the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Retrieval of aerosol microphysical and optical properties over land using a multimode approach

### Otto Hasekamp

Polarimeter retrievals can provide detailed and accurate information on
aerosol microphysical and optical properties. The SRON aerosol algorithm is
one of the few retrieval approaches that can fully exploit this information.
The algorithm core is a two-mode retrieval in which effective radius
(*r*_{eff}), effective variance (*v*_{eff}), refractive index,
and column number are retrieved for each mode; the fraction of spheres for the
coarse mode and an aerosol layer height are also retrieved. Further, land and ocean properties
are retrieved simultaneously with the aerosol properties. In this
contribution, we extend the SRON aerosol algorithm by implementing a
multimode approach in which each mode has fixed *r*_{eff} and
*v*_{eff}. In this way the algorithm obtains more flexibility in
describing the aerosol size distribution and avoids the high nonlinear
dependence of the forward model on the aerosol size parameters. Conversely, the approach depends on the choice of the modes.

We compare the performances of multimode retrievals (varying the number of modes from 2 to 10) with those based on the original (parametric) two-mode approach. Experiments with both synthetic measurements and real measurements (PARASOL satellite level-1 data of intensity and polarization) are conducted. The synthetic data experiments show that multimode retrievals are good alternatives to the parametric two-mode approach. It is found that for multimode approaches, with five modes the retrieval results can already be good for most parameters. The real data experiments (validated with AERONET data) show that, for the aerosol optical thickness (AOT), multimode approaches achieve higher accuracy than the parametric two-mode approach. For single scattering albedo (SSA), both approaches have similar performances.

Aerosols such as dust, smoke, sulfate, and volcanic ash affect the Earth's climate by interaction with radiation (direct effect) and by modifying the properties of clouds (indirect effect). In order to reduce the large uncertainties in aerosol direct and indirect effects, satellite remote sensing is of crucial importance (Lee et al., 2009). Satellite data of intensity and polarization (polarized intensity) that observe a ground pixel under multiple viewing angles contain the richest set of information of aerosols in our atmosphere from a passive remote-sensing perspective (Kokhanovsky, 2015). To acquire useful knowledge based on these data, accurate retrievals of aerosols' microphysical and optical properties are essential. Here, aerosol microphysical properties include the particle effective radius, the effective variance, the refractive index, and the particle shape. Aerosol optical properties mainly include the (multispectral) aerosol optical thickness (AOT) and single scattering albedo (SSA). Accuracy requirements for (a subset of) these parameters are listed in Table 1.

There are currently a number of aerosol retrieval algorithms available (Chowdhary et al., 2001; Hasekamp, 2010; Hasekamp and Landgraf, 2007) based on the use of multi-angle and multispectral measurements of intensity and polarization. These algorithms can be divided in two main groups: approaches based on lookup tables (LUTs) and full inversion approaches. Generally speaking, LUT approaches are faster but less accurate than full inversion approaches because LUT approaches choose the best fitting aerosol model from a discrete LUT. Full inversion approaches are more accurate but slower because they require radiative transfer (RT) calculations as part of the retrieval procedure. The LUT algorithms are, for example, the Laboratoire d'optique atmosphérique (LOA) LUT algorithm over ocean (Deuzé et al., 2000), the LOA LUT algorithm over land (Deuzé et al., 2001; Herman et al., 1997), and the SSA LUT algorithm (Waquet et al., 2016). The full inversion algorithms are, for example, the Generalized Retrieval of Aerosol and Surface Properties (GRASP) algorithm (Dubovik et al., 2011), the SRON aerosol algorithm (Di Noia et al., 2017; Hasekamp and Landgraf, 2007; Hasekamp et al., 2011; Stap et al., 2015; Wu et al., 2015, 2016), the Jet Propulsion Laboratory (JPL) algorithm (Xu et al., 2017), the Goddard Institute for Space Studies (GISS) algorithm (Waquet et al., 2009), and the microphysical aerosol properties from polarimetry (MAPP) algorithm (Stamnes et al., 2018). In addition, some additional aerosol retrieval approaches can be found in Sano et al. (2006), Cheng et al. (2011), Masuda et al. (2000), and Lebsock et al. (2007). It should be noted that of the full inversion approaches only the SRON aerosol algorithm and the GRASP algorithm have been applied at a global scale.

In this study, the SRON aerosol algorithm is used, which is a full inversion
retrieval approach with the first guess generated by LUT retrieval. In the
SRON aerosol algorithm, a damped Gauss–Newton iteration method is used to
solve the nonlinear retrieval problem. Phillips–Tikhonov regularization is
used as the regularization method. In the current version of the algorithm,
it is based on a bimodal description of aerosols in fine and coarse modes, both described by a lognormal size distribution. The
parameters that describe these two modes (for each mode *r*_{eff},
*v*_{eff}, refractive index, and column number and for the coarse mode
additionally the fraction of spheres) are retrieved. A similar approach
has been used by Waquet et al. (2009) and
Stamnes et al. (2018). Other algorithms (GRASP, JPL) do not
retrieve size parameters of each mode but instead describe aerosols with a
larger number of modes with fixed size distribution. The column number of
each mode is then a free parameter in the retrieval.

Both approaches have advantages and disadvantages. The bimodal approach may
not be appropriate in situations in which aerosols contain more than two modes.
Also, the retrieval of *r*_{eff} and *v*_{eff} for each mode
makes the inversion problem highly nonlinear and hence more difficult to
solve. Conversely, multimode approaches are expected to depend
strongly on the assumed size distribution of each mode and the total number
of modes used.

The aim of this paper is to compare the bimodal and multimodal approaches for the retrieval of aerosols from multi-angle polarimeter (MAP) data. For this purpose we extend the SRON algorithm with the capability to perform a multimode retrieval. We then compare the approaches for synthetic measurements and for real measurements of POLDER-3 on PARASOL.

This paper is organized as follows. Section 2 introduces the methodologies of the parametric two-mode retrieval and multimode retrievals. Section 3 describes the data sets and retrieval quality measures used in this study. Section 4 contains the synthetic data experiments. The real data experiments of multimode approaches are discussed in Sect. 5. Finally, the last section summarizes and concludes this study.

## 2.1 Parametric two-mode retrieval

In this section, we first describe the methodology of the original SRON aerosol algorithm, which is referred to as a parametric two-mode retrieval. The inversion retrieval approach is aimed to invert a forward model equation:

Here, ** y** is the
measurement vector containing the multispectral and multi-angle polarimetric
measurements of PARASOL.

*e*_{y}represents the measurement error.

