humidified nephelometer system during the Gucheng
campaign was compared with ambient ALWC calculated from ISORROPIA
thermodynamic model using aerosol chemistry data. A good agreement was
achieved, with a slope and intercept of 1.14 and −8.6 µm^{3} cm^{−3} (*r*^{2} = 0.92), respectively. The
advantage of this new method is that the ambient ALWC can be obtained solely
based on measurements of a three-wavelength humidified nephelometer system,
facilitating the real-time monitoring of the ambient ALWC and promoting the
study of aerosol liquid water and its role in atmospheric chemistry,
secondary aerosol formation and climate change.

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Atmospheric aerosol particles play significant roles in atmospheric environment, climate, human health and the hydrological cycle and have received much attention in recent decades. One of the most important constituents of ambient atmospheric aerosol is liquid water. The content of condensed water on ambient aerosol particles depends mostly on the aerosol hygroscopicity and the ambient relative humidity (RH). Results of previous studies demonstrate that liquid water contributes greatly to the total mass of ambient aerosol particles when the ambient RH is higher than 60 % (Bian et al., 2014). Aerosol liquid water also has large impacts on aerosol optical properties and aerosol radiative effects (Tao et al., 2014; Kuang et al., 2016). Liquid water condensed on aerosol particles can also serves as a site for multiphase reactions which perturb local chemistry and further influence the aging processes of aerosol particles (Martin, 2000). Recent studies have shown that aerosol liquid water serves as a reactor, which can efficiently transform sulfur dioxide to sulfate during haze events, aggravating atmospheric environment in the North China Plain (NCP) (Wang et al., 2016; Cheng et al., 2016). Hence, to gain more insight into the role of aerosol liquid water in atmospheric chemistry, aerosol aging processes and aerosol optical properties, the real-time monitoring of ambient aerosol liquid water content (ALWC) is of crucial importance.

Few techniques are currently available for measuring the ALWC. The humidified tandem differential mobility analyser systems (HTDMAs) are useful tools and widely used to measure hygroscopic growth factors of ambient aerosol particles (Rader and McMurry, 1986; Wu et al., 2016; Meier et al., 2009). Hygroscopicity parameters retrieved from measurements of HTDMAs can be used to calculate the volume of liquid water. Nevertheless, HTDMAs cannot be used to measure the total aerosol water volume, because they are not capable of measuring the hygroscopic properties of the entire aerosol population. With size distributions of aerosol particles in their ambient state and dry state, the aerosol water volume can be estimated. Engelhart et al. (2011) deployed the Dry-Ambient Aerosol Size Spectrometer to measure the aerosol liquid water content and volume growth factor of fine particulate matter. This system provides only aerosol water content of aerosol particles within a certain size range (particle diameter less than 500 nm, for the setup of Engelhart et al., 2011). In addition, in conjunction with aerosol thermodynamic equilibrium models, ALWC can also be estimated with detailed aerosol chemical information. However, simulations of aerosol hygroscopicity and phase state by using thermodynamic equilibrium models are still very complicated even under the thermodynamic equilibrium hypothesis and these models may cause large bias when used for estimating ALWC (Bian et al., 2014).

The idea of using the humidified nephelometer system for the study of aerosol
hygroscopicity had already been proposed early on by Covert et al. (1972).
The instrument measures aerosol light scattering coefficient
(*σ*_{sp}) under dry state and different RH conditions, providing
information on the aerosol light scattering enhancement factor *f*(RH). One
advantage of this method is that it has a fast response time and continuous
measurements can be made, facilitating the monitoring of changes in ambient
conditions. Another advantage of this method is that it provides information
on the overall aerosol hygroscopicity of the entire aerosol population (Kuang
et al., 2017). Measured *σ*_{sp} of aerosol particles in
dry state and *f*(RH) vary
strongly with parameters of particle number size distribution (PNSD), making
it difficult to directly link them with the dry state aerosol particle volume
(*V*_{a}(dry)) and the volume growth factor Vg(RH) of the entire
aerosol population. So far, the ALWC could not be directly estimated based
solely on measurements of the humidified nephelometer system. Several studies
have shown that given the PNSDs at dry state, an iterative algorithm together
with the Mie theory can be used to calculate an overall aerosol hygroscopic
growth factor *g*(RH) based on measurements of *f*(RH) (Zieger et al., 2010;
Fierz-Schmidhauser et al., 2010). In such an iterative algorithm, the *g*(RH)
is assumed to be independent of the aerosol diameter. Thus, ALWC at different
RH levels can be calculated based on derived *g*(RH) and the measured PNSD.
This method not only requires additional measurements of PNSD, but also may
result in significant deviations of the estimated ALWC, because *g*(RH)
should be a function of aerosol diameter rather than a constant value.
Another method, which directly connects *f*(RH) to Vg(RH)
(Vg(RH) = *f*(RH)^{1.5}), is also used for predicting ALWC based on
measurements of the humidified nephelometer system and mass concentrations of
dry aerosol particles (Guo et al., 2015). This method assumes that the
average scattering efficiency of aerosol particles at dry state and different
RH conditions are the same and requires additional measurements of PNSD or
mass concentrations of dry aerosol particles (Guo et al., 2015). However, the
scattering efficiency of aerosol particles varies with particle diameters,
which will change under ambient conditions due to aerosol hygroscopic growth.

