In this study a novel framework for inverse modelling of cloud condensation nuclei (CCN) spectra is developed using Köhler theory. The framework is established by using model-generated synthetic measurements as calibration data for a parametric sensitivity analysis. Assessment of the relative importance of aerosol physicochemical parameters, while accounting for bulk–surface partitioning of surface-active organic species, is carried out over a range of atmospherically relevant supersaturations. By introducing an objective function that provides a scalar metric for diagnosing the deviation of modelled CCN concentrations from synthetic observations, objective function response surfaces are presented as a function of model input parameters. Crucially, for the chosen calibration data, aerosol–CCN spectrum closure is confirmed as a well-posed inverse modelling exercise for a subset of the parameters explored herein. The response surface analysis indicates that the appointment of appropriate calibration data is particularly important. To perform an inverse aerosol–CCN closure analysis and constrain parametric uncertainties, it is shown that a high-resolution CCN spectrum definition of the calibration data is required where single-valued definitions may be expected to fail.

Using Köhler theory to model CCN concentrations requires knowledge of many physicochemical parameters, some of which are difficult to measure in situ on the scale of interest and introduce a considerable amount of parametric uncertainty to model predictions. For all partitioning schemes and environments modelled, model output showed significant sensitivity to perturbations in aerosol log-normal parameters describing the accumulation mode, surface tension, organic : inorganic mass ratio, insoluble fraction, and solution ideality. Many response surfaces pertaining to these parameters contain well-defined minima and are therefore good candidates for calibration using a Monte Carlo Markov Chain (MCMC) approach to constraining parametric uncertainties.

A complete treatment of bulk–surface partitioning is shown to predict CCN spectra similar to those calculated using classical Köhler theory with the surface tension of a pure water drop, as found in previous studies. In addition, model sensitivity to perturbations in the partitioning parameters was found to be negligible. As a result, this study supports previously held recommendations that complex surfactant effects might be neglected, and the continued use of classical Köhler theory in global climate models (GCMs) is recommended to avoid an additional computational burden. The framework developed is suitable for application to many additional composition-dependent processes that might impact CCN activation potential. However, the focus of this study is to demonstrate the efficacy of the applied sensitivity analysis to identify important parameters in those processes and will be extended to facilitate a global sensitivity analysis and inverse aerosol–CCN closure analysis.

Atmospheric aerosols have an influence on the earth's radiation balance, and
thus the climate and its evolution, through many feedback effects and
processes. Aerosols can act to absorb and scatter solar radiation, a process known as the direct
effect

Cloud–aerosol interactions represent the largest uncertainty in current
global radiative forcing estimates

The importance of some physical and chemical properties are certainly
expected to be greater than others

The cloud nucleating potential of aerosols is typically modelled using
Köhler theory

Over the last couple of decades the importance of complex organic aerosols in
determining the activation point has been widely acknowledged

In reality, the transition between aerosol gas and liquid phases is not
stepwise, i.e. the density profile is continuous. To calculate the influence
of bulk–surface partitioning organics,

One method of evaluating predictions made by Köhler theory is to carry
out an aerosol–CCN closure study. Closure is achieved when predicted CCN
concentrations are within the uncertainty bounds of observations typically
collected from CCN counters (CCNCs) at a given supersaturation. Numerous
aerosol–CCN closure studies have been performed to varying degrees of success

Numerous studies have been conducted to examine the sensitivity of both the
activation size and CCN concentrations with respect to relevant
physicochemical parameters of the aerosol population

One of the first applications of inverse modelling to assess the effects of
parametric uncertainty in aerosol–cloud interactions was undertaken by

An inverse modelling framework not only provides a GSA but also facilitates
the conditioning of parametric uncertainties on measurements and prior
uncertainty ranges. Furthermore, such an approach also provides a method of
diagnosing structural inaccuracies within the considered model. Structural
inaccuracies present themselves as statistically significant discrepancies
between optimised parameter values and their corresponding real-world
observed values. By simultaneously matching model input and output, the
technique also provides a method of parameter estimation for parameters which
are not easily measured in situ on the scale of interest, surface tension for
example. These advantages have led to the use of inverse modelling as a
method of model calibration across a broad range of research subjects

In this study, to the best of the authors' knowledge, an inverse modelling framework for CCN spectra is developed for the first time. To diagnose the sensitivity of an entire CCN spectrum to parameter perturbations in a tangible way, an OF is introduced. The OF provides a scalar metric by which the sensitivity of CCN spectra can be quantified with respect to both individual and multiple parameter perturbations.