**contains parameters to be retrieved, which include aerosol properties and land or ocean properties. The forward model**

*x***F**(

**), which describes the dependence between**

*x***and**

*y***, contains two parts: (1) microphysical properties to optical properties and (2) optical properties to the intensity vector (at the top of the atmosphere) through an atmospheric RT model. Nonspherical aerosols are modeled as a size–shape mixture of randomly oriented spheroids (Hill et al., 1984; Mishchenko et al., 1997). We use the Mie–T matrix–improved geometrical optics database by Dubovik et al. (2006) along with their proposed spheroid aspect ratio distribution for computing optical properties for a mixture of spheroids and spheres. For the RT model we refer to Landgraf et al. (2001), Hasekamp and Landgraf (2002), and Hasekamp and Landgraf (2005).**

*x*In the parametric two-mode retrieval algorithm, the fine and coarse modes (denoted
by superscript “f” or “c”) are characterized by the
effective radius ${r}_{\mathrm{eff}}^{\mathrm{f};\mathrm{c}}$, the effective variance
${v}_{\mathrm{eff}}^{\mathrm{f};\mathrm{c}}$, the real and imaginary part of refractive
index ${m}_{\mathrm{r}}^{\mathrm{f};\mathrm{c}}$ and ${m}_{\mathrm{i}}^{\mathrm{f};\mathrm{c}}$, the
aerosol loading *N*^{f;c}, and the fraction of spheres
${f}_{\mathrm{sphere}}^{\mathrm{f};\mathrm{c}}$. The complex refractive index for each
mode is ${m}^{\mathrm{f};\mathrm{c}}={m}_{\mathrm{r}}^{\mathrm{f};\mathrm{c}}+i{m}_{\mathrm{i}}^{\mathrm{f};\mathrm{c}}$. In the latest SRON aerosol algorithm,
*m*^{f;c} values are not directly retrieved (i.e., not in the state vector
** x**), but constructed using ${m}^{\mathrm{f};\mathrm{c}}\left(\mathit{\lambda}\right)={\sum}_{k=\mathrm{1}}^{{n}_{\mathit{\alpha}}^{\mathrm{f};\mathrm{c}}}{\mathit{\alpha}}_{k}^{\mathrm{f};\mathrm{c}}{m}^{k,\mathrm{f};\mathrm{c}}\left(\mathit{\lambda}\right)$, where the mode component coefficients
${\mathit{\alpha}}_{k}^{\mathrm{f};\mathrm{c}}$ ($\mathrm{0}\le {\mathit{\alpha}}_{k}^{\mathrm{f};\mathrm{c}}\le \mathrm{1}$) are
included in the retrieval state vector.

*m*

^{k,f}for the fine mode (or

*m*

^{k,c}for the coarse mode) is the fixed spectral-dependent complex refractive index spectra for some aerosol components, e.g., dust (DUST), water (H

_{2}O), black carbon (BC), and inorganic matter (INORG). In this study, we set ${n}_{\mathit{\alpha}}^{\mathrm{f};\mathrm{c}}=\mathrm{2}$ and assume that the fine mode and the coarse mode are respectively composed by INORG+BC and DUST+INORG. Note that this assumption is flexible and can be updated according to the information content of the measurement. Also, spectra based on principal component analysis (PCA) can be used like in Wu et al. (2015).

To retrieve the state vector from the satellite measurements, a damped Gauss–Newton iteration method with Phillips–Tikhonov regularization is employed (Hasekamp et al., 2011). The inversion algorithm finds the solution $\widehat{\mathit{x}}$, which solves the minimization–optimization problem,

Here, *x*_{a} is the a priori state vector, **W** is a weighting
matrix, *γ* is a regularization parameter, and **S**_{y} is the
measurement error covariance matrix. The weighting matrix **W**
ensures that all state vector parameters range within the same order of
magnitude (Hasekamp et al., 2011) and can be used to give some parameters
more freedom in the inversion than others (similar to the prior covariance
matrix in optimal estimation methods). Since the forward model
**F**(** x**) is nonlinear with respect to

**, the inversion for Eq. (2) is implemented iteratively. For each iteration step (e.g., step**

*x**n*), we approximate the forward model

**F**(

**) with**

*x*
Here, **K** is the Jacobian matrix
(with ${K}_{ij}=\frac{\partial {F}_{i}}{\partial {x}_{j}}\left({\mathit{x}}_{n}\right)$),
which contains the derivatives
of the forward
model with respect to each variable in the state vector ** x**.

Based on the linear approximation (Eq. 3), the optimization problem (Eq. 2) can be reduced to

where $\stackrel{\mathrm{\u0303}}{\mathbf{K}}={\mathbf{S}}_{y}^{-\frac{\mathrm{1}}{\mathrm{2}}}{\mathbf{KW}}^{\frac{\mathrm{1}}{\mathrm{2}}}$, $\stackrel{\mathrm{\u0303}}{\mathit{x}}={\mathbf{W}}^{-\frac{\mathrm{1}}{\mathrm{2}}}\mathit{x}$, and $\stackrel{\mathrm{\u0303}}{\mathit{y}}={\mathbf{S}}_{y}^{-\frac{\mathrm{1}}{\mathrm{2}}}(\mathit{y}-\mathbf{F}({\mathit{x}}_{n}\left)\right)$. The solution of Eq. (4) refers to Rodgers (2000) and Hasekamp et al. (2011) and is iterated by

with the contribution matrix $\stackrel{\mathrm{\u0303}}{\mathbf{G}}=({\stackrel{\mathrm{\u0303}}{\mathbf{K}}}^{T}\stackrel{\mathrm{\u0303}}{\mathbf{K}}+\mathit{\gamma}\mathbf{I}{)}^{-\mathrm{1}}{\stackrel{\mathrm{\u0303}}{\mathbf{K}}}^{T}$ and the averaging kernel matrix $\stackrel{\mathrm{\u0303}}{\mathbf{A}}=\stackrel{\mathrm{\u0303}}{\mathbf{G}}\stackrel{\mathrm{\u0303}}{\mathbf{K}}$. Λ is a filter factor, which limits the step size for each iteration of the state vector. In this way, we use a Gauss–Newton scheme with reduced step size to avoid diverging retrievals (Hasekamp et al., 2011). The filter factor Λ shows values between 0 and 1.

The regularization parameter *γ* and filter factor Λ in
Eqs. (4) and (5) are chosen optimally
(for each iteration) from different values for *γ* (10 values from 0.1
to 5) and for Λ (10 values from 0.1 to 1) by evaluating the
goodness of fit using a simplified (fast) forward model.

## 2.2 Multimode retrieval

We now introduce the multimode SRON aerosol
retrieval approach. In principle, the idea of the multimode approach is that
instead of fitting the size distribution parameters (*r*_{eff} and
*v*_{eff}) of two modes, one aims to fit the size distribution with a
larger number of modes for which *r*_{eff} and *v*_{eff} are
fixed. An expected advantage of this approach is that it makes the inversion
problem more linear (*r*_{eff} and *v*_{eff} tend to make the
inversion problem highly nonlinear). Furthermore, the multimode approach
has more freedom in fitting different shapes of size distribution if the
number of chosen modes is sufficiently large. Conversely, the
multimode approach is expected to depend strongly on the assumed modes.

The performance of the multimode approach is expected to be better and better as the
mode number increases. In this study, we take the 10-mode retrieval as the
maximum mode number retrieval. All the multimode retrieval cases are defined
as in Table 2. For example, for the five-mode retrieval
case, the five modes used for retrieval are actually modes 2, 4, 6, 7, and 9 in the
10-mode retrieval. Here, the five modes correspond to those of
Xu et al. (2017). The abbreviations for different retrieval cases used
in this study are listed in Table 3, in which the parametric
retrieval is denoted with a superscript “p”, i.e.,
2modeRetr^{p}.

