the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Aerosol formation and growth rates from chamber experiments using Kalman smoothing

### Matthew Ozon

### Dominik Stolzenburg

### Lubna Dada

### Aku Seppänen

### Kari E. J. Lehtinen

Bayesian state estimation in the form of Kalman smoothing was applied to differential mobility analyser train (DMA-train) measurements of aerosol size distribution dynamics. Four experiments were analysed in order to estimate the aerosol size distribution, formation rate, and size-dependent growth rate, as functions of time. The first analysed case was a synthetic one, generated by a detailed aerosol dynamics model and the other three chamber experiments performed at the CERN CLOUD facility. The estimated formation and growth rates were compared with other methods used earlier for the CLOUD data and with the true values for the computer-generated synthetic experiment. The agreement in the growth rates was very good for all studied cases: estimations with an earlier method fell within the uncertainty limits of the Kalman smoother results. The formation rates also matched well, within roughly a factor of 2.5 in all cases, which can be considered very good considering the fact that they were estimated from data given by two different instruments, the other being the particle size magnifier (PSM), which is known to have large uncertainties close to its detection limit. The presented fixed interval Kalman smoother (FIKS) method has clear advantages compared with earlier methods that have been applied to this kind of data. First, FIKS can reconstruct the size distribution between possible size gaps in the measurement in such a way that it is consistent with aerosol size distribution dynamics theory, and second, the method gives rise to direct and reliable estimation of size distribution and process rate uncertainties if the uncertainties in the kernel functions and numerical models are known.

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Atmospheric new particle formation and growth are important phenomena when considering global aerosol concentrations. Aerosol number concentration together with their size distribution and chemical composition determine how aerosols affect visibility, health, and climate (Albrecht, 1989; Appel et al., 1985; Daellenbach et al., 2020; Pope and Dockery, 2006; Twomey, 1974). These are determined by atmospheric dynamics and aerosol dynamics such as new particle formation and growth as well as removal rates. Nieminen et al. (2018) reviewed the existing literature on the formation and growth rates ranging from polar sites, with very small aerosol concentrations, to polluted urban sites with extremely high concentrations. The rates have been typically estimated using the methodology reviewed in Kulmala et al. (2012) and adjusted for chamber experiments by Dada et al. (2020). Both are based on rather simple regression or balance equation approaches, and do not permit proper estimation of uncertainties. At the same time, however, instrument development, especially advances in particle detection efficiency and mass spectrometry, has developed rapidly (Kangasluoma et al., 2020). Potentially superior advanced data analysis methods have not been used, and, it is likely that there are significant inaccuracies in the particle formation and growth rates estimated previously (Kürten et al., 2018).

There have been some attempts to estimate aerosol formation and growth rates with different inverse methods (Henze et al., 2004; Kuang et al., 2012; Lehtinen et al., 2004; Sandu et al., 2005; Verheggen and Mozurkewich, 2006; Viskari et al., 2012). We are, however, not aware of any of the above-mentioned methodologies being used widely. The most promising ones in our view, that also include estimations of uncertainties, have been the ones by Kupiainen-Määttä (2016) and Shcherbacheva et al. (2020), who used Markov Chain Monte Carlo methodology to estimate evaporation rates as well as their uncertainties from synthetic cluster dynamics data. In addition, the INSIDE-method by Pichelstorfer et al. (2018), which is based on the numerical solution of the aerosol general dynamic equation and matching the solution optimally to integrated measured concentrations of selected size intervals, has been successfully applied to determine growth rates at the CLOUD (Cosmics Leaving OUtdoor Droplets) experiments at CERN (European Organization for Nuclear Research) (Stolzenburg et al., 2020). Furthermore, the results in a very recent paper by McGuffin et al. (2021), in which nucleation, growth, and emission rates are estimated using techniques from the field of non-linear process control, seem promising.

In a recent paper, Ozon et al. (2021a) presented BAYROSOL, a Julia software package that combines a finite difference solution to the general dynamic equation for aerosols (GDE; Seinfeld and Pandis, 2016) with Bayesian state estimation in order to estimate unknown size-dependent process rates (nucleation, condensation, losses) from the known time evolution of the aerosol size distribution. Bayesian state estimation is a general framework for estimating time-dependent variables (state variables) based on (direct or indirect) noisy observations that are collected sequentially during the temporal evolution of the state variables (Gelb, 1974). The state estimation is based on the so-called state-space representation, which consists of the state evolution model and observation model. In this work, the state variables consist of the particle size distribution – the temporal evolution of which is modelled with GDE – and the nucleation, growth, and deposition rates that are parameters of the GDE. The observation model is the mapping from the size distribution to DMA-train measurements. In Bayesian formulation, both the state variable and the observations are modelled as stochastic processes; their randomness reflects their uncertainty, which decreases when measurement data is accounted for in the state estimation – formally speaking, this is done by conditioning the state variables with respect to measurement data (realised observations) sequentially. The result of Bayesian state estimation is the posterior probability density that reflects the uncertainty of the state variables after accounting for the measurement data.

A large variety of state estimation schemes exits, and the choice between
them depends on (1) the type of the state-space model (linearity,
Gaussianity, etc); (2) the type of data available when computing an estimate
at time *t* (if data is available up to time *k**<**t*, the problem is of
prediction type, while cases where *k*=*t* and *k**>**t* are referred
to as filtering and smoothing, respectively), and (3) the approximations
that are sometimes needed to lower the computational demand of state
estimation. In the case of linear and Gaussian state-space models, a
Bayesian filtering problem can be solved recursively by the well-known
Kalman filter algorithm. In non-linear and non-Gaussian cases, the rigorous
choice is to use so-called particle filters and smoothers
(Särkkä, 2013). However, because these MCMC-based
estimators are highly time consuming in large-dimensional cases,
approximative methods are often used – such as the extended Kalman filter
and smoother adopted in this paper. These recursive algorithms use
sequential linearisation to approximate the non-linear models and
non-Gaussian probability distributions.

