Polar nephelometers are in situ instruments used to measure the angular
distribution of light scattered by aerosol particles. These types of
measurements contain substantial information about the properties of the
aerosol being probed (e.g. concentrations, sizes, refractive indices, shape
parameters), which can be retrieved through inversion algorithms. The
aerosol property retrieval potential (i.e. information content) of a given
set of measurements depends on the spectral, polarimetric, and angular
characteristics of the polar nephelometer that was used to acquire the measurements. To
explore this issue quantitatively, we applied Bayesian information content
analysis and calculated the metric

Aerosols are condensed-phase particles suspended in the air that are distributed ubiquitously in the atmosphere. Aerosol particles have a broad range of sizes spanning orders of magnitudes, and they possess diverse physical and chemical properties. They affect the global climate either by directly scattering or absorbing solar radiation or by influencing cloud formation processes (Boucher et al., 2013). Aerosol particles are also one of the major components of air pollution and aerosol exposure has been linked to cardiovascular and pulmonary diseases and premature deaths (Cohen et al., 2017; Lelieveld et al., 2015).

Due to the importance of aerosol particles for the global atmospheric system and for public health, a multitude of methods have been developed to measure and characterize these particles with both in situ and remote-sensing instruments. Remote-sensing methods rely on the interaction of aerosols with solar radiation or laser light and typically involve detection of the elastically scattered light. In particular, many instruments have been designed to measure the angular dependence of scattered radiance (radiometry), with optional measurement of its dependence on polarization state (polarimetry). Radiometry and polarimetry are the cornerstones of ground-based and space-borne remote-sensing applications because the resulting measurements contain retrievable information on aerosol microphysical properties, which can be obtained via inversion algorithms. Well-known examples of space-borne remote-sensing instruments include the Moderate Resolution Imaging Spectrometer (MODIS; King et al., 1992), the Multi-angle Imaging SpectroRadiometer (MISR; Lee et al., 2009), and the Polarization and Directionality of Earth's Reflectance (POLDER; Deuzé et al., 2001) instrument. In terms of ground-based polarimetric observations, the prime example is the AEosol RObotic NETwork (AERONET; Holben et al., 1998), which is a coordinated network of sun photometers at more than 600 sites worldwide.

Three design aspects are fundamentally important for aerosol polarimetry instruments: (i) the spectral coverage, i.e. the number of wavelengths at which light scattering is measured, (ii) the polarization measurement capability, and (iii) the number and position of the probed angles. Over the last few decades, technological advancements have led to substantial instrumentation improvements in terms of all three of these design aspects (Dubovik et al., 2019). Parallel to these instrumentation developments, advanced aerosol property retrieval algorithms have been designed to better utilize these more informative polarimetric measurements. For example, GRASP-OPEN (Generalized Retrieval of Aerosol and Surface Properties) is a well-established retrieval algorithm that was designed to take advantage of enhanced polarimetric measurements in order to improve the scope and accuracy of aerosol property retrievals (Dubovik et al., 2011, 2014, 2021).

Polar nephelometers, which are in situ instruments for radiometric and polarimetric aerosol measurements, have a rich history dating back to the 1940s (Waldram, 1945). The main physical quantity measured by polar nephelometers is the phase function (denoted here as PF), which is a measure of angular distribution of scattered light radiance by aerosol particles given non-polarized incident light. A subclass of polar nephelometers are also capable of measuring the polarized phase function (denoted here as PPF), which is the angular distribution of the portion of linearly polarized scattered light given non-polarized incident light. Note that some polar nephelometers are capable of measuring more scattering quantities (e.g. Hu et al., 2021), but in this study we only focus on instruments that measure PF and PPF.

