PARASOL provided multi-angle observations (up to 16 viewing angles per ground pixel) in 9 spectral bands (443, 490, 565, 670, 763, 765, 865, 910, 1020 nm) for intensity and 3 spectral bands for Stokes parameters Q and U (490, 670, 865 nm). The mission was operational in the period 2004–2013 (until 2009 as part of the NASA A-Train satellite constellation). The level 1 data are provided on ∼6×6km2 sinusoidally grid. This study uses PARASOL measurements from 6 spectral bands (443, 490, 565, 670, 865, 1020 nm) within latitude ranges from 60° S to 60° N and with at least 14 available viewing angles, as the majority of PARASOL observations contain exactly 14 angles. For measurements with more than 14 available angles, a subset of 14 is selected.

In this study, PARASOL RemoTAP (Remote Sensing of Trace Gas and Aerosol Products) aerosol retrievals provide some of the aerosol and surface properties in the training set and are also used for evaluation of the NN ACA retrievals on real PARASOL measurements. The RemoTAP PARASOL retrievals herein are based on a parametric 3-mode aerosol description characterized by three log-normal size distribution modes (Nmodes=3): one fine mode and two coarse modes (dust and soluble). A detailed overview of RemoTAP can be found in and .

The MODIS cloud phase product used in this work is generated at 1 km (at nadir) spatial resolutions from MODIS-Aqua L2 data product (MYD06_L2, ). Five different cloud flags are categorized in the product: liquid cloud, ice cloud, mixed cloud, uncertain and clear. In this work, a pixel is marked as liquid phase cloud only when the fraction of liquid-cloud-flagged 1 km-resolution MODIS pixels within a 6 km × 6 km PARASOL grid cell is larger than 80 %.

AERO-AC is a global ACA data product from PARASOL measurement, and it is used to compare with the PARASOL-NN ACA retrievals in this paper. In AERO-AC, the ACA properties are only retrieved in case of homogeneous optically thick (COT >3) liquid water clouds. The algorithm proceeds to search for the best-fitting aerosol model among all available models, including six fine modes plus a bimodal non-spherical mineral dust particle model. Pixels with partial cloud coverage and cloud edges are removed. Cirrus above liquid water clouds are also filtered and different quality criteria are applied to improve the products.

The NN training in this study utilizes synthetic measurements of top-of-atmosphere radiance and degree of linear polarization (DoLP), as a function of wavelength and viewing-solar geometries. The synthetic measurements are generated by the RemoTAP forward model , which is a linearized radiative transfer model employed in the RemoTAP retrieval algorithm . In the calculation of the synthetic measurements, liquid clouds are represented by spherical particles with a Gamma size distribution , and the refractive index of water is taken from . For ice clouds, hexagonal crystals with varying aspect ratios and surface distortions are used as proxies for variable-complex-shaped ice crystals . The aerosol size distribution follows three log-normal modes, as described in , where each mode is described by the effective radius (reff), effective variance (veff), complex refractive index (dependent on wavelength), AOT₅₅₀, fraction of spherical particles (fsph) and aerosol layer height (the central altitude of the Gaussian distributed aerosol profile, FWHM, full width at half maximum, fixed at 2000 m). Here we should note that the forward simulation of ACA scenes includes only fine and dust mode aerosols, while the simulation of clear-sky scenes considers also a soluble coarse mode. The spectrally dependent refractive index m(λ) per mode is parameterized by 1m(λ)=∑k=1nααkmk(λ), where mk(λ) are prescribed functions of wavelength, for which we use standard refractive index spectra for different aerosol components, i.e., dust , water-soluble, black carbon , and organic carbon . The Mie- and T-matrix-improved geometrical optics database is used for the computation from aerosol microphysical properties to optical properties. The ocean reflection properties are parameterized based on wind speed as described in , and chlorophyll-a concentration as outlined in . For land surface simulations, the bidirectional reflectance distribution function (BRDF) is parameterized using the Ross-Li model , while the bidirectional polarization distribution function (BPDF) is parameterized as in .