Before performing a GSA and parameter optimisation procedure using an
automated search algorithm, it is deemed judicious to first confirm that the
study is well-posed

The primary goal of this study is to build a framework for inverse modelling of CCN spectra using Köhler theory and to test the suitability of automatic search algorithms as a tool for model calibration and GSA. In constructing the framework, qualitative sensitivity information is presented in the form of OF response surfaces for simultaneous perturbations in two parameters. In addition to considerations of environmentally dependent parameter sensitivities, the role of surface-active organic compounds is also explored. The specific questions to be investigated in this study are the following:

Is it possible to simultaneously match model predictions of CCN spectra with the chosen calibration data and correctly calibrate input parameters using an inverse modelling methodology?

Is inverse modelling of CCN spectra to perform a GSA and parameter uncertainty analysis using an MCMC algorithm feasible?

Qualitatively, how susceptible are CCN concentrations, across a range of atmospherically relevant supersaturations, to simultaneous perturbations in aerosol size distribution and Köhler parameters?

Does the bulk–surface partitioning of surface-active organics play an important role in CCN activity over an atmospherically relevant range of supersaturations, and how sensitive are the associated parameters?

The Köhler equation describes the equilibrium saturation vapour pressure
ratio

The water activity term

A brief overview of the theory behind the bulk–surface partitioning
Köhler model developed by

The interface between bulk liquid and gas phases is not infinitely thin as
Gibbs' surface thermodynamics would suggest

Solving Eqs. (

With the partitioning described by Eqs. (

Modelling CCN concentrations with Köhler theory involves many currently
uncertain parameters, especially with respect to the organic aerosol
fraction. In this study, the Köhler parameters probed in the sensitivity
analysis are

To illustrate the impact of the different partitioning schemes on the CCN
activation point, Fig.

Critical supersaturation

To predict CCN spectra, Köhler theory must be coupled with an aerosol
size distribution. Aerosol size distributions are well represented by a
superposition of log-normal distributions

In order to analyse parameter sensitivity with respect to environmental aerosol characteristics, three distinct log-normal aerosol size distributions are taken from existing literature:

marine average: average global marine measurements from

polluted continental: summertime air mass measurement from the Melpitz station, Germany

rural continental: SMEAR II station, Hyytiälä, Finland

Marine average, rural continental, and polluted continental log-normal aerosol size distributions used to generate calibration data. The distributions are calculated using the true log-normal parameters given in Table 2.

The mixing state of aerosol particles can play an important role in CCN
activation and their optical properties, particularly close to sources of
fresh emissions

Density, molecular weight, and mass fraction of each aerosol
component in all environments. The mass fractions included here are used to
derive true parameter values for

The chemical properties of the MO (Table 1) used in this study are based on
averages calculated from organic acids documented in

Parameter ranges explored in the sensitivity analysis of this study are taken
from literature where possible. Ranges for compositional (

In this section the practicalities of coupling the Köhler model to the
aerosol size distribution in order to calculate the number concentration of
CCN (

Inverse modelling is a methodology often used for finding a set of model input
parameter values that produce model outputs that best represent measurement
data. The optimisation procedure is usually performed using a least squares
or maximum likelihood criterion with respect to some objective function

True parameter values used for calibration data for all environments and their corresponding parameter ranges used for perturbations in the response surface analysis.

The successful application of an inverse modelling approach to any given problem
is reliant on an appropriate definition of both the calibration data,

Real-world measurement data are normally used as calibration data in model
calibration and sensitivity studies. Here, however, synthetic measurements
are numerically generated from the model by using best-estimate parameter
values to represent real-world atmospheric conditions, henceforth referred to
as the “true” input parameter values

CCN spectra calculated from true parameter values (Table 2) for

All 12 sets of calibration data generated from true parameter values for each
partitioning scheme and environment are presented in Fig.