In this paper, we propose a novel method to calculate the ALWC based only on
measurements of a humidified nephelometer system. The proposed method
includes two steps. The first step is calculating *V*_{a}(dry) based
on measurements of the “dry” nephelometer using a machine learning method
called random forest model. With measurements of PNSD and BC, the six
parameters measured by the nephelometer can be simulated using the Mie theory
and the *V*_{a}(dry) can also be calculated based on PNSD. Therefore,
the random forest model can be trained with only the regional historical
datasets of PNSD and BC. In this study, datasets of PNSD and BC measured from
multiple sites are used in the machine learning model to characterise a
regional aerosol and these datasets have covered a wide range of aerosol
loadings. The second step is calculating Vg(RH), based on the
Ångström exponent and *f*(RH) measured by the humidified nephelometer
system. In this step, the
influences of the variations in PNSD and aerosol hygroscopicity are both
considered to derive Vg(RH) from measured *f*(RH). Finally, based on
calculated *V*_{a}(dry) and Vg(RH), ALWCs at different RH points can
be estimated. The used datasets are introduced in Sect. 2. Calculation method
of *V*_{a}(dry) based only on measurements of the nephelometer, which
measures optical properties of aerosols in dry state, is described in
Sect. 3.2. The way of deriving Vg(RH) based on measurements of the humidified
nephelometer system is introduced and discussed in Sect. 3.3. The final
formula of calculating ambient ALWC is described in Sect. 3.4. The
verification of the *V*_{a}(dry) predicted by using the machine
learning method is described in Sect. 4.1. The validation of ambient ALWC
calculated from measurements of the humidified nephelometer system is
presented in Sect. 4.2. The contribution of ambient ALWC to the total ambient
aerosol volume is discussed in Sect. 4.3.

Datasets from six field campaigns were used in this paper. The six campaigns
were conducted at four different measurement sites (Wangdu, Gucheng and
Xianghe in Hebei province and Wuqing in Tianjin) of the North China Plain
(NCP), the locations of these field campaign sites are displayed in Fig. S1
in the Supplement. Time periods and datasets used from these field campaigns
are listed in Table 1. During these field campaigns, aerosol particles with
aerodynamic diameters less than 10 µm were sampled (by passing
through an impactor). The PNSDs in dry state, which range from 3 nm to
10 µm, were jointly measured by a Twin Differential Mobility
Particle Sizer (TDMPS, Leibniz-Institute for Tropospheric Research, Germany;
Birmili et al., 1999) or a scanning mobility particle size spectrometer
(SMPS) and an Aerodynamic Particle Sizer (APS, TSI Inc., Model 3321) with a
temporal resolution of 10 min. The mass concentrations of black carbon (BC)
were measured using a Multi-Angle Absorption Photometer (MAAP Model 5012,
Thermo, Inc., Waltham, MA USA) with a temporal resolution of 1 min during
field campaigns of F1 to F5 and using an aethalometer (AE33) (Drinovec et
al., 2015) during field campaign F6. The aerosol light scattering
coefficients (*σ*_{sp}) at three wavelengths (450, 550, and
700 nm) were measured using a TSI 3563 nephelometer (Anderson and Ogren,
1998) during field campaigns of F1 to F5, and using an Aurora 3000
nephelometer (Müller et al., 2011) during field campaign F6.

Datasets of PNSD, BC and *σ*_{sp} from campaigns F2, F4 and F5
are referred to as D1. Measurements of PNSD and measurements from the
humidified nephelometer system during campaign F6 (Gucheng campaign) are used
to verify the proposed method of calculating the ambient ALWC. Details about
the humidified nephelometer system during the Wangdu and Gucheng campaigns
are introduced in detail in Kuang et al. (2017). During the Gucheng campaign,
an In situ Gas and Aerosol Compositions Monitor (IGAC, Fortelice
International Co.,Taiwan) was used for monitoring water-soluble ions
(Na^{+}, K^{+}, Ca^{2+}, Mg^{2+},
${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$, Cl^{−}) of
PM_{2.5} and their precursor gases: NH_{3}, HCl, and
HNO_{3}. The time resolution of IGAC measurements is
1 h. Ambient air was drawn into the IGAC system
through a stainless-steel pipe wrapped with thermal insulation at a flow rate
of 16.7 L min^{−1}. The ambient RH and temperature were observed using an
automatic weather station with a time resolution of 1 min.

## 3.1 Closure calculations

To ensure the datasets of *σ*_{sp} and PNSD used are of high quality,
a closure study between measured *σ*_{sp} and that calculated based on
measured PNSD and BC with Mie theory (Bohren and Huffman, 2008) is first
performed. Measured *σ*_{sp} bears uncertainties introduced by angular
truncation errors and nonideal light source. To achieve consistency between
measured and modelled *σ*_{sp}, modelled *σ*_{sp} are calculated
according to practical angular situations of the nephelometer (Anderson
et al., 1996). During the *σ*_{sp} modelling process, BC was
considered to be half externally and half core–shell mixed with other aerosol
components. The mass size distribution of BC used in Ma et al. (2012), which was also observed
in the NCP, was used in this research to account for the mass distributions
of BC at different particle sizes. The applied refractive index and density
of BC were 1.80−0.54*i* and 1.5 g cm^{−3} (Kuang et al., 2015). The refractive index of non-light-absorbing aerosol components (other than BC) was set to
$\mathrm{1.53}-{\mathrm{10}}^{-\mathrm{7}}i$ (Wex et al., 2002). For the
Mie theory calculation details please refer to Kuang et
al. (2015).

The closure results between modelled *σ*_{sp} and
*σ*_{sp} measured by TSI 3563 or Aurora 3000 using datasets
observed during six field campaigns (Table 1) are depicted in Fig. 1. In
general, for all six field campaigns, modelled *σ*_{sp} values
correlate very well with measured *σ*_{sp} values. Considering
the measured PNSD has an uncertainty of larger than 10 % (Wiedensohler et
al., 2012), and the measured *σ*_{sp} has an uncertainty of
about 9 % (Sherman et al., 2015), modelled *σ*_{sp} values
agree well with measured *σ*_{sp} values in campaigns F1, F4, F5
and F6, with all points lying near the 1 : 1 line, and most points falling
within the 20 % relative difference lines. For the closure results of
field campaign F2, the modelled *σ*_{sp} values are
systematically lower than measured *σ*_{sp} values. For the
closure results of field campaign F3, most points also lie nearby 1 : 1
line, but points are relatively more dispersed.