In previous work, Ozon et al. (2021a) showed that the fixed interval Kalman smoother (FIKS; Kaipio and Somersalo, 2005) performed very well in estimating the process rates of the GDE in two distinct cases: first, in a case in which continuous nucleation, growth, and losses lead to a nearly steady state size distribution and second, in a case in which there is a growing nucleation mode after a nucleation burst. In the method, the unknowns (such as the discretised particle size distribution) are modelled as random variables, and their prior probability distributions are incorporated in the solution of the inverse problem. One important key feature of the Kalman smoother method is that it also estimates the error covariance matrices of the process rates that are their uncertainties, if the uncertainties of the measurement device are known.

In this article, we show the results of applying BAYROSOL to real experimental data for the first time. We use experiments performed with the differential mobility analyser-train (DMA-train; Stolzenburg et al., 2017) measuring new particle formation and growth at the CERN CLOUD chamber (Duplissy et al., 2016; Kirkby et al., 2011). In addition to testing the method with synthetic DMA-train data (in which the `correct' results are known), we estimate formation and growth rates from three different formation and growth experiments: sulphuric-ammonia (Stolzenburg et al., 2020), alpha-pinene ozonolysis (Heinritzi et al., 2020) and iodic acid (He et al., 2021; Sipilä et al., 2016). We compare the formation rates with results obtained by using the methodology by Dada et al. (2020) based on particle size magnifier (PSM) measurements and the growth rates with the results obtained by the INSIDE method (Pichelstorfer et al., 2018). We chose the DMA-train measurements for three main reasons: first, the high time resolution makes it an ideal instrument for nucleation studies due to a more accurate estimate of dN/dt. Second, the collection efficiencies of the channels have been carefully characterised (Stolzenburg et al., 2017; Wlasits et al., 2020) and yield higher sensitivities to low particle number concentrations (Kangasluoma et al., 2020), which are often faced in atmospherically relevant nucleation studies, and third, the DMA train is, at the same time, an interesting and challenging instrument for detailed data analysis because of the gaps in the measured size spectrum.

## 2.1 Aerosol measurement and evolution models

Typical quantities of interest in chamber experiments studying new particle formation are the particle number size distribution, the formation rate at the critical cluster size and the growth rate of the freshly formed particles. The available measurements to infer these quantities usually size classify the aerosol and measure the size-classified number concentrations. While the retrieval of the particle number size distribution from such measurements is a classical inverse problem (Chambolle and Pock, 2011; Fiebig et al., 2005; Kandlikar and Ramachandran, 1999; Wolfenbarger and Seinfeld, 1990), the estimation of the process parameters (formation and growth rate) is often done by analysis of the time evolution of the retrieved particle size distribution (Dada et al., 2020). Here, we focus on the formulation of the problem within a statistical Bayesian framework, where the state parameters described by a measurement model and an evolution model are treated as multivariate random processes and are estimated from measurements using a FIKS (Ozon et al., 2021a).

A measurement can be described by a vector *y*^{k} representing *m* indirect
observations (channels of the instrument) of the particle size
distribution *n*(*d*_{p},*t*_{k}). The observations are linked to the size
distribution by the so-called instrument transfer (or kernel) functions *H*
such that

The transfer functions *H* can be inferred from calibration experiments and
instrument design considerations. Direct inversion of Eq. (1) for every time
instant *t*_{k} is typically an underdetermined and ill-posed inverse
problem and requires some additional assumptions in order to avoid
oscillatory and unstable solutions (Kandlikar and
Ramachandran, 1999). At the same time, the time evolution of the particle
size distribution *n*(*d*_{p},*t*) can be described by the
aerosol general dynamic equation (GDE):

Here, *g*(*d*_{p},*t*) is the condensation growth/evaporation rate,
*λ*(*d*_{p},*t*) is the particle loss by deposition or
dilution and “CoagSink” and “CoagSrc” are the sink and
source rates due to particle coagulation within the size distribution with
the coagulation coefficients *β*. An exact expression of all terms
can be found in Ozon et al. (2021a) and
Seinfeld and Pandis (2016). The boundary conditions of Eq. (2) are given by
the apparent formation rate ${J}_{{d}_{\mathrm{min}}}\left(t\right)=g({d}_{\mathrm{min}},t)n({d}_{\mathrm{min}},t)$ of newly formed particles at the minimum
detectable size *d*_{min} and a zero numerical flux condition $g\left({d}_{\mathrm{\infty}},t\right)n\left({d}_{\mathrm{\infty}},t\right)=\mathrm{0}$ at very large sizes.
Altogether, the process parameters $g\left({d}_{\mathrm{p}},t\right),J\left(t\right),\mathit{\lambda}\left({d}_{\mathrm{p}},t\right)$ and *β*(*d*_{i},*d*_{j}) as well as
the initial and boundary conditions completely determine the evolution of
the size distribution, but especially *g*(*d*_{p},*t*) and *J*(t) are usually not known. The coagulation coefficients *β*(*d*_{i},*d*_{j}) can often be obtained from theory (and coagulation above
*d*_{min} can be even neglected in many applications with low particle
concentrations) and the loss parameters *λ*(*d*_{p},*t*) are
well quantified for controlled aerosol chamber experiments.

A single measurement of the size distribution *y*^{k} does not depend
explicitly on the process parameters, but as *g*(*d*_{p},*t*) and
*J*(t) determine the temporal evolution of *n*(*d*_{p},*t*_{k}) the
estimation of the process parameters is feasible from a sequence of *l*
measurements *y*^{k} at several time instances.