The surface (land and ocean) properties for the NN training are from randomly picked pixels of RemoTAP global retrieval for the year 2008. The cloud properties are generated randomly. The aerosol properties are randomly generated values or randomly picked from RemoTAP global retrieval in 2008, depending on the task of different NNs (the details are in Appendix Tables , and ). The geometry combination (solar zenith angle, SZA, viewing zenith angle, VZA and relative azimuth angle, RAA) are randomly picked from real PARASOL solar-viewing geometries in 2008. Only the measurements with a minimum of 14 angles are considered (see above) for the NN training, in order to evade from a variable-sized input vector to the NN or, as an alternative, an input vector with missing data.

This work focuses on retrieving the properties of aerosols which are located above a liquid cloud layer, and the retrieval process is depicted in Fig. . Three NNs are used in the process: (1) NN cloud mask, to select pixels covered by a liquid cloud, (2) NN for aerosol retrieval and (3) NN surrogate radiative transfer model (hereafter referred to as NN forward model). The NN forward model is used to efficiently compute the goodness-of-fit at low computational cost, which is essential for identifying cases where the 1D radiative transfer model breaks down-particularly in scenes with low cloud heterogeneity. Under such conditions, the plane-parallel assumption introduces a positive bias in ACAOT retrievals due to errors in polarized radiance modeling in the cloud bow region . These angular inconsistencies are revealed through discrepancies in the fit between forward model and real measurements . Additionally, MODIS cloud phase flags are used to mask cases with thin cirrus above liquid cloud (see above).

The first NN (cloud mask) takes intensity, DoLP, and viewing geometries (SZA, VZA, RAA and scattering angle) as input and outputs liquid cloud fraction and ice cloud fraction separately. The independent pixel approximation (IPA) is used to generate partly cloudy cases in the training set, as described in . The training set consists of 8 million samples including 20 % cloud-free pixels, 10 % fully covered by liquid cloud, 10 % fully covered by ice cloud, and the other 60 % partly covered by a mix of liquid cloud and ice cloud. The total cloud fraction is uniformly distributed in a square space (probability density function: f(x)=x2) with more cloud fractions close to 1. This setting reduces the cloud mask's ability when CF <0.8 but makes it more sensitive at almost fully cloudy cases (cases of interest). The radiative contribution of aerosol and surface properties is also taken into account, as described by . In the training set of this cloud mask NN, 20 % of the samples represent the situation where the aerosol layer is located above the cloud top, in order to improve NN's ability to produce liquid and ice cloud fractions in areas of interest for this study. A pixel will be further processed, if this NN outputs a liquid cloud fraction >0.8 and an ice cloud fraction <0.2, and the MODIS cloud flag also indicates this pixel is covered by liquid cloud. Here, the MODIS cloud flag is important to screen out cases where a thin cirrus is above liquid clouds, which are challenging to be identified by PARASOL measurements alone.

The aerosol retrieval NN takes the input of MAP measurements (i.e., radiance and DoLP), together with the observation geometry. It produces both fine mode and dust mode aerosol properties and underlying liquid cloud properties. Here we use a bi-modal aerosol description, where the size distribution is characterized by two log-normal modes, comprising one fine mode and one coarse mode representing dust. The state vector of the fine mode includes reff, veff, fsph, aerosol column number (Naer), and refractive index coefficients (αk), which correspond to the standard refractive index spectra of inorganic aerosol (real part), black carbon (imaginary part) and organic carbon (imaginary part). The state vector of the dust mode (consisting of non-spherical dust) includes reff, veff, Naer and a coefficient for the imaginary part of the dust refractive index. The parameter fsph is fixed to 0 and αk of the dust refractive index real part is fixed to 1. The liquid cloud properties included in the state vector are COT, cloud layer height (CLH), and the liquid droplet reff and veff. To better represent the real situations, the fine-mode fraction (fraction of fine mode AOT₅₅₀ over the total AOT₅₅₀) is randomly taken from PARASOL-RemoTAP clear-sky retrievals, while the total ACAOT₅₅₀ is randomly generated by a log-uniform distribution between 0 and 2. It should be noted that the coarse soluble mode is not considered in this step as it is usually below the cloud layer. An overview of the distribution for the different state vector elements of the training set are given in Table . The intensity and DoLP, as a function of wavelength and viewing angle, are compressed using a principal component analysis (PCA) before the training. A total of 25 principal components are retained for radiance (which contains 99.99 % explained variance) and 33 for DoLP (which comprise 99.14 % explained variance). Different from the training set of the cloud mask NN, the training set of aerosol retrieval NN only contains ACA samples.

