**Research article**
21 Dec 2021

**Research article** | 21 Dec 2021

# Revisiting matrix-based inversion of scanning mobility particle sizer (SMPS) and humidified tandem differential mobility analyzer (HTDMA) data

Markus D. Petters

**Markus D. Petters**Markus D. Petters

- Department of Marine, Earth, and Atmospheric Sciences, NC State University, Raleigh, NC 27695-8208, USA

- Department of Marine, Earth, and Atmospheric Sciences, NC State University, Raleigh, NC 27695-8208, USA

**Correspondence**: Markus D. Petters (mdpetter@ncsu.edu)

**Correspondence**: Markus D. Petters (mdpetter@ncsu.edu)

Received: 22 Feb 2021 – Discussion started: 17 Mar 2021 – Revised: 27 Sep 2021 – Accepted: 04 Oct 2021 – Published: 21 Dec 2021

Tikhonov regularization is a tool for reducing noise amplification during data inversion. This work introduces RegularizationTools.jl, a general-purpose software package for applying Tikhonov regularization to data. The package implements well-established numerical algorithms and is suitable for systems of up to ∼ 1000 equations. Included is an abstraction to systematically categorize specific inversion configurations and their associated hyperparameters. A generic interface translates arbitrary linear forward models defined by a computer function into the corresponding design matrix. This obviates the need to explicitly write out and discretize the Fredholm integral equation, thus facilitating fast prototyping of new regularization schemes associated with measurement techniques. Example applications include the inversion involving data from scanning mobility particle sizers (SMPSs) and humidified tandem differential mobility analyzers (HTDMAs). Inversion of SMPS size distributions reported in this work builds upon the freely available software DifferentialMobilityAnalyzers.jl. The speed of inversion is improved by a factor of ∼ 200, now requiring between 2 and 5 ms per SMPS scan when using 120 size bins. Previously reported occasional failure to converge to a valid solution is reduced by switching from the L-curve method to generalized cross-validation as the metric to search for the optimal regularization parameter. Higher-order inversions resulting in smooth, denoised reconstructions of size distributions are now included in DifferentialMobilityAnalyzers.jl. This work also demonstrates that an SMPS-style matrix-based inversion can be applied to find the growth factor frequency distribution from raw HTDMA data while also accounting for multiply charged particles. The outcome of the aerosol-related inversion methods is showcased by inverting multi-week SMPS and HTDMA datasets from ground-based observations, including SMPS data obtained at Bodega Marine Laboratory during the CalWater 2/ACAPEX campaign and co-located SMPS and HTDMA data collected at the US Department of Energy observatory located at the Southern Great Plains site in Oklahoma, USA. Results show that the proposed approaches are suitable for unsupervised, nonparametric inversion of large-scale datasets as well as inversion in real time during data acquisition on low-cost reduced-instruction-set architectures used in single-board computers. The included software implementation of Tikhonov regularization is freely available, general, and domain-independent and thus can be applied to many other inverse problems arising in atmospheric measurement techniques and beyond.

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Atmospheric aerosol plays an important role in shaping the microphysics of clouds and the Earth's climate (Farmer et al., 2015; Kreidenweis et al., 2019). To predict the impact of aerosol on the Earth system, the distributions of particle size, chemical composition, hygroscopicity, and morphology must be known. The distribution of these properties across a population of particles formally defines the mixing state of the aerosol (Riemer et al., 2019). Accurate measurements of these distributions are critical for formulating models that link aerosol, cloud, and climate properties.

Differential mobility analyzers (DMAs) select particles as a function of their size, charge, and an applied voltage. DMAs and tandem DMAs are widely used to measure the distributions of size and distributions of aerosol physicochemical properties (Park et al., 2008). For examples, a single DMA can be used to measure the aerosol size distribution by scanning voltage (Wang and Flagan, 1990). Humidified tandem DMAs (HTDMAs) can be used to measure the growth factor or hygroscopicity frequency distribution (Gysel et al., 2009). DMA–particle mass analyzer measurements can be used to resolve particle density distributions (Rawat et al., 2016; Sipkens et al., 2020). Tandem DMAs are important because they are one of only a handful of techniques that can specifically characterize aspects of the aerosol mixing state (Riemer et al., 2019). Unfortunately, particles carrying multiple charges and different sizes transmit through the DMA at a single voltage, which creates artifacts in the raw instrument response that must be removed during post-processing of the data.

Humidified tandem DMAs select a mobility diameter, pass this quasi-monodisperse aerosol through a humidification system, and then measure the humidified mobility response function using a second DMA operated in stepping or scanning mode (Rader and McMurry, 1986; Suda and Petters, 2013; Dawson et al., 2016). The humidified mobility response function is influenced by the particle size distribution, aerosol charge distribution, and growth factor frequency distribution function of the upstream aerosol. Gysel et al. (2009) show that the inversion from the humidified mobility response function to the growth factor frequency distribution is an ill-posed problem.

The inverse solution of ill-posed problems is characterized by strong
sensitivity to noise superimposed on the data. Regularization methods
are needed to relate an observed instrument response to the underlying
physical property of the system under investigation. A common inverse
method is *L*_{2} regularization, developed independently by Phillips (1962),
Twomey (1963), and Tikhonov (1963).
Some examples of *L*_{2} regularization involving atmospheric measurement
techniques include inversion to find aerosol microphysical properties
from measurements of optical properties (Dubovik and King, 2000; Müller et al., 2019),
retrieve trace gas concentrations from remote sensors (Borsdorff et al., 2014),
or estimate fluxes from a combination of measurements and atmospheric
transport models (Krakauer et al., 2004).
Application of *L*_{2} regularization for problems involving DMAs
include the reconstruction of the particle size distribution downstream
of a single DMA (Wolfenbarger and Seinfeld, 1990; Kandlikar and Ramachandran, 1999; Talukdar and Swihart, 2003; Petters, 2018)
and inversion to find size–mass distributions from coupled DMA–particle
mass analyzer measurements (Rawat et al., 2016; Sipkens et al., 2020).

To date, *L*_{2} regularization has not been applied to the inversion
of HTDMA data. However, multiple other approaches have been used to
estimate the growth factor frequency distribution from the humidified
mobility response function. Stolzenburg and McMurry (1988)
introduce the TDMAfit method. TDMAfit assumes a multi-mode normally distributed
hygroscopic growth factor frequency distribution. Parameters of the
growth factor frequency distribution are varied such that the error
between the modeled and observed humidified mobility response functions
is minimized. Cubison et al. (2005) apply
the optimal estimation method (OEM) to derive the growth factor frequency
distribution. This method uses an estimate of the covariance matrix,
the measurements, and the forward model to retrieve the growth factor
frequency distribution. The advantage of the optimal estimation method
over TDMAfit is that it is nonparametric; i.e., it makes no prior
assumption about the functional form of the growth factor frequency
distribution. However, the method sometimes produces oscillatory and
negative solutions. Gysel et al. (2009)
introduce TDMAinv, a piecewise linear version of TDMAfit. The piecewise
method is also nonparametric. Constrained minimization is applied
to find the growth factor frequency distribution; this avoids the
negative solutions encountered in the optimal estimation method. Gysel et al. (2009)
briefly discuss the role of multiple charges in the inversion and
state that “the measured humidified mobility response function is
a superposition of contributions from different dry sizes …
and appropriate data inversion is hardly possible. Unfortunately an
SMPS-style multicharge correction cannot be applied because the relative
contributions from singly and multiply charged particles to every
data point of the MDF cannot be distinguished.” (In the direct quote,
SMPS denotes scanning mobility particle sizer – Wang and Flagan, 1990 –
and MDF denotes mobility distribution function.) Nevertheless, Shen et al. (2021)
compute the contribution of multiply charged particles to the humidified
mobility response function assuming that the larger multiply charged
particles express the mean growth factor. However, they state that
the correction of growth factor frequency distribution for multiply
charged particles “is too complicated” (Shen et al., 2021)
due to the need for multidimensional integration. Finally, Oxford et al. (2020)
introduced a forward model named TAO that corrects for the contribution
of multiply charged particles to the signal when interpreting volatility
tandem DMA measurement.

This work revisits the challenge of performing an SMPS-style inversion
of the humidified mobility distribution to retrieve the growth factor
frequency distribution while also accounting for multiply charged
particles. *L*_{2} regularization is used to find the inverse. The
remainder of the work is structured as follows: Sect. 2 describes
the theory of *L*_{2} regularization and the numerical solution of
the equations. The software package RegularizationTools.jl is
introduced, which is a general domain-independent implementation of
*L*_{2} regularization. Forward models for transfer through the single
DMA and tandem DMA are formulated using the formalism developed in
Petters (2018) and cast into matrix
form using abstractions introduced in RegularizationTools.jl.
Section 3 uses synthetic data to demonstrate that *L*_{2} regularization
can be used to invert the humidified mobility distribution function
to find the growth factor frequency distribution. Section 4 uses real-world
data to showcase improvements for size distribution inversion and
the newly introduced tandem DMA inversion that were added to the freely available
software package DifferentialMobilityAnalyzers.jl (Petters, 2018).
Finally, Sect. 5 summarizes the improvements, advantages, and limitations
of the methodologies introduced in this work.

Section 2.1 and 2.2 use the following linear algebra notation. Capital
bold roman letters denote matrices (**A**), bold italic
letters denote vectors (** x**) and lowercase italic symbols denote
scalars (

*n*).

**A**

^{T}denotes the matrix transpose, and ${\mathbf{A}}^{+}=({\mathbf{A}}^{\mathrm{T}}\mathbf{A}{)}^{-\mathrm{1}}{\mathbf{A}}^{\mathrm{T}}$ is the matrix pseudo-inverse. Section 2.3 uses additional notation described there.