## 3.2 Calculation of *V*_{a} (dry) based on measurements of the “dry” nephelometer

### 3.2.1 Theoretical relationship between *V*_{a} (dry) and *σ*_{sp}

Previous studies demonstrated that the *σ*_{sp} of aerosol particles
is roughly proportional to *V*_{a}(dry) (Pinnick et al., 1980). Here, the quantitative relationship
between *V*_{a}(dry) and *σ*_{sp} is analysed.

The *σ*_{sp} and *V*_{a}(dry) can be expressed
as the following:

where *Q*_{sca}(*m*,*r*) is scattering efficiency for a particle with
refractive index *m* and particle radius *r*, while *n*(*r*) is the aerosol
size distribution. As presented in Eqs. (1) and (2), relating
*V*_{a}(dry) with *σ*_{sp} involves the complex relation
between *Q*_{sca}(*m*,*r*) and particle diameter, which can be simulated
using the Mie theory. According to the aerosol refractive index at visible
spectral range, aerosol chemical components can be classified into two
categories: the light absorbing component and the almost light non-absorbing
components (inorganic salts and acids, and most of the organic compounds).
Near the visible spectral range, the light absorbing component can be
referred to as BC. BC particles are either externally or internally mixed
with other aerosol components. In view of this, *Q*_{sca} at 550 nm,
as a function of particle diameter for four types of aerosol particles, is
simulated using Mie theory: almost non-absorbing aerosol particle, BC
particle, BC particle core–shell mixed with non-absorbing components with
the radii of the inner BC core being 50 and 70 nm, respectively. Same with
those introduced in Sect. 2.2, the refractive indices of BC and light
non-absorbing components used here are 1.80−0.54*i* and $\mathrm{1.53}-{\mathrm{10}}^{-\mathrm{7}}i$,
respectively.

The simulated results are shown in Fig. 2a. Near the visible spectral range,
most of the ambient aerosol components are almost non-absorbing, and their
*Q*_{sca} varies more like the blue line shown in Fig. 2a. In that
case, aerosol particles have diameters less than about 800 nm and
*Q*_{sca} increases almost monotonously with particle diameter and can
be approximately estimated as a linear function of diameter. Figure 2b shows
the simulated size-resolved accumulative contribution to the scattering
coefficient at 550 nm for all PNSDs measured during the Wangdu campaign. The
results indicate that, for continental aerosol particles without influences
of dust, in most cases, all particles with diameter less than about 800 nm
contribute more than 80 % to the total *σ*_{sp}. Therefore,
for Eq. (1) if we express *Q*_{sca}(*m*,*r*) as
${Q}_{\mathrm{sca}}(m,r)=k\cdot r$ then Eq. (1) can be expressed as the
following:

This explains why *σ*_{sp}(550 nm) is roughly
proportional to *V*_{a}(dry). However, the value *k*
varies greatly with particle diameter. The ratio *σ*_{sp}(550 nm) ∕ *V*_{a}(dry)
(hereafter referred to as *R*_{Vsp}) is mostly affected by the PNSD, which
determines the weight of influence different particle diameters have on
*R*_{Vsp}. The discrepancy between the blue line and black line shown in
Fig. 2a indicates that the fraction of externally mixed BC particles and
their sizes has large impact on *R*_{Vsp}. The difference between the black
line and the red line as well as the difference between the solid red line
and the dashed red line shown in Fig. 2a indicate that the way and the amount
of BC mixed with other components also exert significant influences on
*R*_{Vsp}. In summary, the variation of *R*_{Vsp} is mainly determined by
variations in PNSD, mass size distribution and the mixing state of BC. It is
difficult to find a simple function describing the relationship between
measured *σ*_{sp} and *V*_{a}(dry).

Based on PNSD and BC datasets of field campaigns F1 to F6, the relationship
between *σ*_{sp} at 550 nm and *V*_{a}(dry) of
PM_{10} or PM_{2.5} are simulated using the Mie theory. The results are
shown in Fig. 3. The results demonstrate that the *σ*_{sp} at 550 nm is
highly correlated with the *V*_{a}(dry) of PM_{10}
and PM_{2.5}. The square of the correlation coefficient (*r*^{2}) between
*σ*_{sp} at 550 nm and *V*_{a}(dry) of PM_{10}
or PM_{2.5} are 0.94 and 0.99, respectively. A roughly proportional
relationship exists between *V*_{a}(dry) and *σ*_{sp}(550 nm), especially for *V*_{a}(dry) of PM_{2.5}. However, both *R*_{Vsp} of PM_{10} and PM_{2.5}
vary significantly. *R*_{Vsp} of PM_{10} mainly ranges from 2 to 6 cm^{3} (µm^{3} Mm)^{−1}, with an
average of 4.2 cm^{3} (µm^{3} Mm)^{−1}. *R*_{Vsp} of PM_{2.5} mainly ranges from 3 to 6.5 cm^{3} (µm^{3} Mm)^{−1}, with an average of
5.1 cm^{3} (µm^{3} Mm)^{−1}. Simulated size-resolved accumulative contributions to
*σ*_{sp} at 550 nm for all PNSDs measured during campaigns F1 to F6
and corresponding size-resolved accumulative contributions to *V*_{a}(dry) of PM_{10} are shown in Fig. S2. The results indicate
that particles with diameter larger than 2.5 µm usually contribute
negligibly to *σ*_{sp} at 550 nm but contribute about 20 % of the
total PM_{10} volume. Hence *σ*_{sp} at 550 nm is insensitive to
changes in particles mass of diameters between 2.5 and 10 µm. This may
partially explain why *V*_{a}(dry) of PM_{2.5}
correlates better with *σ*_{sp} at 550 nm than *V*_{a}(dry) of PM_{10}.