For multimode retrievals, the state vector ** x** in Eqs. (1)
and (2) is different from that in the parametric two-mode
retrieval. The difference is shown in Table 4,
which specifies the parameters in the state vector. In the multimode
retrieval, since

*r*

_{eff}and

*v*

_{eff}are not retrieved for all modes, they are not included in

**. The aerosol loading ${N}^{j}(j=\mathrm{1},\mathrm{2},\mathrm{\dots},{n}_{\mathrm{mode}})$ for all modes is retrieved and included in**

*x***. In principle, other aerosol parameters like the refractive index coefficients, the fraction of spheres, and the aerosol layer height can be retrieved for each mode independently. However, the measurement vector will not contain sufficient information to extract this information for each mode separately. Therefore, for the retrieval of these parameters we group the modes into two types – fine (i.e., modes 1–6 of Table 2 for the 10-mode case) and coarse (modes 7–10 of Table 2). For the refractive index coefficients, we fit one value for the fine modes and one value for the coarse modes. For the fraction of spheres, we only retrieve one value for the coarse modes and assume the fine modes consist only of spheres. (A recent study by Liu and Mishchenko, 2018, indicates that this assumption becomes unrealistic for an increasing fraction of carbonaceous aerosol in the fine mode.) For the aerosol layer height we fit one value that is assumed representative for all modes. These assumptions are similar to those in the parametric two-mode retrieval. According to Table 4, the number of aerosol parameters for the parametric two-mode retrieval and the multimode retrieval is respectively ${n}_{\mathit{\alpha}}^{\mathrm{f}}+{n}_{\mathit{\alpha}}^{\mathrm{c}}+\mathrm{8}$ and ${n}_{\mathrm{mode}}+{n}_{\mathit{\alpha}}^{\mathrm{f}}+{n}_{\mathit{\alpha}}^{\mathrm{c}}+\mathrm{2}$, where the fine- and coarse-mode component coefficients ${n}_{\mathit{\alpha}}^{\mathrm{f}}$ and ${n}_{\mathit{\alpha}}^{\mathrm{c}}$ are both set to 2 in this study.**

*x*In addition to the aerosol-related parameters, ** x** in multimode retrievals
also includes surface reflectance and polarization parameters in the same
manner as the parametric two-mode retrieval. For surface models of the
bidirectional reflectance distribution function (BRDF), we use the Ross–Li
model (Li and Strahler, 1992; Ross, 1981) for the same settings
as in Litvinov et al. (2011). For modeling surface bidirectional
polarization distribution function (BPDF), a Fresnel model is used as
introduced by Maignan et al. (2009). The surface parameters, to be
retrieved in the state vector (see Table 4),
are scaling parameters for the BPDF model (${x}_{\mathrm{bpdf}}^{\mathrm{scale}}$),
the coefficient of the Li sparse kernel (${x}_{\mathrm{brdf}}^{\mathrm{geo}\mathrm{1}}$),
the coefficient of the Ross thick kernel (${x}_{\mathrm{brdf}}^{\mathrm{geo}\mathrm{2}}$),
and the BRDF scaling parameters at each wavelength band
(${x}_{\mathrm{brdf}}^{iw}$, $iw=\mathrm{1},\mathrm{2},\mathrm{\dots},{n}_{\mathrm{wave}}$). The number of
surface-related parameters in the state vector for all retrieval cases is

*n*

_{wave}+3. Therefore, the length of the state vector (i.e., the total number of aerosol- and surface-related parameters) is ${n}_{\mathit{\alpha}}^{\mathrm{f}}+{n}_{\mathit{\alpha}}^{\mathrm{c}}+{n}_{\mathrm{wave}}+\mathrm{11}$ for the parametric two-mode retrieval and is ${n}_{\mathrm{mode}}+{n}_{\mathit{\alpha}}^{\mathrm{f}}+{n}_{\mathit{\alpha}}^{\mathrm{c}}+{n}_{\mathrm{wave}}+\mathrm{5}$ for the multimode retrieval.

The inversion procedure of multimode retrievals is the same as described by
Eqs. (3), (4), and (5).
**W** is a diagonal matrix and its diagonal values are shown in
Table 5. Note that the prior information of aerosol loading
(*N*) is provided in terms of AOT.

## 2.3 Multimode retrieval of first guess

In the SRON aerosol algorithm, the first guess of ** x** is obtained
before the full inversion retrieval using a LUT, which is based on
tabulated RT calculations for each of the 10 modes listed in
Table 6 separately. The RT
calculations are performed for different combinations of input parameters (as
specified in Table 6), which are, for example, one single
layer height, one value of the refractive index (different for fine and
coarse modes), nine AOT (

*τ*) values, 15 wavelength bands, seven viewing zenith angles (VZAs), 14 solar zenith angles (SZAs), two surface pressures, two values for the scaling parameter for the BPDF model, three values for the coefficient of the Li sparse kernel, four values for the coefficient of the Ross thick kernel, and seven values for the BRDF scaling parameters at each wavelength band.

The precalculated LUT is used as input for an approximate forward model in
the LUT retrieval. Here, the RT multiple scattering
calculations, performed separately for the different modes, are combined
using the method of Gordon and Wang (1994). Single scattering is
computed exactly as its computational cost is negligible. Using the
approximate forward model, a retrieval is performed using the same inversion
method as for the full retrieval
(Eqs. 3–5). The fit parameters are the
aerosol column numbers of the 10 modes and the surface parameters. The result
of the 10-mode LUT retrieval is also used for full retrievals with fewer than
10 modes (e.g., the parametric two-mode retrieval), by fitting the
*n*_{mode} (*n*_{mode}<10) size distribution to the 10-mode
size distribution coming from the LUT retrieval with the *n*_{mode}
aerosol columns for the different modes as fit parameters.

## 3.1 PARASOL data

The satellite data used in this study for aerosol retrievals are from the Polarization and Directionality of Earth Reflectances-3 (POLDER-3) instrument (Deschamps et al., 1994; Fougnie et al., 2007), which was mounted on the PARASOL satellite (retired in 2013). The POLDER-3 instrument in space provided in-orbit multi-angle and multispectral photopolarimetric measurements of intensity and polarization. The PARASOL level-1 Collection 3 product data have been used in this study.

Each PARASOL image including 242 × 274 elements was made on a
charge-coupled device (CCD)
matrix array over a total view of 114^{∘}. Each ground pixel (6 km × 6 km) is measured under up to 16 angles. The intensity
component (Stokes parameter *I*) was measured at nine bands and the polarization
component (Stokes parameters *Q* and *U*) was measured at 490, 670, and
865 nm. PARASOL has a swath width of about 2400 km. The data from PARASOL
have been used for aerosol retrievals in a number of studies
(Dubovik et al., 2011; Hasekamp et al., 2011; Lacagnina et al., 2015, 2017; Stap et al., 2015). In previous
studies using the SRON aerosol algorithm
(Hasekamp et al., 2011; Lacagnina et al., 2015, 2017; Stap et al., 2015),
four bands (i.e., 490, 670, 865, 1020 nm) were used. In this study, two more
bands (440 and 565 nm) are added for retrievals.

