Introduction
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
e.g., and OPCs with other
sizing methods e.g.,
find significant disagreements and in some cases OPCs show systematic
mis-sizing and artificial broadening of size spectra. This highlights that,
although OPCs allow for a fast assessment of qualitative size information,
the task to gain proper particle number size distributions can be
challenging. One reason for this is the measurement principle itself, as
particle size is only indirectly inferred from scattered light intensity.
This intensity, in general, is a non-monotonic function of particle size and
depends also on particle intrinsic properties, such as complex refractive
index and shape .
Especially for particle sizes that are comparable or larger than the
wavelength of the incident light, the size dependency of scattered intensity
tends to be flat and occasionally ambiguous, so that uncertainties in the
particle's intrinsic properties can introduce large sizing uncertainties
. Another reason lies in the
existing methods for OPC calibration and response parametrization. The
available approaches
e.g.,
are not consistent with each other. Further, they do not allow for a
comprehensive description of instrument response and a satisfactory
quantification of corresponding uncertainties.
Figure summarizes the major sources of uncertainty
adjunct to OPC measurements. They can be divided into the last-mentioned
uncertainty in the instrument response and calibration, the uncertainty in
the particle intrinsic properties and the uncertainty in the measured
concentrations themselves, e.g., arising from counting statistics, eventual
particle losses. In order to allow for inter-comparability between
different OPC instruments and the comparison with other measurement
techniques, it is necessary to correct for systematic errors and to quantify
all uncertainties as good as possible, i.e., to improve OPC data accuracy and
assess its precision .
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.
We discuss the advantages of the new approach against the background
of the prevailing concepts and present its superiority by means of
measurement results for two optical particle counters that were used
during the Saharan Aerosol Long-range TRansport and Aerosol-Cloud-Interaction
Experiment (SALTRACE) . Moreover, we
outline a possible way to obtain adequate uncertainties for OPC size
distributions within the new framework.
Methods
OPC measurement principle
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:
Ni=∫0∞κiDFDdD+ΔNi,
with the number of particles Ni counted in bin i, a term
ΔNi accounting for potential counting errors, the corresponding
kernel function κiD giving the probability
for each particle diameter D to be sorted into bin i and the
number size distribution FD .
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 CscatD with
respect to the incident light and OPC scattering geometry, i.e., the
solid-angle range covered by the detector, can be calculated. In case of an
homogeneous sphere, Mie–Lorenz theory provides an
analytical solution. For more complex particle shapes, complementary
frameworks like the T-matrix method or the
discrete dipole approximation can be applied.
Bridging the gap between theoretical calculations and the instrument output,
i.e., finding the instrument-specific parameters linking
CscatD with the measured scattering signal
amplitude,
is the purpose of an OPC calibration. The set of instrument-specific
parameters resulting from the calibration in combination with scattering
theory allows us to predict the OPC output, i.e., to determine the kernel
functions, for any other material with the given optical properties.
Existing concepts for size assignment and calibration evaluation
Though CscatD and, hence, the scattering signal
amplitude generally are non-monotonic functions of particle size (see
Fig. ), the most popular approach of OPC bin size
assignment is to assume or establish monotonicity in order to simplify
Eq. () by allowing for a one-to-one mapping between particle
diameter and bin threshold values. One way to achieve monotonicity is to
replace the correct CscatD by a smoothed monotonic
approximation . Another
option is to simply merge bins in size regions affected by ambiguities in
CscatD, accepting a reduction in resolution
. Following these concepts OPC manufacturers
usually provide their instruments with a table of predefined (polystyrene
latex (PSL) equivalent) diameter bin threshold values. Mathematically, this means
expressing the kernel functions as sharp, adjacent step functions in diameter
space:
κiD=1forD∈Di,Di+10otherwise=∫0DδD̃-Di-δD̃-Di+1dD̃,
with delta functions at Di and Di+1, i.e., the lower and
upper diameter threshold values of bin i. In doing so, Eq. ()
simplifies to
Ni=∫DiDi+1FDdD
and the size distribution can be directly represented by the measured counts
in a discrete way as
FD=NiDi+1-DiforD∈Di,Di+1.
This simplification, however, has fundamental shortcomings:
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 and/or bin threshold
values, this does not automatically hold for particles of different
intrinsic properties (e.g., different refractive index or shape) .