## 2.1 *L*_{2} regularization

### 2.1.1 Theory

The formalisms closely follow the description in Hansen (2000). Consider a system of equations

where ** b** is the measured response,

**A**is the design matrix (which may or may not be square),

**is the true quantity of interest, and**

*x***is the random error. The regular least-squares solution computed using the pseudo-inverse via $\mathit{x}={\mathbf{A}}^{+}\mathit{b}$ is often dominated by contributions from the error, and the thus-obtained estimate for**

*ϵ***is useless. Regularization addresses this issue by solving the minimization problem**

*x*
where *x*_{λ} is the regularized estimate of
** x**, $\Vert \cdot {\Vert}_{\mathrm{2}}$ is the Euclidean
norm,

**L**is a filter matrix,

*λ*is the regularization parameter, and

*x*_{0}is a vector of an a priori estimate of the solution. The a priori estimate can be taken to be

*x*_{0}=0 if no a priori information is known. The filter matrix is often taken to be the identity matrix

**I**or a derivative operator. Common choices are the first and second derivative operator defined as the upper bidiagonal$(-\mathrm{1},\mathrm{1})$ and the upper tridiagonal$(\mathrm{1},-\mathrm{2},\mathrm{1})$ matrix, respectively. For

*λ*=0, the solution is equivalent to ${\mathit{x}}_{\mathit{\lambda}}={\mathbf{A}}^{+}\mathit{b}$. The limit ${lim}_{\mathit{\lambda}\to \mathrm{\infty}}{\mathit{x}}_{\mathit{\lambda}}={\mathit{x}}_{\mathrm{0}}$ applies. Thus the regularization parameter “interpolates” between the noisy ordinary least-squares solution and the a priori estimate

*x*_{0}.

The analytical solution for Eq. (2) is the regularized normal equation

which is derived by taking the derivative of the right-hand
side of Eq. (2) with respect to ** x**, setting it to zero,
and solving for

**. Equation (3) is in standard form if**

*x***L**=

**I**. The optimal regularization parameter can be obtained using a variety of techniques, including the L-curve method (Hansen, 2000) and generalized cross-validation (GCV; Golub et al., 1979). Both methods use metrics that penalize solutions with large variance (amplified noise) or large bias.

The L-curve method involves a plot of $\mathrm{log}\Vert \mathbf{A}{\mathit{x}}_{\mathit{\lambda}}-\mathit{b}{\Vert}_{\mathrm{2}}^{\mathrm{2}}$
vs. $\mathrm{log}\Vert \mathbf{L}({\mathit{x}}_{\mathit{\lambda}}-{\mathit{x}}_{\mathrm{0}}){\Vert}_{\mathrm{2}}^{\mathrm{2}}$. The optimal *λ* occurs at the corner of the resulting L curve,
which can be found algorithmically. However, automating the L-curve
method can be more challenging than other automated methods, as further
discussed below.

The generalized cross-validation estimator presents a mathematical shortcut to compute the leave-one-out cross-validation estimate, which removes one point from the data, creates a model, computes the error between the model and data point not included in the data, and then averages the result over all permutations. It is given by

where **A**_{λ}** b** is

**A**

*x*_{λ},

**A**

_{λ}is the influence matrix, tr is the matrix trace, and

*n*is the size of

**. The optimal**

*b**λ*

_{opt}coincides with the global minimum of

*V*(

*λ*). Equation (4) requires that the system is in standard form. For systems in non-standard form, conversion to standard form is required before computing

*V*(

*λ*). In many cases

*λ*

_{opt}values found by the L-curve method and generalized cross-validation are similar, and the retrieved solutions

*x*_{λ}are nearly indistinguishable. Differences between these two estimates are related to the computational speed to converge and robustness, i.e., that the system converges to the optimal solution.

### 2.1.2 Algorithms

Equation (3) can be solved straightforwardly using any software
that supports linear algebra operations. This brute-force approach,
however, is slow. Efficient algorithms to solve Eqs. (3) and (4) have
been developed. The algorithms used here are briefly described. If **L**≠**I**,
Eq. (3) is transformed to standard form using the generalized singular
value decomposition of **A** and **L** as derived
by Eldén (1982) and summarized
by Hansen (1998). Equation (3) is solved
using Cholesky factorization when possible since it is the computationally
fastest approach (Lira et al., 2016). If Cholesky
factorization fails, one of the fallback solvers selected by the linear
algebra package of the programming language is used. Equation (4)
is solved using the singular value decomposition of **A**
and the iterative algorithm described in Bates et al. (1986).
The optimal *λ*_{opt} for generalized cross-validation is found
by minimizing *V*(*λ*) on a bounded interval using Brent's method
(Mogensen and Riseth, 2018). The optimal *λ*_{opt}
for the L-curve method is found by maximizing Eq. (18) in Hansen (2000)
on a bounded interval using Brent's method.

### 2.1.3 Classification of methods

The inverse problem can be solved using specific methods. Here, method
refers to the content of the filter matrix **L**, whether
an a priori estimate is used, and whether constraints are imposed
on the solution. Methods are encoded through the following expression:

where *L*_{k} denotes the order of the filter matrix **L**,
*x*_{0} denotes whether an a priori estimate is used, *D*_{ϵ}
denotes whether data-based constraints are used (explained further
below), and *B*_{[lb,ub]} denotes whether a lower bound (lb) or upper
bound (ub) is imposed on the solution (explained further below). The
argument (alg) denotes constraints on the search algorithms,
e.g., L curve or GCV and/or the bounded interval over which *λ*
is varied. The expression is composable. For example, the method *L*_{2}
denotes inversion using the second-order derivative without an a priori
estimate, data-based constraints, and lower and/or upper bound constraints.
The method *L*_{0}*B*_{[0,1]}(alg = L curve, ${\mathit{\lambda}}_{\mathrm{1}}=\mathrm{0.01},{\mathit{\lambda}}_{\mathrm{2}}=\mathrm{100})$
denotes inversion with **L**=**I**, imposing that all values of ${\mathit{x}}_{\mathit{\lambda}}\in [\mathrm{0},\mathrm{1}]$,
the use of the L-curve method, and ${\mathit{\lambda}}_{\text{opt}}\in [\mathrm{0.01},\mathrm{100}]$.
If alg is unspecified, defaults of alg = GCV, *λ*_{1}=0.001, and *λ*_{2}=1000
are implied. This approach of method encoding provides a convenient
classification system to enumerate the set of available methods as
well as to specify the method in a high-level application interface
for software function calls. There are eight combinations by which
to compose methods via Eq. (5), *L*, *L**x*_{0}, *L**x*_{0}*B*, *L**x*_{0}*D*,
*L**x*_{0}*B**D*, *L**B*, *L**B**D*, and *L**D*. Combined with the three most
common filter matrices *L*_{0}=**I**, *L*_{1}= upper bidiagonal$(-\mathrm{1},\mathrm{1})$,
and *L*_{2}= upper tridiagonal$(\mathrm{1},-\mathrm{2},\mathrm{1})$, this results in 24 unique
methods.

*Data-based constraints.* Huckle and Sedlacek (2012)
proposed a two-step data-based regularization where the filter matrix
is modified according to

where **L**_{k} is one of the finite-difference
approximations of a derivative, ${\mathbf{D}}_{\widehat{x}}$ is diag$\left(\right|\widehat{{x}_{\mathrm{1}}}|,\mathrm{\dots},|{\widehat{x}}_{\text{n}}\left|\right)$, and
$\widehat{\mathit{x}}$ is the reconstruction of ** x** using

**L**

_{k}. In the case that $\left|\widehat{{x}_{i}}\right|<\mathit{\u03f5}$, those elements are set to be equal to

*ϵ*, where $\mathrm{0}<\mathit{\u03f5}\ll \mathrm{1}$. The method

*L*

_{1}

*D*

_{1e−2}represents a filter matrix with a first-order derivative operator applied to Eq. (6) with $\mathit{\u03f5}=\mathrm{1}e-\mathrm{2}$. Exponential notation is used because subscripts are difficult to superscript.

*Lower/upper bound constraints.* The retrieved *x*_{λ}
from the regularized normal equation can have oscillatory and/or nonphysical
solutions. An alternative approach is to treat Eq. (2) as a constrained
minimization such that the solution is subject to the optional constraint
${\mathit{x}}_{\text{lb}}<\mathit{x}<{\mathit{x}}_{\text{ub}}$. Here, the following procedure
is implemented for the bounded search: first, the optimal *λ*_{opt}
is found using the regularized normal equations. The thus-obtained
solution *x*_{λ} is truncated at the upper and lower
bounds and then passed as an initial condition to a least-squares
numerical solver. Ceres Solver (Agarwal et al., 2020) is used with
the dogleg method and QR solver as implemented in the freely available
LeastSquaresOptim.jl^{1}
library. The net result is an optimized solution that is within the
specified upper and lower bounds. The upper and lower bounds are vectors
of the same size as ** x**.

