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: y=F(x)+ey. Here, y is the measurement vector containing the multispectral and multi-angle polarimetric measurements of PARASOL. ey represents the measurement error. x contains parameters to be retrieved, which include aerosol properties and land or ocean properties. The forward model F(x), which describes the dependence between y and x, 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 . We use the Mie–T matrix–improved geometrical optics database by 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 , , and .

In the parametric two-mode retrieval algorithm, the fine and coarse modes (denoted by superscript “f” or “c”) are characterized by the effective radius refff;c, the effective variance vefff;c, the real and imaginary part of refractive index mrf;c and mif;c, the aerosol loading Nf;c, and the fraction of spheres fspheref;c. The complex refractive index for each mode is mf;c=mrf;c+imif;c. In the latest SRON aerosol algorithm, mf;c values are not directly retrieved (i.e., not in the state vector x), but constructed using mf;c(λ)=∑k=1nαf;cαkf;cmk,f;c(λ), where the mode component coefficients αkf;c (0≤αkf;c≤1) are included in the retrieval state vector. mk,f for the fine mode (or mk,c for the coarse mode) is the fixed spectral-dependent complex refractive index spectra for some aerosol components, e.g., dust (DUST), water (H2O), black carbon (BC), and inorganic matter (INORG). In this study, we set nαf;c=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 .

To retrieve the state vector from the satellite measurements, a damped Gauss–Newton iteration method with Phillips–Tikhonov regularization is employed . The inversion algorithm finds the solution x^, which solves the minimization–optimization problem, x^=minx(||Sy-12(F(x)-y)||2+γ||W-12(x-xa)||2). Here, xa is the a priori state vector, W is a weighting matrix, γ is a regularization parameter, and Sy is the measurement error covariance matrix. The weighting matrix W ensures that all state vector parameters range within the same order of magnitude 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 x, the inversion for Eq. () is implemented iteratively. For each iteration step (e.g., step n), we approximate the forward model F(x) with F(x)≈F(xn)+K(x-xn). Here, K is the Jacobian matrix (with Kij=∂Fi∂xj(xn)), which contains the derivatives of the forward model with respect to each variable in the state vector x.

Based on the linear approximation (Eq. ), the optimization problem (Eq. ) can be reduced to x̃n+1=minx̃(||K̃(x̃-x̃n)-ỹ||2+γ||x̃-x̃a||2), where K̃=Sy-12KW12, x̃=W-12x, and ỹ=Sy-12(y-F(xn)). The solution of Eq. () refers to and and is iterated by x̃n+1=ΛG̃ỹ+Ãx̃n+(I-Ã)x̃a, with the contribution matrix G̃=(K̃TK̃+γI)-1K̃T and the averaging kernel matrix Ã=G̃K̃. 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 . The filter factor Λ shows values between 0 and 1.

The regularization parameter γ and filter factor Λ in Eqs. () and () 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.

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 (reff and veff) of two modes, one aims to fit the size distribution with a larger number of modes for which reff and veff are fixed. An expected advantage of this approach is that it makes the inversion problem more linear (reff and veff 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 . 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 . The abbreviations for different retrieval cases used in this study are listed in Table , in which the parametric retrieval is denoted with a superscript “p”, i.e., 2modeRetrp.

For multimode retrievals, the state vector x in Eqs. () and () is different from that in the parametric two-mode retrieval. The difference is shown in Table , which specifies the parameters in the state vector. In the multimode retrieval, since reff and veff are not retrieved for all modes, they are not included in x. The aerosol loading Nj(j=1,2,…,nmode) 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  for the 10-mode case) and coarse (modes 7–10 of Table ). 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 , 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 , the number of aerosol parameters for the parametric two-mode retrieval and the multimode retrieval is respectively nαf+nαc+8 and nmode+nαf+nαc+2, where the fine- and coarse-mode component coefficients nαf and nαc are both set to 2 in this study.

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 for the same settings as in . For modeling surface bidirectional polarization distribution function (BPDF), a Fresnel model is used as introduced by . The surface parameters, to be retrieved in the state vector (see Table ), are scaling parameters for the BPDF model (xbpdfscale), the coefficient of the Li sparse kernel (xbrdfgeo1), the coefficient of the Ross thick kernel (xbrdfgeo2), and the BRDF scaling parameters at each wavelength band (xbrdfiw, iw=1,2,…,nwave). The number of surface-related parameters in the state vector for all retrieval cases is nwave+3. Therefore, the length of the state vector (i.e., the total number of aerosol- and surface-related parameters) is nαf+nαc+nwave+11 for the parametric two-mode retrieval and is nmode+nαf+nαc+nwave+5 for the multimode retrieval.

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

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  separately. The RT calculations are performed for different combinations of input parameters (as specified in Table ), 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 . 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. –). 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 nmode (nmode<10) size distribution to the 10-mode size distribution coming from the LUT retrieval with the nmode aerosol columns for the different modes as fit parameters.