Due to the involved approximations (e.g., a smoothing of CscatD) nominal manufacturer values
can significantly deviate from reality for certain parts of the instrument
size range. Such deviations are regularly reported .
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.
Some studies stick to the simplified concept of adjacent bins in diameter
space but try to reduce possible sizing deviations.
and use an empirical diameter offset to
uniformly shift the manufacturer threshold values in order to yield best
agreement between measured histogram modes and nominal diameter values of
reference particles. A more universal calibration approach commonly used is
to find the parameters for the linear relationship between measured mean
scattering signal amplitudes and theoretical mean scattering cross sections
for reference particles
. Still assuming a
monotonic CscatD they use the resulting linear fit
parameters, i.e., slope m and intercept c, to derive a size-dependent
scattering signal amplitude UD=m⋅CscatD+c and calculate the bin diameter threshold
values Di from their predefined scattering signal amplitude counterparts
Ui.
presented another way of size assignment
that avoids workarounds for the non-monotonic behavior of
CscatD. Their main new concept is to use the same
calibration parameters m and c and the unmodified/unsmoothed
CscatD to define the kernel functions as diameter
projections of the scattering signal amplitude bins (see
Fig. a):
κiD=1forCscatD∈Cscat,i,Cscat,i+10otherwise=1forCscatD∈Ui-cm,Ui+1-cm0otherwise.
Cscat,i and Cscat,i+1 denote the lower and upper
scattering cross-section threshold values of bin i that are a linear
function of the actual thresholds given by the scattering signal amplitude
values Ui and Ui+1. This means that particle diameters will be
sorted into bin i if CscatD falls within the
limits defined by Ui and Ui+1 scaled with the linear coefficients
m and c. This can be further expressed as
κiD=∫0DδCscatD̃-Ui-cm-δCscatD̃-Ui+1-cmdD̃.
To simplify the inverse problem of Eq. () and, again,
directly gain size distribution information from OPC histogram data, they use
the kernel functions to calculate so-called “perfect” (mean) diameters
Dp,i and widths Wi to characterize all bins:
Wi=∫0∞κiDdD.Dp,i=Wi-1∫0∞D⋅κiDdD
With these values a discrete representation for the size distribution is
given by
FDp,i=NiWi.
The
uncertainties in the calibration parameters m and c are used to derive
instrument-related uncertainties for Dp,i, Wi and,
therewith, the resulting size distribution values
FDp,i. Although this approach supersedes
workarounds for the ambiguities in CscatD, it still
has shortcomings. One conceptual inadequacy is that representing the bins by
their perfect diameter Dp,i is ultimately not appropriate, as
it will only match with the real mean diameter of particles sorted into bin
i in the unrealistic case of a flat size distribution. If, for instance,
the size distribution is (strongly) dropping towards larger particles, the
occurrence of smaller particle diameters is more likely, meaning that the
real mean diameter of particles falling into bin i would be (much) smaller
than Dp,i. As a result, this causes a sizing bias between the
real and calculated size distribution.
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 nr=1.585 at the SkyOPC wavelength of 655 nm –
is represented by the black curve. The upper graph (a) shows an
ideal case without instrumental broadening of size spectra, whereas the lower
graph (b) shows a more realistic case where the effect of signal
broadening is considered. The broadening is parametrized by a constant
relative Gaussian uncertainty on the scattering cross-section bin threshold
values.
offer yet another approach to directly
estimate size distributions from measured histograms. They translate the
range of possible scattering geometries seen by individual
particles into a range of possible
UD∝CscatD, a so-called “Mie
band”. From this Mie band they calculate the a priori probabilities for
discrete (equidistant) particle diameter intervals to contribute to the count
rate of each bin. This approach potentially allows particle diameters to be
sorted into more than one bin, i.e., overlapping OPC bins in diameter space.
According to the derived contribution probabilities they then distribute the
measured bin counts to the discrete diameter intervals. The major shortcoming
of this method is similar to the one discussed above. Even if the a priori
probabilities for two diameter intervals (of equal width) to contribute to a
certain bin's count rate are the same, the true particle abundance in these
intervals will not be the same for the realistic case of a non-flat size
distribution. Therefore, the inverse direction, i.e., to equally distribute
the counted particles to the two intervals, is generally incorrect.
