Optical particle counters (OPCs) are common tools for the in situ measurement of aerosol particle number size distributions. As the actual quantity measured by OPCs is the intensity of light scattered by individual particles, it is necessary to translate the distribution of detected scattering signals into the desired information, i.e., the distribution of particle sizes. A crucial part in this challenge is the modeling of OPC response and the calibration of the instrument – in other words, establishing the relation between instrument-specific particle scattering cross-section and measured signal amplitude. To date, existing methods lack a comprehensive parametrization of OPC response, particularly regarding the instrument-induced broadening of signal amplitude distributions. This deficiency can lead to significant size distribution biases. We introduce an advanced OPC response model including a simple parametrization of the broadening effect and a self-consistent way to evaluate calibration measurements using a Markov chain Monte Carlo (MCMC) method. We further outline how to consistently derive particle number size distributions with realistic uncertainty estimates within this new framework. Based on measurements of particle standards for two OPCs, the Grimm model 1.129 (SkyOPC) and the DMT Passive Cavity Aerosol Spectrometer Probe (PCASP), we demonstrate that residuals between measured and modeled response can be substantially reduced when using the new approach instead of existing methods. More importantly, for the investigated set of measurements only the new approach yields results that conform with the true size distributions within the range of model uncertainty. The presented innovations will help improving the accuracy of OPC-derived size distributions and the assessment of their precision.

The size distribution of aerosol particles is a key property to understand
the impact of aerosols on human health and Earth's climate. To measure
aerosol size distributions, optical particle counters (OPCs) are widely used
in air quality programs and atmospheric studies. However, several studies
directly comparing size distributions from different OPC instruments

Sources of uncertainty for size distributions derived from OPC measurements. The main purpose of this study is to introduce an advanced description of OPC response and to offer improved estimates for the corresponding uncertainties (red-rimmed box).

In the following paper we focus on the central aspect of OPC
response modeling and calibration and present a new approach that

allows for a more accurate description of OPC instrument response and

yields realistic associated uncertainty estimates.

The basic principle behind OPC measurements is that particles passing through
a sampling volume illuminated by a light source – usually a monochromatic
laser – scatter light into a photosensitive detector. The amplitudes of the
detected scattering signal pulses are a function of particle size.
An OPC counts these pulses and typically sorts them into different bins according to their amplitudes.
Therefore the resulting measurement data are histograms of scattering signal amplitudes.
The mathematical problem of retrieving number size distributions from recorded scattering
signal amplitude histograms is of inverse nature and is described by a set of
so-called Fredholm integral equations of the first kind:

Connecting the OPC output, i.e., the particle count histograms, and the
desired information, i.e., particle number size distribution, the kernel
functions are the key aspect of every OPC measurement. Deriving the kernel
functions requires knowledge of the scattering signal amplitude threshold
values defining the bin limits, the instrument-specific relationship between
scattering signal amplitude and particle scattering cross section and the
theoretical relationship between scattering cross section and particle size.
The latter is subject to intrinsic particle properties such as complex
refractive index and shape. For given intrinsic properties the size-dependent
particle scattering cross section

Bridging the gap between theoretical calculations and the instrument output,
i.e., finding the instrument-specific parameters linking

Though

Even if quasi-monotonicity between particle size and (discretized) scattering signal amplitude
can be established for particles of certain intrinsic
properties (e.g., polystyrene latex spheres) by a smart choice of OPC
collecting optics

Due to the involved approximations (e.g., a smoothing of

The instrument response can change over time, e.g., due to degradation of OPC light source intensity, pollution or misalignment of optical elements. Such changes usually do not induce a uniform shift in the apparent size distributions but rather cause a complicated deformation.

No uncertainty estimates are provided for the nominal diameter threshold values. This lack entails an underestimation of size distribution uncertainties.