### 2.1.4 Software implementation

*L*_{2} regularization, as described in the previous sections, is
implemented in a freely available software package RegularizationTools.jl
that is written by the author and provided as a supplement to this
work. The implementation is in the Julia programming language (Bezanson et al., 2017).
The package has a similar name and some overlap with the package Regularization
Tools by Hansen (2007). However, the
packages differ in software architecture, programming language, and
scope. RegularizationTools.jl provides a simple high-level
interface to compute *x*_{λ} using a single function
call; for example

where Eq. (7) is in a form that would be directly used as computer code. In Eq. (7) A denotes the design matrix **A**, b denotes the observation vector ** b**, and L

_{k}x

_{0}B is a parameterized algebraic data type that encodes the specific method in accordance with Eq. (5). The arguments of L

_{k}x

_{0}B are k, which specifies the order, and x

_{0}, which specifies the vector of the a priori estimate

*x*_{0}; lb and ub are vectors that specify the lower and upper bounds. Other methods can be specified according to Eq. (5). Examples are provided in the documentation of the package.

## 2.2 Computing the design matrix

The design matrix can be obtained from a forward model

where ** y** is a vector representing the error-free
observations,

**is the vector of true inputs,**

*x***is a vector of controlling parameters, and**

*c**F*is the linear forward model function that maps over

**to compute**

*x***subject to the constraint of**

*y***. The matrix of the linear transformation**

*c***=**

*y***A**

**is then given by**

*x*where ${\mathit{e}}_{\mathrm{1}},\mathrm{\dots},{\mathit{e}}_{\text{n}}$ is the standard basis. RegularizationTools.jl also provides an abstract generic interface that simplifies computation of the design matrix from arbitrary forward models of linear processes. Examples demonstrating how to use this generic interface are provided in the documentation of the package. The examples include the solution for transit through the tandem DMA described further below, the solution of the Fredholm integral equation of the first kind given by Baart (1982), the optical convolution that underlies size distribution retrieval from scattering and absorption properties (Müller et al., 2019), and the 2D Gaussian blur function encountered in image processing.

## 2.3 Design matrices for differential mobility analyzers

Differential mobility analyzers consist of two electrodes held at
a constant or time-varying electric potential. Cylindrical (Knutson and Whitby, 1975)
and radial (Zhang et al., 1995; Russell et al., 1996)
electrode geometries are the most common. Charged particles in a flow
between the electrodes are deflected to an exit slit and measured
by a suitable detector, usually a condensation particle counter. The
fraction of particles carrying *k* charges is described by a statistical
distribution that is created by the charge conditioner used upstream
of the DMA. The functions governing the transfer through bipolar charge
conditioners, single DMAs, and tandem DMAs are well understood (Knutson and Whitby, 1975; Rader and McMurry, 1986; Reineking and Porstendörfer, 1986; Wang and Flagan, 1990; Stolzenburg and McMurry, 2008; Jiang et al., 2014).

The DMA selects particles by electrical mobility. The relationship between mobility and mobility diameter is well known and well defined. The relationship is given, for example, in Eq. (2) in Petters (2018). This work also makes use of the “apparent + 1 mobility diameter”. It is defined as the conversion from mobility to diameter assuming singly charged particles using the mobility grid scanned by either DMA 1 or DMA 2. The apparent + 1 mobility diameter represents the natural diameter axis of a DMA response function, i.e., a plot of the raw detector response versus the nominal DMA setpoint diameter. It is an equivalent measure of mobility. The apparent + 1 mobility diameter is ambiguous. Larger particles carrying more than one charge may have the same apparent + 1 mobility diameter as smaller particles carrying fewer charges. The “apparent growth factor” is defined as the apparent + 1 mobility diameter scanned by DMA 2 divided by the nominal selected dry diameter in the DMA.

The traditional mathematical formulation of transfer through the DMA
is summarized in Stolzenburg and McMurry (2008)
and references therein. Briefly, the integrated response downstream
of the DMA operated at voltage *V*_{1} is given by a single integral
that includes a summation over all selected charges. The size distribution
is measured by varying voltage *V*_{1}, which produces the raw response
function defined as the integrated response downstream of the DMA
as a function of upstream voltage. The size distribution is found
by inversion. The basic mathematical problem associated with inverting
the response function to find the size distribution is summarized
by Kandlikar and Ramachandran (1999). The integral is
discretized by quadrature to find the design matrix that maps the
size distribution to the response function. *L*_{2} regularization
is one of several methods to reconstruct the size distribution from
the response function (Voutilainen et al., 2001; Kandlikar and Ramachandran, 1999).

The integrated response downstream of a tandem DMA that is operated
at voltages *V*_{1} and *V*_{2} requires evaluating integrals of
the upstream particle size distribution over size and the grown particle
size distribution over size. The integration must be repeated for
each charge state. Scanning over a range of voltages *V*_{2} results
in the raw tandem DMA response function. For the forward calculation, the
objective is to find a design matrix that maps the growth factor frequency
distribution to the raw TDMA response function.

Petters (2018) introduced a computational approach to model transfer through the DMA. The main idea of the approach is to provide a domain-specific language comprising a set of simple building blocks that can be used to algebraically express the response functions intuitively through a form of pseudo-code. The main advantage of this approach is that the expressions simultaneously encode the theory governing the transfer through the DMA and the algorithmic solution to compute the response function. The resulting expressions are concise. They are easily identified within actual source code when working through the examples provided with the package documentation. This makes the code easily modifiable by non-experts to change existing terms or add new convolution terms without the need to develop algorithms.

A disadvantage of the computational approach compared to the traditional mathematical approach is that computation lacks standardization of notation. This can blur the line between general pseudo-code and language-specific syntax. Some of the applied computing concepts may be less widely known when compared to standard mathematical approaches. Nevertheless, the author believes that the advantages of the computational approach outweigh the drawbacks. Therefore, this work builds upon the expressions reported in Petters (2018). Updates and clarifications to the earlier work are noted where appropriate.

The computational language includes a standardized representation
of aerosol size distributions, operators to construct expressions,
and functions to evaluate the expressions. Size distributions are
represented as a histogram and internally stored in the form of the
*SizeDistribution* composite data type. Composite data types
combine multiple arrays into a single symbol for ease of use, thus
facilitating faster experimental design and analysis. The size distribution
data type SizeDistribution includes vectors of the selected
mobility bins considered by the DMA, +1 mobility diameter bin edges
and +1 mobility diameter bin midpoints computed from the mobility
grid, number concentration, log-normalized spectral density, and logarithmic
bin widths. SizeDistributions are denoted in blackboard bold
font (e.g., 𝕟, 𝕣). SizeDistributions
are the building block of composable algebraic expressions through
operators that evaluate to transformed SizeDistributions. For
example, 𝕟_{1}+𝕟_{2} is the superposition of
two size distributions, *a*⋅𝕟 is the uniform scaling
of the concentration fields by factor *a*, **A**⋅𝕟 is
the matrix multiplication of **A** and concentration fields of
the size distribution, and
*T*⋅𝕟 is the elementwise scaling of the diameter field
by factor *T*. (Note that Petters, 2018, used *T*. ⋅ 𝕟
as the elementwise scaling. The extra dot has been dropped to stay
consistent with the current software implementation.)

Generic functions are used to evaluate expressions. The function $\sum (f,m)$
evaluates the function *f*(*x*) for $x=[\mathrm{1},\mathrm{\dots},m]$ and sums the results.
If *f*(*x*) evaluates to a vector, the sum is the sum of the vectors.
The function map(*f*,*x*) applies *f*(*x*) to each element
of vector ** x** and returns a vector of results in the same order.
The function foldl(

*f*,

*x*) applies the bivariate function

*f*(

*a*,

*x*) to each element of

**and accumulates the result, where**

*x**a*represents the accumulated value. If no initial value is provided, as is the case in this paper, foldl applies the function to the first two elements of the list to compute the first

*a*. For example $\mathrm{foldl}(-,[\mathrm{1},\mathrm{2},\mathrm{3}\left]\right)$ evaluates the function $-(a,x)$ and yields $\mathrm{1}-\mathrm{2}-\mathrm{3}=-\mathrm{4}$. The function $\mathrm{mapfoldl}(f,g,x)$ combines map and foldl. It applies function

*f*to each element in

**such that**

*x***=**

*y**f*(

*x*) and then reduces the result using the bivariate function

*g*(

*a*,

*y*), where

*a*represents the accumulated value. For example, $\text{mapfoldl(sqrt},-,[\mathrm{4},\mathrm{16},\mathrm{64}])$ evaluates to $\mathrm{foldl}(-,[\mathrm{2},\mathrm{4},\mathrm{8}\left]\right)=\mathrm{2}-\mathrm{4}-\mathrm{8}=-\mathrm{10}$. The function vcat(

*x***,**

**) concatenates arrays**

*y***and**

*x***along the first dimension in Julia. However, other programming languages may concatenate along a different dimension as the definition of horizontal and vertical is arbitrary. Passing vcat to foldl (or mapfoldl) will result in a concatenated array. Anonymous functions are used as arguments of reducing functions. Anonymous functions are denoted as**

*y**x*→expression, where

*x*is the argument consumed in the evaluation of the expression. These functions are generic and represent widely used computing concepts. They are implemented in most modern programming languages.

DMA geometry, dimensions, and configuration are abstracted into composite
types Λ (configuration comprising flow rates, power supply
polarity, and thermodynamic state) and *δ* (DMA domain defined
by a mobility–size grid). Each DMA is fully described by a pair Λ,*δ*.
Subscripts and superscripts are used to distinguish between different
configurations in chained DMA setups, e.g., with *δ*_{1} and *δ*_{2}
denoting the first and second DMA, respectively. Application of size
distribution expressions to transfer functions constructs a concise
model of the transmitted DMA mobility distribution, denoted as the
DMA response function. Implementation of the language is distributed
through a freely available and independently documented package DifferentialMobilityAnalyzers.jl, written in the Julia language. Expressions in the text are
provided in general mathematical form for readability.