In summary, none of the existing concepts for OPC calibration and bin size
assignment prove completely satisfactory. The simplifications to the inverse
problem of Eq. () and the attempts to directly gain size
information from OPC histogram data are always accompanied by systematic
errors. Further, some approaches, especially the use of the manufacturer-provided set of bin diameter threshold values, do not offer
instrument-related (sizing) uncertainty estimates.
Instrumental broadening of size spectra
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 e.g.,. An inhomogeneity of
incident light intensity leads to differences in scattering signal amplitudes
for particles passing the sampling volume at slightly varying locations. This
means that due to the intrinsic nature of real OPCs, even spherical and
homogeneous identical particles appear to vary in size. So far this signal
broadening has, if at all, been treated separately from the basic instrument
calibration, although it is an instrument-specific property, meaning it is
always present in OPC measurements. Depending on the degree of
instrument-induced signal broadening in relation to the actual size
distribution width, this effect may lead to significant measurement biases
being most pronounced for narrow size distributions (or size distribution
modes). In atmospheric research, such narrow size spectra are, for example, met
during ice residual measurements in (contrail) cirrus
or aerosol chamber experiments
e.g.,.
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 and
coincident count events .
The latter becomes relevant for very high particle concentrations when
average inter-particle distances are not larger than the size of the
sampling volume anymore. In such a case, the probability of erroneously
interpreting the sum of several scattering signals from multiple particles
as a single particle's signal increases. In addition to an artificial
deformation of the size distribution towards larger sizes this entails
an underestimation of total particle number concentration.
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 .
The resulting inverse matrix equation is then solved for the true
size distribution. One disadvantage of such methods based on empirical
matrices is that their elements might not be universally valid, as
for instance the magnitude of broadening that is related to varying
particle orientations depends on the degree of particle asphericity.
Moreover, the number of uncertainty-afflicted parameters becomes quite
large. Assuming an OPC with K bins, the number of parameters required
to describe signal broadening is K2.
Therefore, for the inversion of OPC histogram data it is advantageous to
treat signal broadening in a more universal way. In
Sect. we present a new approach that includes the
instrument-specific part of signal broadening within the basic
parametrization of OPC response. In addition, signal broadening resulting
from different orientations of aspherical particles can be included in the
inversion process via a set of possible size-to-scattering cross-section
relations as outlined in Sect. .
Uncertainty in particle properties
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
e.g.,,
making it difficult to find representative complex refractive indices
for the bulk aerosol. In any case, the quality of OPC-derived size
distributions depends on the quality of information on the optical
particle properties.
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
e.g.,.
However, the size distribution uncertainty induced by the uncertainties in
the particle properties can be substantially size dependent. To yield
improved size distribution uncertainty estimates one needs realistic
estimates for the set of possible size-to-scattering cross-section relations
and a proper way to propagate these estimates (as, for example, outlined in
Sect. ).
New approach
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.
Parametrization of the instrument response
Let a particle of intrinsic properties ϑ (e.g., complex refractive
index) and diameter D have the scattering cross section
Cscat,ϑ with respect to the incident light and OPC
scattering geometry. Usually, the scattering signal amplitude in the detector Uϑ
is assumed to be a linear function of Cscat,ϑ. Hence, for an ideal instrument Uϑ
is completely defined by Cscat,ϑ and the linear coefficients m and c.
However, as
explained in Sect. , instrument-induced signal broadening
causes the single signal amplitude Uϑ to be replaced by a
probability density function (PDF) for a set of possible Uϑ. Under the assumption that the nonuniformity of light
intensity is a primary reason for the broadening, the relative variance of
this PDF is independent of the absolute value of Uϑ. In a
simplified approach, the Uϑ PDF is, thus, approximated by a
Gaussian distribution with a constant relative standard deviation b. This
is equivalent to a “blurring” of the initially sharp OPC scattering
cross-section bin threshold values (resulting from the predefined scattering
signal amplitude bin thresholds) with the same relative standard deviation
b. Replacing the delta functions in Eq. () (i.e.,
Gaussian functions of vanishing standard deviation) by Gaussian functions of
constant relative standard deviation yields the new kernel functions:
κiD∣b,m,c=12πb∫0D1Uiexp-Cscat,ϑD̃-Ui-cm22b2Ui2-1Ui+1exp-Cscat,ϑD̃-Ui+1-cm22b2Ui+12dD̃,
with the new instrument-specific parameter triplet (b,m,c) and the
scattering signal amplitude threshold values Ui and Ui+1 defining
bin i. Figure illustrates the difference in OPC kernel
functions between an ideal instrument that follows
Eq. () and an instrument with a finite relative
Gaussian signal broadening. While the assumption of a Gaussian broadening is
an adequate approximation for the OPCs used in this study, it is possible to
customize the shape of the broadening effect for other OPC types (e.g.,
open-path geometries), if necessary.