An example subset of kernel functions for the SkyOPC describing the
probabilities for particle diameters and corresponding scattering
cross sections to be sorted into the predefined OPC scattering signal
amplitude histogram bins, visualized by the different colors. The theoretical
relationship between particle diameter and scattering cross section for non-absorbing PSL
– with a refractive index of

In contrast to the instruments presented in this study, this is an important aspect for open-path OPCs where particle positions with respect to the optics vary considerably.

into a range of possibleIn summary, none of the existing concepts for OPC calibration and bin size
assignment prove completely satisfactory. The simplifications to the inverse
problem of Eq. (

A shortcoming common to all available methods is that they do not consider
the artificial broadening of size distributions in the basic parametrization
of OPC response. One primary cause for the increase in apparent size
distribution width is the nonuniformity of light intensity inside the OPC
sampling volume

Signal broadening can be further enhanced by other effects such
as varying orientation of aspherical particles with respect to the
direction of the incident light

To correct for artificial broadening of size spectra the common procedure
is to define a matrix that contains the probabilities (associated
with the broadening effect) to find a particle of a certain size class
in adjacent size classes in its elements

Therefore, for the inversion of OPC histogram data it is advantageous to
treat signal broadening in a more universal way. In
Sect.

So far, we discussed the inverse nature of the OPC measurement principle,
challenges and shortcomings of the parametrization of basic OPC response
and the artificial broadening of size spectra. An aspect that further
complicates OPC measurements is that, in most situations, the (size-dependent)
optical properties of the aerosol particles are a priori
unknown or at least subject to a considerable degree of uncertainty.
Externally or internally mixed individual particles can be combinations
of different non-homogeneously distributed materials

In order to derive a size distribution uncertainty estimate from
uncertainties in the particle properties, however, most studies follow the
pragmatic approach and report the maximum impact on the size distribution as
a conservative estimate

In this section, we introduce an approach to the parametrization of OPC response that involves instrument-specific signal broadening and overcomes the shortcomings of existing methods. We further propose a way to evaluate calibration measurements and to obtain aerosol particle number size distributions with realistic uncertainty estimates from OPC data.

Let a particle of intrinsic properties

Taking account of signal broadening, the new parametrization allows for
an extension of the classical OPC calibration evaluation approach
that is restricted to the determination of the linear coefficients

Given a set of particle standards with known intrinsic properties and size
distributions, the forward solution of Eq. (

For stable measurement conditions, i.e., constant OPC volumetric sample
flow, the uncertainties of the measured particle counts follow
the Poisson counting statistics. With increasing number of counts,
the relative uncertainty hence decreases with

Comparing measured particle standard histograms with the corresponding model
results for a suitable instrument parameter tuple

For a given instrument parameter tuple

A way to meet the challenge of model parameter probability maximization
under initially unknown model uncertainties is to make use of Bayesian
statistics and Markov chain Monte Carlo (MCMC) methods

Flow chart demonstrating a possible pathway for the retrieval of size distribution information from OPC histogram data within the new framework.

The new instrument parametrization, including instrument-specific signal
broadening and the parameter PDFs resulting from the MCMC-based calibration
evaluation, now permits us to derive size distributions from OPC measurements in
a self-consistent way. Propagating the parameter uncertainties yields
improved estimates for the corresponding size distribution uncertainties.
Figure

Other uncertainty-afflicted instrument properties as, for instance, sample flow rate or size-dependent aspiration efficiency can be randomly sampled in a comparable manner.

In addition, a random relative shift from the chosenApart from a thorough and transparent derivation of size distribution uncertainties, the proposed retrieval method has further advantages. For one, the Monte Carlo sampling enables a one-to-one mapping between each size distribution solution (ensemble member) and the corresponding initial parameter picks, thereby facilitating, for example, parameter sensitivity studies. This can help to identify dominating initial parameter uncertainties and even allow us to confine the initial parameter estimates by comparing the traceable solutions to size distribution results obtained by independent measurements. It should be clarified, however, that the proposed retrieval method alone simply propagates the initial parameter PDFs and cannot provide information on their adequacy; i.e., it cannot judge the value of individual parameter picks. A big advantage of the method is that the solution ensemble itself allows for a simple yet appropriate further propagation of size distribution uncertainties. This is explicitly useful when calculating quantities depending on the size distribution. For instance, PDFs for the effective particle diameter or the aerosol extinction coefficient can easily be (numerically) derived by collecting the individual results obtained for each size distribution solution (ensemble member). In light of all benefits, we recommend using the proposed retrieval method (or equivalent approaches) for every occasion, despite the additional effort involved.