### 3.2.2 Machine learning

Based on analyses in Sect. 3.2.1, *R*_{Vsp} varies a lot with PNSD being the
most dominant influencing factor. The “dry” nephelometer provides not only
one single *σ*_{sp} at 550 nm, it measures six parameters including
*σ*_{sp} and back scattering coefficients (*σ*_{bsp}) at three
wavelengths (for TSI 3563: 450, 550 and 700 nm). The
Ångström exponent calculated from spectral
dependence of *σ*_{sp} provides information on the mean predominant
aerosol size and is associated mostly with PNSD. The variation of the
hemispheric backscattering fraction (HBF), which is the ratio between
*σ*_{bsp} and *σ*_{sp}, is also essentially related to the PNSD.
HBFs at three wavelengths (450, 550 and 700 nm) and the
Ångström exponents calculated from *σ*_{sp}
at different wavelengths (450–550, 550–700 and 450–700 nm) for typical
non-absorbing aerosol particles with their diameters ranging from 100 nm to
3 µm are simulated using the Mie theory. The results are shown
in Fig. 4a and b. HBF values at three different wavelengths and their
differences are more sensitive to changes in PNSD of particle diameters less
than about 400 nm. Ångström exponents calculated
from *σ*_{sp} at different wavelengths almost decrease monotonously
with particle diameter when particle diameter is less than about 1 µm; however, they differ distinctly when particle diameter is
larger than 300 nm. These results indicate that HBFs at three wavelengths
and Ångström exponents calculated from *σ*_{sp} at different wavelengths are sensitive to different diameter ranges
of PNSD.

Thus, all six parameters measured by the “dry” nephelometer together can
provide valuable information about variations in *R*_{Vsp}. However, no
explicit formula exists between these six parameters and *V*_{a}(dry). How to use these six optical parameters is a
problem;
machine learning methods that can handle many input parameters are capable
of learning from historical datasets and then make predictions, and strict
relationships among variables are not required. Machine learning methods are
powerful tools for tackling highly nonlinear problems and are widely used in
different areas. In the light of this, predicting *V*_{a}(dry) based on six optical parameters measured by the “dry”
nephelometer might be accomplished by using a machine learning method. In
this study, random forest is chosen for this purpose.

Random forest is a machine learning technique that is widely used for
classification and non-linear regression problems (Breiman, 2001). For
non-linear regression cases, random forest model consists of an ensemble of
binary regression decision tress. Each tree has a randomised training scheme,
and an average over the whole ensemble of regression tree predictions is used
for final prediction. In this study, the function RandomForestRegressor from
the Python Scikit-Learn machine learning library
(http://scikit-learn.org/stable/index.html, last access: 16 May 2018)
is used. This model has several strengths. First, through averaging over an
ensemble of decision trees there is a significantly lower risk of
overfitting. Second, it involves fewer assumptions about the dependence
between inputs and outputs when compared with traditional parametric
regression models. The random forest model has two parameters: the number of
input variables (*N*_{in}) and the number of trees grown
(*N*_{tree}). In this study, *N*_{in} and *N*_{tree} are
six and eight, respectively. The six input parameterises the three scattering
coefficients, three backscattering coefficients.

The quality of input datasets is critical to the prediction accuracy of the
machine learning method. As discussed in Sect. 3.1, modelled
*σ*_{sp} during some field campaigns are not completely
consistent with measured *σ*_{sp}, large bias might exist between
them due to the measurement uncertainties of PNSD and *σ*_{sp}.
To avoid the uncertainties in measurements of PNSD, aerosol optical properties are propagated in the training processes of the random forest model. In this study, both the
required datasets of six optical parameters which corresponding to
measurements of TSI 3563 and *V*_{a}(dry) for training the random
forest model are calculated or simulated based on measurements of PNSD and BC
from field campaigns F1 to F4 and F6. Datasets of PNSD and six optical
parameters measured by the nephelometer during campaign F5 are used to verify
the prediction ability of the trained random forest model. The performance of
this random forest model on predicting both *V*_{a}(dry) of
PM_{10} and PM_{2.5} are investigated. A schematic diagram of
this method is shown in Fig. 5.

## 3.3 Connecting *f*(RH) to Vg(RH)

### 3.3.1 *κ*-Köhler theory

*κ*-Köhler theory is used to describe the hygroscopic
growth of aerosol particles with different sizes, and the formula expression
of *κ*-Köhler theory can be written as follows (Petters and Kreidenweis, 2007):

where *D* is the diameter of the droplet, *D*_{d} is the dry diameter, *σ*_{s∕a} is the surface tension of solution/air interface, *T* is the temperature,
*M*_{water} is the molecular weight of water, *R* is the universal gas
constant, *ρ*_{w} is the density of water, and *κ* is the
hygroscopicity parameter. By combining the Mie theory and the
*κ*-Köhler theory, both *f*(RH) and Vg(RH) can be simulated. In
the processes of calculations for modelling *f*(RH)
and Vg(RH), the treatment of BC is same
with those introduced in Sect. 2.2. As aerosol particle grows due to aerosol
water uptake, the refractive index will change. In the Mie calculation,
impacts of aerosol liquid water on the refractive index are considered based
on volume mixing rule. The used refractive index of liquid water is
$\mathrm{1.33}-{\mathrm{10}}^{-\mathrm{7}}i$ (Seinfeld and Pandis, 2006).

### 3.3.2 Parameterization schemes for *f*(RH) and Vg(RH)

The *f*(RH) is defined as *f*(RH) = *σ*_{sp}(RH,550 nm) ∕ *σ*_{sp}(dry,550 nm), where *σ*_{sp}(RH,550 nm) and *σ*_{sp}(dry,550 nm) represents *σ*_{sp} at wavelength 550 nm under certain RH and dry
conditions. Additionally, Vg(RH) is defined
as Vg(RH) = *V*_{a}(RH) ∕ *V*_{a}(dry), where
*V*_{a}(RH) represents total volume of aerosol particles under
certain RH conditions.