Calibration evaluation
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
m and c.
Given a set of particle standards with known intrinsic properties and size
distributions, the forward solution of Eq. () using
Eq. () for the kernel functions yields the model count
histograms, i.e., the parametrized theoretical instrument response:
Mij=∫0∞κiD∣b,m,cFjDdD,
with the model counts Mij for OPC bin i and particle standard
j and the corresponding number size distribution FjD.
With the real measured particle counts Nij the task of a calibration
within the new framework is now to inversely find the values for the
parameters that bring Mij and Nij into best agreement.
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 Nij-1/2. Naturally,
the simplified model will not be able to reproduce the calibration
measurements perfectly because there will be additional deviations
that are not parametrized. Provided sufficiently high numbers of counts
in the course of the sampling, the relative bin count uncertainties
due to Poisson counting statistics will become negligible compared
to these additional deviations. As a consequence, bringing model and
measurement into agreement corresponds to maximizing the probability
of the model counts Mij afflicted with a priori unknown uncertainties
σij, which cover the additional model deviations, to occur
given the measured Nij. To ensure that the modeled instrument
response later agrees with reality within its margin of uncertainty,
it is necessary to find a good representation of the unknown uncertainties
σij and quantify them in the course of the calibration,
too.
Comparing measured particle standard histograms with the corresponding model
results for a suitable instrument parameter tuple b,m,c
reveals that remaining deviations between the two mainly appear as
(nonuniform) small shifts, meaning that compared to the model histograms
some measured histograms are shifted to smaller while others are shifted to
larger scattering signal amplitudes. It thus seems natural to treat these
deviations as a remaining uncertainty of the modeled scattering signal
amplitudes. Apart from the experimental finding, there are also theoretical
explanations for the observed shifts. OPC light source intensity fluctuations
around a temporal average induce time-dependent scattering signal amplitude
variations, causing measured histograms to move up and down slightly with
time. In contrast to the instantaneous signal broadening discussed in
Sect. , these fluctuations act on greater timescales,
leading to histograms shifts that are differently pronounced for samples
recorded with certain time lags. Other possible sources for such
time-dependent shifts are changes in detector sensitivity or signal
background noise. Due to the linear relationship, time-dependent signal
amplitude fluctuations can equivalently be thought of as relative
fluctuations in the scattering cross sections, assuming a fixed light source
intensity. Therefore, we express the model count uncertainties σij
in terms of a relative uncertainty of the theoretical particle scattering
cross sections Cscat,ϑD.
For a given instrument parameter tuple b,m,c the set of model
bin counts Mij results from Eqs. () and
() with the (best estimate) theoretical function
Cscat,ϑD. A relative shift in the
theoretical scattering cross sections corresponding to a multiplication of
Cscat,ϑD by a factor ε≠1
leads to a different set of model bin counts Mij,ε. Assuming
the PDF of the possible relative shifts ε to be a Gaussian
function centered at 1 and having a standard deviation of
σε one can derive the respective PDFs for the model bin
counts Mij,ε. For the sake of convenience and simplicity the
resulting model bin count PDFs can themselves be approximated by Gaussian
PDFs, which is usually an adequate approximation. This leads to the following
expression for the unknown model bin count uncertainties:
σij2=12πσε2∫0∞(Mij,ε-Mij)2exp-ε22σε2dε,
with Mij,ε defined by Eqs. () and
(), replacing Cscat,ϑD
with Cscat,ϑ,εD=ε⋅Cscat,ϑD. In summary, the new calibration
evaluation should yield the set of model parameters
b,m,c,σε composed of the instrument-specific
parameter tuple b,m,c and, according to the above
considerations, the remaining relative uncertainty of the theoretical
scattering cross sections σε.