Care should be taken when choosing the functional form of the OF. The
functional description should reflect the characteristics of measurement
errors seen in the relevant observation data set. Common definitions of the
OF include the simple least squares (SLS) or a particular maximum likelihood
estimator. Definitions such as SLS or root mean square error (RMSE) are valid
when the measurement errors are believed to be equal throughout the data set
(homoscedastic) and uncorrelated. More generally, a weighted RMSE definition
can be applied:

To illustrate how the OF behaves in relation to perturbations in a single
parameter, Fig.

Rural continental CCN spectra for partitioning scheme

Typically, studies provide one-at-a-time (OAT) sensitivity analyses of model
outputs, e.g

Consider a fractional perturbation

Global variability in updraft velocities has considerable importance for the
aerosol indirect effect as it can lead to the development of different cloud
types and a range of supersaturations

It is clear that the surface tension

For

Sensitivity curves for marine average CCN concentrations as a
function of supersaturation. Selected parameters are perturbed individually
by 10 % for all partitioning schemes;

A GSA is preferred over traditional OAT analyses as it provides a
comprehensive analysis that spans the entirety of the parameter space

Traditionally, in 2-D sensitivity analyses the surface illustrates the
response in a single model output variable; for example,

Consideration of the behaviour of the OF in 2-D planes of the full parameter
space is also instructive for testing whether aerosol–CCN closure is an
appropriate problem for an investigation using inverse methods. While the
response surfaces only suggest how the OF may evolve when traversing the full
parameter space, if the surfaces do not show a single well-defined minimum, then it may certainly be expected that inverse parameter optimisation may be
unsuccessful

Figure

In what follows, parameter sensitivities for all four partitioning schemes in
the marine average environment are analysed in Sect. 5.2.1–5.2.4 before
considering environmental dependencies in Sect. 5.2.5. The focus is on the
marine environment due to the extensive spatial coverage, a high surface–cloud
albedo contrast

Parameters of interest are perturbed across ranges of values that reflect
uncertainties found in existing observations that include both laboratory and
in situ measurements. These ranges are documented in Table 2 and discussed in
Sect. 3.1. Blue crosses indicate the true parameter values and collectively
correspond to the full true parameter set

Figure

Response surfaces were recalculated for the inclusion of bulk–surface
partitioning effects in the Raoult term

Response surfaces for

The sensitivity to perturbations in solution ideality is shown in
Fig.

Response surfaces for

Here, the effects of a concentration-dependent surface tension,
Eq. (

Response surfaces for this partitioning scheme are shown in
Fig.

Response surfaces for

Here the full partitioning framework is considered. The surface tension is
calculated using the partitioning parameters

Summary of qualitative sensitivities and parameter interactions
observed in response surfaces for all parameters used in the complete
partitioning scheme

Figure

The ability of simple Köhler theory, when the surface tension of water is
used, to approximately replicate the CCN concentrations generated from the
full partitioning treatment is in agreement with existing literature

For the full partitioning scheme considered here, the relative sensitivity of each parameter, and both their linear and non-linear interactions, are summarised in Table 3. Parameters that are indicated to have high or very high sensitivities are good candidates for a future study using automated search algorithms to provide a quantitative GSA and parameter optimisation with an appropriate definition of calibration data (Sect. 5.3).

Response surfaces for

Relative parameter sensitivities were not found to vary a significant amount
between environments, and therefore we have not included response surfaces for
all environments in Sects. 5.2.1–5.2.4. In Fig.

Parameters exhibiting well-defined minima in OF response surfaces can be
considered identifiable. However, if the parameter does not exhibit well-defined minima in several parameter pairs, and in particular if these
surfaces are relatively flat, then automatic search algorithms will likely
struggle to converge on unique parameters values. Response surfaces that are
flat with respect to perturbations in a particular parameter indicate that
such a parameter is insensitive and thus accurate calibration is unnecessary
for the model under consideration. Insensitive parameters can be removed from
the optimisation procedure and replaced with a fixed value. In GCMs
parameters such as surface tension and the hygroscopicity parameter

Figure

In the absence of direct measurements of CCN spectra for real-world
calibration data sets, model predictions of the activation point could be
used to derive a pseudo-synthetic CCN spectrum from an aerosol size
distribution measured with a differential mobility analyser (DMA). DMA
instrumentation can vary substantially in size resolution.
Figure

The importance of information content is particularly evident when
considering the resolution and range of supersaturations spanned by the
calibration data. Multiple definitions of calibration data are shown in
Fig.