A physically based single-parameter representation is proposed by Brock et al. (2016) to describe *f*(RH). The parameterization
scheme is written as follows:

where *κ*_{sca} is the parameter which fits *f*(RH) best. Here, a
brief introduction is given about the physical understanding of this
parameterization scheme. For aerosol particles whose diameters larger than
100 nm, regardless of the Kelvin effect, the hygroscopic growth factor for
an aerosol particle can be approximately expressed as
*g*(RH) ≅ ${\left(\mathrm{1}+\mathit{\kappa}\frac{\mathrm{RH}}{\mathrm{100}-\mathrm{RH}}\right)}^{\frac{\mathrm{1}}{\mathrm{3}}}$ (Brock et al.,
2016). Enhancement factor in volume can be expressed as the cube of *g*(RH).
Aerosol particles larger than 100 nm contribute the most to
*σ*_{sp} and *V*_{a}(dry) (as shown in Fig. S2). If a
constant *κ* which represents the overall aerosol hygroscopicity of
ambient aerosol particles is used as the *κ* of different particle
sizes, then Vg(RH) can be approximately expressed as Vg(RH) = $\mathrm{1}+\mathit{\kappa}\frac{\mathrm{RH}}{\mathrm{100}-\mathrm{RH}}$. In addition, *σ*_{sp} is
usually proportional to *V*_{a}(dry), which indicates that the
relative change in *σ*_{sp} due to aerosol water uptake is
roughly proportional to relative change in aerosol volume. Therefore, *f*(RH)
might also be well described by using the formula form of Eq. (5). Previous
studies have shown that this parameterization scheme can describe *f*(RH)
well (Brock et al., 2016; Kuang et al., 2017).

During processes of measuring *f*(RH), the sample RH
in the “dry” nephelometer (RH_{0}) is not zero. According to Eq. (5), the measured *f*(RH)${}_{\mathrm{measure}}=\frac{f\left(\mathrm{RH}\right)}{f\left({\mathrm{RH}}_{\mathrm{0}}\right)}$ should be fitted using the
following formula:

Based on this equation, *κ*_{sca} can be calculated from measured
*f*(RH) directly. The typical value of RH_{0}
measured in the “dry” nephelometer during the Wangdu campaign is about 20 %.
The importance of the RH_{0} correction changes under different aerosol
hygroscopicity and RH_{0} conditions. The parameter *κ*_{sca} is
fitted with and without consideration of RH_{0} for *f*(RH) measurements during the Wangdu campaign, and the results
are shown in Fig. S3. The results demonstrate that, overall, the *κ*_{sca} will be underestimated if the influence of
RH_{0} is not considered, and the larger the *κ*_{sca}, the more that the *κ*_{sca} will be underestimated.

In addition, based on discussions about the physical understanding of Eq. (5), the Vg(RH) should be well described by the following equation:

where *κ*_{Vf} is the parameter which fits Vg(RH) best. To
validate this conclusion, a simulative experiment is conducted. In the
simulative experiment, average PNSD in dry state and mass concentration of BC
during the Haze in China (HaChi) campaign (Kuang et al., 2015) are used.
During HaChi campaign, size-resolved *κ* distributions are derived from
measured size-segregated chemical compositions (Liu et al., 2014) and their
average is used in this experiment to account the size dependence of aerosol
hygroscopicity. Modelled results of *f*(RH) and Vg(RH) are shown in
Fig. S4. Results demonstrate that modelled *f*(RH)
and Vg(RH) can be well parameterized using the formula form of Eqs. (5) and
(7). Fitted values of *κ*_{sca} and *κ*_{Vf} are
0.227 and 0.285, respectively. This result indicates that if linkage between
*κ*_{sca} and *κ*_{Vf} is established, measurements
of *f*(RH) can be directly related to Vg(RH).

### 3.3.3 Bridge the gap between *f*(RH) and Vg(RH)

Many factors have significant influences on the relationships between *f*(RH)
and Vg(RH), including PNSD, BC mixing state and the size-resolved aerosol
hygroscopicity. To gain insights into the relationships between
*κ*_{sca} and *κ*_{Vf}, a simulative experiment
using Mie theory and *κ*-Köhler theory is designed. In this
experiment, all PNSDs at dry state along with mass concentrations of BC from
D1 are used, characteristics of these
PNSDs can be found in Kuang et al. (2017). As to size-resolved aerosol
hygroscopicity, a number of size-resolved *κ* distributions were
derived from measured size-segregated chemical compositions during HaChi
campaign (Liu et al., 2014). Results from other research also show similar
size dependence of aerosol hygroscopicity (Meng et al., 2014). In view of
this, the shape of the average size-resolved *κ* distribution during
HaChi campaign (black line shown in Fig. S5) is used in the designed
experiment. Other than the shape of size-resolved *κ* distribution, the
overall aerosol hygroscopicity, which determines the magnitude of *f*(RH),
also has a large impact on the relationship between *κ*_{sca} and
*κ*_{Vf}. In view of this, ratios ranging from 0.05 to 2, with an
interval of 0.05, are multiplied with the average size-resolved *κ*
distribution (the black line shown in Fig. S5) to produce a number of
size-resolved *κ* distributions which represent aerosol particles from
nearly hydrophobic to highly hygroscopic. During simulating processes, each
PNSD is modelled with all produced size-resolved *κ* distributions. In
the following, the ratio *κ*_{Vf}∕*κ*_{sca}, termed as
*R*_{Vf}, is used to indicate the relationship between
*κ*_{sca} and *κ*_{Vf}.