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 e.g.,.
Following Bayes' theorem the (posterior)
probability P for a set of model bin counts Mij
to occur under a set of measured bin counts Nij
can be expressed as
PMij∣Nij∝PNij∣Mij⋅Pb,m,c,σij,
i.e., the product of the likelihood function determining the probability of
the Nij to occur given the Mij
and the so-called prior probability Pb,m,c,σij,
including all prior knowledge on the model parameters for instance from
physical constraints or invariance considerations
e.g.,. The proportionality factor equating both
sides of Eq. () can be thought of as a normalization constant.
Upon the assumption of Gaussian model bin count PDFs the likelihood function
can be expressed as
PNij∣Mij=∏ij12πσij2exp-Nij-Mij22σij2,
with Mij and σij defined by Eqs. () and
() respectively. MCMC methods allow us to efficiently
sample the model parameter space utilizing the forward solution to the
problem to find the region of maximum probability according to
Eq. (). This way, the PDFs for the instrument parameters
b,m,c and the relative uncertainty of the theoretical particle
scattering cross sections σε are obtained together with
all correlations between the individual parameters. In this study we utilize
the Python-based sampler tool emcee
.
Retrieval of size distributions within the new framework
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 illustrates a possible workflow within the
proposed framework to go from measured OPC count histogram data to PDFs in
size distribution solutions. Similar to what has been proposed by
, the basic idea is to start with random Monte
Carlo samples drawn from the model parameter PDFs and a set of possible
theoretical particle diameter to scattering cross-section relationships
Cscat,ϑD, e.g., given by
the likely range of aerosol particle complex refractive indices, their shape
and orientation. In addition, a random
relative shift from the chosen Cscat,ϑD is
picked according to its potential (time-dependent) systematic deviation,
e.g.,
induced by light source intensity fluctuations. This relative shift is drawn
from the Gaussian PDF N1,σε2
parametrized by σε, which is derived as part of the
calibration evaluation. With the resulting (shifted)
Cscat,ϑ,εD and the instrument
parameter tuple b,m,c, the set of OPC kernel functions
κiD can be calculated following
Eq. (). Given this set of bin kernel functions the aerosol
size distribution FD is adjusted such that the
(uncertainty-weighted) deviations between modeled and measured bin counts are
minimized. The result is then either one best solution for FD
or an ensemble of possible solutions for each iteration, depending on the
respective inversion algorithm see, e.g.,and the references
herein. By repeating this
procedure multiple times one finally acquires a collective solution ensemble,
representing members of the size distribution solution PDF, considering all
uncertainties in instrument-specific parameters and the theoretical diameter
to scattering cross-section relationships. In this work (see
Sect. ) we use a parametrized size distribution and, again,
a MCMC method for the inversion. The corresponding size distribution
parameter solutions obtained for each Monte Carlo iteration are merged into a
final size distribution parameter solution ensemble.
Apart 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.
Results and discussion
In this section, we present the results for the evaluation of the PSL
calibration measurements following the new method proposed in
Sect. . We compare these results with the theoretical
instrument response for nominal manufacturer diameter bin thresholds and
results obtained for the approach of ,
representing a state-of-the-art conventional method. Hereafter, we abbreviate
these approaches as MFR and R12, respectively. We further demonstrate the
impact of method choice on size distribution inversion results for
measurements of DEHS samples.
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 (a) shows the theoretical
instrument response according to the manufacturer-provided set of nominal
diameter threshold values in gold (MFR), panel (b) the results
following the calibration and instrument parametrization approach by
in red brown (R12) and panel (c) the results of the new approach in blue.
Same as Fig. but for the PCASP and a different set of PSL particle standards. Instead
of the PCASP (low gain stage) default binning, a custom high-resolution linear
partitioning is applied here to better highlight the differences between the
approaches.