A variety of polar nephelometer designs has been introduced over the years. The first design class of these instruments is the goniometer-type polar nephelometer, which measures angularly resolved light scattering using a rotating detector (e.g. Waldram, 1945). The major disadvantage of these types of instruments is the long sampling time required to measure a full phase function, which severely limits their usability in field applications. A second design class is the multi-detector-type instrument, which enables more rapid measurements by employing detectors at fixed positions to probe PF or PPF at discreet angles simultaneously. Multi-detector nephelometers can have either very basic designs (e.g. four angles and only one wavelength; Li et al., 2019), or more complex designs, such as the instruments introduced by Barkey et al. (2007) and Nakagawa et al. (2016), which measure both PF and PPF at 21 and 17 angles, respectively. A third design class is the laser imaging nephelometer (Dolgos and Martins, 2014; Ahern et al., 2022). The laser imaging nephelometer illuminates aerosol particles using laser light and collects the scattered light on a charged coupled device (CCD). The angularly resolved scattering measurements are then extracted from the image collected by the CCD. An example of a laser imaging nephelometer is the Polarized Imaging Nephelometer introduced by Dolgos and Martins (2014), which measures PF and PPF over 174 angles and at three different wavelengths.

The angularly resolved measurements provided by polar nephelometers enable the retrieval of a large number of aerosol properties via inversion algorithms such as GRASP-OPEN (which can be applied in a versatile manner to both remote-sensing and in situ measurement data: Espinosa et al., 2019; Dubovik et al., 2014, 2021). Furthermore, since these instruments provide measurements that are in principle quite similar to remote-sensing data, they can be used as well-constrained test beds for evaluation and improvement of the inversion algorithms used by the remote-sensing community (Schuster et al., 2019). However, the high dimensionality of polar nephelometer measurements with respect to spectral, polarimetric, and multi-angle measurement options also presents an instrument design challenge. In particular, it raises the following questions. (i) Which types of measurements should be prioritized and optimized in order to maximize the information content concerning the underlying aerosol properties? (ii) Can relatively non-informative measurement types be identified in order to prevent overdesign and complexities that could hamper robustness or cost-effectiveness?

Bayesian information content analysis is a tool that has been used extensively by the remote-sensing community to guide such instrument design challenges by enabling the calculation of quantitative information content metrics for specific instrumentation configurations, settings, and applications (Rodgers, 2000). For example, the framework has been used to demonstrate the importance of polarized and multi-angle scattering measurements in the context of space-borne satellite measurements and ground-based sun photometers (Knobelspiesse and Nag, 2018; Chen et al., 2017; Ding et al., 2016; Xu and Wang, 2015; Ottaviani et al., 2013; Knobelspiesse et al., 2012). However, to the best of our knowledge, the framework has not yet been applied to in situ light scattering measurements in order to tackle the design challenges posed above. As well as being used to assess information content, the Bayesian theoretical framework (Rodgers, 2000) can also be used as an inversion method. In this context, the Bayesian approach can be regarded as a particular case of the more general retrieval method employed in GRASP-OPEN, as discussed by Dubovik et al. (2021), where it is also noted that Bayesian-based retrievals can be hampered by the requirement of directly specifying a priori constraints for all aerosol state parameters.

In this study, we use a Bayesian framework purely as a method for assessing the information content of synthetic measurement data in order to investigate the challenges associated with polar nephelometer instrument design. Specifically, we assess and compare the aerosol property retrieval potential of different polar nephelometer instrument configurations given different target applications and assumed prior knowledge. Furthermore, within the theoretical framework, we use a case study to demonstrate how a well-known optimization algorithm (a reductive greedy algorithm) can be used to determine the optimal placements of the angular sensors in a polar nephelometer.

The paper is structured in the following manner. In Sect. 2, we provide general information on the Bayesian information content framework, while in Sect. 3 we introduce the specific methodological aspects used to construct the information content analysis used in this study. In Sect. 4 we present the results of the analysis, including an overview of the information content of different polar nephelometer instrument designs, an assessment of the value of spectral and polarimetric measurements, an investigation of measurement artefacts associated with scattered light truncation, and the results of the angular sensor placement optimization case study. Finally, we summarize the conclusions of the study in Sect. 5.

In scientific and engineering applications it is typically not possible to
measure quantities of interest directly. Instead, the values of such
quantities must be inferred indirectly from measurements of causally related
intermediary quantities. These types of inference problems are known
generically as inverse problems. Inverse problems can be expressed
mathematically by considering a so-called state vector,

The main objective in an inverse problem is to retrieve a best estimate of
the state vector (

By defining

To quantify the overall information content of a measurement, the total
DOFS for all retrievable state parameters is defined according to the following sum:

It must be stressed that DOFS values depend intrinsically on the assumed a
priori variables and measurement uncertainties when applying Bayesian
inference. Therefore, one must be careful not to overinterpret absolute
DOFS values without considering their full context. These issues can be
partially avoided by using DOFS values in a relative manner, e.g. by ranking different designs with respect to achievable information content. As an
alternative approach to circumventing the influence of assumed prior
knowledge, Alexandrov and Mishchenko (2017) proposed the use of

Summary of the information content metrics used in this study.