In the SRON aerosol algorithm, we do not directly use *Q* and *U* in the
measurement vector but use the degree of linear polarization (DoLP) as the
polarization component (together with the intensity component *I*) in the
measurement vector. Here, DoLP equals $\frac{\sqrt{{Q}^{\mathrm{2}}+{U}^{\mathrm{2}}}}{I}$. For our
retrievals on PARASOL measurements, we assume an intensity error
*I*_{err}=0.01 and the polarization error
DoLP_{err}=0.007, in the diagonal matrix
${\mathbf{S}}_{y}^{\frac{\mathrm{1}}{\mathrm{2}}}$ in Eq. (4). Here the
intensity error is the relative error, and the polarization error is the
absolute error. *I*_{err}=0.01 holds for all POLDER bands except
for the band 440 nm, where ${I}_{\mathrm{err}}^{\mathrm{440}}$ is set at 0.03
because the intensity measurements at 440 nm are usually considered less
accurate than those at other bands
(Dubovik et al., 2011; Fougnie et al., 2007). Note that in our study
in principle the 0.01 for *I*_{err} and 0.007 for DoLP_{err} used underestimating the PARASOL errors but in our inversion approach only the
relative dependence between intensity errors and DoLP errors is important. The absolute value is compensated for by
the regularization parameter.

It should also be noted that higher-accuracy aerosol retrievals are to be expected for all parameters from instruments that have higher polarimetric accuracy, more scattering angles, and/or more spectral bands (e.g., Hasekamp and Landgraf, 2007; Mishchenko and Travis, 1997). Examples of such improved instruments are GLORY-APS (Mishchenko et al., 2007), MAIA (Diner et al., 2018), SPEXone (Hasekamp et al., 2018), and HARP-2 (Martins et al., 2017).

## 3.2 Meteorological data

During retrievals, some atmospheric and meteorological inputs are needed to be interpolated to each pixel (where there is a PARASOL measurement) at a specified time and a geographical location. The required atmospheric parameters and inputs are humidity, temperature, pressure, and height. In this study, we obtain this information from National Centers for Environmental Prediction (NCEP) reanalysis data (Kalnay et al., 1996).

## 3.3 AERONET data

In this study we focus on aerosol retrievals over land. We validate the retrieved AOT with AERONET (AErosol RObotic NETwork) level 2.0 data (quality assured) of AOT (Holben et al., 2001). The retrieved SSA is validated with AERONET level 1.5 (cloud screened and quality controlled) Almucantar retrieval inversion products (Dubovik et al., 2002) of SSA.

## 3.4 Retrieval measures

In a retrieval, it is a common approach to apply
the goodness of fit (*χ*^{2}) to decide whether the retrievals have
successfully converged. The goodness of fit *χ*^{2} for each pixel is
calculated by

Here, *n*_{meas} is the total number of measurements (multi-angle
and multispectral intensity and polarization) for each pixel. *y*_{i}
represents the measurement (synthetic or real) and *F*_{i} represents the
simulated measurement through the forward model. *S*_{y}(*i*,*i*) is the diagonal
value of the measurement error covariance matrix, corresponding to the
*i*th measurement.

We consider retrievals with ${\mathit{\chi}}^{\mathrm{2}}<{\mathit{\chi}}_{\mathrm{max}}^{\mathrm{2}}$ as valid retrievals.
This filter rejects cases in which the forward model is not able to fit the
measurements, i.e., because of cloud-contaminated pixels
(Stap et al., 2015, 2016), corrupted measurements
(Hasekamp et al., 2011), and cases in which the first guess state vector
deviates too much from the truth. Based on *χ*^{2}, we define the pass rate
${r}_{\mathrm{pass}}=\frac{{n}_{\mathrm{pass}}}{{n}_{\mathrm{pix}}}$ to be the number
of successful pixels (*n*_{pass}) over the number of all pixels
(*n*_{pix}).

To evaluate the retrieved aerosol properties, two measures are used, which
are the RMSE and the bias. The two measures are both
with respect to the differences between the retrieved values and the
reference values (AERONET for real measurements and the truth for synthetic
measurements). Here the difference ${x}_{\mathrm{ipix}}^{\mathrm{diff}}\left[j\right]$ (at the
*i*th pixel (ipix) for the *j*th variable in the state vector
** x**) is computed by ${x}_{\mathrm{ipix}}^{\mathrm{diff}}\left[j\right]={x}_{\mathrm{ipix}}^{\mathrm{retr}}\left[j\right]-{x}_{\mathrm{ipix}}^{\mathrm{true}}\left[j\right]$, where
${x}_{\mathrm{ipix}}^{\mathrm{retr}}$ represents the retrieved aerosol property for
the pixel ipix, while ${x}_{\mathrm{ipix}}^{\mathrm{true}}$ represents
the reference aerosol property.

For each aerosol property, the RMSE counts the overall retrieval errors for all pixels with $\sqrt{\frac{\mathrm{1}}{{n}_{\mathrm{pass}}}{\sum}_{\mathrm{ipix}=\mathrm{1}}^{{n}_{\mathrm{pass}}}\left({x}_{\mathrm{ipix}}^{\mathrm{diff}}\right[j]{)}^{\mathrm{2}}}$. The bias is calculated by $\frac{\mathrm{1}}{{n}_{\mathrm{pass}}}{\sum}_{\mathrm{ipix}=\mathrm{1}}^{{n}_{\mathrm{pass}}}\left({x}_{\mathrm{ipix}}^{\mathrm{diff}}\right[j\left]\right)$. The bias can be positive or negative, meaning the overestimation or the underestimation.

## 4.1 Synthetic measurements

To investigate the capability of multimode retrievals of aerosol microphysical and optical properties, we first perform synthetic data experiments. We can assess the capability of different retrieval setups by comparing the result of the retrieval to the truth that was used to create the synthetic measurement. The synthetic measurements are computed for the PARASOL wavelengths and 14 viewing angles, which is representative for PARASOL (Sect. 3.1).

The synthetic measurements are created pixel by pixel with two steps.
(1) We generate aerosol modes based on assumed true aerosol properties of the
effective radius *r*_{eff}, the effective variance
*v*_{eff}, the fraction of spheres *f*_{sphere}, the
aerosol loading *N*, the mode component coefficients *α*_{k}, and the aerosol
height *z*. In this study, two sets of synthetic measurements are created.
One set is created based on 10 aerosol modes. Each mode has fixed
*r*_{eff}, *v*_{eff} as shown in
Table 2. The other set is two-mode based. For this set,
${r}_{\mathrm{eff}}^{\mathrm{f}}$ and ${r}_{\mathrm{eff}}^{\mathrm{c}}$ are
perturbed within [0.1, 0.3] and [0.65, 3.4], respectively.
${v}_{\mathrm{eff}}^{\mathrm{f}}$ and ${v}_{\mathrm{eff}}^{\mathrm{c}}$ are
perturbed within [0.1, 0.3] and [0.4, 0.6], respectively. (2) Based on the
generated aerosol modes, the forward model as discussed in
Sect. 2.1 is used to generate the synthetic
measurements. The assumed true aerosol properties for each pixel are
generated stochastically.