Considering that values of the Ångström exponent contain information
about PNSD (Kuang et al., 2017) and values of *κ*_{sca} represent
overall hygroscopicity of ambient aerosol particles, and that both of these
parameters can be directly calculated from measurements of a three-wavelength
humidified nephelometer system (Kuang et al., 2017), simulated
*R*_{Vf} values are spread into a two-dimensional gridded plot. The
first dimension is the Ångström exponent with an interval of 0.02 and
the second dimension is *κ*_{sca} with an interval of 0.01.
Average *R*_{Vf} value within each grid is represented by colour and
shown in Fig. 6a. Values of the Ångström exponent corresponding to
used PNSDs are calculated from simultaneously measured *σ*_{sp}
values at 450 and 550 nm from the TSI 3563 nephelometer. Results shown in
Fig. 6a exhibit that both PNSD and overall aerosol hygroscopicity have
significant influences on *R*_{Vf}. Simulated values of
*R*_{Vf} range from 0.8 to 1.7, with an average of 1.2. Overall, the
*R*_{Vf} value is lower when the value of the Ångström
exponent is larger. The percentile value of standard deviation of
*R*_{Vf} values within each grid, divided by its average, is shown in
Fig. 6b. In most cases, these percentile values are less than 10 % (about
90 %) which demonstrates that *R*_{Vf} varies little within each
grid shown in Fig. 6a. Figure 6 shows the influence of aerosol size and
chemistry on *R*_{Vf}. For an Ångström exponent less than
∼ 1.1, *R*_{Vf} varies strongly with *κ*_{sca}.
However, for an Ångström exponent values greater than ∼ 1.1,
the *R*_{Vf} relative standard deviation exhibits a higher variability
with the Ångström exponent, thus showing the sensitivity of
*R*_{Vf} to changes in aerosol size for small particles. In general,
results shown in Fig. 6 imply that results of Fig. 6a can serve as a lookup
table to estimate *R*_{Vf} and thereby *κ*_{Vf}, such that
these values can be directly predicted from measurements of a
three-wavelength humidified nephelometer system.

For the lookup table shown in Fig. 6a, a fixed size-resolved *κ*
distribution is used, which might not be able to capture variations of
*R*_{Vf} induced by different types of size-resolved *κ*
distributions under different PNSD conditions. A simulative experiment is
conducted to investigate the performance of this lookup table. In this
experiment, the following datasets are used: PNSDs and mass concentrations of
BC from D1 (the number of used PNSD is 11996), and size-resolved *κ*
distributions from HaChi campaign (Liu et al., 2014), which are presented in
Fig. 7a (the number is 23). Results shown in Fig. 7a imply that the shape of
size-resolved *κ* distribution is highly variable, yet has no apparent
correlation with aerosol loading. During the simulating processes for each
PNSD, it is used to simulate *R*_{Vf} values corresponding to all used
size-resolved *κ* distributions; therefore, 275 908 *R*_{Vf}
values are modelled. Also, modelled values of *κ*_{sca} and
corresponding values of the modelled Ångström exponent are used
together to estimate *R*_{Vf} values using the lookup table shown in
Fig. 7a. Results of relative differences between estimated and modelled
*R*_{Vf}, values under different pollution conditions are shown in
Fig. 7b. Overall, 88 % of points have absolute relative differences less
than 15 % and 68 % of points have absolute relative differences less
than 10 %. This lookup table performs better when the air is relatively
polluted.

## 3.4 Calculation of ambient ALWC

According to the equation Vg(RH) = $\mathrm{1}+{\mathit{\kappa}}_{\mathrm{Vf}}\frac{\mathrm{RH}}{\mathrm{100}-\mathrm{RH}}$, ALWC refers to volume concentrations of aerosol liquid water at different RH points and can be expressed as the following:

According to discussions of Sect. 3.2, *V*_{a}(dry) can
be predicted based only on measurements from the “dry” nephelometer by
using a random forest model. The training of the random forest model
requires only regional historical datasets of simultaneously measured PNSD
and BC. The *κ*_{sca} is directly fitted from *f*(RH) measurements. The *R*_{Vf} can be estimated using the lookup table
introduced in Sect. 3.3. Thus, based only on measurements from a
three-wavelength humidified nephelometer system, ALWCs of ambient aerosol
particles at different RH points can be estimated. If both measurements from
the humidified nephelometer system and ambient RH are available, ambient
ALWC can be calculated. The flowchart of calculating ambient ALWC based on
measurements of the humidified nephelometer system is shown in Fig. 8. The
nephelometer used, corresponding to this flowchart, should be TSI 3563. If
nephelometer of the used humidified nephelometer system is Aurora 3000,
wavelengths in this flowchart will change but other steps are totally the
same.

## 4.1 Validation of the random forest model for predicting *V*_{a}(dry) based on measurements of the “dry” nephelometer

The machine learning method, random forest model, is proposed to predict
*V*_{a}(dry) based only on *σ*_{sp} and *σ*_{bsp} at three wavelengths measured by the “dry” nephelometer.
Datasets of PNSD and BC from field campaigns F1 to F4 and F6 are used to
train the random forest model. Datasets of PNSD and optical parameters
measured by the “dry” nephelometer from field campaign F5 are used to
verify the trained random forest model. The schematic diagram of this method
is shown in Fig. 5. The comparison results between calculated and predicted
*V*_{a}(dry) of PM_{10} and PM_{2.5} are shown in
Fig. 9. The square of correlation coefficient between predicted and
calculated *V*_{a}(dry) of PM_{10} is 0.96, and almost all
points lie between or near 20 % relative difference lines. The square of
correlation coefficient between predicted and calculated *V*_{a}(dry)
of PM_{2.5} is 0.997, and almost all points lie between or near
10 % relative difference lines. The standard deviations of relative
differences between predicted and calculated *V*_{a}(dry) of
PM_{10} and PM_{2.5} are 10 and 4 %, respectively. These
results indicate that *V*_{a}(dry) of PM_{2.5} can be well
predicted by using the machine learning method. While *V*_{a}(dry) of
PM_{10} predicted by using the machine learning method has a
relatively larger bias.

Machine learning methods do not explicitly express relationships between
many variables; however, they learn and implicitly construct complex
relationships among variables from historical datasets. Many different and
comprehensive machine learning methods are developed for diverse
applications and can be directly used as a tool for solving a lot of
nonlinear problems which may not be mathematically well understood. We
suggest using a machine learning method for estimating *V*_{a}(dry) based on measurements of the “dry” nephelometer. The
way of estimating *V*_{a}(dry) with machine learning
method might be applicable for different regions around the world if used
estimators are trained with corresponding regional historical datasets.