In this study, the Bayesian framework described in Sect. 2 was used to investigate the information content and aerosol property retrieval potential of different polar nephelometer measurement configurations. Figure 1 summarizes the implementation of this analysis, while the subsequent subsections explain inputs, calculation steps, and outputs in more detail.

Non-linearity of light scattering as a function of aerosol state parameters is a central aspect of the inverse problem of aerosol polarimetry. One effect is that the information content of a polarimetric measurement also depends on the properties of the aerosol under investigation (besides dependence on instrument features and a priori knowledge of the properties of the aerosol sample). Therefore, our general approach is to investigate different aerosol test cases, e.g. fine- versus coarse-mode aerosol or non-absorbing versus absorbing aerosol.

A schematic overview of the implementation of the Bayesian information content analysis.

The measurement capabilities of polar nephelometers differ in three main
aspects: the spectral characteristics (i.e. the number of measurement
wavelengths

The general angular geometry of the problem is shown in Fig. 2. The polar
scattering angle (

Schemes of scattering measurement geometry:

Probing scattering angles very close to the incident light beam is a general
problem in all types of nephelometers (Moosmüller and Arnott, 2003) due
to, e.g., blockage of this beam or interference between the incident and
scattered light. This leads to truncation of the probed angle range at
extreme values representing the forward and backward directions. The
difference between the smallest measurable scattering angle

We are currently testing and validating a laser imaging nephelometer similar in
design to the instrument by Dolgos and Martins (2014), which suffers from a
gap in measurements near 90

For an instrument covering

Simulated

We considered three different spectral measurement settings:

single-wavelength measurement at 532 nm (1

three-wavelength measurement at 450, 532, and 630 nm (3

four-wavelength measurement at 450, 532, 630, and 1020 nm (4

The measurement error covariance matrix,

For PF and PPF at any given wavelength, we chose measurement errors based on
those reported by Dolgos and Martins (2014) for their laser imaging polar
nephelometer. Specifically, we took the average values of the errors
reported by these authors for aerosols with high and low scattering
coefficient cases. That is,

In the general case of a multi-wavelength configuration (with

As an aerosol model, we generally considered aerosol particles to be
homogeneous spheres with log-normally distributed size distribution modes
and uniform composition within each size mode (however, a small number
of simulations are also performed for non-spherical particles, as explained
below). The assumption of log-normally distributed size modes is very
typical in aerosol science (Seinfeld and Pandis, 2006). Furthermore, it is
common to employ multi-modal log-normal functions to construct volumetric
particle size distributions for a complete aerosol. This is the method
adopted in this study, where Eq. (12) describes the general equation we used
for a multi-modal log-normal volume particle size distribution (VPSD).

We assumed that aerosol particles have uniform complex refractive index
values

Particle shape is one of the more challenging properties to incorporate in forward models of aerosol light scattering. It is commonplace to regard aerosols as spherical particles in both remote-sensing and in situ light scattering applications, since relatively simple forward models can then be constructed based on Mie theory. However, for intrinsically non-spherical particles especially in the coarse mode, such as dust particles, the spherical-aerosol assumption fails to properly capture the scattering properties of aerosol particles, which results in systematic retrieval errors. Due to the lack of analytical solutions for non-spherical particles, the aerosol light scattering has to be computed through computationally expensive numerical methods. Moreover, consideration of complex particle morphologies necessitates the use of additional state parameters to describe non-spherical particles theoretically, and at some point the total number of considered state parameters can become too high relative to the information content of the measurements.

To overcome such complexities while incorporating non-spherical-particle
properties, Dubovik et al. (2006) proposed the use of theoretical spheroid
particles with a fixed shape distribution that is representative of the
shape distribution of Saharan dust particles. Furthermore, the authors
expanded the model to encompass both spherical and non-spherical particles
and defined a single additional aerosol state parameter called the

Due to the non-linear nature of the aerosol light scattering problem, the sensitivity of scattering observations varies over different values of state parameters. Therefore, for gaining proper insights through information content analysis, multiple synthetic aerosol test cases with a variety of state parameters have to be assessed. These synthetic aerosol test cases should emulate generic aerosol models that are expected to be generated in real-world environments.