For synthetic data experiments, we only consider noise-free retrievals; i.e., no noise is added to the generated synthetic measurements. In this way we focus the experiment on errors related to inconsistencies between the synthetic measurement and retrieval (i.e., different modes), and the capability of the retrieval algorithm itself (for consistent retrievals).

## 4.2 AOT

The synthetic retrievals for AOT are first evaluated. The abbreviations for
different retrieval cases are summarized in Table 3. For
synthetic retrievals, ${\mathit{\chi}}_{\mathrm{max}}^{\mathrm{2}}=\mathrm{0.5}$ is chosen as the
threshold for *χ*^{2} to define the successfully converged retrievals. Both
consistent retrievals and inconsistent retrievals are tested. Consistent
retrievals are retrievals for which the mode number for retrievals equals the
mode number for creating synthetic measurements. Inconsistent retrievals are
the cases when both mode numbers are not equal. Here, although the synthetic
measurements do not contain noise, we use the values assumed in the retrieval
procedure to compute the *χ*^{2}. Note that for consistent retrievals, in
principle the *χ*^{2} should be much smaller than 0.5 and should even be
very close to 0 when the global minimum has been reached. This does obviously
not hold for inconsistent retrievals in which a different number of modes
have
been used in the retrieval than in the creation of the synthetic
measurements.

Figure 1 shows synthetic retrievals of AOT with the
parametric two-mode retrieval (2modeRetr^{p}) and the 10-mode
retrieval (10modeRetr). Both consistent retrievals (i.e.,
Fig. 1a and d) and inconsistent
retrievals (i.e., Fig. 1b and c)
are performed. To quantitatively evaluate the performances of different
retrieval cases, RMSE and bias are indicated. For a fair comparison, RMSE and
bias should be calculated for the same number of points. Thus a constant
number *n*_{validate} (*n*_{validate}<*n*_{pass}) of
points are selected to calculate RMSE and bias. In each retrieval case, the
selected *n*_{validate} (*n*_{validate} is chosen at 150 here)
points correspond to the points with the smallest *n*_{validate}
number of *χ*^{2}. The total number of retrievals is *n*_{pix}
(*n*_{pix}=200 here).

We first look at the performance of the consistent 10-mode synthetic
retrieval, which is shown in Fig. 1d. The case is
named 10modeRetr+10modeSyn. It shows that the retrieved AOT matches very well
with the true AOT. The retrievals at all pixels can pass the strict filter
*χ*^{2}<0.5. Another consistent retrieval case, i.e., parametric two-mode
retrieval on two-mode synthetic measurements
(2modeRetr^{p}+2modeSyn^{p}), is shown in
Fig. 1a, in which the AOT retrieval for *r*_{pass}
(i.e., 98.5 %) pixels is also very accurate. Figure 1a
and d show that both the 10-mode and the
parametric two-mode retrievals have good capabilities of retrieving AOT for
consistent synthetic measurements.

In addition to consistent retrievals, it is interesting to test the performances of
inconsistent retrievals of AOT. This is because in reality,
it is unknown
how many modes the true atmosphere contains. For this purpose,
inconsistent retrievals are also shown:
parametric two-mode retrieval on 10-mode synthetic measurements (2modeRetr^{p}+10modeSyn)
in Fig. 1b
and 10-mode retrieval on two-mode synthetic measurements
(10modeRetr+2modeSyn^{p}) in Fig. 1c.
Although AOT retrievals in both inconsistent cases are not as good as those in consistent
cases, there is still a good agreement between the retrieved total AOT and the true
total AOT over different mode numbers. This shows that
inconsistent retrievals are also capable of retrieving AOT.

Next, we check the performances of other multimode
(i.e., two-,three-, …, nine-mode) retrievals.
Figure 2 shows the RMSE and the bias for all retrieval cases
in the synthetic tests.
The *x* axis in each subplot represents the parametric two-mode retrieval (2modeRetr^{p})
and different multimode retrieval cases (i.e., 2modeRetr, 3modeRetr, …,
10modeRetr).
Figure 2a and c are for the cases
on the two-mode measurements.
It confirms that the parametric two-mode retrieval as the consistent case
has the smallest RMSE and
the smallest absolute bias (i.e.,
closest to zero) compared to inconsistent retrieval cases.
Figure 2b and d show the cases on
the 10-mode measurements. It can be found that the inconsistent retrieval
for which $\mathrm{5}<{n}_{\mathrm{mode}}<\mathrm{10}$
has as good of a performance as the consistent retrieval
(10modeRetr+10modeSyn). Actually, although three-, four-, and five-mode retrievals
on the 10-mode measurements function a bit worse
than the multimode retrievals with *n*_{mode}>5, their accuracy is better than
the parametric two-mode retrieval on the 10-mode synthetic
measurements.
Therefore, we can conclude that multimode retrievals have more freedom to be compatible
with inconsistent multimode measurements. Conversely, for inconsistent retrievals on
two-mode synthetic measurements, the biases are larger than for the parametric
two-mode retrieval on the 10-mode measurements.

## 4.3 AOT of the fine and coarse modes

It has been investigated that the multimode retrievals are capable of retrieving AOT (the total AOT over all modes) for both consistent and inconsistent cases. Since each retrieval case and each measurement case include two types of modes (i.e., the fine and coarse types), it is interesting to test multimode retrievals on the AOT over all fine modes (${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{f}}$) and the AOT over all coarse modes (${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{c}}$).

Figure 3 shows ${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{f}}$.
For consistent cases (Fig. 3a
and d), the retrievals are accurate and nearly unbiased.
For inconsistent cases (Fig. 3b
and c), there are clear underestimations.
This generally happens in inconsistent retrievals on the two-mode measurements,
which can be seen in Fig. 3g (in which all the
inconsistent retrievals show a negative bias).
This does not happen for inconsistent retrievals on the 10-mode measurements,
for which
the parametric two-mode retrieval (2modeRetr^{p}), the fixed two-mode retrieval (2modeRetr),
and the three-mode retrieval (3modeRetr)
underestimate ${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{f}}$;
the four-mode retrieval (4modeRetr) and the five-mode retrieval (5modeRetr)
slightly overestimate ${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{f}}$;
retrievals are almost unbiased for
${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{f}}$ if *n*_{mode}>5.
By checking the RMSE of all retrieval cases on
the two-mode measurements (Fig. 3e) and the RMSE on
the 10-mode measurements
(Fig. 3f), retrievals
have quite acceptable
accuracies on both types of measurements if *n*_{mode}>3.

The total AOT of the coarse modes (${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{c}}$) is shown in
Fig. 4.
Compared
to the underestimation in
Fig. 3b and c for
${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{f}}$, there is an overestimation for
${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{c}}$,
as shown in Fig. 4b and c.
The reverse bias between ${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{f}}$ and
${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{c}}$ results in total AOT over all modes that is
almost unbiased, as shown in Fig. 1.
These offset effects can also be seen by comparing
Figs. 4g and 3g or by
comparing Figs. 4h and 3h.
According to the RMSE shown in Fig. 4e
and f, retrievals with *n*_{mode}>3
have good retrieval
accuracy of ${\mathit{\tau}}_{\mathrm{550}}^{\mathrm{c}}$ on both synthetic measurements.