## 2.2 State estimation with Kalman smoothing

Following Ozon et al. (2021a), we formulate the
problem as a Bayesian state estimation problem. After discretisation of the
problem in size space, i.e. particle diameter ($i=\mathrm{1},\mathrm{\dots},q)$ and time
($k=\mathrm{1},\mathrm{\dots}l)$, we can define the state variable ${\mathit{X}}^{k}={\left[{\mathit{N}}^{k}\phantom{\rule{2em}{0ex}}{\mathit{g}}^{k}\phantom{\rule{2em}{0ex}}{\mathit{\lambda}}^{k}\phantom{\rule{2em}{0ex}}{\mathit{J}}^{k}\right]}^{T}$ for each time step *k* with the particle
concentrations ${N}_{i}^{k}$ per size discretisation bin *i*, the
condensation and loss terms ${g}_{i}^{k}$ and ${\mathit{\lambda}}_{i}^{k}$,
respectively, for each size discretisation bin *i*, and the nucleation
rate *J*^{k}. Here, we have denoted ${\mathit{N}}^{k}=\left[{N}_{\mathrm{1}}^{k},\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{N}_{q}^{k}\right],\phantom{\rule{0.125em}{0ex}}{\mathit{N}}^{k}=\left[{g}_{\mathrm{1}}^{k},\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{g}_{q}^{k}\right]$ and ${\mathit{\lambda}}^{k}=\left[{\mathit{\lambda}}_{\mathrm{1}}^{k},\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{\mathit{\lambda}}_{q}^{k}\right]$.
The problem can then be formulated as

Equation (3) represents the discretised non-linear evolution model, which is
based on the general dynamic equation for *N*^{k}, on second-order processes
for *g*^{k} and *J*^{k}, and a random walk evolution for *λ*^{k} (see
Sect. 2.3, Eqs. 5–6). Equation (4) represents the discretised linear
observation model. The terms *w*^{k} and *v*^{k} are the error terms, which
are approximated as normally distributed $\mathcal{N}(\mathbf{0},{\mathbf{\Gamma}}_{w}^{k})$ and
$\mathcal{N}(\mathbf{0},{\mathbf{\Gamma}}_{e}^{k})$ with the covariance matrices ${\mathbf{\Gamma}}_{w}^{k},\phantom{\rule{0.125em}{0ex}}{\mathbf{\Gamma}}_{v}^{k}$, which not only
include stochastic noise, but also errors due to discretisation, model, and
parameter uncertainties.

We note that the above description of the state-space model Eqs. (3)–(4) is
slightly simplified for the sake of notational convenience. Namely, two
additional features – both described in detail by Ozon et al. (2021a) – are
included in the model: first, we assume that the process rates are positive
quantities and incorporate this positivity constraint into the evolution
model by reparametrising these quantities in the model. For example, for the
nucleation rate *J*^{k}, we write ${\mathit{J}}^{k}=\frac{\mathrm{1}}{\mathit{\alpha}}\mathrm{ln}\left(\mathrm{1}+{e}^{\mathit{\alpha}{\mathit{\xi}}_{J}^{k}}\right)$, where ${\mathit{\xi}}_{J}^{k}$ is an unconstrained
random variable and *α* is a scaling constant. Respective
parametrisations are written for ${g}_{i}^{k}$ and ${\mathit{\lambda}}_{i}^{k}$.
Secondly, as noted above, second-order models are written for rates *J*^{k}
and *g*^{k}. More specifically, we consider the respective state parameters
${\mathit{\xi}}_{J}^{k}$, ${\mathit{\xi}}_{g}^{k}$ as second-order Markov
processes; for example ${\mathit{\xi}}_{J}^{k}={\mathit{\psi}}_{\mathrm{1}}{\mathit{\xi}}_{J}^{k-\mathrm{1}}+{\mathit{\psi}}_{\mathrm{2}}{\mathit{\xi}}_{J}^{k-\mathrm{2}}+\mathit{\eta}$, where *ψ*_{1} and *ψ*_{2} are model
parameters and ** η** is Gaussian state noise. The second-order models are
written, because they promote temporal smoothness in the processes. When the
positivity constraint and the second-order models are included in the model,
the state variable

*X*^{k}in the state-space model Eqs. (3)–(4) is rewritten in the form ${\mathit{X}}^{k}={\left[{\mathit{N}}^{k}\phantom{\rule{2em}{0ex}}{\mathit{\xi}}_{g}^{k}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\mathit{\xi}}_{g}^{k-\mathrm{1}}\phantom{\rule{2em}{0ex}}{\mathit{\lambda}}^{k}\phantom{\rule{2em}{0ex}}{\mathit{\xi}}_{J}^{k}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\mathit{\xi}}_{J}^{k-\mathrm{1}}\right]}^{T}$, and at each time step the above logarithmic functions are used for mapping the unconstrained variables ${\mathit{\xi}}_{J}^{k}$, ${\mathit{\xi}}_{g}^{k}$ and ${\mathit{\xi}}_{\mathit{\lambda}}^{k}$ to respective quantities

*J*

^{k}, ${g}_{i}^{k}$ and ${\mathit{\lambda}}_{i}^{k}$. For the details on the above modifications as well as discretisation of the GDE model, we refer to Ozon et al. (2021a).

The GDE, i.e. the non-linear evolution model for *N*^{k} (Eq. 2), is
similar to an advection equation. Therefore, its numerical discretisation
schemes are often unstable and must be treated carefully to avoid
oscillation and divergence or to minimise numerical diffusion
(Shen et al., 2007; Smolarkiewicz, 1984).
Thus, we show detailed considerations on the magnitude of the different
error terms in the Supplement.

Considering this structure of the problem, a non-linear extension to the
Kalman Filter (Extended Kalman Filter; EKF) is a well-suited algorithm for
solving the unknown size distribution and process parameters
(Gelb, 1974; Kaipio and Somersalo, 2005). It is a two-stage recursive procedure, where in the first stage the future state and
propagation of error is predicted based on the state evolution model (Eq. 3). In the second stage, the state variable and its covariance are
estimated by updating the predicted state variable and covariance. This
so-called measurement update accounts for the discrepancy between the
realised measurements at time *t*_{k} and modelled measurements
corresponding to the predicted state variable. This procedure is repeated
until the final measurement *k*=*l*. After finishing the EKF recursions, we
utilise a fixed interval Kalman smoother (FIKS), which consists of a
backward recursion from a backward gain matrix and smooths the results by
backwards recursion from *l* to 1. The workflow of the EKF and FIKS are
illustrated in Table 1 and more details on this algorithm can also be found
in Ozon et al. (2021a).

## 2.3 Adaption to chamber experiments

The state-space model has been adjusted to best represent the evolution of an aerosol system during new particle formation experiments in an atmospheric simulation chamber like CLOUD. For the time evolution of the process parameters, we assume a rather smooth evolution for the nucleation and growth rates, approximated by a second-order process (Ozon et al., 2021a):

The constants *r*_{1},*r*_{2} depend on the characteristic time of change,
discretisation time, and a dampening factor, and their definition can be taken
from Ozon et al. (2021a). The corresponding
values for our experiments are listed in Table S1 in the Supplement.