The majority of our simulations are performed for unimodal, spherical-aerosol test cases (i.e. spherical aerosols that are assumed to lie in a
single size distribution mode). These aerosol test cases represent scenarios
where particles are generated in a laboratory environment using, for
example, size-classification instruments such as an aerodynamic aerosol
classifier (Tavakoli and Olfert, 2013). Two different aerosol materials
were considered for the unimodal cases: diethylhexyl sebacate (DEHS) and
brown carbon (BrC). DEHS is a non-absorbing oil that is used extensively to
generate spherical-aerosol particles and to test optical instruments. BrC is
class of organic particulate matter that is moderately light absorbing at
shorter visible wavelengths and that occurs naturally in the atmosphere
(Laskin et al., 2015). It is possible to nebulize surrogate BrC solutions in
a laboratory to generate spherical BrC particles. For simplicity, we refer
to the aerosol test cases with DEHS and BrC aerosols as non-absorbing- and
absorbing-aerosol test cases, respectively. The refractive index values that
were used for the non-absorbing- and absorbing-aerosol test cases are
presented in Table 3. For non-absorbing DEHS, we used the

State parameters (

Aerosol particles can take a very broad range of different sizes and can
therefore exhibit very distinct light scattering properties. Aerosol
particles with radii below 0.5

Ambient aerosol particles in the atmosphere typically have more complex size
distributions containing multiple modes. One common simplification in
aerosol remote sensing is to use a bimodal VPSD to describe ambient aerosol
size distributions, which is typically a reasonable compromise between
maintaining a sufficient level of complexity while keeping the length of the
state parameter vector reasonably small. We employed this bimodal method
here in order to investigate information content for more atmospherically
relevant aerosol test cases. We defined one mode with a median radius below
0.5

To construct the bimodal aerosol test cases we used size distribution and
refractive index data from a study conducted by Espinosa et al. (2019). In
that study, aerosol particles were measured over wide regions of the
continental USA by an aircraft-borne laser imaging nephelometer. Aerosol
state parameters (size distribution, refractive index, Sph%) were then retrieved
from the measurements using GRASP-OPEN. The authors reported seven
distinctive classes of aerosol particles. Amongst these classes we selected
the Urban and Colorado (CO) Storm (hereafter referred to as “dust”) cases
for our study, since these two cases represented the extreme values in terms
of the fine-to-coarse-mode volume concentrations and the parameter based on the fraction of
non-spherical particles. Table 4 shows the specific state
parameters that were used for the bimodal aerosol test cases. Since in the
original study only single

State parameters (

The majority of the results reported in Sect. 4 of the present paper correspond to the unimodal-aerosol test cases. Bimodal test case results are only presented in Sect. 4.5, where the focus is only on insights that go beyond those obtained from the unimodal analyses.

The a priori covariance matrix

The uncertainties (

We set the non-diagonal elements of

To simulate the polarimetric measurement data (PF (

In this part of the study, we investigated to what extent particular choices
of sensor placement affect the information content of polarimetric
measurements. Building up on this, we introduced a method to identify
optimal sensor placement for exemplary target aerosol states and instrument
configurations. We constrained this analysis to unimodal, non-absorbing-aerosol models measured with single-wavelength instruments with either
radiometric or polarimetric capabilities. A total of

Monte Carlo simulations were first conducted to randomly generate different
angular sensor placements. Specifically, 1000 random draws of

Here we assess how the information content increases with an increasing number of angular data points for the angular configurations of four polar nephelometer designs previously reported in the literature (Li et al., 2019; Nakagawa et al., 2016; Dolgos and Martins, 2014; Dick et al., 2007). For each of these angular configurations, different spectral and polarimetric capabilities are considered (specifically, all four combinations of data sets that would be obtained with single/triple wavelength and radiometric/polarimetric measurements). The metric nDOFS (Table 1) was calculated for the fine, unimodal, non-absorbing-aerosol test case using the atmospherically based a priori covariance matrix given in Table 5.