The measurements of PSL particle standards, carried out as described in
Sect. , are utilized to calibrate the OPCs following
both the new and the R12 approach (introduced in
Sect. ). Figures and
contrast the resulting modeled relative bin count
histograms and the measured relative histograms for the SkyOPC and the PCASP
(low gain stage) respectively. The model histograms are calculated by means
of Eq. () with the well-defined Gaussian PSL size
distributions and the kernel functions given by
Eq. () for the R12 (shown in red brown colors) and
Eq. () for the new approach (shown in blue colors). The
best estimate model histograms, i.e., the model histograms for the maximum
probability model parameter tuple – m,cbest for the
R12 and b,m,c,σεbest for the new
approach – are represented by the color-framed white histogram bars. For the
SkyOPC, additionally the model histograms for the MFR approach following
Eq. () and using the manufacturer-supplied set of
nominal values are displayed in golden colors. The underlying measured
histograms are depicted by the gray bars. For the new and the R12 approach
the parameter PDFs resulting from the evaluation of the calibration
measurements (see Figs. and ) are
sampled using a Monte Carlo method to yield the corresponding PDFs of the
model histogram bin counts that are visualized by error bars spanning the
range between the 16 and 84th percentiles. In each panel of
Figs. and the mean diameter and
standard deviation of the Gaussian PSL size distribution is displayed in the
left upper corner. Figure supplements the SkyOPC
histogram comparisons with scatter plots showing all modeled and measured
relative bin counts for the different approaches. Finally,
Fig. quantitatively compares the total sum of residuals,
∑ijRij,best=∑ijNij-Mij,best,
between the measured N and best estimate model relative bin counts M for
the two instruments and the different approaches. The subscripts i and j
represent the different OPC bins and used particle standards respectively.
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. ,
golden colors) exhibit significant deviations from the underlying measured
histograms. They offer much smaller widths than their measured counterparts.
In addition, absolute offsets between the histogram modes are apparent for
both SkyOPC and PCASP (not shown). Deviations are largest for the SkyOPC
because it was operating under dusty conditions during SALTRACE over a longer
period previous to the presented measurements, presumably causing a pollution
of optical elements. In consequence, the scatter plots for the MFR approach
in the upper row of Fig. show the largest
discrepancy between model and measurements. This becomes also obvious for
both instruments when looking at the total sums of residuals in
Fig. . The residuals for the MFR approach are
substantially enhanced compared to the others.
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. and . Here, and especially
in Fig. , it is also apparent that the R12
approach remains unable to reproduce the measurements within the margins of
model uncertainty for most of the relative bin counts. Particularly for the
smaller relative count values the absence of a parametrization of signal
broadening leads to large model deviations. However, in comparison to the MFR
approach total residuals for the model best estimates are reduced by 25 and
35 % for the SkyOPC and PCASP (low gain stage, default binning)
respectively. Beyond that, an estimate for the model uncertainty is
established.
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. and
rightmost columns). As a result, the total residuals between measured and
modeled relative bin counts for the model's best estimates decrease by 82 and
77 % compared to the MFR approach for the SkyOPC and PCASP (low gain stage,
default binning) respectively. With respect to the R12 approach total
residuals for the SkyOPC, the PCASP default binning and the finer PCASP
custom binning are lowered by 77, 64 and 76 %. Further,
Fig. shows that the new approach proves capable
to correctly reproduce the measured histograms within the margins of model
uncertainty over the complete range of relative bin counts.
Figure shows the SkyOPC counting efficiency curve
obtained by parallel measurements with the UHSAS as a reference counter
during the PSL calibration measurements. These measurements offer another
perspective on the comparison between the two approaches. The mean total
concentration fractions measured by the SkyOPC,
fmsm,jDj=∑iNij,SkyOPC∑kNkj,UHSAS⋅φUHSASφSkyOPC,
are calculated from the respective total number of counts and the volumetric
instrument flow rates φ for each particle standard j and are
depicted by the red diamond markers. The associated 68 % confidence
intervals (approximately corresponding to ± 1 standard deviation) that
result from error propagation involving count rate scatter and instrument
sample flow uncertainties are represented by the red error bars. The modeled
concentration fractions are derived from the bin kernel functions
κi as
fmdlD=∑iκiD
and are visualized by the solid lines for the model best estimates, again in
red brown for the R12 and in blue for the new approach. The shaded areas show
the range between the 16 and 84th percentiles derived from the model
parameter PDFs. The R12 approach predicts a sharp drop-off to smaller
particle diameters in contrast to the measurements. The new approach is able
to correctly model both shape and absolute values of the observed sigmoidal
behavior of the counting efficiency curve.
