We present a new correction scheme for filter-based absorption photometers based on a constrained two-stream (CTS) radiative transfer model and experimental calibrations. The two-stream model was initialized using experimentally accessible optical parameters of the filter. Experimental calibrations were taken from the literature and from dedicated experiments for the present manuscript. Uncertainties in the model and calibration experiments are discussed and uncertainties for retrieval of absorption coefficients are derived. For single-scattering albedos lower than 0.8, the new CTS method and also other correction schemes suffer from the uncertainty in calibration experiments, with an uncertainty of about 20 % in the absorption coefficient. For high single-scattering albedos, the CTS correction significantly reduces errors. At a single-scattering albedo of about 0.98 the error can be reduced to 30 %, whereas errors using the Bond correction (Bond et al., 1999) are up to 100 %. The correction scheme was tested using data from an independent experiment. The tests confirm the modeled performance of the correction scheme when comparing the CTS method to other established correction methods.

Absorption of solar radiation by particles plays an important role in the
Earth's radiative balance. The sign of the forcing, which determines whether the
particles cause a warming or a cooling of the Earth, strongly depends on the
particles' ability to absorb and scatter radiation. Absorption coefficients
can vary by many orders of magnitude. For instance, Delene and Ogren (2002)
reported yearly averaged values of 0.38 Mm

A recently published review article (Moosmüller et al., 2009) gives an overview of methods for measuring aerosol light absorption. Pros and cons for two categories of methods, filter-based and in situ, are discussed. In this context, in situ denotes methods where the analysis is made while the particles are suspended in air, e.g., photoacoustic photometers, extinction-minus-scattering measurements, and cavity ring-down techniques. In contrast, all filter-based instruments measure particle-related optical parameters after the particles have been deposited on a filter. In situ methods have the advantage of avoiding the contact of particles with the surfaces of the fibers of a filter that might change the absorbing and scattering properties of the particles. Filter-based methods measure transmittance or reflectance, or even both, of the particle-filter system. A few methods use the transmittance as a measure for the particle absorption. These methods suffer from a cross sensitivity to particle scattering (Bond et al., 1999; Weingartner et al., 2003; Virkkula et al., 2005; Arnott et al., 2005; Müller et al., 2011), as explained later in this section, but the instruments are generally less expensive and easier to operate than in situ instruments. Consequently, all of the long-term data sets of aerosol light absorption to date have been obtained with filter-based instruments.

The most widely used fiber-filter-based absorption measurement techniques are developed from the integrating plate method (IPM), although the IPM introduced by Lin et al. (1973) does not make use of fiber filters but instead uses polycarbonate membrane filters. The measurement principle to relate the transmittance to the particle absorption and not to the particle extinction was also used for filter-based absorption photometers, since fiber filters act as angular integrating media through multiple scattering of light inside the filter. There are several instruments based on this idea, e.g., the Aethalometer (Hansen et al., 1984), the Particle Soot Absorption Photometer (PSAP; Radiance Research, Seattle, WA), and the Continuous Soot Monitoring System (COSMOS; Miyazaki et al., 2008). Since particles are embedded in a multiple-scattering medium, corrections have to be developed to derive the particle absorption from measurements of transmittance. Another instrument, the Multi-Angle Absorption Photometer (MAAP; Petzold and Schönlinner, 2004), also measures reflection to correct for a further artifact of particle scattering.

Section 2 gives an overview of often-used measurement systems and correction methods. The two-stream radiative transfer model for relating the particle absorption to the transmittance of the filter is introduced in Sect. 3. Calibration experiments for deriving model parameters are given in Sect. 4. An error analysis of the new correction scheme and comparison to existing correction methods are given in Sects. 5 and 6, respectively.

Filter-based absorption photometers measure the relative transmittance, which
is the decrease in the light transmittance, while sample air is drawn through
a filter and particles are deposited on the filter. A schematic of this
technique is shown in Fig. 1.
Inside a system of filter and deposited particles multiple
scattering of light occurs. Nevertheless, the filter attenuation
coefficient is calculated using an equation
of the form of the Beer–Lambert law, which is not strictly valid
if multiple scattering is non-negligible.
The filter attenuation coefficient

Functional principle of filter-based absorption photometers.