The two central OPCs examined in this study are the Grimm model 1.129
(SkyOPC) and the DMT Airborne Passive Cavity Aerosol Spectrometer Probe
(PCASP-100X with an upgraded signal processing package SPP-200, abbreviated
PCASP hereafter). Both aerosol spectrometers were part of the airborne in
situ instrumentation used in the SALTRACE campaign. The SkyOPCs were operated
inside the cabin of the German Aerospace Center's Falcon research aircraft
behind an isokinetic aerosol inlet, while the PCASP was mounted in one of the
under-wing stations. Detailed descriptions of the instruments can be found in

During SALTRACE and the lab measurements presented here the SkyOPC was
operated in the fast mode for smaller sizes, covering a nominal diameter range
of 0.25 to about 3

As the intensity of scattered light intensity over the PCASP size range covers more than 6 orders of magnitude, the PCASP optical detection system is divided into three amplification stages, called the high gain, mid-gain, and low gain stage.

In order to better study differences between the approaches discussed in Sect.The DMT Ultra-High Sensitivity Aerosol Spectrometer (UHSAS)

The calibration measurements were performed using monodisperse aerosols
of PSL spheres. For the SkyOPC, the data set is
complemented by di(2-ethylhexyl) sebacate (DEHS) aerosol samples.
The complex refractive indices for PSL and DEHS are approximately

For mean particle diameters up to 800 nm the aerosol was additionally
filtered with the aid of a differential mobility analyzer (Grimm Vienna-type
L-DMA, abbreviated DMA hereafter;

In this section, we present the results for the evaluation of the PSL
calibration measurements following the new method proposed in
Sect.

Comparison of modeled relative histograms (colored) and measured
counterparts (gray, hatched) for the SkyOPC and different PSL particle
standards (rows) for different approaches of OPC kernel function
parametrization (columns). The colored histogram bars represent each model's
best estimate and the error bars are the range between the 16 and 84th percentiles
of the corresponding PDFs. Panel

Same as Fig.

The measurements of PSL particle standards, carried out as described in
Sect.

Scatter plots of all modeled relative SkyOPC bin counts for the PSL standards versus their measured counterparts for the three different approaches (rows). The comparisons are shown on linear and logarithmic scales on the left- and right-hand side respectively. The markers represent the model best estimates and the error bars are the range between the 16 and 84th percentiles of the corresponding PDFs. The black lines follow the one-to-one relationship. Significant model underestimations, i.e., vanishingly small model values where non-vanishing bin counts are measured, occur in the two upper rows. The number fraction of significantly underestimated values is noted in the upper left corner of the logarithmic scale plots and the corresponding values are shown with triangular markers in the linear scale plots.

The model histograms for the MFR approach (e.g., Fig.

Total sum of residuals between measured relative bin counts and the corresponding model best estimates including all PSL calibration measurements for the SkyOPC and PCASP. In addition to the absolute residual values (solid bars), the arrows and percentage numbers demonstrate the relative reduction by changing the approach.

The R12 approach allows for the correction of the absolute shifts of the
histogram modes. Nevertheless, instrument-specific signal broadening is still
ignored. The modeled histograms, thus, continue to underestimate the widths
of the actually measured histograms, which is visible in the histogram plots
in Figs.

By introducing a simple parametrization of instrument-specific signal
broadening and a self-consistent way of evaluating OPC calibration
measurements, the new method succeeds in modeling the measured histogram
widths correctly (see Figs.

Figure

Measurements of DEHS samples, as outlined in Sect.

Comparison between modeled and measured SkyOPC (bins 1–15) counting efficiency. The measured mean counting efficiency values are plotted with red diamond markers and their associated 68 % confidence intervals with red error bars. The solid lines represent the model best estimates for the different approaches. The shaded areas correspond to the range between the 16 and 84th percentiles.

Parametrized size distribution retrieval results for two DEHS
samples with mean diameters of 0.4 (upper row, graphs

It should be noted, though, that the standard deviations of the DEHS size distributions used here are quite small. When size distributions become broader the impact of instrument-specific signal broadening on the width of the recorded histograms decreases and, hence, differences between the methods will become less pronounced. Besides, uncertainties in aerosol properties like complex refractive index and shape might be the dominant source of size distribution uncertainty in many situations. However, this example demonstrates that the new method is able to retrieve even narrow size distributions correctly and, hence, to provide access to realistic uncertainty estimates for all situations. The results also imply that even for the same data and OPC instrument, calibrated with the same set of measurements, retrieved size distributions can be contradictory solely due to different instrument response parametrizations and calibration evaluation approaches.