Measurements of DEHS samples, as outlined in Sect. ,
allow us to test the possible implication of the choice of method for size
distribution inversion results using an independent material. As proposed in
Sect. and illustrated in
Fig. , the inversion of measured OPC histogram data is
based on the parametrization of instrument response, the respective parameter
PDFs derived from the calibration and
Cscat,ϑD for the new material. The use of
DEHS spherical droplets guarantees that this latter relationship is
well-defined for the given scattering geometry as complex refractive index
and shape of the aerosol particles are known, thus adding no further
complexity to the retrieval. Moreover, the size distribution of the filtered
DEHS samples approximately follows a Gaussian distribution, simplifying the
inversion in this case to the determination of the size distribution
parameters, mean diameter μsd and standard deviation
σsd. The inversion algorithm used here to solve
Eq. () for the parametrized size distribution is based on a
MCMC method . To
obtain adequate size distribution parameter PDFs, 10 000 Monte Carlo samples
are drawn from the corresponding instrument parameter PDFs.
Figure shows the inversion results for two DEHS samples
and the two methods, i.e., the R12 approach (red brown) and the new one
(blue). The theoretical (true) size distributions and the corresponding
values for the means and standard deviations are depicted by the red lines
and markers. For both the new and R12 approach the retrieved size
distribution means agree with the theoretical values within their range of
uncertainty, meaning that both methods allow for a correct (mean) sizing.
This finding additionally proves the validity of the used
Cscat,ϑD (for DEHS and the calibration
material PSL). Upon closer inspection the retrieved means tend to slightly
underestimate the true values, which could imply minor deviations between the
true and the used OPC scattering geometry and/or refractive index values for
PSL and DEHS. The parameter PDFs for the size distribution means
μsd are almost identical for the two methods concerning both
PDF median values and widths, i.e., uncertainty ranges. This agreement
disappears for the size distribution standard deviations
σsd. The new method again agrees with the theoretical values
within the range of parameter uncertainty and, hence, successfully predicts
the full shape of the size distribution. The R12 approach attributes the
width of a measured histogram completely to the width of the size
distribution, thus overestimating σsd significantly. For the
examples shown here, the R12 approach overestimates the true values for
σsd by 714 and 302 % with respect to the medians of the
retrieved parameter PDFs. The widths of the σsd parameter
PDFs, i.e., the estimated range of uncertainty in this parameter, also differ
for the two methods. With respect to the distance between the 16 and 84th
percentiles the R12 approach yields 285 and 224 % higher PDF widths than
the new method leading to greater overall uncertainties in the retrieved size
distributions, which are nonetheless unable to encompass the true ones.
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 a1 to
a3) and 0.5 µm (lower row, graphs b1 to
b3). The (normalized) Gaussian size distributions are shown in
graphs (a1) and (b1). For both the R12 and the new approach
the size distribution retrieval PDFs are represented by their diameter-wise
medians (solid lines) and 2nd, 16, 84 and 98th percentiles (shaded areas).
The theoretical (true) size distributions are indicated by the red lines. The
corner plots display the solution PDFs for the size distribution parameters,
i.e., mean and standard deviation for the R12 approach in graphs (a2)
and (b2) and the new approach in graphs (a3) and
(b3). The dashed lines in the 1-D histograms represent the parameter
PDF medians, 16 and 84th percentiles (in µm). The median values and
their distances to the percentiles are noted on top of each histogram. The
2-D correlation plots show the solution scatter (black points) superposed
with color-coded 2-D histograms and smoothed Gaussian contours at 0.5, 1.0,
1.5 and 2σ. The true parameter values are again indicated by the red
lines and markers.
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.
Conclusions
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 exhibit clear improvements
in sizing and, therewith, a reduction in total residuals between modeled
and measured bin histograms. The introduction of instrument parameter
uncertainties that go along with the calibration evaluation allows us
to derive related size distribution uncertainty estimates. However, these
estimates fail to explain the remaining differences between modeled and
measured instrument response for the presented data. The main reason
for this is the absence of a parametrization of instrument-induced
signal broadening. This artificial increase in apparent size distribution
width, which is stronger the narrower the actual size distribution is
compared to the degree of broadening, may involve significant systematic
OPC measurement biases (in atmospheric research) when disregarded.
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.