Petters (2018) gives a simple expression
that models transfer through the DMA. The function ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(k,{z}^{\text{s}})$
evaluates to a vector representing the fraction of particles carrying
*k* charges that exit DMA^{Λ,δ} as a function of mobility

where *z*^{s} is the centroid mobility selected by the DMA (determined
by the voltage and DMA geometry), ** Z** is a vector of particle mobilities,
Ω is the diffusing DMA transfer function (Stolzenburg and McMurry, 2008),

*T*

_{c}is the charge frequency distribution function (Wiedensohler, 1988), and

*T*

_{l}is the diameter-dependent penetration efficiency function (Reineking and Porstendörfer, 1986). The diameter ${\mathit{D}}_{p,\mathrm{1}}={D}_{p}(\mathit{Z},k=\mathrm{1})$ and evaluates to a vector of diameters. The function Ω has been updated from Petters (2018). The version in Petters (2018) computed the shape of the transfer function for the mobility diameter corresponding to singly charged particles and then applied the same shape of the transfer function and diffusional loss to the multiply charged particles. The functional Ω depends on three arguments $\mathrm{\Omega}(\mathit{Z},{z}^{\text{s}},k)$ and implicitly on the DMA configuration Λ (i.e., Eq 13 in Stolzenburg and McMurry, 2008). The output is a vector along the mobility grid

**. The maximum transmission occurs at $\mathit{Z}/{z}^{\text{s}}=\mathrm{1}$. The last argument denotes the number of charges. It is used to compute the mobility diameter from**

*Z**z*

^{s}and in turn the diffusion coefficient which is required to account for diffusional broadening of the transfer function. The output of ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(k,{z}^{\text{s}})$ is the transmission of particles through the DMA in terms of the true particle mobility diameter. This is achieved by passing ${z}^{\text{s}}/k$ as an argument of Ω, which corresponds to the centroid mobility setting for the DMA to transmit particles with

*k*charges under the assumption that they carry only a single charge. The net result is that ${\mathit{D}}_{p,\mathrm{1}}={D}_{p}(\mathit{Z},k=\mathrm{1})$ becomes equal to the true mobility diameter axis. As a consequence the charge fraction ${T}_{\text{c}}(k,{\mathit{D}}_{p,\mathrm{1}})$ and penetration efficiency

*T*

_{l}(

*D*_{p,1}) are evaluated at the correct diameter. The function ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(\mathrm{1},{z}^{\text{s}})$ evaluates to a vector of the same length as

**. Performing an elementwise sum over all ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(k,{z}^{\text{s}})$ (where the sum is over all charges**

*Z**k*) produces the net transmission probability function. Multiplication of the transmission probability function with the input distribution results in the mobility distribution transmitted by the DMA. Examples for ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(\mathrm{1},{z}^{\text{s}})\cdot \mathbb{n}$, ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(\mathrm{2},{z}^{\text{s}})\cdot \mathbb{n}$, and ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(\mathrm{3},{z}^{\text{s}})\cdot \mathbb{n}$ are shown in Fig. 2, right panel, in Petters (2018). Note that Eq. (10) can be evaluated using arbitrarily discretized

**vectors.**

*Z*Petters (2018) also gives an expression that evaluates to the convolution matrix for passage through a single DMA that is valid in the context of the size distribution measurement system, e.g., SMPS. Since the expression includes a summation over all charges, the information on the particle physical diameter of multiply charged particles is lost.

where *m* is the upper number of multiply charged particles, ^{T}
is the transpose operator, and *Z*_{s} is a vector of centroid mobilities
scanned by the DMA. The matrix is square if *Z*_{s}=** Z** in Eq. (11).
However, this is not a necessary restriction. Equation (11) evaluates to
the same as Eq. (8) in Petters (2018), but the notation is revised
to be more general by removing the Julia-specific splatting construct
and replacing it with more widely used generic functions.

To help with parsing the expression, ${T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(k,{z}^{\text{s}})$
evaluates to a vector of transmission for *k* charges and setpoint
centroid mobility *z*^{s} as a function of the entire mobility grid
(e.g., 120 bins discretized between mobility *z*_{1} and *z*_{2}).
The function $\mathrm{\Sigma}[k\to {T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(k,{z}^{\text{s}}),m]$
superimposes the vectors for all charges. Mapping ${z}^{\text{s}}\to \mathrm{\Sigma}[k\to {T}_{\text{size}}^{\mathrm{\Lambda},\mathit{\delta}}(k,{z}^{\text{s}}),m]$
over the centroid mobility grid *Z*_{s} produces an array of vectors,
each corresponding to the transmission for a single size bin. Transposing
the vectors and reducing the collection through concatenation produces
the design matrix that links the mobility size distribution to the
response function; i.e.,

where 𝕣 is the response distribution, 𝕟 is
the true mobility size distribution, and ** ϵ** is a vector denoting
the random error that may be superimposed as a result of measurement
uncertainties. By design 𝕟 and 𝕣 are SizeDistribution
objects, which represent the distribution as a histogram in both spectral
density units ($\mathrm{d}N/\mathrm{d}\mathrm{ln}D$) and concentration-per-bin units. The latter
is the raw response function, where each element corresponds to the
integrated response downstream of DMA 1 for a set upstream voltage
(or corresponding

*z*

^{s}or apparent + 1 mobility diameter but not true physical diameter for multiply charged particles). Note, however, that the response function is not a true particle size distribution in the scientific sense since information about multiply charged particles is lost. The representation of 𝕣 as a SizeDistribution object is to allow response functions to be used in the expression-based framework used here.

The mobility distribution exiting the humidity conditioner and before entering DMA 2 in the humidified tandem DMA is evaluated using the expression

where ${g}_{\mathrm{0}}={D}_{\text{wet}}/{D}_{\text{dry}}$ is the diameter growth factor, *D*_{dry}
is the selected diameter by DMA 1, *D*_{wet} is the diameter after
the humidifier, ${T}_{\text{size}}^{{\mathrm{\Lambda}}_{\mathrm{1}},{\mathit{\delta}}_{\mathrm{1}}}(k,{z}^{\text{s}})$ is as
in Eq. (10), and 𝕟 is the mobility size distribution upstream
of DMA 1. Subscripts are used to differentiate DMA 1 and 2 which possibly
have different geometries, flow rates, thermodynamic states, and mobility
grids, e.g., Λ_{1}, Λ_{2} and *δ*_{1}, *δ*_{2}.
To help parse Eq. (13), the product ${T}_{\text{size}}^{{\mathrm{\Lambda}}_{\mathrm{1}},{\mathit{\delta}}_{\mathrm{1}}}(k,{z}^{\text{s}})\cdot \mathbb{n}$
evaluates to the transmitted mobility distributions of particles carrying
*k* charges at the setpoint mobility *z*^{s} in DMA 1. The size
distribution is grown by the growth factor *g*_{0}, which is achieved
by applying the ⋅ operator to the product ${T}_{\text{size}}^{{\mathrm{\Lambda}}_{\mathrm{1}},{\mathit{\delta}}_{\mathrm{1}}}(k,{z}^{\text{s}})\cdot \mathbb{n}$.
Equation (13) assumes that *g*_{0} applies to all particle sizes.

The total humidified apparent + 1 mobility diameter distribution ${\mathbb{m}}_{\text{t}}^{{\mathit{\delta}}_{\mathrm{2}}}$ exiting DMA 2 is given by

where *m* is upper number of charges on the multiply charged particles
and

is the convolution matrix for transport through DMA 2 and particles
carrying *k* charges. In Eq. (15), *Z*_{s,2} is a vector of centroid
mobilities scanned by DMA 2. Note that the choice of ** Z** inside Ω
is up to the user. Sensible choices are $\mathit{Z}={\mathit{Z}}_{s,\mathrm{1}}$ or

**=**

*Z*

*Z*_{s,2}, the implications of which are further discussed later. Equations (14) and (15) have been modified from those in Petters (2018) in the following manner. The convolution matrix

**O**

_{k}is computed individually for each charge. The version in Petters (2018) computed the matrix corresponding to singly charged particles and then applied the same matrix to multiply charged particles. Since

**O**

_{k}is now charge resolved, it is moved into the summation in Eq. (14). Computation of

**O**

_{k}through Eq. (15) has been revised to be more general by removing a Julia-language-specific construct.

**O**

_{1}computed by Eq. (15) produces the same matrix as in Petters (2018). The resulting ${\mathbb{m}}_{\text{t}}^{{\mathit{\delta}}_{\mathrm{2}}}$ size distribution represents the apparent + 1 mobility diameter scanned by DMA 2. Equations (13)–(15) relax an approximation made in a similar treatment in Petters (2018). There it was assumed that the apparent + 1 mobility diameter (and thus apparent growth factor) for particles carrying multiple charges is the same as for singly charged particles. This is incorrect. Particles carrying more than a single charge alias at a smaller particle size (Gysel et al., 2009; Shen et al., 2021). The effect is due to the size dependence of the slip-flow correction factor and is captured by the revised charge-resolved convolution matrices

**O**

_{k}.

If the aerosol is externally mixed, the humidified apparent growth factor distribution function exiting DMA 2 is given by

where *P*_{g} is the growth factor probability density function and
the diameters resulting from the intermediate calculation ${\sum}_{k=\mathrm{1}}^{m}\left({\mathbf{O}}_{\text{k}}\cdot {\mathbb{M}}_{\text{k}}^{{\mathit{\delta}}_{\mathrm{1}}}\right)$
are normalized by *D*_{dry}. ${\mathbb{m}}_{\text{t}}^{{\mathit{\delta}}_{\mathrm{2}}}$ in Eq. (16) is the forward model through the tandem DMA. Using the notation
in Sect. 2.2,

where ** x** is the discrete representation of the true

*P*

_{g}and the vector

**of constraining parameters comprises the DMA setups ${\mathrm{\Lambda}}_{\mathrm{1}},{\mathrm{\Lambda}}_{\mathrm{2}},{\mathit{\delta}}_{\mathrm{1}}$, and**

*c**δ*

_{2}and upstream size distribution 𝕟. Computer code that creates a forward model for tandem DMAs has been added to the DifferentialMobiltyAnalyzers.jl package and is annotated in the documentation of the package.