## 4.2 Comparison between ambient ALWC calculated from ISORROPIA and measurements of the humidified nephelometer system

So far, widely used tools for prediction of ambient ALWC are thermodynamic
models. ISORROPIA-II thermodynamic model
(http://nenes.eas.gatech.edu/ISORROPIA/index_old.html, last access:
16 May 2018) is a famous one and is widely used in research for predicting pH
and ALWC of ambient aerosol particles (Guo et al., 2015; Cheng et al., 2016;
Liu et al., 2017; Fountoukis and Nenes, 2007). Water-soluble ions and gaseous
precursors are required as inputs of thermodynamic model. During the Gucheng
campaign, measurements from both the humidified nephelometer system and IGAC
are available. Thus, the ambient ALWC can be calculated through two
independent methods: thermodynamic model based on IGAC measurements and the
method proposed in Sect. 3.4, which is based on measurements of the
humidified nephelometer system. In this study, the forward mode in
ISORROPIA-II is used and water-soluble ions in PM_{2.5} and gaseous
precursors (NH_{3}, HNO_{3}, HCl) measured by the IGAC
instrument along with simultaneously measured RH and *T* are used as inputs.
The aerosol water associated with organic matter is not considered in the
method of ISORROPIA model, due to the lack of measurements of organic aerosol
mass. However, results from previous studies indicate that organic matter
induced particle water only account for about 5 % of total ALWC (Liu et
al., 2017). For the ALWC calculated from the humidified nephelometer system,
the needed *V*_{a}(dry) of PM_{2.5} in Eq. (7) is calculated
from simultaneously measured PNSD.

The comparison results between ambient ALWC calculated from these two
independent methods are shown in Fig. 10a. The square of correlation
coefficient between them is 0.92, most of the points lie within or nearby
30 % relative difference lines. The slope is 1.14, and the intercept is
−8.6 µm^{3} cm^{−3}. When ambient RH is higher than 80 %,
the ambient ALWCs calculated from measurements of the humidified
nephelometer system are higher relative to those calculated based on
ISORROPIA-II. When ambient RH is lower than 60 %, the ambient ALWCs
calculated from measurements of the humidified nephelometer system are
lower relative to those calculated based on ISORROPIA-II. Overall, a
good agreement is achieved between ambient ALWC calculated from measurements
of the humidified nephelometer system and ISORROPIA thermodynamic model.

Guo et al. (2015) conducted the
comparison between ambient ALWC calculated from ISORROPIA model and ambient
ALWC calculated from measurements of the humidified nephelometer system by
assuming Vg(RH) = *f*(RH)^{1.5}. Thus, the comparison results between ambient ALWC
calculated based on ISORROPIA and ambient ALWC calculated by assuming
Vg(RH) = *f*(RH)^{1.5} are also shown in Fig. 10b. The square of the correlation
coefficient between them is also 0.92. However, the slope and intercept are
1.7 and −21 µm^{3} cm^{−3}, respectively. When the ambient
RH is higher than about 80 %, calculated ambient ALWC will be
significantly overestimated if it is assumed that Vg(RH) = *f*(RH)^{1.5}. This method assumes
that average scattering efficiency of aerosol particles at dry state and
different RH conditions are the same. When ambient RH is high, the particle
diameters changes a lot. As the results shown in Fig. S6, for non-absorbing
particle, when diameter of aerosol particle in dry state is less than 500
nm, the aerosol scattering efficiency increase almost monotonously with
increasing RH especially when RH is higher than 80 %. Therefore, it is not
suitable to assume that average scattering efficiency of aerosol particles
at dry state and different RH conditions are the same.

## 4.3 Volume fractions of ALWC in total ambient aerosol volume

During the Wangdu campaign, *κ*_{sca} ranged from 0.05 to 0.3 with an
average of 0.19. Estimated values of *R*_{Vf} ranges from 0.86 to 1.47, with
an average of 1.15. Estimated values of *κ*_{Vf} ranges from 0.05 to
0.35, with an average of 0.22. The calculated volume fractions of water in
total volume of ambient aerosols during the Wangdu campaign are shown in
Fig. 11a. The results indicate that during the Wangdu campaign, when ambient RH
is higher than 70 %, the *κ*_{Vf} values are relatively higher. The
volume fractions of water are always higher than 50 % when ambient RH is
higher than 80 %.

During the Gucheng campaign, *κ*_{sca} ranges from 0.008 to 0.22 with an
average of 0.1, *κ*_{Vf} ranges from 0.01 to 0.21 with an average of
0.12. The aerosol hygroscopicity during the Gucheng campaign is much lower than
aerosol hygroscopicity during the Wangdu campaign. The calculated volume
fractions of water in total volume of ambient aerosols during the Gucheng
campaign are shown in Fig. 11b. During the Gucheng campaign, the maximum volume
fraction of water in ambient aerosol is 42 % when ambient RH is at
80 %. On average, when ambient RH is higher than 90 %, the volume
fraction of water in ambient aerosols reaches higher than 50 %.

## 4.4 Discussions about the applicability of the proposed method

The method proposed in this research is based on datasets of PNSD, *σ*_{sp} and size-resolved *κ* distribution, which are measured on the
NCP without influences of dust events and sea salt. Caution should be
exercised if using the proposed method to estimate the ALWC when the air
mass is significantly influenced by sea salt or dust. The way of estimating
*V*_{a}(dry) with machine learning method might be
applicable for different regions around the world. However, the used
predictor from machine learning should be trained with corresponding
regional historical datasets of PNSD and BC. The way of connecting *f*(RH) to Vg(RH) might also
be applicable for other continental regions. Still, we suggest that the used
lookup table is simulated from regional historical datasets.