A peculiarity of this method is that more light is absorbed by the deposited
particles compared to particles suspended in air. This can be an advantage
because the sensitivity to absorption is increased but a disadvantage because
the enhancement factor is not constant. This

Another filter-based instrument, the Multi-Angle Absorption Photometer (MAAP, Thermo Electron Corporation), measures the transmittance and the reflectance of the particle-laden filter at two angles, and derives the particle absorption using a radiative transfer model (Petzold et al., 2002, 2005; Petzold and Schönlinner, 2004; Hyvärinen et al., 2013). No additional measurement of aerosol scattering is needed. However, in Petzold et al. (2005) and Müller et al. (2011) a remaining cross sensitivity to scattering (defined as the ratio of apparent absorption and scattering coefficient) in the range of 0–3 % was found.

In recent years several laboratory studies have been conducted to test available
photometer corrections for various aerosols. From Müller et al. (2011) it
can be concluded that correction methods to account for the apparent
absorption need to be revised. It was shown that the cross sensitivity to
particle scattering changes while the filter was loaded with scattering
particles, and the cross sensitivity was higher at larger wavelengths. For
example, when data from PSAP were corrected using the method given in Bond et
al. (1999), the cross sensitivities for wavelengths 467, 531, and 650 nm were
on average 0.3, 0.4, and 0.7 %, respectively. The apparent absorption
became smaller as the filter loading increased, and at a relative
transmittance of 70 % the apparent absorption was about

Since both the enhancement of absorption and the apparent absorption are functions of the transmittance, it is unlikely that a simple correction can be found with the relative transmittance being the only parameter for the loading state of particle-laden filters. In order to explain measured transmittances, a model for particle-loaded filters was developed. The model includes a two-stream radiative transfer model and parameterizations for the apparent absorption and the absorption enhancement. Since the two-stream model is constrained by the parameterizations, the model is called constrained two-stream (CTS) model. The CTS model is basically a forward calculation to simulate optical properties of the filter with known particle loading. An inversion algorithm is presented for deriving the particle absorption coefficient from transmittances, wherein the effects of particle loading are considered using the CTS model. The combination of the inversion algorithm and CTS model is called the CTS algorithm.

Parameterizations for apparent absorption and absorption enhancement were derived from calibration experiments. Experiments with non-absorbing particles led to a new parameterization of the apparent absorption. Parameterizations of the enhancement effect for absorbing particles were taken from Bond et al. (1999) and Virkkula et al. (2005). The CTS algorithm was compared with the widely used corrections given in Bond et al. (1999) and Virkkula et al. (2005), which are referred to as B1999 and V2005 corrections throughout the rest of the manuscript. The CTS algorithm was developed for the PSAP. However, one can adopt the correction to other types of filter-based absorption photometers.

In the following sections the development of the CTS algorithm is explained in detail. A schematic diagram summarizing the main steps from model initialization to application of the model for deriving absorption coefficients is shown in Fig. 8. The scheme is helpful when reading the sections on the development of the CTS algorithm.

The radiative transfer of pristine and particle-loaded filters can be
described by two-stream models (Bohren, 1987; Arnott et al., 2005; Moteki et
al., 2010). In Arnott et al. (2005) such a model was used to derive
absorption coefficients from Aethalometers. That model is based on a
two-layer system, a composite layer of homogeneously distributed particles in
a filter matrix and a particle-free layer of the filter matrix. In Moteki et
al. (2010) a two-stream model for a system of many layers is given. With that
model a more realistic particle concentration profile following from sampling
theory (Lee and Mukund, 2001) can be incorporated into the radiative transfer
calculations. Particle concentration profiles depend on many parameters,
e.g.,
particle size, face velocity (average velocity of aerosol perpendicular to
the filter) and efficiency coefficients for different collection mechanisms.
In Nakayama et al. (2010) it was shown for the PSAP that the absorption
enhancement factor of 0.1

Two-stream models are approximations to estimate the intensities transmitted
through (

Inside the filter, light is scattered and absorbed by deposited particles or
fibers of the filter. We define the extinction optical depth

This means that particle scattering and absorption optical depths in a multiple-scattering environment differ from the optical depths for the same particle population in an airborne state because of interference effects. First, there is an interference because the particles are deposited on fibers and do not scatter light independently. Second, the particles are deposited in a multiple-scattering environment because of the high number of light-scnattering fibers in the vicinity of the individual particles. Then the path length of photons passing through the filter becomes larger and the probability of being absorbed increases. Furthermore the particle concentration in the filter increases with time and particles may interfere among each other. There is no theoretical solution describing such interference interactions that we can include in the radiative transfer model. Fortunately, this model uncertainty is implicitly compensated by the CTS model. The calibration experiments are subject to these interferences, which means that the parameterizations of the apparent absorption and absorption enhancement implicitly contain the interference effects. The CTS correction thereby inherits a compensation for the interference effects.

Scattering conserves energy but changes the direction of the propagation of
light. The particle asymmetry parameter

Therein the particle asymmetry parameter is the average weighted asymmetry
parameter of all particles with

In an optically
thick layer with multiple scattering the transmittance

The value of

The single-scattering albedo

The solution for a two-layer system (subscript 2L) is given in Gorbunov et
al. (2002) to be

For non-absorbing layers with

Filters used for the PSAP are made up of two different layers. Layer 1 is
made of very fine fibers for collecting particles, and layer 2 is a backing
layer for mechanical strength. The relative particle penetration depth

Transmitted and reflected intensities of a filter. The filter
consists of two parts: the fiber layer of thickness L and the backing
layer. Particles are collected homogeneously in the upper part of the fiber
layer (layer 1a) with the relative thickness

The relative optical depth of a particle-loaded, two-layer system is defined
by

The two-stream model is subject to a few assumptions. The parameter

The PSAP uses fiber filters of the type Pallflex E70-2075W (Pall Corp., Ann
Arbor, USA). The solution of the radiative transfer problem requires the
scattering and absorption optical depths and the asymmetry parameter of the
pristine filter, which is a system of two layers. The first
layer, layer 1 in our convention, consists of glass fibers and collects
almost all particles. The second layer, layer 2, or the backing layer, is
important for the mechanical stability but is not effective in terms of
particle collection. In Moteki et al. (2010), absorption and scattering cross
sections and asymmetry parameters of fibers of the same filter type were
calculated by scattering theory using a code for infinite cylinders and
oblique incidence of light (Bohren and Huffman, 1983). Fiber diameters were
estimated from scanning electron images to be 0.5

Setup of measurements inside the integrating sphere. The filter is placed in the center of the sphere and illuminated through the entrance port. Reflected and transmitted light is scattered many times on the walls of the sphere before measured at the exit port. The radiant fluxes are indicated by arrows and explained in the text.

The filter scattering and absorption optical depths were calculated from
measurements of the absorbance using an integrating sphere and from
transmittance and reflectance measured with a polar photometer. Measurements
were done for both the total filter and for the backing layer. For the
latter, the backing layer (layer 2) was isolated by removing the fluffy fiber
layer (layer 1). The filter was fixed in the center of an integrating sphere
(see Fig. 3). The orientation of the filter was the same as for the PSAP. The
filter was illuminated with a spectrally broad UV–VIS light source via an
open input port of the sphere. A spectral photometer at the output port of
the sphere measured the spectral intensity

The value of

The angular intensity

Polar photometer. Left: the filter is illuminated with intensity

Measurements of

The scattering optical depths of the fiber layer at the PSAP wavelengths of
467, 530, and 660 nm are 7.76, 7.69, and 7.34, respectively. The absorption
optical depths for these wavelengths are 0.033, 0.038, and 0.018,
respectively. While the values for scattering differ by only

Overview of parameters of fibrous filters. For comparison, values are given for wavelength 550 nm and for the wavelengths of PSAP.

Uncertainties in the scattering and absorption optical depth were derived from the reproducibility of the experimental results with a set of filters. Systematic errors were estimated to be smaller than 5 %. In total, the uncertainties in both the scattering and absorption filter optical depths are about 10 %.

Modeled relative optical depth.

The relative particle penetration depth is a simplified model of the true
sampling mechanism. According to Moteki et al. (2010), the penetration depth
can greatly influence the sensitivity to absorbing particles. We have chosen
a relative penetration depth of

The relative optical depth of the two-layer system (layers 1a and 1b) is
simulated using Eq. (12) for a range of particle scattering
(0 <

This chapter introduces the concept of the CTS correction scheme.

A two-layer model can be used for modeling the optical properties of
particle-loaded filters qualitatively, but not quantitatively. The basic
concept for using the model quantitatively is to constrain modeled results to
agree with experimental results. The relative optical depth

Equations (18) and (19) are the sensitivities to pure black and white
particles, respectively. For Eq. (18) the asymmetry parameter is not defined
since there is no scattering. In reality, scattering always occurs, so the
definition of

Then the relative optical depth can be written as

A fundamental property of

Mixed-term

The simulation provides a qualitative insight into the radiative transfer of a system of fibers and particles. However, a model-based correction is limited to assumptions made for simplification. First, the model assumes that particles are uniformly deposited within the top layer of the filter, whereas in a real filter the deposition profile would decrease exponentially (cf. Moteki et al., 2010). Second, the scattering of light by particles and fibers is assumed to be independent, which means that no interference between particle and fiber scattering occurs. Another model simplification is the assumption of a perfectly diffuse illumination. In addition, there might be more artifacts that are not considered in the model. Based on these considerations, an experimental calibration of the theoretical model is indispensable.

From calibration experiments (superscript exp), one can calculate the
sensitivities

The concept for constraining the two-stream model is to combine sensitivity
functions for black and white particles derived from experiments and the
modeled mixed-term function for grey particles. The relative optical depth
then can be written as

The meaning of Eq. (23) is that the model is bound to experimental
calibrations for pure black and pure white aerosols, and the mixed-term
function can be interpreted as a modeled interpolation for cases when both
absorption and scattering occur. The parameters describing the physical
properties of the filter are solely used in the modeled mixed-term function.
In the following, Eq. (23) is abbreviated as

It is assumed that the sensitivity functions from calibration experiments for absorbing and scattering particles implicitly include filter sampling artifacts. The radiative transfer model is not able to handle these artifacts, but the CTS algorithm inherently compensates for sampling artifacts from the experimental calibration corrections.

Overview of model implementation, constraints to experimental sensitivity functions, and retrieval of absorption from measurements with absorption photometers and measurements of scattering properties.

For deriving the absorption optical depth from Eq. (24), the particle
scattering optical depth and the particle asymmetry parameter must be known.
In Sect. 4.2 we will show how these parameters are derived from a total and
backscattering integrating nephelometer. For calculating the particle
absorption optical depth

The iteration is stopped when the difference between measured and calculated
relative optical depth

The absorption coefficient is calculated from two consecutive absorption
optical depths by

The basic difference between the CTS correction and the B1999 and V2005
corrections is that the CTS correction first corrects the relative optical
depth for the scattering and filter artifacts. The result is the absorption
optical depth, from which the absorption coefficient is calculated from the
change in the optical depth with time. This is a kind of a

The relative transmittance

Equation (27) also shows the connection between relative and total optical depths.

The ratio of particle absorption coefficient and filter attenuation
coefficient (

For black particles the filter transmission function accounts for the enhancement effect due to scattering of light in the fiber matrix. Filter transmission functions for the PSAP were derived during calibration experiments leading to different correction methods, i.e., the B1999 and V2005 corrections. For convenience, the time dependence is omitted throughout the rest of the manuscript.

Calibration of filter-based absorption photometers necessarily requires a
reference method for measuring particle absorption. In Bond et al. (1999) the
PSAP was calibrated using the difference of extinction minus scattering of
airborne particles (unaltered/not collected on filter or other substrate) as
the reference. The filter transmission function of the B1999 correction
scheme is given by

Filter transmission functions based on “coefficients” are
unhandy since radiative transfer models need optical depths. Equation (30)
can be written in terms of relative optical depths by

During the Reno Aerosol Optics Study (RAOS; Sheridan et al., 2005), different
reference methods with photoacoustic spectrometers, extinction coefficient
minus scattering coefficient, and cavity ring-down instruments were
available. Results from the RAOS experiment revealed agreement between
photoacoustic and extinction-minus-scattering methods of

In Virkkula et al. (2005) and Virkkula (2010), the filter transmission
function is given by

Equation (33) expressed as relative optical depth is

There are obvious differences between the filter transmission functions of the B1999 and V2005 corrections (Fig. 9) that are not yet understood. However, the purpose of this manuscript is not to present a new calibration with black particles but to introduce a new concept based on a radiative transfer model. We will evaluate the sensitivity of the CTS method to these two filter transmission functions.

Comparison of the unconstrained two-stream model and experimental calibrations in B1999 and V2005. Shown are the relative optical depths versus the particle absorption optical depth.

The response of PSAPs to white particles was measured in the laboratory. A
solution of NaCl was nebulized, passed through a cyclone, and dried.
Effective particle sizes and thus particle asymmetry parameters were varied
by changing the operating conditions of the nebulizer. The response to NaCl
particles was measured by two three-wavelength PSAPs and a three-wavelength,
total and backscattering, integrating nephelometer (TSI, model 3563).
Scattering coefficients were corrected for angular non-idealities using the
parameterization given in Anderson and Ogren (1998). The scattering optical
depth was calculated for the PSAPs by

The summation starts with the beginning of loading the filter. Asymmetry
parameters for each wavelength were determined from the measured hemispheric
backscattering fraction using a relation given in Andrews et
al. (2006). Asymmetry parameters

Equation (36) accounts for two particle size effects. Smaller particles usually have smaller asymmetry parameters. Furthermore, smaller particles penetrate deeper into the filter, which influences the transmittance as discussed in the context of a size-dependent absorption enhancement. Thus, there seems to be a coupling between asymmetry parameter and particle penetration depth. No measurements with a particle size spectrometer were done. Thus we can not separate these two effects from the measurements. Nevertheless, because of the coupling, the effect of a size-dependent particle penetration depth is implicitly considered by the particle asymmetry parameter. A detailed analysis of the strength of the coupling is beyond the scope of this manuscript and requires more experiments.

In Fig. 10a and b, two families of curves can be seen. One family occurred for small particles with scattering Ångström exponents of 2.5 and asymmetry parameters between 0.615 and 0.476. The other family of curves is for larger particles with scattering Ångström exponents between 2.0 and 1.0 and asymmetry parameters ranging from 0.684 to 0.613. The relative deviation between fitted and measured curves indicates that the branch with the higher scattering Ångström exponents is not well described by the parameterization since the deviation is up to 20 %. A satisfactory explanation for this behavior was not found. Measurement uncertainties can be excluded, since nephelometers typically have errors smaller than 5 %, and the deviations between the two PSAPs used for these experiments were smaller than 2 %. A possible explanation could be an invalid model assumption that the asymmetry parameter and the scattering optical depth solely describe the scattering artifact. Furthermore the calculation of the asymmetry parameter from the measured backscatter fraction might be inaccurate for particle populations with high scattering Ångström exponents. However, the maximum deviation of about 20 % is much smaller than the 100 % uncertainty in the scattering correction given in Bond et al. (1999). Thus a deeper investigation of this issue was not done.

The error of the scattering parameterization is calculated by the root mean
square of the relative deviation between parameterization and measurement:

Mathematically this is identical to the standard deviation of the relative
deviation. The error is about 9 % for all data and 7 % when excluding
noisy data at low loadings (

The errors of the CTS algorithm due to one or more erroneous input parameters
are investigated by means of error propagation. Generally, if

In this section we investigate the influences of the wavelength dependencies
of the filter scattering and absorption optical depths on the retrieval of
the particle absorption optical depth. Equation (25) is used as the

In this section we investigate the error for predicting the relative optical depth. Discussion of this error is easy since the iterative solver is not needed. Uncertainties influencing the prediction of the relative optical depth (Eq. 24) are uncertainties in the filter parameters and the uncertainties in the particle optical depths.

Relative uncertainties in the relative optical depth caused by
uncertainties in

The filter parameters (and relative uncertainties) are

Relative uncertainties in the relative optical depth caused by
uncertainties in the particle characterizations:

The error in predicted relative optical depths due to uncertainties in the particle optical depths and asymmetry parameters is discussed in the following. Uncertainties in correction schemes for black particles are on the order of 20 % (Bond et al., 1999), including uncertainties in the PSAP and reference instruments. In Virkkula et al. (2005), the uncertainty in the parameterization is given to be about 3 %, whereas uncertainties in the reference absorption were not considered. A comparison of B1999 and V2005 corrections given in Baumgardner et al. (2012) affirmed that the error is of the order of 20 %. Errors of the scattering optical depth result from calibration uncertainties of about 3 % (Heintzenberg et al., 2006) and from uncertainties in the truncation error. For investigating the scattering artifacts we used the truncation correction given in Anderson and Ogren (1998). With this method the error in the scattering coefficient is about 2 % for a wide range of atmospheric aerosols and can be up to 5 % for strongly absorbing particles (Bond et al., 2009). In the following the total uncertainty in the scattering optical depth is assumed to be 7 %. Asymmetry parameters can be estimated from a parameterization of the backscatter fraction measured with nephelometers (Andrews et al., 2006). Fiebig and Ogren (2006) compared asymmetry parameters derived with this method and asymmetry parameters retrieved by an inversion algorithm from data of monitoring stations with aerosols classified as being arctic, continental, and marine. Differences of 3 to 4 % and in extreme cases up to 14 % were attributed to a large part to assumptions made in calculating the asymmetry parameter from the measured backscatter fraction. For further error analysis we used an uncertainty in the asymmetry parameter of 10 %, which lies between the expected and extreme values. The error of the relative optical depth is shown in Fig. 12, for 7 % error of the particle scattering (panel a), for 20 % of the particle absorption (panel b), and for 10 % of the particle asymmetry parameter (panel c).

The total error of the relative optical depth including uncertainties in the filter and particle parameters is shown in Fig. 13a. The same data are plotted as a function of the single-scattering albedo in Fig. 13b. There is no large variation of the uncertainty between single-scattering albedos from 0.2 to 0.95. In this range the error is dominated by the error of the absorption measurement. For single-scattering albedos above 0.95, the errors of the scattering and asymmetry parameters dominate. Comparing Figs. 11 and 12, it can be seen that the model uncertainties are small compared to uncertainties in the calibration measurements.

The error of the retrieved absorption optical depth was derived similarly.
The absorption optical depth is calculated using Eq. (25). The error is
calculated considering all uncertainties, including the uncertainties in
filter parameters, particle scattering optical depth and asymmetry parameter,
and uncertainties in the PSAP calibration. The black particle calibration was
taken from the B1999 correction. The relative uncertainty

Recalculated PSAP absorption coefficients of the RAOS campaign. The left four plots show absorption coefficients for the two CTS corrections and the B1999 and V2005 corrections divided by the reference absorption. In each subplot, median values (open squares) and 25th and 75th percentiles (whiskers) are shown for ranges of single-scattering albedos. The right plot shows the apparent absorption coefficient measured by the three PSAP correction schemes divided by the scattering coefficient for experiments with white particles.

In this chapter we show results of a re-evaluation of data from the RAOS experiment (Sheridan et al., 2005), in which absorption coefficients were measured using a PSAP (Virkkula et al., 2005) and scattering and backscattering were measured with a nephelometer. Scattering coefficients were corrected for the truncation error using the method shown in Anderson and Ogren (1998). Furthermore, the scattering coefficients were adjusted to the wavelengths of the PSAP by use of the scattering Ångström exponents. Asymmetry parameters were derived from the backscatter fraction as shown in Andrews et al. (2006). The reference absorption was determined from the average of photoacoustic spectrometers and extinction minus scattering (Sheridan et al., 2005). Values from the PSAP were corrected using the B1999, V2005, and CTS methods.

Data from several experiments were classified according to the single-scattering albedo. The ranges of single-scattering albedos (number of experimental runs, and the total time in minutes) are 0.98 to 0.97 (7 runs, 332 min), 0.95 to 0.94 (2 runs, 129 min), 0.9 to 0.86 (9, 400 min), 0.84–0.7 (9, 415 min), black (2, 104 min), and white (13 runs, 755 min). Black denotes a range of single-scattering albedos from 0.25 to 0.6, and white stands for a single-scattering albedo of unity. For the CTS correction, black particle loading corrections from both B1999 (Eq. 31) and V2005 (Eq. 34) were used. The corrections are denoted as CTS-B1999 and CTS-V2005, respectively. Ratios of corrected absorption coefficients and the reference absorption were calculated. Figure 15 shows median values and 25th and 75th percentiles for all classes of single-scattering albedos. For experiments with white particles, apparent absorption coefficients divided by the scattering coefficient are shown. It can be seen that the ratios for CTS-V2005 and V2005 scheme are close to unity for black particles and single-scattering albedos between 0.7 and 0.84. This is not surprising since the V2005 correction and accordingly also the black particle correction for CTS-V2005 were derived from the same data set from the RAOS study. For higher single-scattering albedos between 0.95 and 0.98, the ratios are 1.52 for B1999; 1.16 for V2005; and 1.07 and 1.04 for CTS-B1999 and CTS-V2005, respectively. For single-scattering albedos above 0.9, the span between the 25th and 75th percentiles is significantly smaller for the CTS corrections compared to B1999 and V2005. This and the better ratio at high single-scattering albedos can be explained by considering a parameterization for the scattering and asymmetry parameter in the new loading correction. For white particles, the ratios of apparent absorption and scattering for CTS, B1999, and V2005 corrections are 0.0008, 0.015, and 0.005, respectively. The spans of 25th and 75th percentiles for CTS are remarkably smaller compared to B1999 and V2005. The method comparison shows that the concept of CTS significantly reduces uncertainties in the particle absorption at high single-scattering albedos.

A new method, the constrained two-stream (CTS) method, for correcting filter-based absorption photometers was developed. The method is basically applicable to any instrument whose measurement principle is based on the measurement of the light attenuation of a particle-loaded fiber filter. For applying the CTS method, simultaneous measurements of the particle scattering coefficient and asymmetry parameter are needed, which can be derived from total and backscattering nephelometers. In the present paper the CTS method is introduced for PSAPs. However, the method can be implemented for any other filter-based absorption photometer that measures light transmittance, e.g., the Aethalometer, after measuring the optical properties of the filter and the responses to absorbing and non-absorbing particles.

The CTS method is based on a two-stream radiative transfer model. The model was initialized using measured optical properties of the particle filter and bound to experimentally based calibrations. Calibrations for particles with low single-scattering albedos were taken from the literature, whereas a new calibration for reducing artifacts due to particle scattering was developed. A dependence on the particle asymmetry parameter was found and included in the CTS method. Uncertainties in the model were simulated and discussed and the total uncertainty for retrieving particle absorption coefficients was derived. The CTS method shows significantly smaller uncertainties for single-scattering albedos larger than 0.9 compared to the well-established correction by Bond et al. (1999). For example, the uncertainties in the B1999 and CTS corrections at a single-scattering albedo of 0.98 are 100 and 30 %, respectively. The uncertainties in both methods for single-scattering albedos below 0.8 are very similar with values of about 20 %. A comparison of correction methods with data from an independent experiment, the Reno Aerosol Optics Study, confirmed the significant improvements for high single-scattering albedos.

The rather high uncertainties at low single-scattering albedos follow from comparison of results of two calibration experiments that differ by 20 %. A further reduction of uncertainties in filter-based absorption photometers requires better experimental calibrations. Additionally, a size-dependent particle penetration depth, and thus a size-dependent sensitivity, should be included in upcoming correction methods. A coupled model of sampling theory and radiative transfer could be a significant step towards explaining differences between different calibration experiments and reducing uncertainties for filter-based absorption photometers.

The filter transmission function for the B1999 correction without particle
scattering is given by

First, the independent variable

Integration of Eq. (A2) in the interval

The right-hand side may be rewritten as

Substituting

With boundary conditions for an initially unloaded filter

Further reformulation to separate the filter optical depth yields

The filter transmission function for non-scattering particles is given by

The independent variable

Equation (B2) is solved by integration over the interval

The solution of the integral equation is

Rearranging of Eq. (B4) and using the boundary conditions for an initially unloaded filter

The second solution of the quadratic equation does not fulfil the conditions for initially unloaded filters.

We gratefully thank all participants of the RAOS experiment for providing us with data.Edited by: J.-P. Putaud