In contrast to the growth and formation rates, the loss rates in a chamber
experiment do not depend on time, but can be described by time-independent
wall and dilution losses $\mathit{\lambda}\left({d}_{\mathrm{p}}\right)={\mathit{\lambda}}_{\mathrm{dil}}+{\mathit{\lambda}}_{\mathrm{wall}}\left({d}_{\mathrm{p}}\right)$.
These loss rates are well characterised by dedicated wall loss experiments
(Stolzenburg et al., 2020) and the dilution rate of the
chamber ${\mathit{\lambda}}_{\mathrm{dil}}={Q}_{\mathrm{tot}}/{V}_{\mathrm{chamber}}$, where *Q*_{tot} is the total
flow rate to the chamber to maintain constant pressure, and *V*_{chamber} is
the chamber volume. The time evolution is described by a random walk with a
small stochastic noise term ${\mathit{w}}_{\mathit{\lambda}}^{k}$, and the expectation of the
initial state (see Table S1 in the Supplement) is set to the experimentally
determined value with a standard deviation of ± 10 %:

For fully defining the problem (Eqs. 3–4), an estimate of the covariance
matrices corresponding to the error terms *v*^{k} and *w*^{k} is needed. The
definition of the covariance matrices corresponding to the state noise
*w*^{k} on the size-distribution evolution ${\mathit{w}}_{N}^{k}$, the growth rate
${\mathit{w}}_{g}^{k}$, the formation rate ${\mathit{w}}_{J}^{k}$, and the wall loss rate
${\mathit{w}}_{\mathit{\lambda}}^{k}$ follow the consideration of
Ozon et al. (2021a). The covariances of the wall
losses, the growth rate, and the size distribution are dominantly diagonal
with some additional off-diagonal terms in order to account for a
correlation in size. The formulation given by
Ozon et al. (2021a) was slightly altered to give a
stronger correlation between the closest size bins due to the sparser
size resolution of the DMA-train compared to the simulated SMPS system
(values for the different experiments are given in Table S1 in the
Supplement):

For the size-distribution evolution, we find that ${\mathit{\sigma}}_{i,N}^{\mathrm{2}}=\left({\mathit{\delta}}^{k}{)}^{\mathrm{2}}\mathrm{Var}\right({W}_{i}^{k})$, with *δ*^{k} as
the discretisation time-step and ${W}_{i}^{k}$ as the error of the discretisation
of the size-distribution evolution. A detailed derivation of ${W}_{i}^{k}$ is
given in the Supplement.

The modelling error of the observation model and the measurement noise both
contributing to *v*^{k} are assumed to be mutually independent. For this
reason, the covariance of the error term ${\mathbf{\Gamma}}_{v}^{k}$ in the
measurement model is written as a sum of the covariances of these two random
variables, i.e. ${\mathbf{\Gamma}}_{v}^{k}={\mathbf{\Gamma}}_{\mathrm{mod}}^{k}+{\mathbf{\Gamma}}_{y}^{k}$. For a detailed derivation we refer to
the Supplement, where we also show that the discretisation error is
negligible compared to model and measurement error if a fine enough size
discretisation is chosen. We approximate ${\mathbf{\Gamma}}_{v}^{k}$ with
uncorrelated processes, and hence the covariance matrices are of the
diagonal form. For the measurement error, the variance is given by Poisson
counting statistics in the case of a single-particle counting detector such
as a condensational particle counter (CPC):

For the model uncertainty, we assume the variance of the kernel
Var(Δ*H*_{i,j}) is composed of an uncertainty
proportional to *H*_{i,j} (for example due to an offset in the absolute
calibration of the instrument) and a shifting size information error (for
example discrepancy between set and actual classified size in a mobility
spectrometer). It can then be formulated as (detailed values for the
experiments under investigation can be found in Table S1 in the Supplement)

We use experimental data from the CERN CLOUD experiment (Duplissy et al., 2016; Kirkby et al., 2011) where we measured particle size distributions in the sub-10 nm range with a DMA-train (Stolzenburg et al., 2017). The raw data obtained from the DMA-train is used as input for the analysis of three different sets of experiments performed in the atmospheric simulation chamber. The DMA-train instrument kernels are also used for modelling an instrument response to simulated size-distribution data in order to verify the general performance of the FIKS on DMA-train-like data.

## 3.1 DMA-train

The DMA-train is an electrical mobility spectrometer specifically designed
to measure sub-10 nm size distributions (Stolzenburg et
al., 2017). Six identical DMAs are applied in parallel, i.e. they sample
through the same inlet. They are set to six distinct but fixed voltages and
hence classified sizes. The charging state of the aerosol is pre-conditioned
in two TSI Inc. Advanced Aerosol Neutralizers (Model 3088), each supplying
three DMAs at 5.5 L min^{−1} total flow. We use the Wiedensohler
approximation (Wiedensohler, 1988) to describe the
steady-state charge distribution at the DMA inlets.
Kallinger and Szymanski (2015) showed that for the used
neutralizers the steady-state charge distribution is still achieved for flow
rates up to 5 L min^{−1} and we assume that this holds true for 5.5 L min^{−1} flow too.
After size classification, the aerosol is detected in condensation particle
counters. Four channels are equipped with TSI Inc. Model 3776 ultrafine CPCs
for detection of aerosols down to 2.5 nm. They were operated at reduced
temperature settings in order to increase the detection efficiency of the
smallest particles, achieving a 50 % detection efficiency for particles as
small as 2 nm (Wlasits et al., 2020). Two channels of the
DMA-train were operated with particle counters specifically designed for
sub-2 nm particle detection using diethylene glycol (DEG), an Airmodus Ltd.
particle size magnifier (Model A10, PSM) and a TSI Inc. nano enhancer (Model 3777). Each is used as a booster stage to activate the particles, which are
subsequently detected by either an Airmodus Ltd. CPC (Model A20, for the
PSM) or a TSI Inc. CPC (Model 3772, for the nano enhancer). Both channels
have a higher aerosol flow rate of 2.5 L min^{−1} resulting in a broader transfer
function and higher transmission at the DMA compared to the 1.5 L min^{−1} sample
flow in the other four channels. The sheath flow at the DMAs is kept
constant at 15 L min^{−1} for all six channels.