The satellite data used in this study for aerosol retrievals are from the Polarization and Directionality of Earth Reflectances-3 (POLDER-3) instrument , 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 . In previous studies using the SRON aerosol algorithm , 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 Q2+U2I. For our retrievals on PARASOL measurements, we assume an intensity error Ierr=0.01 and the polarization error DoLPerr=0.007, in the diagonal matrix Sy12 in Eq. (). Here the intensity error is the relative error, and the polarization error is the absolute error. Ierr=0.01 holds for all POLDER bands except for the band 440 nm, where Ierr440 is set at 0.03 because the intensity measurements at 440 nm are usually considered less accurate than those at other bands . Note that in our study in principle the 0.01 for Ierr and 0.007 for DoLPerr 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., ). Examples of such improved instruments are GLORY-APS , MAIA , SPEXone , and HARP-2 .

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 .

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 . The retrieved SSA is validated with AERONET level 1.5 (cloud screened and quality controlled) Almucantar retrieval inversion products of SSA.

In a retrieval, it is a common approach to apply the goodness of fit (χ2) to decide whether the retrievals have successfully converged. The goodness of fit χ2 for each pixel is calculated by χ2=1nmeas∑i=1nmeas(Fi-yi)2Sy(i,i). Here, nmeas is the total number of measurements (multi-angle and multispectral intensity and polarization) for each pixel. yi represents the measurement (synthetic or real) and Fi represents the simulated measurement through the forward model. Sy(i,i) is the diagonal value of the measurement error covariance matrix, corresponding to the ith measurement.

We consider retrievals with χ2<χmax2 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 , corrupted measurements , and cases in which the first guess state vector deviates too much from the truth. Based on χ2, we define the pass rate rpass=npassnpix to be the number of successful pixels (npass) over the number of all pixels (npix).

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 xipixdiff[j] (at the ith pixel (ipix) for the jth variable in the state vector x) is computed by xipixdiff[j]=xipixretr[j]-xipixtrue[j], where xipixretr represents the retrieved aerosol property for the pixel ipix, while xipixtrue represents the reference aerosol property.

For each aerosol property, the RMSE counts the overall retrieval errors for all pixels with 1npass∑ipix=1npass(xipixdiff[j])2. The bias is calculated by 1npass∑ipix=1npass(xipixdiff[j]). The bias can be positive or negative, meaning the overestimation or the underestimation.

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. ).

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 reff, the effective variance veff, the fraction of spheres fsphere, the aerosol loading N, the mode component coefficients αk, 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 reff, veff as shown in Table . The other set is two-mode based. For this set, refff and reffc are perturbed within [0.1, 0.3] and [0.65, 3.4], respectively. vefff and veffc 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.  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).

The synthetic retrievals for AOT are first evaluated. The abbreviations for different retrieval cases are summarized in Table . For synthetic retrievals, χmax2=0.5 is chosen as the threshold for χ2 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 χ2. Note that for consistent retrievals, in principle the χ2 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  shows synthetic retrievals of AOT with the parametric two-mode retrieval (2modeRetrp) and the 10-mode retrieval (10modeRetr). Both consistent retrievals (i.e., Fig. a and d) and inconsistent retrievals (i.e., Fig. b 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 nvalidate (nvalidate<npass) of points are selected to calculate RMSE and bias. In each retrieval case, the selected nvalidate (nvalidate is chosen at 150 here) points correspond to the points with the smallest nvalidate number of χ2. The total number of retrievals is npix (npix=200 here).

We first look at the performance of the consistent 10-mode synthetic retrieval, which is shown in Fig. d. 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 χ2<0.5. Another consistent retrieval case, i.e., parametric two-mode retrieval on two-mode synthetic measurements (2modeRetrp+2modeSynp), is shown in Fig. a, in which the AOT retrieval for rpass (i.e., 98.5 %) pixels is also very accurate. Figure a 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 (2modeRetrp+10modeSyn) in Fig. b and 10-mode retrieval on two-mode synthetic measurements (10modeRetr+2modeSynp) in Fig. c. 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  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 (2modeRetrp) and different multimode retrieval cases (i.e., 2modeRetr, 3modeRetr, …, 10modeRetr). Figure a 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 b and d show the cases on the 10-mode measurements. It can be found that the inconsistent retrieval for which 5<nmode<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 nmode>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.

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 (τ550f) and the AOT over all coarse modes (τ550c).

Figure  shows τ550f. For consistent cases (Fig. a and d), the retrievals are accurate and nearly unbiased. For inconsistent cases (Fig. b and c), there are clear underestimations. This generally happens in inconsistent retrievals on the two-mode measurements, which can be seen in Fig. g (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 (2modeRetrp), the fixed two-mode retrieval (2modeRetr), and the three-mode retrieval (3modeRetr) underestimate τ550f; the four-mode retrieval (4modeRetr) and the five-mode retrieval (5modeRetr) slightly overestimate τ550f; retrievals are almost unbiased for τ550f if nmode>5. By checking the RMSE of all retrieval cases on the two-mode measurements (Fig. e) and the RMSE on the 10-mode measurements (Fig. f), retrievals have quite acceptable accuracies on both types of measurements if nmode>3.