## 4.4 SSA

We also tested multimode retrievals of SSA. Figure 5 shows the parametric two-mode retrieval and the 10-mode retrieval for SSA while Fig. 6 shows the RMSE and the bias for different retrieval cases for the difference between the retrieved SSA and the true SSA.

By comparing SSA for consistent retrieval cases
(Fig. 5a and d), for the *n*_{pass}
pixels (marked as red points), the match between the retrieved SSA and the true SSA
in Fig. 5a is slightly worse than the match in Fig. 5d.
This demonstrates the challenge in retrieving *r*_{eff} and *v*_{eff} in
the parametric two-mode approach – even for a consistent setup – since
*r*_{eff} and *v*_{eff} affect the derived SSA.
For inconsistent retrievals however, we see that the parametric two-mode retrieval on
the 10-mode synthetic measurements works better than vice versa.
Although Fig. 5 shows different performances between
the parametric two-mode retrieval (2modeRetr^{p})
and the 10-mode retrieval (10modeRetr),
the accuracy and the bias in the four cases are quite good.

Figure 6b and d show the RMSE
and bias comparisons among all retrieval cases on the 10-mode synthetic measurements.
All retrievals for SSA except the fixed two-mode case are shown to be accurate and have small bias.
On the two-mode synthetic measurements,
the RMSEs (Fig. 6a) of multimode retrievals are
a bit worse than the consistent
SSA retrieval.
For the bias on the two-mode synthetic measurement shown in
Fig. 6c,
it varies between 0 and −0.005. For the 10-mode synthetic measurements, multimode retrievals (if *n*_{mode}>4) and the parametric two-mode retrieval
are virtually unbiased.

## 4.5 Refractive index

### 4.5.1 Real part of refractive index

As described in Sect. 2.2, for multimode retrievals we also use a separate refractive index for the fine and coarse modes. In this case, the fine-mode refractive index corresponds to mode numbers 1–6 in Table 2 and the coarse-mode refractive index to modes 7–10. Here we first test the retrievals of the real part of the refractive index for the fine modes and the coarse modes, i.e., ${m}_{\mathrm{r}}^{\mathrm{f}}$ and ${m}_{\mathrm{r}}^{\mathrm{c}}$ (at wavelength 550 nm), as respectively shown in Figs. 7 and 8.

For the consistent retrievals (2modeRetr^{p}+2modeSyn^{p} and
10modeRetr+10modeRetr), ${m}_{\mathrm{r}}^{\mathrm{f}}$ is retrieved with a small RMSE
and nearly unbiased, as shown in Fig. 7a
and d. Similarly,
${m}_{\mathrm{r}}^{\mathrm{c}}$ is also well retrieved in the consistent retrievals, which are
shown in Fig. 8a
and d. Actually, ${m}_{\mathrm{r}}^{\mathrm{c}}$ retrieval is shown
better than ${m}_{\mathrm{r}}^{\mathrm{f}}$ retrieval, and 10-mode retrieval on 10-mode synthetic
measurements is shown
better than parametric two-mode retrieval on two-mode synthetic measurements.

For inconsistent retrieval cases, we first check the performances on the 10-mode
measurements, i.e.,
the right panel of Figs. 7 and 8.
It shows that the parametric two-mode retrieval and
the multimode retrievals with *n*_{mode}>4 are capable of retrieving
${m}_{\mathrm{r}}^{\mathrm{f}}$ and ${m}_{\mathrm{r}}^{\mathrm{c}}$.
However, this is not the case for retrievals on the two-mode measurements,
i.e., the left panel in Figs. 7 and 8.
${m}_{\mathrm{r}}^{\mathrm{f}}$ is retrieved with
overestimation, as shown in
Fig. 7b and g.
For the retrieval of ${m}_{\mathrm{r}}^{\mathrm{c}}$
(see Fig. 8b and g)
an underestimation can be observed.
It can be concluded that the parametric two-mode retrieval works better for
the
fine-mode real part of the refractive index than the multimode retrievals.

### 4.5.2 Imaginary part of refractive index

Next, we test the retrievals of the imaginary part of the refractive index. The fine-mode and coarse-mode cases (i.e., ${m}_{\mathrm{i}}^{\mathrm{f}}$ and ${m}_{\mathrm{i}}^{\mathrm{c}}$) are respectively shown in Figs. 9 and 10.

For consistent retrievals, ${m}_{\mathrm{i}}^{\mathrm{f}}$ is shown to be well retrieved for both the parametric two-mode case and the 10-mode case; see Fig. 9a and d. This is a result similar to that of the consistent retrievals of ${m}_{\mathrm{r}}^{\mathrm{f}}$ (Fig. 7a and d). However, for the coarse-mode case, the consistent retrievals of ${m}_{\mathrm{i}}^{\mathrm{c}}$ (Fig. 10a and d) do not look as good as the consistent retrievals of ${m}_{\mathrm{r}}^{\mathrm{c}}$ (Fig. 8a and d), especially for the consistent parametric two-mode case (Fig. 10a), in which there are some clear outliers. Based on these results, we conclude that for the consistent cases, (1) ${m}_{\mathrm{i}}^{\mathrm{f}}$ retrieval is better than ${m}_{\mathrm{i}}^{\mathrm{c}}$ retrieval; (2) 10-mode retrieval of ${m}_{\mathrm{i}}^{\mathrm{f}}$ and ${m}_{\mathrm{i}}^{\mathrm{c}}$ looks better than the parametric two-mode retrieval.

For inconsistent retrieval cases,
the performances on the 10-mode synthetic
measurements (see panels c, f, and h of Figs. 9
and 10) show that
${m}_{\mathrm{i}}^{\mathrm{f}}$ and ${m}_{\mathrm{i}}^{\mathrm{c}}$ can be well retrieved
in the parametric two-mode and multimode retrievals with *n*_{mode}>4. This result is similar to
what was shown for the inconsistent retrievals of
${m}_{\mathrm{r}}^{\mathrm{f}}$ and ${m}_{\mathrm{r}}^{\mathrm{c}}$
(see panels c, f, and h of Figs. 7
and 8), except for one difference; i.e., the parametric
two-mode retrieval has clear overestimation when retrieving ${m}_{\mathrm{i}}^{\mathrm{f}}$,
as shown in Fig. 9c or h.
Now we check inconsistent retrievals on the two-mode measurements.
For ${m}_{\mathrm{i}}^{\mathrm{f}}$ (see panels b, e, and g of Fig. 9),
clear overestimation can be observed.
For ${m}_{\mathrm{i}}^{\mathrm{c}}$ (see panels b, e, and g of Fig. 10),
the multimode retrievals with *n*_{mode}>4
are quite accurate and only slightly underestimate
${m}_{\mathrm{i}}^{\mathrm{c}}$.
We can therefore conclude that
the multimode retrievals with *n*_{mode}>4 work slightly better than
the parametric two-mode retrieval for
the fine- and coarse-mode imaginary part of the refractive index.

## 4.6 Height

The retrievals of the central height *z* of the aerosol layer are shown in
Fig. 11. It can be seen that *z* can be well retrieved in the
consistent retrievals (RMSE < 50 m,
bias ≈ −5 m), as shown in Fig. 11a
and d.