Variation in nDOFS for spectral polar, polarimetric, and angular configurations. The angular configurations are based on instruments available in the literature. The dotted line corresponds to nDOFS equalling 0.7 and is included for reference.

Figure 3 displays the comparison of the information content of data sets acquired by the above instruments. As expected, the nDOFS (ordinate) values generally increase with an increasing number of angular sensors (abscissa). However, this increase is not strictly monotonic. Specifically, data from the four-sensor instrument has comparable or even slightly higher nDOFS compared to the seven-sensor instrument depending on which spectral/polarimetric capabilities are considered. This suggests that implementing more sensors on its own is not necessarily enough to achieve more informative data and that the particular placement of the sensors may also be important. Comparable performance of the four- and seven-sensor instruments for the test aerosol case assessed here should not be interpreted in terms of better or poorer instrument design. Instead, the four-sensor instrument may be optimized for probing aerosols similar to this specific test aerosol case (unimodal, non-absorbing, fine-mode aerosol), whereas the seven-sensor instrument may be optimized for different target aerosol properties. The significance of sensor placement for different test aerosol properties and variable sensor number is explored in more detail in Sect. 4.8.

Figure 3 also shows that the addition of spectral and/or PPF measurement capabilities can result in DOFS increases that are comparable to the increases obtained by simply adding more angular sensors. For example, it can be seen that for this aerosol test case, the addition of polarization and/or multi-wavelength measurements will make the nDOFS values of instruments with four and seven angular sensors roughly equivalent to those of a single-wavelength instrument with 21 sensors without polarimetric measurement capabilities. In the following sections the effect of polarimetric, spectral, and angular measurement on the information content of a variety of retrievable state parameters is explored in more detail.

DOFS

Polarimetry may be applied to a wide range of different aerosol types and in
applications with different prior knowledge of the aerosol under
investigation. Qualitatively speaking, the more prior knowledge is available
on the state parameters of a certain aerosol, the more challenging it
becomes to gain additional information through probing this aerosol with a
polarimetric measurement. The DOFS-based information content metrics as
introduced in Sect. 2 are one approach to quantify differences in
information gain as a function of prior knowledge, instrument configuration,
measurement error, and reference state of the probed aerosol (Fig. 1). Here
we focus on the effect of prior knowledge of DOFS, which comes in through the a
priori variance matrix (Eqs. 3 and 4), and how to interpret and compare
resulting DOFS values. For this purpose we choose two distinct variants of a
priori knowledge as detailed in Sect. 3.4. As for the measurement
capabilities, we computed DOFS

To address the measurement of largely unconstrained atmospheric aerosols,
DOFS

The main reason that the DOFS

Results from Fig. 4 further suggest that the DOFS

Using low percentages of the test case state parameter values as
corresponding a priori ranges sets tighter bonds on the a priori knowledge
compared to the wider atmospheric a priori range. This results in lower
information content of identical measurements when expressed with
DOFS

Figure 5 is equivalent to Fig. 4 but shows DOFS

DOFS

Figure 5 also indicates that when the a priori variances are chosen as fixed
percentage of the corresponding fine and coarse

In Sect. 4.2 we explained that DOFS

Figure 5 indicates that increasing the number of measurement wavelengths can
lead to noticeable information gain for all three size distribution
parameters (mean radius, GSD, and volume concentration). For each of these
parameters, DOFS

The results further indicate that although the addition of infrared
wavelength in 4

For the refractive index parameters, Fig. 5 suggests that the addition of
multi-wavelength measurements only weakly increases information content with
respect to the

Comparing the curves with red circles and red crosses in Fig. 5 provides
an insight into the information content benefit of adding PPF measurements with an
absolute error of 0.056 (Eq. 9). This level of absolute PPF measurement error is
based on the laser imaging nephelometer of Dolgos and Martins (2014). For
this case, the addition of PPF results in an only minor DOFS

In the context of remote-sensing instrumentation, PPF measurement errors can be
considerably lower than the default noise level of 0.056 that is considered
here. For example, Xu and Wang (2015) consider an absolute error of 0.01 for
polarized measurements made by AERONET sun photometers. The blue curves in
Fig. 5 demonstrate the information content benefit of adding PPF measurements
with such a reduced absolute error (e.g. 0.01 rather than 0.056) while
retaining the PF noise level unchanged. This change results in a substantial
information content increase over all state parameters and wavelength
numbers, with the exception of

It should be noted that all of the results shown so far are valid for probing rather simple aerosol systems, i.e. unimodal size distributions of spherical particles with homogeneous optical properties. For more complex aerosols the information content benefit associated with the addition of polarimetric measurements can be even higher, as we demonstrate below in Sect. 4.5.