Note that the humidified nephelometer usually operates with RH less than
95 %. However, aerosol water increase dramatically with increasing RH
when RH is greater than 95 %. Such high RH conditions can occur during
the haze events. This may limit the usage of the proposed method when ambient
RH is extremely high. As discussed in Sect. 3.3, the proposed way of
connecting *f*(RH) and Vg(RH) is based on the *κ*-Köhler theory. If
*κ* does not change with RH, the proposed method should be applicable
when RH is higher than 95 %, even if the measurements of humidified
nephelometer system are conducted when RH is less than 95 %. Many studies
have done research about the change of *κ* with the changing RH (Rastak
et al., 2017; Renbaum-Wolff et al., 2016), their results demonstrate that the
*κ* changes with increasing RH. However, few studies have investigated
the variation of *κ* of ambient aerosol particles with changing RH when
RH is less than 100 %. Liu et al. (2011) have measured *κ* of
ambient aerosol particles at different RHs (90, 95 and 98.5 %) on the
NCP. Their results demonstrated that *κ* at different RHs differs
little for ambient aerosol particles with different diameters. Results of
Kuang et al. (2017) indicated that *κ* values retrieved from *f*(RH)
measurements agree well with *κ* values at RH of 98 % of aerosol
particles with diameter of 250 nm. In this respect, the proposed method
might be applicable even when ambient RH is extremely high for ambient
aerosol particles on the NCP. Moreover, for calculating the ambient ALWC, the
measured ambient RH is required. If the ambient RH is higher than 95 %,
the measured ambient RH with current techniques is highly uncertain. Given
this, cautions should be exercised if the ambient ALWC is calculated when the
ambient RH is higher than 95 %.

In this paper, a novel method is proposed to calculate ALWC based on
measurements of a three-wavelength humidified nephelometer system. Two
critical relationships are required in this method. One is the relationship
between *V*_{a}(dry) and measurements of the “dry”
nephelometer. Another one is the relationship between Vg(RH) and *f*(RH). The ALWC can be
calculated from the estimated *V*_{a}(dry) and
Vg(RH).

Previous studies have shown that an approximate proportional relationship
exists between *V*_{a}(dry) and corresponding *σ*_{sp},
especially for fine particles (particle diameter less than 1 µm).
However, PNSD and other factors still have significant influences on this
proportional relationship. It is difficult to directly estimate
*V*_{a}(dry) from measured *σ*_{sp}. In this paper, a
random forest predictor from machine learning procedure is used to estimate
*V*_{a}(dry) based on measurements of a three-wavelength nephelometer.
This random forest predictor is trained based on historical datasets of PNSD
and BC from several field campaigns conducted on the NCP. This method is then
validated using measurements from the Wangdu
campaign. The square of correlation coefficient between measured and
estimated *V*_{a}(dry) of PM_{10} and PM_{2.5} are 0.96
and 0.997, respectively.

The relationship between Vg(RH) and
*f*(RH) is investigated in Sect. 3 by conducting a
simulative experiment. It is found that the complicated relationship between
Vg(RH) and *f*(RH)
can be disentangled by using a lookup table, and parameters required in the
lookup table can be directly calculated from measurements of a
three-wavelength humidified nephelometer system. Given that the *V*_{a}(dry) can be estimated from a three-wavelength “dry”
nephelometer, the ambient ALWC can be estimated from measurements of a
three-wavelength humidified nephelometer system in conjunction with measured
ambient RH. We have conducted the comparison between ambient ALWC calculated
from ISORROPIA and ambient ALWC calculated from measurements of the
humidified nephelometer system. The square of correlation coefficient
between them is 0.92, and most of the points lie within or nearby 30 %
relative difference lines. The slope and intercept are 1.14 and −8.6 µm^{3} cm^{−3}, respectively. Overall, a good
agreement is achieved between ambient ALWC calculated from measurements of
the humidified nephelometer system and ISORROPIA thermodynamic model.

Results introduced in this research have bridged the gap between *f*(RH) and Vg(RH). The
advantage of using measurements of a humidified nephelometer system to
estimate ALWC is that this technique has a fast response time and can
provide continuous measurements of the changing ambient conditions. The new
method proposed in this research will facilitate the real-time monitoring of
the ambient ALWC and further our understanding of roles of ALWC in
atmospheric chemistry, secondary aerosol formation and climate change.

The data used in this study are available from the corresponding author upon request (zcs@pku.edu.cn).

RH | relative humidity |

PM_{2.5} |
particulate matter with aerodynamic diameter of less than 2.5 µm |

PM_{10} |
particulate matter with aerodynamic diameter of less than 10 µm |

f(RH) |
aerosol light scattering enhancement factor at 550 nm |

ALWC | aerosol liquid water content: volume concentrations of water in ambient aerosols |

V_{a}(dry) |
total volume of ambient aerosol particles in dry state |

Vg(RH) | aerosol volume enhancement factor due to water uptake |

NCP | North China Plain |

HTDMA | humidified tandem differential mobility analyser system |

PNSD | particle number size distribution |

BC | black carbon |

g(RH) |
hygroscopic growth factor |

APS | Aerodynamic Particle Sizer |

SMPS | scanning mobility particle size spectrometer |

σ_{sp} |
aerosol light scattering coefficient |

σ_{bsp} |
aerosol back scattering coefficient |

σ_{ext} |
aerosol extinction coefficient |

R_{Vsp} |
σ_{sp}(550 nm) ∕ V_{a}(dry) |

F1 to F6 | referred as to five field campaigns listed in Table 1 |

D1 | PNSD, BC and nephelometer measurements from F2, F4 and F5 |

The supplement related to this article is available online at: https://doi.org/10.5194/amt-11-2967-2018-supplement.

The authors declare that they have no conflict of interest.

This work is supported by the National Natural Science Foundation of China
(41590872 and 41505107), the National Key R&D Program of China
(2016YFC020000: Task 5) and the National Research Program for Key Issues in
Air Pollution Control (DQGG0103).

Edited by: Mingjin Tang

Reviewed by: three anonymous referees

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