For purposes of the forward model, the mobility grid for DMA 1 is
discretized at a resolution of *i* bins by specifying the ** Z** vector
in Eq. (10). If the

**vector does not match that of the aerosol size distribution 𝕟, the size distribution bins are interpolated onto the diameter bins corresponding to the**

*Z***bins. Transmission through DMA 1 is computed for a specified**

*Z**z*

^{s}(the dry mobility) and

*g*

_{0}(the growth factor) via Eq. (13). The resulting ${\mathbb{M}}_{\text{k}}^{{\mathit{\delta}}_{\mathrm{1}}}$ lies on the same

**grid with**

*Z**i*bins. Any mismatches between the apparent growth factor and the underlying

**grid are resolved via interpolation implicit in the ⋅ operator. (**

*Z**f*⋅𝕟 is the uniform scaling of the diameter field of the size distribution by factor

*f*. If the resulting diameters are off the original diameter grid, the result is interpolated onto the grid defined within 𝕟.) The mobility grid for DMA 2 is represented by the vector

*Z*_{s,2}in Eq. (15) and discretized at a resolution of

*j*bins over a custom mobility range. If the vector

**inside the square brackets of Eq. (15) $\left[{\mathrm{\Omega}}^{{\mathrm{\Lambda}}_{\mathrm{2}},{\mathit{\delta}}_{\mathrm{2}}}\right(\mathit{Z},{z}^{\text{s}}/k,k).\cdot {T}_{l}^{{\mathrm{\Lambda}}_{\mathrm{2}},{\mathit{\delta}}_{\mathrm{2}}}({\mathit{D}}_{p,\mathrm{1}}\left)\right]$ equals that of DMA 1, the product ${\mathbf{O}}_{\text{k}}\cdot {\mathbb{M}}_{\text{k}}^{{\mathit{\delta}}_{\mathrm{1}}}$ will map the**

*Z**i*bins from DMA 1 to the

*j*bins in DMA 2. Alternatively, if the

**vector inside the square brackets of Eq. (15) is taken to be equal to**

*Z*

*Z*_{s,2}, the matrices

**O**

_{k}are square and of dimension

*j*×

*j*. In that case, the transmitted and grown distribution from DMA 1 (

*i*bins along the mobility axis of DMA 1) is interpolated onto the mobility grid of DMA 2 prior to evaluating ${\mathbf{O}}_{\text{k}}\cdot {\mathbb{M}}_{\text{k}}^{{\mathit{\delta}}_{\mathrm{1}}}$. The advantage of this approach is that for

*j*<

*i*, the matrices

**O**

_{k}are smaller and subsequent calculations are faster. The forward model, defined by Eq. (14), can be evaluated for arbitrary

*g*

_{0}values. Thus the growth factor probability distribution

*P*

_{g}in Eq. (17) can be discretized into

*n*arbitrary growth factor bins. A natural choice is to accept growth factor values that coincide with the mobility grid of DMA 2; i.e., the bins align with $\mathit{g}={\mathit{D}}_{p,\mathrm{1}}/{D}_{\text{d}}$, where

*D*

_{d}is the nominal diameter selected by DMA 1,

*D*_{p,1}is equal to ${D}_{p}(\mathit{Z},k=\mathrm{1})$. However, this is not required for evaluating Eq. (17). Equation (17) is cast into matrix form such that the humidified mobility distribution function is given by

where the matrix **B** is understood to be computed
for a specific input aerosol size distribution and ** ϵ** is
a vector that denotes the random error that may be superimposed as
a result of measurement uncertainties. If the grids used to represent
the growth factor distribution and that of DMA 2 do not align, interpolation
is used to map the growth factor bins from the growth factor distribution
onto those corresponding to the DMA 2 grid. The choice of

*i*,

*j*, and

*n*, the ranges of mobility grids for DMA 1 and DMA 2 and the range of the growth grid for

*P*_{g}, is only constrained by computing resources and a physically reasonable representation of the problem domain. Reasonable choices are

*i*=120 and $j=n=\mathrm{60}$, where the range in apparent growth factor spans $\mathrm{0.8}<\mathit{g}<\mathrm{5.0}$. The size of

**B**is

*j*×

*n*. Uncertainties in the size distribution propagate into

**B**. The main influence of the error will be the relative fraction of +1, +2, and +3 charged particles. Assuming a random error of ± 20 % in concentration, the overall effect on ${\mathbb{m}}_{\text{t}}^{{\mathit{\delta}}_{\mathrm{2}}}$ is expected to be small. Note that interpolation is widely used in this framework. Interpolation may affect how errors propagate through the model. Interpolation in Eq. (13) is unavoidable. However, interpolation can be minimized by working with non-square

**O**

_{k}and matching the grid of

*P*_{g}to that of DMA 2. Informal tests working with different binning schemes suggest that the influence of interpolation choices on the final result is smaller than typical experimental errors.

Figure 1 shows an example application of Eq. (18) for an input growth
factor frequency distribution where all particles are assumed to have
the same growth factor of ∼ 1.6. The frequency distribution
is evaluated along a discrete growth factor grid with 120 bins with the range
$\mathrm{0.8}<\mathit{g}<\mathrm{5}$. Note that the size grid (or apparent growth factor
grid) must be extended to large sizes to capture the growth of multiply charged
particles computed via Eq. (13). The assumed input size distribution
is bimodal with mode diameters of 60 and 140 nm, geometric standard
deviations of 1.4 and 1.6, and number concentrations of 1300 and 2000 cm^{−3} in modes 1 and 2, respectively. The assumed sheath-to-sample
flow ratios are 5:1 in both DMAs. The product **B***P*_{g}
is the raw response that would be measured by a condensation particle
counter at the exit of the instrument. The contribution of +1, +2,
and +3 charged particles to the total can be computed via ${\mathbf{O}}_{\text{k}}\cdot {\mathbb{M}}_{\text{k}}^{{\mathit{\delta}}_{\mathrm{1}}}$.
Although the nominal growth factor is the same for all sizes, the
apparent mode of the growth factor decreases with increasing particle
charge (see also Gysel et al., 2009; Shen et al., 2021).
Therefore the axis is denoted as the apparent growth factor. Summing
the partial distributions results in **B***P*_{g}, demonstrating
that the matrix equation correctly maps *P*_{g} to the response,
including multiply charged particles.

Figure 2 shows the relationship between four illustrative growth factor
frequency distributions and the modeled apparent mobility distribution
functions. The apparent mobility distribution function represents
the raw particle concentration that would be measured by a detector
as a function of the apparent + 1 mobility diameter. The diameter axis
is normalized by the dry diameter selected by DMA 1. The selected
examples comprise a testbed to evaluate the feasibility of an SMPS-style
matrix-based inversion to recover *P*_{g}. The *Populations*
example consists of an external mixture with compositions corresponding
to four unique growth factors. The *Bimodal* example is the superposition
of two Gaussian distributions with 70 % of particles in the less hygroscopic
mode. The *Truncated* example is a Gaussian distribution truncated
at *g*=1.0. The *Uniform* example is a uniform distribution
over a fixed interval. All frequency distributions integrate to unity,
thus accounting for 100 % of the particle population. The dry diameter
and assumed input size distribution to compute the matrix **B**
are the same as in Fig. 1. However, unlike in Fig. 1, the frequency
distribution and matrix are evaluated along a coarser discrete growth
factor grid with 60 bins with the range $\mathrm{0.8}<\mathit{g}<\mathrm{5}$. Note that the
growth factor bin width is not constant, with wider bins at larger
growth factors. This is due to the evaluation of the humidified size
distribution along a geometrically stepped mobility grid. As will
be shown next, 60-bin resolution is a suitable compromise between
speed, accuracy, and resolution when computing the matrix-based inversion
to infer *P*_{g} from noise-perturbed apparent growth factor frequency
distributions.

Simulated examples are used to test if Eq. (18) is invertible. Figure 3 shows an example simulation for the Bimodal growth factor
distribution test case. The humidified apparent growth factor distributions
are calculated using Eq. (18). The noise-free example corresponds
to ** ϵ**=0 and represents the idealized measurement. Poisson
counting statistics are simulated by converting concentration to the
expected number of counts for a typical particle counter flow rate
and bin integration time. Counts in each bin are computed by drawing
a pseudo-random number from a Poisson distribution and converting
the result back to concentration. Lower flow rates and shorter integration
times increase the noise perturbation of the apparent growth factor
distribution. The apparent growth factor distribution is then inverted
using the ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ and

*L*

_{2}

*x*

_{0}

*B*

_{[0,1]}method. Here

*B*

_{[0,1]}is shorthand for setting all lower bounds equal to zero and all upper bounds equal to 1. The a priori estimate

*x*_{0}is taken to be the normalized apparent growth factor distribution derived from the measured response function, where the normalization ensures that the sum over all bins is unity. Note that the inversion is performed treating the growth factor distribution in units of frequency instead of frequency density. This choice enables the upper bound constraint of unity. Since the true noise-free input growth factor frequency distribution is known, the fidelity of the inversion can be evaluated by computing the root mean square error between the noise-free solution and the regularized solution. Evaluating the root mean square error in frequency rather than frequency density space results in more comparable values when contrasting narrow and broad probability distribution functions. The figure shows that both inversion methods produce a root mean square error between 0.002 and 0.003. Values less than 0.01 are typical of the reconstruction of Bimodal distributions at this bin resolution (see Supplement). Visual evaluation of the agreement between the reconstruction and the input suggests that either method is suitable for inversion.