The constant sampling at fixed sizes allows for either a higher time resolution at large aerosol number concentrations or a higher sensitivity towards low number concentrations due to longer signal averaging times compared to a scanning or stepping differential mobility spectrometer. To increase the number of measured particle sizes, one DMA is still operated in an alternating mode, switching between 6.2 and 8 nm every ten seconds. The other DMAs are set to classify particles of 4.3, 3.2, 2.55, 2.2, and 1.8 nm. The instrument kernels are obtained from calibration experiments, where we use the DMA transfer function and sampling loss characterisation from Stolzenburg et al. (2017), the CPC activation efficiencies from Wlasits et al. (2020), and the charging efficiency was tested to follow the Wiedensohler approximation in Tauber et al. (2020). The kernel functions for all seven classified sizes are shown in Fig. 1 for an instrument averaging time of 120 s, sulphuric acid-like test particles (using the Ammonium Sulfate detection efficiencies from Wlasits et al., 2020) and including the detector flow rates of each condensation particle counter. Therefore, the kernels can be used to convert raw particle counts at the detecting CPCs into a particle size distribution (within an inverse problem) and vice versa. Note that for different chemical composition of the input particles, the CPC response might be different. Therefore, the kernels used for analysing experiments where particles were formed from oxidised organics (Kirkby et al., 2016) or from iodic acid (He et al., 2021) are different and approximated by the calibration curves for oxidised beta-caryophyllene and sodium chloride from Wlasits et al. (2020).

## 3.2 CLOUD

We use experimental data from the DMA-train measuring new particle formation
in nucleation experiments at the CERN CLOUD chamber. The 26.1 m^{3}
stainless-steel chamber provides a high-purity, temperature-controlled
environment in order to perform experiments under atmospherically relevant
conditions, where trace gases can be added precisely at pptv (parts per
trillion by volume) level and sunlight can be simulated by UV-illumination
of the chamber. We use three different sets of experiments of varying
chemical composition, in order to demonstrate the performance of the FIKS in
reconstructing formation and growth rates. See also Kirkby
et al. (2011) and Duplissy et al. (2016).

First, a nucleation experiment using sulphuric acid and ammonia was
performed at 5^{∘} by adding SO_{2} and O_{3} to the chamber and
through the photo-dissociation of ozone, the formation of OH radicals and
sulphuric acid was induced, which resulted in strong new particle formation
(Stolzenburg et al., 2020). Second, we performed
nucleation and growth experiments at 5^{∘} using oxidised organics
from dark (i.e. no UV-illumination) ozonolysis of alpha-pinene
(Kirkby et al., 2016;
Stolzenburg et al., 2018). Both experiments resulted in moderate new
particle formation rate and thus, in the Kalman smoother, we used DMA-train
data that was averaged over 120 s time intervals. Third, we studied
nucleation from iodine oxides at 10^{∘} (He et al.,
2021), which resulted in high particle formation rates and fast growth. For
the third experiment, we reduced the DMA-train averaging time down to 20 s, while keeping high counting statistics over the averaging interval.

## 3.3 PSM-derived formation rates

Particle formation rates (*J*_{dp}) are calculated from the time derivative
of the total particle number concentration larger than 1.7 nm following the
method introduced in Dada et al. (2020). The
particle number size distribution is measured with the particle size
magnifier (PSM) coupled with a condensation particle counter (1.5–2.5 nm), a TSI nano-SMPS (3–65 nm) and home-built long-SMPS (10–800 nm).
The formation rates are corrected for the size dependent wall and
coagulation losses. Additionally, since the chamber is operated in
continuous flow mode, the particle concentrations are corrected for dilution
losses. For more information on the PSM-derived formation rates, see
Dada et al. (2020). The uncertainty on
*J*_{1.7} was assumed to be 30 % for the CLOUD chamber derived from the
repetition of the same experiment. A procedure as described in Dada et al. (2020) using propagation of error in the concentration measurement,
dilution, coagulation, and wall losses as well as the error on the
time derivative of the total particle concentration within a Monte-Carlo
simulation could be used if such repetition experiments were not available.
It needs to be noted, that for comparison of the formation rate value at the
arbitrary minimum detectable size *d*_{min}=1.7 nm with a system
inherent nucleation rate at the critical cluster size, additional
sophisticated approaches might be necessary (Kürten et
al., 2015).

## 3.4 Growth rates using INSIDE

In order to compare the particle growth rates derived by Kalman smoothing,
we use the size- and time-dependent growth rate analysis tool INSIDE
(Pichelstorfer et al., 2018). It uses input particle
size distributions at time *t*_{1} in order to simulate the known aerosol
dynamics (coagulation, wall losses, and dilution) until a time *t*_{2}
(typically separated by one measurement cycle of an instrument, i.e. the 120 s averaging time mentioned above). At *t*_{2}, the simulated aerosol
size distribution is compared to the measured size distribution and by
evaluating the general dynamics equation above a certain diameter
*d*_{eval} the growth term ${\left(\frac{\mathrm{d}{d}_{\mathrm{p}}}{\mathrm{d}t}\right|}_{{d}_{\mathrm{eval}}}=\frac{{\left(\right)}_{\frac{\mathrm{d}{N}_{\mathrm{\infty}}}{\mathrm{d}t}}^{}{d}_{\mathrm{eval}}\mathrm{\infty}}{-}{\left(n({d}_{\mathrm{p}},t)\right|}_{{d}_{\mathrm{eval}}}$ can be computed. Evaluating this equation for
several evaluation diameters *d*_{eval} and at all measurement
times *t*_{i}, this results in time- and size-resolved growth rates from a
size-distribution measurement. This is distinct from most other
integrative growth rate analysis approaches, which can only derive one
growth rate value for a specified size interval in a single run
(Dada et al., 2020;
Kulmala et al., 2012; Lehtipalo et al., 2014; Paasonen et al., 2018). Lastly,
it should be noted that for INSIDE, compared to the Kalman smoothing, each
time step is analysed individually, and the growth rate analysis framework
relies on already-inverted size distributions. For this inversion, only a
simple point-by-point inversion procedure is used for the DMA-train data of
this work, assuming narrow DMA transfer functions and little variation of
the size distribution across it (Stolzenburg and McMurry,
2008). Moreover, the INSIDE method does not provide an uncertainty estimate
on the growth rate calculation and hence the Kalman smoothing will provide
valuable insights on the uncertainty related to growth rate measurements.