The total AOT of the coarse modes (τ550c) is shown in Fig. . Compared to the underestimation in Fig. b and c for τ550f, there is an overestimation for τ550c, as shown in Fig. b and c. The reverse bias between τ550f and τ550c results in total AOT over all modes that is almost unbiased, as shown in Fig. . These offset effects can also be seen by comparing Figs. g and g or by comparing Figs. h and h. According to the RMSE shown in Fig. e and f, retrievals with nmode>3 have good retrieval accuracy of τ550c on both synthetic measurements.

We also tested multimode retrievals of SSA. Figure  shows the parametric two-mode retrieval and the 10-mode retrieval for SSA while Fig.  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. a and d), for the npass pixels (marked as red points), the match between the retrieved SSA and the true SSA in Fig. a is slightly worse than the match in Fig. d. This demonstrates the challenge in retrieving reff and veff in the parametric two-mode approach – even for a consistent setup – since reff and veff 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.  shows different performances between the parametric two-mode retrieval (2modeRetrp) and the 10-mode retrieval (10modeRetr), the accuracy and the bias in the four cases are quite good.

Figure b 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. a) of multimode retrievals are a bit worse than the consistent SSA retrieval. For the bias on the two-mode synthetic measurement shown in Fig. c, it varies between 0 and -0.005. For the 10-mode synthetic measurements, multimode retrievals (if nmode>4) and the parametric two-mode retrieval are virtually unbiased.

The retrievals of the central height z of the aerosol layer are shown in Fig. . It can be seen that z can be well retrieved in the consistent retrievals (RMSE < 50 m, bias ≈ -5 m), as shown in Fig. a 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. b, 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. c (bias = -396.2 m), but multimode retrievals with nmode>5 perform very well with high accuracy (RMSE < 20 m) and little bias, as shown in Fig. f 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 nmode>5 are capable of retrieving the central height of the aerosol layer.

The pass rate rpass(χ2<χmax2,χmax2=0.5) for the parametric two-mode retrieval (2modeRetrp) and the multimode retrievals are shown in Fig. . In both Fig. a and b, the fixed two-mode retrieval (2modeRetr) has the smallest pass rate (rpass ≈ 55 %) compared to the other retrieval cases. This is an indication that two fixed modes are not enough. Figure a shows retrievals on two-mode measurements (2modeSynp). The rpass for retrievals with nmode>2 are about 75 % to 90 %. The highest pass rate (up to 98.5 %) in Fig. a is reached by the parametric two-mode retrieval on two-mode synthetic measurements. Figure b 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.

The synthetic experiments above have shown that multimode retrievals with nmode>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. .

To validate PARASOL (satellite) retrievals, AERONET (ground-based) AOT and SSA data are used, as introduced in Sect. . AERONET measurements at 20 stations (listed in Table ) 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.

In this section, the performances of multimode retrievals for AOT are compared to that of the parametric two-mode retrieval. Figure  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. .

We first focus on the performances at 675 nm, i.e., Fig. b (2modeRetrp), e (5modeRetr), and h (10modeRetr). The total number of PARASOL retrievals for the 20 AERONET stations is npix (npix=63488 here). For real data retrievals, χmax2=5.0 is used as the filter for goodness of fit. The total number of pixels at which the retrieval passes χ2<χmax2 is npass. The number of red points shown in each figure is not npass, but naveraged (naveraged ≈ 1100 here), which represents the number of npass retrievals after daily averages. The magenta points represent the nvalidate (nvalidate=1000 here) best retrievals corresponding to the smallest χ2, which is needed if we want to compare the different retrieval setups for the same number of measurements.

For real data retrievals, we set χmax2 at 5.0, which means that we actually underestimated the assumed errors in the retrieval (otherwise the χ2 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. b, 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. . From Fig. a, it is seen that multimode retrievals generally have better agreement with AERONET than parametric two-mode retrieval, especially for the multimode retrievals with nmode>4. From Fig. b, 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.

Section  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. , e.g., Fig. g (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 (2modeRetrp), 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. . 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.

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

Similarly to what was shown for AOT (Fig. ), Fig.  shows SSA comparisons for the same retrieval setups as above (2modeRetrp, 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.  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. , 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. . Note that for the parametric two-mode retrieval, the RMSE at 675 nm (0.0601) in Fig.  is actually smaller than the RMSE at 440 nm (0.0629), but the difference is small.

Comparing RMSE in each column of Fig. , it can hardly be concluded which one among the different retrieval setups (2modeRetrp, 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. a, 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. b), 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 .