For the inconsistent retrievals on the two-mode synthetic measurements,
4-, 5-, 6-, 7-, 9-, and 10-mode retrievals
(RMSE ≈ 200 m, bias ≈ −200 m) perform better than other
inconsistent cases, which
are shown in Fig. 11b, e,
and g. For inconsistent retrievals on the 10-mode synthetic measurements, the
parametric two-mode retrieval performs with clear underestimation as shown in
Fig. 11c (bias = −396.2 m), but
multimode retrievals with *n*_{mode}>5
perform very well with high accuracy (RMSE < 20 m) and little bias, as
shown in Fig. 11f and g.
To summarize, for inconsistent retrievals, the RMSE is typically
around 500 m and the bias is around 300 m.

Based on the results above, we conclude that
the multimode retrievals with *n*_{mode}>5
are capable of
retrieving the central height of the aerosol layer.

## 4.7 Pass rate of synthetic retrievals

The pass rate
${r}_{\mathrm{pass}}({\mathit{\chi}}^{\mathrm{2}}<{\mathit{\chi}}_{\mathrm{max}}^{\mathrm{2}},{\mathit{\chi}}_{\mathrm{max}}^{\mathrm{2}}=\mathrm{0.5})$ for
the parametric two-mode retrieval (2modeRetr^{p}) and the multimode retrievals
are shown
in Fig. 12.
In both Fig. 12a and b,
the fixed two-mode retrieval (2modeRetr)
has the smallest pass rate (*r*_{pass} ≈ 55 %)
compared to the other retrieval cases.
This is an indication that two fixed modes are not enough.
Figure 12a shows
retrievals on two-mode measurements (2modeSyn^{p}).
The *r*_{pass} for
retrievals with *n*_{mode}>2
are about 75 % to 90 %. The highest pass rate (up to 98.5 %) in
Fig. 12a is reached by the
parametric two-mode retrieval on two-mode synthetic measurements.
Figure 12b shows
the retrievals on 10-mode measurements (10modeSyn).
The pass rates are
high (95 % to 100 %) for all retrieval cases except for the two-fixed-mode retrieval.

## 5.1 Experimental setup

The synthetic experiments above have shown that multimode retrievals
with *n*_{mode}>5 have the capability
to retrieve aerosol optical and microphysical properties.
Next, we test the performances of multimode retrievals
on real data, i.e.,
PARASOL satellite data, as introduced in Sect. 3.1.

To validate PARASOL (satellite) retrievals, AERONET (ground-based) AOT and SSA data are used, as introduced in Sect. 3.3. AERONET measurements at 20 stations (listed in Table 7) in the year 2006 are used in this study to validate multimode retrievals on real data. To make PARASOL retrievals and AERONET data comparable, only the PARASOL retrievals within 20 km around each AERONET station are selected. The AERONET data are averaged within 2 h from PARASOL.

## 5.2 AOT: multimode retrievals versus parametric two-mode retrieval

In this section, the performances of multimode retrievals for AOT are compared to that of the parametric two-mode retrieval. Figure 13 shows real data retrievals of AOT among the parametric two-mode retrieval, the five-mode retrieval, and the 10-mode retrieval at three different wavelengths, i.e., 440, 675, and 870 nm, which are represented by the three columns in Fig. 13.

We first focus on the performances at 675 nm, i.e.,
Fig. 13b (2modeRetr^{p}),
e (5modeRetr),
and h (10modeRetr).
The total number of PARASOL retrievals for the 20 AERONET stations
is *n*_{pix} (*n*_{pix}=63 488 here).
For real data retrievals, ${\mathit{\chi}}_{\mathrm{max}}^{\mathrm{2}}=\mathrm{5.0}$ is used as the filter for
goodness of fit.
The total number of pixels at which the retrieval passes ${\mathit{\chi}}^{\mathrm{2}}<{\mathit{\chi}}_{\mathrm{max}}^{\mathrm{2}}$
is *n*_{pass}. The number of red points shown in each figure is not
*n*_{pass}, but *n*_{averaged} (*n*_{averaged} ≈ 1100 here), which represents the number of
*n*_{pass} retrievals after daily averages. The magenta points represent the
*n*_{validate} (*n*_{validate}=1000 here) best retrievals corresponding
to the smallest *χ*^{2},
which is needed if we want to compare the different retrieval setups
for the same number of measurements.

For real data retrievals, we set ${\mathit{\chi}}_{\mathrm{max}}^{\mathrm{2}}$ at 5.0,
which means that we actually underestimated the assumed errors in the retrieval
(otherwise the *χ*^{2} would be around 1.0).
The pass rates for the parametric two-mode retrieval and the multimode retrievals
are between 32.5 % and 40.8 %.
Comparing Fig. 13b, e,
and h, the 10-mode retrieval performs the best with the
smallest RMSE (0.1230) and the smallest absolute bias (0.0048).
However, the parametric two-mode retrieval has the largest RMSE (0.1624) and
the five-mode retrieval has the largest absolute bias (0.0301).

In addition to these three retrieval cases, we also perform multimode
retrievals with different numbers of modes. The RMSE and
the bias for all the retrieval cases (2–10 modes) are shown in Fig. 14.
From Fig. 14a, it is seen that
multimode retrievals
generally have better agreement with AERONET than
parametric two-mode retrieval, especially for
the multimode retrievals with *n*_{mode}>4.
From Fig. 14b, it can be found that
the parametric two-mode retrieval has an overestimation (0.019) and
all the multimode retrievals
show an underestimation. The 10-mode retrieval is almost unbiased, with the smallest
underestimation. Other multimode retrievals show larger underestimation
(from 0.0235 to 0.0429).

Based on the results above, we can conclude that multimode retrievals generally work better for retrieving AOT than the parametric two-mode retrieval. However, multimode (except for 10-mode) retrievals have larger absolute bias than the parametric two-mode retrieval.

## 5.3 AOT: multimode retrievals for different wavelengths

Section 5.2 discussed the retrieval performances at 675 nm. It is interesting to see how the retrievals perform at other wavelengths. For this purpose, 440 and 870 nm are chosen to evaluate the results.

We compare the three sub-figures in each row of Fig. 13,
e.g.,
Fig. 13g (440 nm),
h (675 nm),
and i (870 nm).
It can be observed that for the 10-mode retrieval (10modeRetr) at 440, 675, and 870 nm,
the RMSEs are
respectively 0.1654, 0.1230, and 0.1188.
For the five-mode retrieval (5modeRetr), the RMSEs are respectively 0.2152, 0.1513, and 0.1305.
For the parametric two-mode retrieval (2modeRetr^{p}),
the RMSEs are respectively 0.2209, 0.1624, and 0.1403.
It can be therefore found that
as the wavelength increases, the retrieval accuracy improves
in an absolute sense. However, this is mainly caused by the fact that
the AOT value itself decreases with wavelength.

Second, we check at 440 and 875 nm whether the conclusions at 675 nm hold. For this purpose, we look at the first and the third columns of Fig. 13. It can be seen that the RMSE decreases from the parametric two-mode retrieval to the five-mode retrieval to the 10-mode retrieval. This means that the retrieval accuracy at 440 nm (or 875 nm) improves as the mode number increases. Therefore, the conclusion at 675 nm also holds for other wavelengths.