## 4.1 Numerical simulation test

First, we simulated a data set representing a typical nucleation experiment
performed in an atmospheric simulation chamber like CLOUD and modelled the
DMA-train response according to the above Kernel functions and then applied
the FIKS to this synthetic data set. We used the same framework as in
Ozon et al. (2021a) to simulate a nucleation
experiment with formation rate at 1 nm *J*_{1.0}=5.25 cm^{−3} s^{−1}, and
size-independent growth rate GR = 2.5 nm h^{−1} and the loss rates equal to the CERN CLOUD
experiment (Stolzenburg et al., 2020). The evolution of
the simulated size distribution is shown in Fig. 2a. The measurement data
*y*^{k} are then simulated using the kernel functions from Fig. 1a and
altered with a Poisson-distributed random counting error. The FIKS is then
applied to the measurement data with the input parameters given in Table S1
in the Supplement. We use a resolution of 32 bins from 1.7 to 10 nm for the
FIKS to keep the computational effort low. We also tested 16 to 64 size
discretisation bins, but higher resolution required additional adjustments
in the size correlation of the covariance given in Eq. (7), which would
result in significant differences compared to the original work of Ozon et
al. (2021a) without providing significantly more accuracy. Figure 2 also
shows the Kalman smoother estimates for the size distribution, growth rate,
and formation rate at 1.7 nm. The reconstructed size distribution (Fig. 2b)
is very similar to the true size distribution (Fig. 2a), especially taking
into account the sparser discretisation of the former. Also the estimated
growth and nucleation rates agree well with true values of the respective
process rates specified in the simulation. Moreover, the uncertainty
estimates are feasible: for both quantities, the true values are within the
uncertainty limits given by FIKS. The reconstructed size distribution and
especially, growth rate (Fig. 2b and c, respectively) show some temporal
oscillations, which are related to periods in which the particle population
grows from the size range visible for one DMA-train channel to the next one.
That is, the oscillation is a result of an insufficient coverage of the
size range by the DMA-train kernels. The problem could be approached by
application of a regularisation scheme in the measurement model
(Voutilainen and Kaipio, 2001) or by adjusting the
kernels improving the overlap, which will be discussed in more detail in
Sect. 4.5. Nevertheless, the overall good retrieval of the simulated size
distribution and process rates demonstrates that the Kalman smoothing approach
is well suited for analysing DMA-train data from chamber nucleation
experiments.

## 4.2 Sulphuric acid-ammonia experiment

We applied the FIKS to experimental data from a sulphuric acid-ammonia
nucleation and growth experiment performed at 5^{∘} at the CERN CLOUD
chamber. The raw data measured with the DMA-train were averaged in
120 s time intervals in order to increase the counting statistics per
channel and then used as input for BAYROSOL. The details of parameter
choices in FIKS are given again in Table S1 in the Supplement. The results
of applying the Kalman smoothing to this experimental data are shown in Fig. 3.

The size and time dependence of the FIKS estimate for the true size distribution (Fig. 3a) is very smooth, and it is also able to bridge the information gaps between the largest size-distribution channels. This is a significant improvement from the traditional point-by-point inversion, where the data of each DMA-train channel is inverted independently: here the reconstructed total number concentration is significantly more noisy. The smoother FIKS reconstructed size distribution and total number concentration (Fig. 3d) still show some small temporal fluctuations. The fluctuations are even more clearly visible in the reconstructed evolution of the formation rate (Fig. 3b). In contrast to the oscillations found in the growth rate for the simulated case, the fluctuations do not only occur when the size distribution reaches a new DMA-train size channel. Furthermore, the same fluctuations are also recovered when the formation rate is inferred from the PSM and nano-SMPS, i.e. an entirely different set of instrumentation and different type of data analysis. This result suggests that the fluctuation of the particle formation truly occurs physically in the experiment and is not a reconstruction error caused by instrument noise or a bias caused by the inversion method. The absolute values of the two independent formation rate estimates agree upon a factor of 1.5. Furthermore, large portions of the uncertainty intervals of these two estimates overlap with each other, which is another indicator of the feasibility of FIKS for analysing DMA-train data. In addition, the inferred growth rates also agree within the systematic uncertainties for both approaches (Fig. 3c for one time instant). It is worth noting, however, that both growth rate estimates rely on data from the same instrument and are hence more interdependent than the formation rate estimates. Nevertheless, the good agreement between them corroborates the feasibility of Kalman smoothing for reconstructing nanoparticle growth rates from experimental data.

## 4.3 Alpha-pinene ozonolysis experiment

Third, we applied the FIKS to experimental data obtained by the DMA-train from an alpha-pinene ozonolysis experiment. Besides the different chemical composition of the growing particles (resulting in different assumptions for the DMA-train transfer functions) in comparison to the experimental results used in Sect. 4.2, this experiment is characterised by a different size dependency of the growth rate, due to the increased condensation of low- and semi-volatile organics with increasing particle size (Simon et al., 2020; Stolzenburg et al., 2018; Tröstl et al., 2016). The formation rate, however, remains rather similar to the sulphuric acid-ammonia experiment under these specific experimental conditions, but the slower initial growth rates result in generally lower produced particle concentrations. The results of applying FIKS to the data from this experiment are shown in Fig. 4. Again, the size distribution given by FIKS is much smoother than that obtained with a standard inversion procedure; see Heinritzi et al. (2020) and Fig. 4d for the total number concentration evolution retrieved from the standard inversion and the FIKS.