Light absorbing aerosols are found ubiquitously throughout the atmosphere.
Retrieving information on light absorption by an aerosol from polarimetric
light scattering data is known to be difficult and can require the use of
additional independent measurements as constraints (e.g. Espinosa et al.,
2019). Nevertheless, the results shown in Fig. 5 for DOFS

Figure S5 displays the same results shown in Fig. 5 but for the absorbing-aerosol test case. Comparing the two figures, it can be seen that the
DOFS

A further result seen in both Figs. 5 and S5 is that there are considerable
systematic differences in DOFS

DOFS

To explore these findings in more detail, we further assess the sensitivity
of phase function to

Although

The results presented here demonstrate that polarimetric measurements can be
very informative on

The more complex an aerosol, the more state parameters are required to
describe it. Retrieving the state parameters of more complex aerosols from
polarimetric measurements will, accordingly, require data with overall
higher information content. This may amplify the benefit of performing
measurements with instruments that have more comprehensive capabilities or
smaller measurement errors. To address this hypothesis, we used the bimodal
non-spherical-aerosol model with two test cases based on the results from
Espinosa et al. (2019). These two cases represent “urban” and
“dust-dominated” aerosols measured over the USA. The DOFS values are calculated
using the atmospherically based a priori variance assumptions and the default
noise level for PPF (i.e. 0.056). The results are shown in Fig. 6. It should be
reiterated that for these analyses the a priori variances assumed for coarse-
and fine-size distribution parameters are not identical. For example, the a
priori variance for a coarse median radius is

DOFS

The DOFS

The results further show strong DOFS

As described in Sect. 3.1, angular truncation in nephelometry refers to
the inability of measuring light scattering at certain angles due to physical
design limitations. To investigate the effect of truncation on aerosol
property retrieval potential, we used two simplified aerosol tests cases
(fine and coarse unimodal, non-absorbing, spherical particles) and two
measurement configurations: a basic instrument (

Figure 7 demonstrates the effect of extreme-angle truncation

Variation in DOFS

In contrast to the fine test case, Fig. 7 indicates that DOFS

This information content analysis of the extreme-angle truncation effect has implications for polar nephelometer design. For example, if an instrument is being designed to measure both fine and coarse aerosols, then minimizing extreme-angle truncation as much as technically possible is a beneficial aim. However, if an instrument is being designed to measure only simple, fine-aerosol particles (i.e. without complex shapes), then one should aim to include multi-wavelength and polarimetric measurement capabilities rather than expending considerable design effort on minimizing extreme-angle truncation.

As explained in Sect. 3.1.2, some polar nephelometers suffer from side
angle truncation (i.e. the loss of scattering information at scattering
angles around 90

Variation in DOFS

Variation in DOFS

The number of angular measurements,

Figure 9 displays DOFS

Finally, it should be noted that for some of the instrument configurations
considered in this

In the previous Sect. 4.7,

Figure 10a and b show the results of the Monte Carlo simulations as grey
markers for both the fine and coarse test cases, respectively. At a given

Panels

While the Monte Carlo method suggests a brute force approach to the task of
identifying optimal detection angle placements, we also present here a proof
of concept analysis showing how a conventional optimization method could be
applied to this problem. For this purpose we employ the reductive greedy
algorithm described in Sect. 3.5. The nDOFS values corresponding to the
detection angle placements determined by the greedy algorithm are also
displayed in Fig. 10. The algorithm was applied separately to the fine-mode
(red open markers) and coarse-mode (blue open markers) test cases. To place
these results in further context, the corresponding nDOFS values for
equidistantly placed detection angles are also displayed in Fig. 10a and b as purple open markers. The greedy algorithm performs excellently for
both the fine and coarse test cases: it is able to identify detection angle
placements that produce nDOFS values that are greater than the corresponding
values for equidistantly placed detection angles and that are similar to
the upper bounds of Monte Carlo-simulated configurations, regardless of the
given

Panels

The question then arises as to how the optimal detection angle locations
determined for the fine-mode aerosol test case compare with those determined
for the coarse-mode case. Figure 11a and b display these optimal detection
angle locations for

Given the differences in the optimal detection angle locations for the fine- and coarse-mode aerosol test cases, it can be expected that using the optimally determined placements for one aerosol test case will produce suboptimal results when applied to the other aerosol test case. Figure 10a and b indicate that this is indeed the case. The optimal coarse-mode detection angle placements produce nDOFS values that are substantially lower than the optimal values when applied to the fine-mode aerosol test case (Fig. 10a) and vice versa (Fig. 10b). In both cases, even the equidistantly placed detection angle configuration yields higher nDOFS values than using the optimal detection angle placement of the wrong aerosol test case.

To mitigate this problem, one might attempt to combine the optimal detection
angle placements for both cases to create an instrument that has similarly
high information content with respect to the measurement of both fine- and
coarse-mode particles. To investigate this approach we created combined
optimal configurations at each value of

In this section we have used a case study to demonstrate that DOFS is a useful
metric for optimizing the detection angle placement in a polar nephelometer.
Furthermore, we have shown that the optimal placements derived for the
measurement of one type of aerosol will not necessarily coincide with the
optimal placements determined for another aerosol type. As a final point, it
should be stressed that these findings are only relevant at low values
of

Polar nephelometer designs can vary greatly in terms of spectral and
polarimetric characteristics, as well as the number of angular measurements,
their possible truncation, and their specific locations. These variations
affect how much information can be retrieved from polar nephelometer
measurements about the value of specific aerosol properties. To quantify
these effects, we conducted a Bayesian sensitivity analysis to calculate an
information content metric, DOFS, for a range of polar nephelometer instrument
configurations, target aerosol cases, and assumed levels of measurement
uncertainty and a priori knowledge. The majority of our analysis was focused
on simulating the measurement of unimodal spherical aerosols described by
three size distribution parameters (VMR, GSD,

To assess the benefit of polarimetric measurements for experiments with high
prior knowledge of aerosol state parameters and to ensure consistent
comparison of the size distribution parameters across the fine and coarse
cases, we additionally employed an a priori covariance matrix which was more
stringent than the corresponding atmospherically based a priori matrix. This led
to a noticeable reduction in DOFS

Comparing absorbing vs. non-absorbing unimodal-aerosol test cases revealed
the unique nature of the state parameter

By considering a more complex bimodal non-spherical-aerosol model, we showed
that conducting more comprehensive spectral and PPF measurements can
substantially improve the information content of different state parameters,
such as

We investigated the dependence of information content on angular truncation.
For truncation in extreme forward and backward direction, DOFS

Finally, as a proof of concept nDOFS was employed as a metric for optimizing
angular sensor placement using a greedy algorithm. It was demonstrated that
for a given aerosol test case and

The results from this study provided insights on how different components involved in a Bayesian-based information content analysis, such as the a priori covariance and aerosol model, could affect the outcome and interpretability of the data. Moreover, the results from this study can help guide the future polar nephelometer designs and improve existing prototypes. Potential follow-up studies could further expand the analysis to include more complex aerosol models (e.g. binned size distributions), to investigate and compare specific measurement wavelengths (i.e. by varying the chosen measurement wavelengths rather than simply their number), and to use more advanced forward models (e.g. models based on the discrete dipole approximation) to simulate non-spherical aerosols such as soot.

The original data contributions presented in this study are included in the
article and attached Supplement. The GRASP-OPEN model used to
perform forward calculations is publicly available on the official GRASP
website (

The supplement related to this article is available online at:

RLM, MG-B, and AM conceptualized the study. AM, TL, AL, DF, and OD developed the software code for performing the simulations. AM curated the data. MG-B acquired the funding and administered the project and together with RLM performed supervisory duties. AM, RLM, and MG-B prepared the paper with contributions from all co-authors.

At least one of the (co-)authors is a member of the editorial board of

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

This research has been supported by MeteoSwiss through a science project in the framework of the Swiss contribution to the global atmosphere watch programme (GAW-CH) as well as the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (BISAR project; grant no. 200021_204823).

This paper was edited by Charles Brock and reviewed by Reed Espinosa and Adam Ahern.