Figure 4 is similar to Fig. 3, showing an example simulation for aerosol with uniform composition; i.e., all particles have the
same growth factor. Although the *L*_{2}*x*_{0}*B*_{[0,1]} approach correctly
infers the most probable growth factor, the predicted distribution
is incorrect. Multiple modes to the left and right of the main mode
are observed. The *L*_{2}*x*_{0} method produces an oscillatory solution
with negative values (not shown). The small modes are the residual
of this oscillatory solution that is truncated by the enforced [0,1]
bound and the inability of the least-squares solver to converge on
a better solution. A large root mean square error of 0.107 results.
In contrast, the data-constrained method ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$
leads to better reconstruction of the true input. The main advantage
of the ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ inversion method over *L*_{2}*x*_{0}*B*_{[0,1]}
is that it is better able to reconstruct inputs with sharp edges.

The total number of composable regularization methods according to
Eq. (5) is 24. Half of these methods do not include lower and upper
bounds, and these are not suitable for tandem DMA inversion due to
the negative and oscillatory solutions for narrow inputs. The remaining
12 methods have been systematically tested using Monte Carlo analysis
described in detail in the supporting information. Briefly, 60 000
inversions were performed on synthetic data similarly to the examples
shown in Figs. 3 and 4. The total number concentration, dry diameter,
number of bins, and random seeds were varied, and the root mean square
error was evaluated for each simulation. Results compiled in Fig. S1 show that all of the methods perform equally well for the Bimodal,
Uniform, and Truncated examples shown in Fig. 3. Method
${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ outperforms the other methods for grids
with <60 growth factor bins with the range $\mathrm{0.8}<\mathit{g}<\mathrm{5}$ and test cases
with either one (e.g., Fig. 4) or two discrete populations. However,
even ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ can lead to results similar to the
example *L*_{2}*x*_{0}*B*_{[0,1]} shown in Fig. 4 for some random seeds.
Higher-resolution grids generally lead to poor performance for discrete
populations even for method ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$.

An alternative approach to fit single-component data is to perform
a nonlinear least-squares fit to match the apparent growth factor
distribution using the forward model while restricting the number
of compositions to either one or two. This corresponds to a two- or
four-parameter fit. Results from this procedure are either one or
two growth factors and one or two fractions. The corresponding methods
are denoted as LSQ_{1} and LSQ_{2}, respectively. In the example
shown in Fig. 4, LSQ_{1} has the smallest root mean square error
and is the best method to reconstruct the true growth factor. The
LSQ_{1} method is most suitable for inferring the growth factor for
laboratory measurements when it is known that the aerosol is internally
mixed and only a single growth factor is expected.

Which method, however, should be selected when inverting real-world
data and when the number of components is unknown? Since the true solution
is also unknown, the root mean square error between the truth and
reconstruction is unavailable. It is, however, possible to compute
the residual between the measured apparent growth factor distribution
and the predicted apparent growth factor distribution from different
reconstructions. A large residual can be used to flag truncated oscillatory
solutions such as *L*_{2}*x*_{0}*B*_{[0,1]} for narrow/single-composition
cases. Similarly, the residual is high if the true input is a broad
growth factor frequency distribution that is attempted to be fitted
using LSQ_{1}. For example, the red spectrum in the left graph
of Fig. 4 shows poor agreement with the input and results in a much
larger residual than LSQ_{1} (values not shown). Therefore, a proposed
unsupervised inversion scheme is to compute the solution of LSQ_{1},
LSQ_{2}, and ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ and then select the solution
with the lowest residual relative to the apparent growth factor distribution.

Note, however, that the low residuals between the apparent growth factor distribution and the model do not automatically ensure that the algorithm has a good or adequate solution. Additional tests should be performed to validate the physical plausibility of the solution. For example, the retrieved growth factors should be physically plausible at the applied relative humidity. The mode of the apparent growth factor distribution and the mode of the inverted growth factor distribution should be similar. A histogram of the root mean square error between can be plotted for a large dataset. Visual inspection of fits for large root mean square error can be used to derive a threshold above which reconstructions are automatically rejected. The integrated probability density function of the reconstructions should be near unity. Deviations from unity may occur due to concentration errors between the size distribution measurement and the growth factor distribution measurement, unaccounted transmission losses, and errors from the inversion. Reconstructions deviating significantly from unity should be flagged and rejected.

A limitation of the above approach is that the forward model (and
thus matrix **B**) assumes that the larger multiply charged
particles have the same growth factor frequency distribution as the
smaller singly charged particles. This limitation can in principle
be eliminated by specifying a 2D probability frequency distribution
that also depends on the dry diameter. Constructing an appropriate forward
model that adds another integration dimension to Eq. (17) is straightforward.
An inversion that solves for the 2D frequency distribution, similarly
to those performed elsewhere (Rawat et al., 2016; Sipkens et al., 2020),
is feasible using the algorithms in RegularizationTools.jl and
has been attempted by the author. In practice, however, this approach
proved impractical. For example, using 10 dry diameters and a 30-bin
size resolution results in a large inversion matrix. Adding an integration
dimension to the forward model and recomputing this matrix for each
scan significantly slows the inversion. Furthermore, interpolation
is needed to estimate the growth factor frequency distribution for
the multiply charged particles. The physical size of the multiply
charged particles depends on their charge. For example, +2 charged
particles are approximately 1.5 times larger than +1 charged particles.
The diameter of the multiply charged particles will therefore not
necessarily coincide with any of the 10 dry diameters selected for
direct measurement. This introduces additional uncertainty due to
assumptions that need to be made in the interpolation scheme. Errors
from scans with low non-zero concentration at the edge of the size
distribution propagate back into the inversion at other dry sizes.
Finally, only a single size distribution can be used to compute the
matrix **B**. Collecting data for 10 dry sizes can take
20 min or longer, during which the aerosol size distribution may change,
thus invalidating the use of a single inversion matrix. In situations
where the temporal evolution of the size distribution is predictable,
e.g., environmental chamber measurements, Kalman smoothing might be
used to predict the in-between states (Ozon et al., 2021b, a).
Although no exhaustive analysis was performed, the compounding errors
during a 2D inversion seem to outweigh the benefits of relaxing the
assumption that +2 and +3 charged particles have the same growth factor
frequency distribution as the +1 charged particles.

## 4.1 Data sources

### 4.1.1 Bodega Marine Laboratory

Aerosol size distribution data to contrast inversion schemes were
obtained from measurements taken at Bodega Marine Laboratory (39^{∘}18^{′}25^{′′} N,
123^{∘}3^{′}58^{′′} W) between 16 January and
8 March 2015 as part of the CalWater 2/ACAPEX campaign. A subset of
the data have been published by Atwood et al. (2019).
Sample flow was brought into a mobile laboratory using an inlet, dried
to 10±2 % relative humidity using a Nafion membrane drier, and
brought to charge equilibrium using an X-ray source (TSI 3088, TSI
Inc., Shoreview, MN, USA) prior to entering a cylindrical DMA column
(TSI 3081). The DMA was configured to measure the size distribution
in scanning mobility particle sizer mode. Voltage was scanned exponentially
from 10 kV to 10 V over 300 s. A condensation particle counter (TSI 3771, flow rate 1 L min^{−1}) and a cloud condensation nuclei
counter (DMT Model 100, Droplet Measurement Technologies, Boulder,
CO, USA, flow rate 0.3 L min^{−1}) were used to measure particle
concentration downstream of the DMA. The sheath-to-sample flow rate
in the DMA was 5:1.3 L min^{−1}. Raw DMA response distributions
comprising concentration measured by a condensation particle counter (CPC) vs. apparent + 1 mobility diameter were
constructed along a 120-bin, geometrically stepped mobility grid.
Response distributions are denoted as 𝕣. The apparent + 1 mobility diameter is computed from the centroid mobility selected
by the DMA assuming that all particles are singly charged. The dynamic
diameter range for this setup is from 12 to 550 nm. The inversion
matrix **A** is computed using Eq. (11) for the diffusionally broadened
transfer function (Stolzenburg and McMurry, 2008)
and transmission loss correction through the DMA (Reineking and Porstendörfer, 1986).
Inclusion of these terms results in a more ill-posed inverse problem
due to increasing overlap between the kernels (Kandlikar and Ramachandran, 1999).
The DMA response functions were inverted using the *L*_{0}*x*_{0}*B*_{[0,∞]}
and *L*_{2}*B*_{[0,∞]} methods. The a priori estimate for
*L*_{0}*x*_{0}*B*_{[0,∞]} was taken to be ${\mathit{x}}_{\mathrm{0}}={\mathbf{S}}^{-\mathbf{1}}\mathbb{r}$,
where **S** is obtained by summing the rows of **A**
and placing the results on the diagonal of **S** (Talukdar and Swihart, 2003).
The method *L*_{0}*x*_{0}*B*_{[0,∞]} with ${\mathit{x}}_{\mathrm{0}}={\mathbf{S}}^{-\mathbf{1}}\mathbb{r}$
is essentially equivalent to the method used by Petters (2018),
where it was shown that the thus-inverted spectra are similar to those
output by the inversion algorithm employed by the commercial TSI Aerosol
Instrument Manager software suite. Small differences between *L*_{0}*x*_{0}*B*_{[0,∞]}
employed here and the approach of Petters (2018)
include the use of generalized cross-validation instead of the L-curve
method to search for the optimal regularization parameter and the
method to eliminate negative values after inversion. Petters (2018)
truncated negative values instead of using a least-squares numerical
solver as described in Sect. 2.1.3.