Because in the alpha-pinene ozonolysis experiment the nucleation and growth rates were known to be similar to those in the sulphuric acid-ammonia experiment, the parameters of their evolution models were selected as in Sect. 4.2 (Table S1 in the Supplement). The formation rate is lower by a factor of 2.5 than the one obtained from the PSM. Also the formation rate retrieved with the method from Dada et al. (2020) but using the DMA-train data is significantly higher, but in between the two estimates. The possible deviation has hence two plausible reasons: the instrumental differences can be caused by different calibration procedures for the DMA-train and PSM (Dada et al., 2020; direct cross-calibration using NAIS versus Wlasits et al., 2020 using beta-caryophyllene ozonolysis surrogates). The methodological differences could also arise from the very low counting statistics in the DMA-train during this experiment compared to the other two, which will cause the inherent Gaussian assumption of the FIKS to fail. As the deviation in the reconstructed total number concentration of the DMA-train data using two inversion procedures is only a factor of ∼ 1.3 (Fig. 4d), the formation rate discrepancies could be largely due to the more difficult and uncertain calibration procedures. Considering the fact that inter-instrument deviations in sub-10 nm size-distribution measurements can be as large as one order of magnitude (Kangasluoma et al., 2020), the achieved agreement is remarkable, especially as some fluctuations within the chamber can again be reconstructed in both approaches. The retrieved growth rates from the FIKS estimate and the INSIDE method agree remarkably well and both show the increasing growth rates with increasing particle size up to 4.3 nm, an indication for a strong Kelvin effect in organic condensation (Stolzenburg et al., 2018; Tröstl et al., 2016). The decreasing Kalman smoother estimate above 4.3 nm is related to the fact that the FIKS searches for a smooth estimate of the growth rate (Eq. 6), but at that point (2 h 30 min after experiment start), practically no information from the size distribution above 4.3 nm is available, resulting in a slow decrease towards zero. This is due to the strong smoothness used a priori in the FIKS algorithm (Table S1 in the Supplement). The INSIDE method does not report growth rates from regions with no information (Pichelstorfer et al., 2018) from the size distribution, and hence the estimate stops at 4.3 nm.

## 4.4 Iodic acid experiment

Finally, we analysed a more dynamic experiment of nucleation and growth from
iodic acid (He et al., 2020, 2021). The experiment
is characterised by extremely high nucleation rates, which are two orders of
magnitude higher than in the sulphuric acid-ammonia and alpha-pinene
ozonolysis experiments. However, the growth rate is only half an order of
magnitude higher compared to the other two example cases. Figure 5 shows
that, in spite of the highly dynamic experiment, the formation rate
recovered by Kalman smoothing agrees with the estimate obtained from the PSM
(Fig. 5b). The usage of the DMA-train data with a time resolution of 20 s allows for the precise recovery of the spike in formation rate in
the beginning of the experiment. This causes the build-up of a high
condensation sink and vapour/cluster depletion almost shutting off any
further nucleation during the continuation of the experiment. The
four-minute time resolution of the data for the calculation of the
nucleation rate from the PSM is limited in that respect. The reconstructed
growth rates (Fig. 5c) again show agreement between the FIKS estimate and
the INSIDE method, indicating a clear decreasing trend with size, which is
expected for condensation at the kinetic limit if the vapour molecular size
is taken into account (He et al.,
2021; Lehtinen and Kulmala, 2003; Nieminen et al., 2010; Stolzenburg et al.,
2020). The lower values towards 1.8 nm could be caused by a biased estimate
of the PSM detection efficiency, because neither a calibration for iodic
acid clusters nor for sodium chloride particles (which was used for the
other detectors in the DMA-train) was available. The instabilities in the
size distribution at the smallest sizes and fluctuations of the formation
rate are expected, considering the highly dynamic process of this experiment.
Overall, the good agreement for the inferred process parameters of the
aerosol general dynamics equation, i.e. the formation and growth rates, is
still remarkable. However, the reconstructed size distribution from the FIKS
estimate shows some discontinuities, especially during growth above 3 nm. This is because the available instrument information from the DMA-train
starts to get very sparse in that size range given the dynamic processes
involved in the iodic acid nucleation and growth. This also causes the
overshooting in the total number concentration using the standard inversion
compared to the FIKS result (Fig. 5d), as the larger size channels cover a
very broad range at too-high resolution for the linear interpolation of the
standard inversion used to obtain *N*_{tot}. More available size channels
(hence more DMAs in the case of the DMA-train) would help to resolve such
discontinuities. We will therefore provide an instrument design
recommendation based on simulated data in the next section.

## 4.5 Instrument design recommendation from a signal processing point of view

The size range covered by the seven DMA-train channels was chosen
semi-arbitrarily based on some external constraints: the lowest measured
centroid diameter was supposed to be as close as possible to 1.7 nm where
the formation rate is typically measured for experiments performed at the
CLOUD chamber. In order to cover the sub-10 nm range, the largest channels
were set to 8 nm. The other channel diameters were chosen to yield sufficient
coverage of the sub-3 nm range. The width of the transfer functions was
fixed by the detector sample flow rates and the 15 L min^{−1} critical orifices
provided by Grimm Aerosol for the DMA sheath air supply (also the standard
flow rate used by the manufacturer for this type of DMA). However, the
chosen centroid diameters (and hence fixed voltages at the DMAs) and
selected sheath flow rates could easily be altered.

In order to numerically study the effect of choice for the DMA-train
channels, we constructed a model corresponding to a channel choice different
from that in the previous sections: the kernel for the DMA-train with seven
channels having centroid diameters of 1.8, 2.2, 2.8, 3.6, 4.6, 6.1, and 8.0 nm is illustrated in Fig. 6a. Moreover, channels 3 to 7 have been altered to
use a 2.5 L min^{−1} sample flow rate (could be achieved by a make-up flow at each
CPC) and a reduced sheath flow rate of 10 L min^{−1}, only providing a theoretical
non-diffusive resolution of ∼ 4, which is significantly lower.
However, this permits covering the entire size range between 1.8 and 8 nm by
overlapping channels. We revisited the numerical simulation described in
Sect. 4.1 using the DMA-train model corresponding to this configuration. The
resulting size distribution given by FIKS is shown in Fig. 6b. Comparison
between the two reconstructions in Figs. 2b and 6b reveals that the new
choice of DMA-channels stabilises the size-distribution estimate, which is
supported by the comparison of the estimated total number concentrations
(Fig. 6d). While with the original design the size distribution evolves in
step-wise manner when the growing particle mode reaches larger
size-channels, the size distribution retrieved with the adjusted kernels is
smoother, also temporally, which is especially visible in the evolution of
the total number concentration. The improvement is also significant for the
retrieval of the growth rate, where the estimate no longer overshoots as
illustrated by the snapshot size dependence of the growth rate for a single
instant in time, shown in Fig. 6c. It should be noted that
regularisation schemes in the measurement model as proposed by
Voutilainen and Kaipio (2001) could also provide
smoother estimates, but this would need a significant adjustment of the
algorithm provided by Ozon et al. (2021a) and is hence not implemented here.