The NN for forward calculation is designed to reproduce the MAP measurements from the viewing geometries and the retrieved properties, including aerosol properties of both fine mode and coarse mode and the liquid cloud properties. To make the forward model flexible in viewing geometries, it is trained separately per viewing direction and with the uniformly random-generated SZA, VZA and RAA. For each aerosol retrieval, the NN should be applied 14 times to simulate a MAP measurement at 14 viewing angles. The goodness-of-fit criterion is calculated as: 2χ2=1n∑in(yi-Fi)2σi2, where n is the total channel of measurements, and yi, Fi respectively stands for the satellite measurements and the NN reproduced measurements at the ith channel. For the PARASOL measurements in this study, a total of 126 channels are used including 6 wavelengths for intensity and 3 wavelengths for DoLP with 14 viewing angles per wavelength. The noise σi is the estimated absolute noise of each channel. Here we use a relative noise of 0.02 for the intensity and an absolute noise of 0.012 for DoLP.

It should be noted that the NN forward model is not a complete forward model. It only works for pixels fully covered by a liquid cloud without any radiative contribution from the surface and is designed only for the purpose of goodness-of-fit assessment for ACA retrievals. The performance of NN forward model on holdout set is shown in Fig. . The bias of both intensity and DoLP is close to zero. The rstd (relative standard deviation) of intensity is 0.7 % and the std (standard deviation) of DoLP is 0.0025. Both of them are below the instrument measurement noise, which suggests the NN forward model is good enough to replace the full physical model (RemoTAP) in estimating goodness-of-fit.

To increase numerical efficiency and reduce memory usage during the training process, we choose the “neural network ensemble” approach . In our approach, the whole training set is equally and randomly divided into several parts (further separated into training set, 90 % samples, and holdout set, 10 % samples), and an individual NN is trained on each part of the training set. The final output is the average of the outputs from all the ensembles. Here, three ensembles are used for liquid cloud mask NN, 16 ensembles for the aerosol retrieval NN, and six ensembles for the NN forward model. The number and size of ensembles is determined by the performance on synthetic validation sets.

For the cloud mask and retrieval NN, we add measurement noise to the training set as a form of regularization . The measurement noise is modeled as a Gaussian random number with a zero mean and a standard deviation of 1 %–3 % relative noise for intensity and 0.012 absolute noise for DoLP.

In this study, Pytorch (version 1.11.0, https://pytorch.org/, last access: 11 October 2021) is used to implement the NNs, which are structured as multi-layer perceptrons (MLPs). The training process employs the backpropagation (BP) algorithm and batch training with a batch size of 12 000. The performance of NNs in this paper shows little sensitivity to batch size, so a larger batch size is chosen for larger convergence rate . The Adam optimizer is used to minimize the root mean square error (RMSE) loss function. The architecture of the NN used in this work consists of three hidden layers. We used the settings (γ=0.001, β1=0.9, β2=0.999, ϵ=10-8) suggested by , where γ is the initial learning rate. For computational efficiency, ReLU is chosen to be the activation function. The liquid cloud mask NN has 64 neurons in each layer, the aerosol retrieval NN has 128 neurons and the NN forward model has 192 neurons. The detailed statistical distribution of the training sets can be found in the Appendix Tables , and .

The ACA retrievals are evaluated with nearby RemoTAP clear-sky aerosol retrievals in 2008 within the same 1°×1° grid cell. If a grid cell contains at least 3 ACA retrievals and at least 3 clear-sky aerosol retrievals, then the comparisons are made by taking the average of the retrieved aerosol properties for both ACA and clear-sky aerosol retrievals, respectively. Figure  shows the PARASOL-NN ACA and RemoTAP clear-sky aerosol retrievals in mid-Africa, 4 August 2008. In general, it shows large ACAOT₅₅₀ (∼1) of strongly-absorbing (SSA550<0.85), fine-mode-dominated aerosols (AE440–670>1.5), which is typical in this region, as is also observed in and . In the PARASOL-NN retrieval, the ACAOT₅₅₀ is smaller than the adjacent clear-sky AOT₅₅₀, because part of the aerosols are located below the clouds. The ACA seems to be slightly smaller in size (larger AE_440–670) and more absorbing (lower SSA₅₅₀) than the nearby clear-sky retrievals. This is expected, because the total column aerosol (as retrieved in the clear-sky case) is more influenced by non-absorbing coarse sea salt particles, which are mostly located below the cloud.

The whole year global comparison between the ACA retrievals and the matching clear-sky retrievals is shown in Fig. . For AOT₅₅₀, there is a root-mean-square difference (RMSD) between the ACA and clear-sky aerosol retrievals of 0.155. This is larger than the RMSE for the synthetic experiment (∼0.10) but we should keep in mind that the clear-sky RemoTAP retrievals do not provide an exact reference. In the first place, the retrieval error in the RemoTAP clear-sky retrievals (based on AERONET validation) is ∼0.10 over land and ∼0.05 over ocean . Second, we will in general expect a lower ACAOT₅₅₀ than the adjacent clear-sky AOT₅₅₀, because part of the aerosol may be located below the cloud, which explains the negative bias in the ACAOT₅₅₀. However, we also find cases where the ACAOT₅₅₀ is higher than the adjacent clear-sky AOT₅₅₀, which suggests the ACA retrievals may still be contaminated by cirrus, despite the NN cloud mask and the MODIS cloud phase mask. It can also be noticed that the RMSD of fine mode AOT₅₅₀ is smaller than the total AOT₅₅₀, and there is less overestimated pixels as well. This may be explained by the fact that coarse sea-salt, that has largest concentrations below the cloud, are excluded in the fine mode comparison. For AE_440–670, the RMSE (0.429) in Fig.  is similar to the RMSE found in the synthetic experiment (Fig. ), despite the fact that the AE_440–670 error on the clear-sky retrievals is ∼0.37 over land and ∼0.25 over ocean . For SSA₅₅₀, the RMSD (0.0586) is somewhat larger than in the synthetic experiment, but in general, the results suggest that the intrinsic aerosol properties (AE and SSA) are more comparable for ACA and adjacent clear-sky aerosol retrievals than the AOT, although the correlation of SSA₅₅₀ is low (0.37). To demonstrate the necessity of the goodness-of-fit mask, the comparison without goodness-of-fit mask is shown in Fig. S4 in the Supplement, it can be seen the performance of ACAOT₅₅₀, AE_440–670 and SSA₅₅₀ become substantially worse.