Real-world CCN measurements are subject to significant natural variability, which dominates errors associated with instrumentation, such as counting
errors. It is therefore instructive to assess the impact of natural
variability on the appropriateness of a given definition of the calibration
data with respect to resolution. In doing so, an appropriate calibration data
set can be defined for future aerosol–CCN spectra closure and parametric
uncertainty studies using statistically robust Bayesian methods such as MCMC
simulation. In order to represent natural variability, each data point of the
calibration data is corrupted with a synthetic error, and the residuals in
the OF (Eq.

Here the natural variability of the

The feasibility of performing an aerosol–CCN closure analysis when confronted
with real-world observations in an inverse modelling framework can be
assessed by comparing OF response surfaces with and without a synthetic
corruption. The true model (i.e. neglecting natural variability) response
surfaces shown in Fig.

In this study the information content consists of the deviation of model predictions from three synthetic calibration data sets corresponding to different environments. When adding information content in a synthetic study such as the one performed here, it is important to be mindful that such information content could be retrieved from field observations as the end goal is to compare model predictions with an observational data set rather than synthetic measurements.

A methodology that is able to scrutinise the sensitivity of CCN spectra to perturbations in aerosol physicochemical parameters across a range of atmospherically relevant supersaturations has been constructed. The response surface analysis provides a visualisation of pairwise parameter sensitivity while simultaneously confirming aerosol–CCN spectrum closure as a well-posed inverse modelling exercise for appropriately defined calibration data. Across all partitioning schemes and environments, a total of 543 response surfaces were calculated.

In agreement with

For all partitioning cases, model sensitivity to surface tension, solution
ideality, and compositional fractions is on the order of that of the
log-normal aerosol size distribution parameters

At this stage, results show that there are many parameter interactions present in CCN modelling. In addition, it is also clear that log-normal distribution parameters, compositional fractions, surface tension, and solution ideality are all parameters that exhibit high sensitivity, and as a community we must seek to reduce uncertainties in these parameters for effective global climate modelling. Herein it has been demonstrated that inverse modelling of CCN spectra may indeed be an effective methodology for constraining these uncertainties under an appropriate definition for the calibration data. Both the resolution and range of the calibration data are important not only for diagnosing parametric sensitivities but also for a simultaneous minimisation of the OF and correct parameter calibration, i.e. ascertaining the feasibility of conducting an aerosol–CCN closure analysis in an inverse modelling framework. In particular, the constraint of parameter uncertainty using an inverse modelling framework seems challenging when using single-valued definitions of calibration data. Through a thorough consideration of the importance of calibration data resolution and the influence of natural variability, the application of MCMC to perform a GSA and parameter uncertainty analysis seems promising when employing a corrupted high-resolution or uncorrupted CCNC-like synthetic spectral definition for the calibration data, the former of which will form the focus of a future study. As the end goal is to confront the model with real-world observations, this result should serve as a recommendation for the development of instrumentation that can be used in situ to measure CCN spectra at higher resolutions. In the absence of such instrumentation, functional fitting of CCN spectra obtained from current instrumentation can be used.

The raw data used to generate all response surfaces, i.e. the value of the OF (Eq. 19) as a function of the input parameter values over the ranges specified in Table 2, are available on request from the corresponding author, D. G. Partridge (dan.partridge@aces.su.se).

This work was supported by the UK Natural Environment Research Council grants NE/I020148/1 (Aerosol-Cloud Interactions – A Directed Programme to Reduce Uncertainty in Forcing) and NE/J024252/1 (Global Aerosol Synthesis And Science Project). P. Stier would like to acknowledge funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) ERC project ACCLAIM (grant agreement no. FP7-280025). Edited by: V.-M. Kerminen Reviewed by: two anonymous referees