Retrieving aerosol particle number size distributions and associated uncertainties from OPC histogram data is a challenging task. Scattered light intensity (the measurand) generally is a non-monotonic function of particle size (the quantity of interest) and depends also on particle intrinsic properties such as complex refractive index. Besides, due to the non-ideal behavior of real OPCs, measured intensity distributions are artificially broadened. To realistically model OPC response, i.e., to find suitable OPC bin kernel functions defining the probabilities for particle diameters to be sorted into the instrument's discrete scattering signal amplitude bins, is thus a crucial requirement.

We have introduced a new approach to model OPC response and, within this framework, a self-consistent way for the evaluation of calibration measurements. Two OPCs involved in the SALTRACE campaign, the SkyOPC and the PCASP, and measurements of PSL particles have been utilized to compare the new approach with existing concepts. The results lead to the following conclusions.

The manufacturer-provided set of (PSL-equivalent) nominal diameter threshold values for the OPC bin borders should be treated with caution and the resultant size distributions should be considered as rather qualitative measures. Not only can the concept of adjacent continuous bins in diameter space be problematic given the non-monotonic relation between particle size and scattering signal amplitude, but the values are also material-dependent and drifts in size assignment, e.g., due to pollution of OPC optics or light source intensity drifts, can occur over time. We have shown that the corresponding size distributions can significantly deviate from reality, even for the reference material. Furthermore, no uncertainty estimates are provided for the nominal diameter values that could be used to infer instrument-related size distribution uncertainties.

Calibrating the instrument can remove absolute sizing offsets. The
results for a state-of-the-art OPC calibration and response parametrization
approach

By introducing a simple (one parameter) approach to describe this ever-present broadening of size spectra, the new method leads to substantial improvements. Residuals between modeled and measured OPC response are considerably reduced compared to the other methods. The new method further correctly predicts the size dependency of OPC counting efficiency. Most importantly, the measurements are successfully reproduced within the range of model uncertainty.

In the context of the new method we have also outlined a self-consistent way to thoroughly propagate parameter uncertainties and gain realistic size distribution PDFs without avoiding to address the actual inverse problem underlying OPC measurements. Besides the advanced uncertainty assessment, a benefit of the proposed Monte Carlo retrieval procedure is the facilitation of subsequent uncertainty propagation for quantities calculated from the size distribution (e.g., the effective diameter). When this procedure is combined with the new OPC response model, exemplary results for measurements of DEHS samples demonstrate that even narrow size distributions are retrieved correctly. For the conventional method the same retrieval procedure, propagating the corresponding parameter uncertainties, yields larger size distribution uncertainties and significantly overestimated size distribution widths.

In summary, the new method has the following major advantages over
existing concepts for OPC bin size assignment:

The inevitable instrument-specific broadening of measured size spectra is parametrized for the first time, leading to a more accurate modeling of OPC response.

The model parameter PDFs resulting from the evaluation of calibration measurements allow for realistic uncertainty estimates for this response and, as a consequence, provide a basis for proper size distribution uncertainties.

Calibration data shown in this paper are available on request to bernadett.weinzierl@univie.ac.at and walser.adrian@web.de.

The research leading to these results has received funding from the Helmholtz
Association under grant number VH-NG-606
(Helmholtz-Hochschul-Nachwuchsforschergruppe AerCARE) and from the European
Research Council under the European Community's Horizon 2020 research and
innovation framework program/ERC grant agreement number 640458 (A-LIFE). The
SALTRACE campaign was mainly funded by the Helmholtz Association, DLR,
TROPOS, and LMU. We further acknowledge funding from the LMU Munich's
Institutional Strategy LMUexcellent within the framework of the German
Excellence Initiative, and from the European Union through the European
Seventh Framework Programme (FP7 2007-2013) under grant agreement number
607905 (Marie Curie Initial Training Network VERTIGO). We also would like to
thank Matthias Richter (GRIMM Aerosol
Technik GmbH & Co. KG, Ainring, Germany) for fruitful discussions and
information about their instruments. We acknowledge the use of the software
module providing the corner plots by