Figure  depicts the comparison of ACAOT₆₇₀ and AE between PARASOL-NN and AERO-AC at 1°×1° grid for the whole year 2008. The RMSD on ACAOT is 0.094, which is close to 0.107 from synthetic experiments. However, the correlation coefficient on ACAOT is relatively low (∼0.5), and especially at large ACAOT₆₇₀ values from AERO-AC, PARASOL NN algorithm retrieves much lower values. The RMSD on AE is 0.8, which is much greater than in synthetic experiments (∼ 0.4) and the comparison to adjacent clear-sky retrievals (∼0.6). For large AE (>1.5, as predicted by PARASOL-NN), the two data products agree well, but for smaller AE (<1.5 predicted by PARASOL-NN) the overall agreement is poor. Specifically, there is a group of pixels where AERO-AC predicts values close to ∼1.7). This group of pixels can be explained by a low ACAOT₈₆₅ (<0.1, retrieved by AERO-AC), where the AERO-AC algorithm assumes only fine-mode aerosols in the retrieval. Panel c of Fig.  shows a comparison where we filter out cases with AERO-AC ACAOT865<0.1 (in addition to the filter ACAOT550<0.2 already applied for both AERO-AC and PARASOL-NN). For this comparison, the RMSD is reduced from 0.8 to 0.5 and the correlation coefficient improved from 0.5 to 0.75. Also, clearly the lower limit of ∼ 0.4 in the AERO-AC AE is visible. Besides the reasons mentioned above, the discrepancy may also be caused by the fact that the AE from PARASOL-NN is calculated between 440 and 670 nm while that from AERO-AC is between 670 and 865 nm. To further interpret the differences between our PARASOL-NN algorithm and AERO-AC, we also compared AERO-AC to nearby RemoTAP clear-sky retrievals (see Fig. S8). From this comparison it follows that the ACOAT₆₇₀ from AERO-AC is in general larger than the nearby clear-sky AOT₆₇₀, with some very large ACAOT₆₇₀ values (>2) when the clear-sky AOT₆₇₀ is <0.5. This seems to suggest a tendency in AERO-AC to overestimate ACAOT₆₇₀, given that the ACAOT cannot be larger than the total column AOT. The comparison for the AERO-AC AE to clear-sky retrievals shows a similar pattern as the comparison with the above-cloud AE from the PARASOL-NN, although at larger AE the latter agreement is better than the agreement with clear-sky AE. The relatively large AE differences between AERO-AC and NN ACA retrievals (as well as the large AE differences between AERO-AC and PARASOL-RemoTAP clear-sky retrievals) may be related to differences in aerosol model assumptions. AERO-AC relies more on specific aerosol model assumptions under certain conditions, whereas PARASOL-NN and PARASOL-RemoTAP use the same continuous range of aerosol properties for all retrievals. On the other hand, the PARASOL-NN seems to slightly underestimate AE in fine mode dominated cases, based on the synthetic experiments (Fig. ). Moreover, it should be kept in mind that the different wavelength pairs are used for the AE calculation, which may cause discrepancies in the AE value (see Fig. S9).

Figure  shows the global seasonal average of ACAOT₅₅₀ and the number of ACA events in spring (March–May), summer (June–August), autumn (September–November) and winter (December, January and February) on the 1°×1° grid. The average of ACAOT₅₅₀ is calculated only when at least 25 valid PARASOL retrievals are found in the grid cell. The number of ACA events in a cell is defined as the total number of “good” retrievals where ACAOT is larger than 0.1.

The results in Fig.  agree well with the major ACA regions from previous studies , which include: (1) Tropical Southeast Atlantic, primarily consisting of biomass burning aerosols. (2) North Pacific, mainly containing industrial pollutants. (3) “Dust Belt” (5–40° N), where mineral dust particles are commonly detected above clouds in this latitudinal band.

The spatial occurrence of ACA events varies largely among each season. In the western coast of mid-Africa, the ACA events occur more in summer and autumn, while in spring and winter, not many events are observed. In the western coast of North America, although the events are detected for all the seasons, fewer events occurred in autumn and winter compared with the other seasons. The events in southeastern China can also be observed for almost all the seasons with somewhat less events in summer and autumn.

When looking into the global seasonal average of ACAOT₅₅₀, we can find two regions with significantly heavy ACA load: the western coast of mid-Africa (mainly summer and autumn, ACAOT550>0.5), western coast of Morocco in north Africa (during summer, ACAOT550>0.5) and northeastern China (during spring, ACAOT550∼0.2), and these regions are also observed to have a large number of ACA events. In contrast, for some regions with frequent ACA events, such as the western coast of North America, the seasonal average ACAOT is relatively low (ACAOT ∼0.1). This agrees well with the analyses by in the same year 2008.

We also investigated the annual average of AE and SSA, as is shown in Fig. . The AE and SSA are calculated where ACAOT550>0.2. Compared with ACA events in other areas, events around the western coast of mid-Africa exhibit a different characteristic: aerosols have a high AE (indicating smaller particles) and a low SSA (indicating more absorbing components). The high AE and low SSA is an expected feature of the smoke in mid-Africa . We have to remark that our AE in regions between 45–60° N and 45–60° S is ∼0.8, which differs largely from ∼1.8 in , despite the good agreement of our above cloud AE with the adjacent clear-sky AE in these latitudes. This is because in regions between 45–60° N and 45–60° S, the ACAOT₈₆₅ retrieved by the AERO-AC algorithm are likely too low to support reliable aerosol type identification, and only fine-mode ACAOT and AE retrievals are performed.