## 5.4 SSA

Next we validate PARASOL retrievals of SSA with the AERONET-based SSA (described in Sect. 3.3). The AERONET SSA itself is not a result from a direct measurement but from an inversion procedure with different kinds of assumptions (Dubovik et al., 2002). The error in the AERONET SSA is at least 0.03 (Dubovik et al., 2002). The comparisons shown in this section should be interpreted taking this uncertainty into account.

Similarly to what was shown for AOT (Fig. 13),
Fig. 15 shows
SSA comparisons for the same retrieval setups as above (2modeRetr^{p},
5modeRetr, and 10modeRetr) at 440, 675, and 870 nm.
For SSA, it is usually difficult to retrieve it when AOT is small; thus
in Fig. 15 the SSA retrievals when AOT is
larger than 0.3 at the corresponding wavelength are shown.

We first check the tendency of the SSA accuracy for different wavelengths. By comparing RMSE in each row of Fig. 15, it can be found that the RMSE increases as the wavelength increases for all setups. Thus, PARASOL retrievals of SSA have a “decreasing accuracy” tendency as the wavelength increases. The reason is (again) that the AOT decreases with wavelength and the SSA retrieval becomes less accurate for decreasing AOT. The reverse is true for AOT retrievals as discussed in Sect. 5.3. Note that for the parametric two-mode retrieval, the RMSE at 675 nm (0.0601) in Fig. 15 is actually smaller than the RMSE at 440 nm (0.0629), but the difference is small.

Comparing RMSE in each column of Fig. 15,
it can hardly be concluded which one among
the different retrieval setups (2modeRetr^{p}, 5modeRetr, and
10modeRetr) compares best against AERONET.
For example, the 10-mode retrieval performs better
at 440 and 675 nm, but the parametric two-mode retrieval performs better at 870 nm.
Different retrieval setups for SSA seem to have similar accuracies.
This can be confirmed by
Fig. 16a, in which RMSE values vary within a small
interval (0.0577 to 0.0611)
for most retrieval cases except for the fixed two-mode retrieval, the three-mode
retrieval, and the five-mode retrieval. As for the bias (Fig. 16b),
all the setups show an overestimation and the bias values
in all the retrieval cases are quite similar (except for the fixed two-mode
retrieval).
Based on the comparison above, we can conclude that multimode retrievals have performances similar
to those of the parametric two-mode retrieval for SSA.

For the PARASOL retrievals in this paper we did not retrieve the aerosol layer height but used a fixed value of 1 km. This resulted in better AOT retrievals. The reason for poor performance of aerosol height retrieval from PARASOL is probably the absence of near-UV polarization measurements in combination with the relatively poor polarimetric accuracy (Wu et al., 2016).

In this study we compared aerosol retrievals from Multi-Angle Polarimeter (MAP) data for different definitions of the retrieval state vector: (1) a two-mode definition in which the state vector includes aerosol properties for fine–coarse modes and land or ocean surface properties; (2) a multimode definition in which the state vector excludes the effective radius and the effective variance and only retrieves the aerosol column of each mode. For the purpose of this study we extended the SRON aerosol algorithm – which was based on a parametric two-mode approach – to include capability of a multimode retrieval. To evaluate the retrieval capability for different state vector definitions, the performances between multimode approaches and the parametric two-mode retrieval approach were compared on both synthetic measurements and real (PARASOL) measurements.

In synthetic experiments, the consistent retrievals (when the number of modes for retrievals
equals the number of modes for creating synthetic measurements) show both
the multimode and parametric two-mode approaches
can reach high accuracy for most of the parameters, e.g.,
the AOT, the SSA, the
refractive index, and the aerosol height. For inconsistent retrievals
on 10-mode synthetic measurements,
the multimode retrievals with *n*_{mode}>5 were
shown to be capable of
retrieving aerosol properties with sufficient accuracy, and they perform similar
to the parametric two-mode retrievals. The good
performances of
multimode approaches indicate that multimode retrievals
have good compatibility with different kinds of measurements.

It should be noted that the geometry used for the synthetic study in this paper is quite favorable
as it assumes measurements in the principal plane. We also performed the same synthetic
study for a much less favorable geometry (SZA = 20^{∘},
relative azimuth angle = 60^{∘}$/-\mathrm{120}$^{∘}).
Although for the latter
geometry, the performance is somewhat worse,
the main conclusions from the synthetic study still hold for this geometry.

After synthetic experiments, real (PARASOL) data experiments were performed. Multimode retrievals of AOT were shown to compare better to AERONET than the parametric two-mode retrieval (e.g., RMSE 0.1230 over 0.1624). Here, we found that the agreement with AERONET improves with an increasing number of modes, with the 10-mode retrieval showing the best agreement with AERONET for AOT. For real data retrievals of SSA, both multimode and parametric two-mode retrievals have similar performances.

When comparing retrievals among different algorithms, it is important to realize that the performance of a given algorithm depends on a number of factors, the definition of the aerosol state vector being one of them. Other factors are the inversion approach (cost function, regularization strength, multiple versus single pixel), the accuracy of the forward model, and the surface reflection model. It is important to study the abovementioned aspects with an individual algorithm. However, now that the SRON algorithm has been extended to include an arbitrary number of fixed modes, it has become easier to compare to other algorithms using a similar state vector definition (Dubovik et al., 2011; Xu et al., 2017). This would be an important topic for future research.

The multimode approach provides an opportunity to make aerosol retrievals more computationally efficient. This is due to the fact that the effective radius and the effective variance are not retrieved in the multimode retrievals, thus the Mie–T matrix calculation for each mode can be fixed and precomputed as a function of refractive index. Then, there is no need to integrate over size distribution during the retrieval. Therefore, the most time-consuming part (as it is called many times) of the retrieval can be significantly accelerated.

The PARASOL level-1 data can be downloaded from the website http://www.icare.univ-lille1.fr/parasol/products (last access: 13 December 2018) (ICARE Data and Services Center, 2018). The AERONET data can be downloaded from the website https://aeronet.gsfc.nasa.gov/ (last access: 13 December 2018) (NASA, 2018). The meteorological NCEP data can be accessed through the website http://www.cdc.noaa.gov/ (last access: 13 December 2018) (NOAA/OAR/ESRL PSD, 2018). The retrieval results will be made available on SRON's FTP site.

GF and OH designed the experiments, analyzed the results, and finalized the paper.

The authors declare that they have no conflict of interest.

This work is funded by a NWO–NSO project ACEPOL: Aerosol Characterization
from Polarimeter and Lidar under project number ALW-GO/16-09. We thank
PARASOL team and AERONET team for maintaining the data. NCEP reanalysis data
were provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their website
at https://www.esrl.noaa.gov/psd/, last access: 13 December 2018. We would also like to thank the
Netherlands Supercomputing Centre (SURFsara) for providing us with the
computing facility, the Cartesius cluster. We are very grateful to the
editor, Michael Mishchenko, and Ruediger Lang for their reviews and insightful
comments.

Edited by: Alexander
Kokhanovsky

Reviewed by: Ruediger Lang and Michael Mishchenko

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