The result of this additional numerical study thus demonstrates that in the chosen conditions of the simulation, the reconstruction quality improves when the resolution of individual channels is lowered. This seemingly counterintuitive effect stems from the fact that FIKS estimates do not rely only on the measurements, but are also advised by the GDE model, which makes the problem of optimising the measurement design a somewhat cumbersome task. A rigorous investigation of the optimal experimental design is out of the scope of this paper, but the above observation is worth noticing – especially because a lot of recent experimental effort in the sub-10 nm range has been devoted to improving the instrument resolution (Kangasluoma et al., 2020). While on the individual channel level this might reduce systematic uncertainties, as discussed in Kangasluoma et al. (2020), signal processing requires more a broad coverage of the size distribution than high resolution. However, the ideal instrument would combine both: full coverage of the size distribution but achieved with more high-resolution channels. For the DMA-train principle, this would require new ideas in instrument design in order to incorporate more DMAs without the instrument becoming impractically bulky. Extending the measurement size range above 10 nm would require even more DMAs, but this range is usually well-covered by commercially available instruments, which could easily be added to the FIKS measurement model, facilitating the measurement of experiments where particles grow well beyond 10 nm.

A recently developed methodology (BAYROSOL), applying the fixed interval Kalman smoother (FIKS) to a finite difference solution to the aerosol GDE was used to analyse DMA-train measurements of aerosol dynamics at the CLOUD chamber facility at CERN. The overall aim of this methodology is to estimate unknown aerosol microphysical process rates as well as their uncertainties from size-distribution evolution measurements. In a previous paper, the methodology was shown to be able to predict new particle formation, growth, and loss rates from synthetic computer-generated aerosol size distribution evolution data, while here the method has been applied to real experimental data for the first time. Four experimental cases with particle formation and growth were tested: (1) a computer-generated synthetic case, (2) sulphuric acid-ammonia, (3) alpha-pinene ozonolysis, and (4) iodic acid.

The DMA-train was selected for two main reasons: first, the instrument kernel functions are well characterised, giving rise to reliable estimation of the instrument uncertainties and second, new particle formation rates have not been estimated directly from DMA-train measurements before. In addition, as the current version of the DMA-train is designed in such a way that the individual DMAs have a rather narrow collection of kernels for the channels, with significant gaps between some of the channels, the FIKS can reconstruct the size distributions from the measured signals in such a way that the distributions are rather smooth and consistent with the GDE.

We compared the growth rates, which with BAYROSOL can be estimated as functions of both size and time, with those obtained by INSIDE, a method applied earlier to CLOUD data, and the agreement was remarkably good for all studied cases. INSIDE is also based on matching the GDE solution to measured size distribution dynamics, however without the capability of estimating uncertainties of the estimations. The FIKS-based estimates for the particle formation rates were compared with those estimated from data obtained by a separate instrument, the particle size magnifier (PSM), based on the rate of change of the total number concentration measured by the instrument corrected by coagulation and wall losses. Again, the agreement was very good, especially considering the fact that instrument uncertainties are large at the very lowest end of the measured size spectrum. For the iodic-acid case, the FIKS estimate of the formation rate was even able to capture rapidly changing dynamics of the experiment. It was remarkable for all cases that some fluctuations in the formation rates were recovered by both methods independently, indicating that these are physical variations during the experiment.

Finally, we utilised the FIKS from an instrument development point of view. Typically, an as-high-as-possible resolution for the different measurement channels has been the aim when measuring nanometer-sized particles. This aim, however, gives rise to gaps in the measured size range, as is the case in the DMA-train studied. Thus we studied whether a better coverage of the size spectrum, but lower resolution of the individual channels would be advantageous for size distribution estimation.

Summarising, we believe that Bayesian state estimation methods such as FIKS can be very useful in the field of aerosol science in many aspects. As mentioned above, they can be used to fill gaps in measurements in such a way that not only the obtained size distributions but also unknown process rates are consistent with theory describing aerosol size-distribution dynamics. In addition, the methodology provides estimations of uncertainties both for size distributions as well as process rates based on uncertainty estimations in the measurements and the used models, which is unfortunately not common when reporting results of aerosol measurements. Finally, it is conceivable that the methodology presented here will be superior to several previous approaches when combining measurement data obtained with several different instruments that operate at different size ranges. This will be a topic of our forthcoming studies.

The version of the implementation of the estimation method (BAYROSOL1.1) (https://doi.org/10.5281/zenodo.4450492, Ozon et al., 2021b) is available under the MIT Expat License; it is the version used to generate the results discussed in this paper. The package also contains the code used to obtain the results discussed in Ozon et al. (2021a). It is possible to generate the simulated data described in Sect. 4.1. The experimental data (DMA-train kernels) and raw data, are also included in the repository.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-21-12595-2021-supplement.

MO, DS, and LD analysed the experimental data, MO developed the software, and all authors were involved in the scientific discussion and in the writing of the manuscript.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This publication has been produced within the framework of the EMME-CARE project, which has received funding from the European Union's Horizon 2020 Research and Innovation Programme (under grant agreement no. 856612) and the Government of Cyprus. This research has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 895875 (“NPF-PANDA”) as well as Academy of Finland Project #325647. The sole responsibility of this publication lies with the author. The European Union is not responsible for any use that may be made of the information contained therein.

This research has been supported by the H2020 Marie Skłodowska-Curie Actions (grant no. 895875), the Academy of Finland (grant no. 325647), and the Horizon 2020 (grant no. EMME-CARE (856612)).

This paper was edited by David Topping and reviewed by Christina Williamson and one anonymous referee.

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