### 4.1.2 Southern Great Plains site

Aerosol size distribution and humidified tandem DMA data to illustrate
the tandem DMA inversion schemes were taken from measurements made
by the US Department of Energy (DOE) Atmospheric Radiation Measurement
(ARM) program. The Southern Great Plains (SGP) site is located in
Lamont, OK, USA (36^{∘}36^{′}26.4^{′′} N, 97^{∘}29^{′}15.5^{′′} W), in a rural continental setting that is surrounded by agricultural
activity as well as oil and gas production. The aerosol evolution
at the site is influenced by frequent new-particle-formation events
(Hodshire et al., 2016; Chen et al., 2018; Marinescu et al., 2019).
Number concentrations fluctuate in response to the nitrate and organic
aerosol cycle on short timescales and synoptic weather variability
on longer timescales. During winter months, the inorganic aerosol
composition at the site is dominated by nitrate aerosol (Jefferson et al., 2017; Mahish et al., 2018),
and hygroscopicity derived from scattering measurements is largest
during those months (Jefferson et al., 2017).

The instruments and measurements are part of the Aerosol Observing System
(AOS; Uin and Smith, 2020). The instruments are operated
by DOE personnel, and data are distributed through a publicly accessible
archive. Size distributions were measured with a scanning mobility
particle sizer (TSI Model 3936; Kuang, 2016).
Data in the archive have already been inverted and are reported at 5 min intervals.
Humidified DMA response functions were measured using a humidified
tandem DMA (Model 3100, Brechtel Manufacturing, Inc., Hayward, CA,
USA). The first and second DMA are operated at a 5:0.63 L min^{−1}
and 5:1 L min^{−1} sheath-to-sample flow ratio, respectively (Janek
Uin, personal communication, 2021). The instrument measures the humidified
mobility distribution function at 85 % relative humidity for 50, 100,
150, 200, and 250 nm dry-diameter particles. Typical data density
results in 228 scans per day, with equal coverage for the five dry
sizes. Pre-processing that has already been applied to the archived data
accounts for conversion between mobility and apparent mobility diameter,
the size-dependent detector counting efficiency, and number count
smearing during the scan resulting from insufficient particle counter
response time. When divided by the dry diameter, the archived data
correspond to the apparent growth factor distribution evaluated by
the forward model in Eq. (17).

The matrix **B** was evaluated for each scan using the
flow rates given above, the dimensions of the DMAs given in Lopez-Yglesias et al. (2014),
and the aerosol size distribution measured by the co-located SMPS
with the timestamp closest to the scan of the humidified tandem DMA.
Typical time differences between the two instruments' scan times are
between 1 and 3 min. The humidified size distribution was interpolated
onto a discrete growth factor grid with 60 bins with the range $\mathrm{0.8}<\mathit{g}<\mathrm{5.0}$ to match the matrix **B**. The data were then inverted
using the ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ method. The method ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$
was further constrained such that growth factors (gf's) <1 are disallowed.
This is achieved by setting the upper bound to zero for bins with
gf < 1. Growth factors of less than unity can occur due to particle
restructuring upon humidification (Mikhailov et al., 2004; Shingler et al., 2016)
or evaporation during transit through the humidifier and second DMA.
Both effects are assumed to be less important for ambient aerosol
compared to the desire to constrain the inversion. In addition, the
efficacy of the LSQ_{1} and LSQ_{2} methods for inverting the
data was tested. For each scan, the root mean square error between
the measured apparent growth factor distribution and the predicted
growth factor distribution was evaluated for all three inversion approaches
(${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$, LSQ_{1}, and LSQ_{2}). The method
that resulted in the smallest residual was taken to be the inverted
growth factor frequency distribution.

## 4.2 Results

### 4.2.1 Inversion of size distribution data (Bodega Marine Laboratory site)

Figure 5 shows a real-world example size distribution response function
gridded into 120 size bins. The total particle concentration is ∼2000 cm^{−3}. The ragged structure is typically explained by random
noise due to Poisson counting statistics. However, in this example
the noise level is larger than Poisson counting statistics alone,
which is thought to be due to the processing of raw data internal
to the specific CPC model that was used to collect the data. At this
diameter resolution and with inclusion of the diffusion and loss
terms in the forward model, the unregularized matrix inverse is entirely
dominated by amplified random noise and is useless. The *L*_{0}*x*_{0}*B*_{[0,∞]}
method converges to the solution with slight amplification of the
random noise presented in the raw response function. The random noise
is carried over into the a priori estimate ${\mathit{x}}_{\mathrm{0}}={\mathbf{S}}^{-\mathbf{1}}\mathbb{r}$,
which roughly represents the noise visible in the reconstructed solution.
Nevertheless, *L*_{0}*x*_{0}*B*_{[0,∞]} is highly robust and unlikely
to go astray because *x*_{0} is an excellent approximation
of the solution at diameters of less than 100 nm where singly charged
particles dominate and is a good initial estimate for larger particles.
Second-order inversion using *L*_{2}*B*_{[0,∞]} produces a smooth,
denoised solution due to application of the derivative operator in
the regularization filter matrix. The solution converges even though
no a priori estimate is used; i.e., *x*_{0}=0. Inclusion
of an a priori in the form of *L*_{2}*x*_{0}*B*_{[0,∞]} is
possible. However, noise in the a priori propagates into the
solution, thus negating the intended benefit of the second-order Tikhonov
matrix. The algorithms specified in Sect. 2.1.2 significantly
speed up the inversion relative to previous versions of the software
(Petters, 2018). Wall-clock times
on an i7-8559U CPU for the inversion of a single spectrum are 5 and
2 ms for *L*_{0}*x*_{0}*B*_{[0,∞]} and *L*_{2}*B*_{[0,∞]}, respectively.
This contrasts to the 500 to 1000 ms required by the brute-force algorithm
– approximately equivalent to *L*_{0}*x*_{0}*B*_{[0,∞]} – used
previously. Finding the global minimum of *V*(*λ*) to identify
the optimal regularization parameter also eliminates the occasional
failure to converge when the L-curve algorithm is used (Petters, 2018).
Either *L*_{0}*x*_{0}*B*_{[0,∞]} or *L*_{2}*B*_{[0,∞]}, combined
with generalized cross-validation, is suitable for use in routine
unsupervised inversion of size distribution data.

Figure 6 shows the time evolution of the normalized particle size
distributions over a 7-week period. The normalization is to highlight
changes in the mode diameter(s). In general, the aerosol at the site
is dominated by continental rural background conditions and the land–sea
breeze circulation (Atwood et al., 2019).
The time series is punctuated by aerosol transported from the California
Central Valley to the site through the Petaluma Gap (Martin et al., 2017).
Periods of low particle concentration occurred during the passage
of an atmospheric river on 7–9 February 2015 and a marine inflow
event on 27–28 February 2015. The atmospheric river brought heavy
precipitation and marine air masses from the southwest direction, while
the marine inflow event brought strong winds and precipitation-free
maritime air from the northwest direction. Several periods of prolonged
modal growth were observed starting, e.g., 11 and 24 February and 1 March 2015. Figure 6 demonstrates the influence of inversion
noise on visualizing the dynamic evolution of the size distribution.
The denoised *L*_{2}*B*_{[0,∞]} solution significantly improves
visualization of modes without the need to reduce the size resolution
in the inversion. The signal is especially improved during low-concentration
periods during the atmospheric river passage and marine inflow event.

### 4.2.2 Inversion humidified tandem DMA data (DOE ARM SGP site)

Figure 7 shows real-world examples of growth factor frequency distributions
for five dry sizes. Also shown for context is the evolution of the
normalized aerosol number size distribution. Figure 7 shows dynamic
evolution of the size distribution with sudden changes in mode diameter,
several apparent new particle formation events, and several prolonged
modal growth events. The distribution of the methods selected for
best inversion was LSQ_{1} (∼ 5 % of spectra), LSQ_{2}
(∼ 50 % of spectra), and ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$
(∼ 45 % of spectra). In Fig. 7, the LSQ_{2} inverted
frequency distributions show a clean bimodal structure (two colors per
scan), while the ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ spectra appear more smeared.
The 250 nm dry-diameter data show a dominant contribution of more
hygroscopic particles with gf ∼ 1.5–1.6 and a
small contribution of less hygroscopic particles with gf ∼ 1.05–1.2.
Similar trends are observed for 200, 150, and 100 nm particles. However,
the hygroscopicity of the dominant mode decreases with decreasing
diameter. The fraction of cases where a broad hygroscopicity frequency
distribution is observed is larger than for the 250 nm particles.
Notably, time periods with broad growth factor frequency distributions
are observed at multiple sizes. For example, the period of 9–11
February 2020 shows a broad frequency distribution at 100, 150, 200,
and 250 nm dry diameters. Occasionally temporal trends in the hygroscopicity
of the less hygroscopic mode are observed. For example, the growth
factor of the less hygroscopic mode systematically increases on 20
February 2020 for 150, 200, and 250 nm particles, indicative of a
chemical transformation of some, but not all, of the particles. The
50 nm dry-particle hygroscopicity frequency distributions are also
predominantly bimodal. However, the overall growth factor is significantly
smaller, with most gf < 1.2.

RegularizationTools.jl is a general-purpose software package
to invert data using *L*_{2} regularization. It is included as a
supplement to this work and published as free software through the
GNU General Public License. The package implements well-established
numerical algorithms (Golub et al., 1979; Eldén, 1982; Bates et al., 1986; Hansen, 1998, 2000; Mogensen and Riseth, 2018)
and filter matrices (Huckle and Sedlacek, 2012). Systems
with up to ∼ 1000 equations can be inverted. The upper
limit is determined by the need to compute the generalized singular
value decomposition of the design matrix and filter matrix, which
has at minimum *O*(*n*^{2}) time complexity. The time to compute the
generalized singular value decomposition exceeds several tens of seconds
for systems exceeding 1000 equations. Iterative methods to support
inversion of large-scale systems have been formulated (e.g., Lampe et al., 2012),
but these are currently not implemented.

The software package can be used to simplify the prototyping of a
wide variety of inverse problems that arise in science and engineering
applications. Although the package does not add any novel regularization
methods, it provides a systematic method to categorize inversion methods
via the expression in Eq. (5). A total of 24 basic permutations can
be combined with a set of hyperparameters to attempt the inversion
of ill-posed problems. Hyperparameters include boundary constraints,
values for a priori estimates, and the lower-bound *ϵ*
for the Huckle and Sedlacek (2012) two-pass inversion
approach. Users can define custom filter matrices and thus are able
to further extend the number of methods. Equation (7) provides an
example of a simplified interface that allows testing of different
permutations with a simple function call. Furthermore, a generic interface
is provided to translate arbitrary linear forward models defined by
a computer function into the corresponding matrix of linear transformation.
This obviates the need to explicitly write out the Fredholm integral
equation and discretize it using the quadrature or the Galerkin method.
For example, the forward model for transfer of a growth factor frequency
distribution through the tandem DMA in Eq. (17) represents a triple
integral and also contains a sum term for the multiple charges. Explicit
discretization of this model would be tedious compared to the method
employed here. As demonstrated in the documentation of the package,
the generic interface can readily be used to solve other common inversion
problems. Only a few lines of new code are needed to reproduce the
essential core of the algorithm used in the unsupervised inversion
of lidar data (Müller et al., 2019), which involves
the retrieval of a size distribution from multi-wavelength scattering
and absorption data (see package documentation for code).

*L*_{2} regularization is one of several techniques that is suitable
for inverting size distribution data (e.g., Voutilainen et al., 2001; Kandlikar and Ramachandran, 1999).
The technique has been used previously for size distribution inversion
(e.g., Wolfenbarger and Seinfeld, 1990; Talukdar and Swihart, 2003; Petters, 2018).
An advantage of this method is that data can be inverted when the
number of data channels becomes large (Talukdar and Swihart, 2003).
In contrast, Bayesian inversion schemes, which are not further discussed
here, are suitable for uncertainty quantification (Voutilainen et al., 2001).
To the author's knowledge the package DifferentialMobilityAnalyzers.jl
is the only publicly available free software for size distribution
inversion from DMA data. This work extends the capabilities of that
package. The *L*_{0}*x*_{0}*B*_{[0,∞]} and *L*_{2}*B*_{[0,∞]}
methods can be used with generalized cross-validation to perform fast
unsupervised inversion of size distribution data. Convergence issues
resulting from the use of the L-curve method used previously (Petters, 2018)
are resolved by switching to the generalized cross-validation approach
to find the optimal regularization parameter. Higher-order inversions
resulting in smooth, denoised solutions are now available. It is expected
that such denoised spectra will benefit unsupervised machine-learning
approaches that seek to extract features from such datasets (e.g., Joutsensaari et al., 2018; Atwood et al., 2019),
although this hypothesis has not been tested by the author. Revision
of the numerical algorithms improves the speed of inversion by a factor
of ∼ 200. The millisecond inversion speed for a single
scan permits rapid inversion of large datasets and facilitates inversion
in real time during data acquisition on low-cost and low-computational-power hardware platforms. For example, the inversion has been tested
on ARM Cortex A72/A53 64 bit reduced-instruction-set architecture
used by the ROCKPro64 single-board computer. The Julia language provides
tier-1 support for this architecture. Julia binaries are available;
DifferentialMobilitityAnalyzers.jl and RegularizationTools.jl
compile and run without any modification. Inversion speeds on the
order of several tens of milliseconds are fast enough on this inexpensive
but relatively low powered platform to permit embedding the inversion
into the data acquisition and display software and running the inversion
before each display update.

To the author's knowledge this is the first time *L*_{2} regularization
has been applied to the inversion of tandem DMA data. Inversion of
simulated data shows that an SMPS-style matrix-based inversion
is possible while also accounting for multiply charged particles.
Application of solution constraints fixes the issue of oscillatory
and negative solutions that were encountered with the matrix-based
optimal estimation method used by Cubison et al. (2005).
The 12 methods that include boundary constraints were systematically
tested against five test cases. All of the methods performed similarly
well when inverting frequency distributions. However, poor results
were obtained when inverting narrow distributions or data produced
by single compositions. The method ${L}_{\mathrm{0}}{D}_{\mathrm{1}e-\mathrm{3}}{B}_{[\mathrm{0},\mathrm{1}]}$ is often,
but not always, able to invert these data. For narrow distributions
a nonlinear least-squares fit with either one or two growth factors,
termed LSQ_{1} and LSQ_{2}, can fill this gap. Ambient data
can be inverted by applying all three methods and then selecting the
inversion with the smallest root mean square error between the data
and the prediction. In contrast to previous inversion routines (Stolzenburg and McMurry, 1988; Cubison et al., 2005; Gysel et al., 2009),
explicit knowledge of the aerosol size distribution is needed. These
data can be obtained either using a co-located scanning mobility particle
sizer or by configuring the tandem DMA to also measure the size distribution
every few scans. The resulting algorithm is unsupervised and nonparametric;
i.e., it can be fully automated and does not require any a priori assumption about the functional form of the growth factor frequency
distribution. The speed of the inversion algorithm is much slower
than for size distribution inversion for several reasons. For each
scan, the matrix **B** must be recomputed to account for
changes in the size distribution. This requires recomputing the generalized
singular value decomposition for **B** and
**L**, which is slow. Furthermore, three inversions are
computed for each scan. The LSQ_{1} and LSQ_{2} methods use
a gradient descent algorithm together with the forward model, which
is slower than the matrix inverse. Nevertheless, a single day's worth
of data can be inverted on a regular personal computer within a few
minutes.

Application of the inversion to a 16 d dataset demonstrates that the thus-obtained growth factor frequency distribution data can reveal significant details about the mixing state of the aerosol. The inverted dataset is suitable as input to carry out common analyses made with growth factor frequency distributions. Examples include the characterization of the evolution of the aerosol mixing state as a function of time, characterization of changes in the growth factor with the dry diameter and its relationship to chemical composition, or characterization of the growth factor at the mode diameter of particles during modal growth events (Park et al., 2008; Wu et al., 2013; Jung and Kawamura, 2014). Additional examples include the decomposition of the hygroscopicity frequency distributions into distinct growth factor classes (Swietlicki et al., 2008), evaluation of the temporal trends of spectral concentration for hygroscopicity-resolved data (Royalty et al., 2017), evaluation of the accuracy of (organic) mass concentration measured by aerosol mass spectrometers through hygroscopicity constraints (Jimenez et al., 2016), and inclusion of growth factor frequency distributions to account for the mixing state in aerosol hygroscopicity to cloud condensation nuclei closure (Mahish et al., 2018).

Current and future versions of the DifferentialMobilityAnalyzers.jl and RegularizationTools.jl are also hosted on GitHub. Details about the SGP HTDMA data and the SMPS data are provided in the references. Source code to reproduce the figures, derived datasets, and archived versions of the software packages is available via Zenodo: https://doi.org/10.5281/zenodo.5550382 (Petters, 2021).

The supplement related to this article is available online at: https://doi.org/10.5194/amt-14-7909-2021-supplement.

The contact author has declared that there are no competing interests.

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

Data from the SGP site were obtained from the Atmospheric Radiation Measurement (ARM) program sponsored by the US Department of Energy, Office of Science, Biological and Environmental Research, Climate and Environmental Sciences Division. I thank Janek Uin for providing additional information about the data. Size distribution data at Bodega Marine Laboratory were collected with support from the National Science Foundation grant AGS-1450690. I thank Nicholas Rothfuss, Sam Atwood, and Hans Taylor for help operating the SMPS at Bodega Marine Laboratory. I thank Kimberly Prather, Sonia Kreidenweis, and Paul DeMott for logistical support during the field campaign. I thank Sarah Petters for helpful discussions. I thank Mark Stolzenburg for exceptionally helpful referee comments.

This research has been supported by the US Department of Energy, Office of Science, Biological and Environment Research (grant no. DE-SC 0021074) and NASA (grant no. 80NSSC19K0694).

This paper was edited by Mingjin Tang and reviewed by Mark Stolzenburg, Christopher Oxford, and one anonymous referee.

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^{1}

https://github.com/matthieugomez/LeastSquaresOptim.jl (last access: 10 December 2021).

- Abstract
- Introduction
- Theory
- Matrix inversion of the humidified mobility distribution function using synthetic data
- Inversion of real-world data
- Discussion, summary, and conclusions
- Code and data availability
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement

- Abstract
- Introduction
- Theory
- Matrix inversion of the humidified mobility distribution function using synthetic data
- Inversion of real-world data
- Discussion, summary, and conclusions
- Code and data availability
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement