There are two ways to describe an optical aerosol model: using optical properties or using the microphysical properties. The optical properties (scattering and absorbing coefficients, phase matrix) are derived from the following microphysical properties: the size distribution (which gives the diameter distribution of the aerosol population), the complex refractive index (which gives characteristics of the light scattered by the particle for the real part and the absorbing quality of the particle for the imaginary part), and the sphericity (which describes the aerosol shape and non-sphericity aspect) (Hansen and Travis, 1974; Van der Hulst, 1981; Lenoble, 1993; Liou, 2002; Mishchenko et al., 2002; Lenoble et al., 2013; Bohren et al., 2016). Thus, to avoid losing information about the microphysical properties (i.e., the aerosol composition), we prefer to describe the aerosol model using its microphysical properties rather than its optical properties (knowing that it will give us the possibility of computing the optical properties from the microphysical ones). The size distribution characterization may be variable in its chemical or optical description, i.e., mass and numbers, respectively. This results in a different shape and description of the size distribution. For an optical approach, the Gaussian distribution is widely used as the most appropriate model for the aerosol size distribution (Whitby, 1978; Shettle and Fenn, 1979; amongst other subsequent studies). In order to design an optical aerosol size distribution in its vertical description, a combination of a Gaussian law for each aerosol mode is suitable (the fine mode and the coarse mode identified hereafter by f and c), even if it can be much more complex at a small scale (Liou, 2002; Hsu et al., 2004; Roger et al., 2009; Dubovik and King, 2000; Dubovik et al., 2011; Lee et al., 2015). In this way, the particle volume size distribution can be described by the derivative of the particle volume at a specific radius V(r) by the natural logarithm of the radius: 1dV(r)dln⁡r=Cvf2πσfexp⁡-ln⁡r-ln⁡rvf‾22σf2+Cvc2πσcexp⁡-ln⁡r-ln⁡rvc‾22σc2, where the six microphysical parameters that described this model are Cvf (the particle volume concentration of the fine mode), Cvc (the particle volume concentration of the coarse mode), rvf‾ and rvc‾ (the particle median volume radius of the fine and coarse mode, respectively), and σf and σc (the standard deviation of the Gaussian law of the fine and coarse mode).

The phase function of aerosols is usually normalized (Lenoble, 1985); thus the size distribution does not need to be defined in an absolute manner. We then may define the relative volume concentration %Cvf and %Cvc (scaled between 0 and 1) rather than Cvf and Cvc (discussed latter in this paper). The complex refractive index of the aerosol, n=nr+ini, is the second required microphysical parameter. The real part (nr) describes the scattering properties of the aerosol, while the imaginary one (ni) describes absorption properties. Both parts have to be known for a given wavelength. Finally, the percentage of sphericity %Sph can be considered as well to account for the non-sphericity of aerosols (Mishchenko et al., 2000; Dubovik et al., 2002b; Herman et al., 2005) in contrast to a “spherical approach” (Mie, 1908). This non-sphericity mostly applies the coarse mode.

Aerosol microphysical property data were extracted from the AERONET measurements (Holben et al., 1998; Dubovik and King, 2000; Dubovik et al., 2000; Sinyuk et al., 2007; Gilles et al., 2019). We used Level 2.0 (quality-assured) of the “Version 3 direct sun” and of the “Version 3.0 inversions”, except for the percentage of sphericity %Sph, for which we used Level 1.5 (in July 2021, Level 2.0 was not yet available for this parameter).

From these datasets, we selected all records corresponding to (1) aerosol optical thicknesses at four wavelengths (440, 675, 870, and 1020 nm); (2) aerosol Ångström coefficients between 440 and 870 nm, which allows us to determine the aerosol optical thickness at 550 nm; and (3) microphysical properties Cvf, Cvc, %Cvf, %Cvc, rvf, rvc, σr, σc, nr440​​​​​​​, nr650, nr850, nr1020, ni440, ni650, ni850, ni1020, and %Sph.

A minimum threshold of 50 measurements of the inversion product was used to exclude all sites without a sufficient number of measurements. We also ensure that all seasons are represented in the dataset. As mentioned above, one possible application of our aerosol microphysical model is the validation of satellite products in an operational context, whereby the atmospheric correction is performed when the aerosol loading is not too high. Thus, we decided to limit the dataset to aerosol optical thicknesses at 550 nm lower than 0.8.

Out of 1139 available AERONET sites, we selected 851 globally distributed sites (Fig. 1), resulting in ∼ 1.3 million retrievals of aerosol microphysical properties. To characterize the representativeness of these sites, we analyzed the type of land cover surface around the selected AERONET sites. As shown in Fig. 2, urban (24 %), cropland (22 %), forest (17 %), grassland and shrubland (16 %), and coastal areas and islands (16 %) are more or less equally represented.

For the measurements, AERONET instruments consist of two detectors mounted on robots, a system developed by Cimel: one for the measurement of solar (and now lunar) extinction, which provides the aerosol optical thicknesses (and then the Ångström coefficients) and the water vapor content, the other detector measuring the luminance of the day sky using two protocols – the almucantar and the principal plane (see Tables 1 and 2 in Holben et al., 1998). The almucantar procedure and measurements were used by Dubovik and King (2000) to derive the aerosol microphysical properties. Nevertheless, due to the observation protocol, the atmospheric condition (particularly its turbidity and homogeneity), the processing, and the retrieval purpose, the aerosol microphysical property retrievals are not provided within a single retrieval. There are three different sets of retrievals:

The size distribution (see Eq. 1) of Cvf, Cvc, %Cvf, %Cvc, rvf, rvc, σr, and σc is always available when aerosol microphysical property retrievals are performed by AERONET. For this study, this first block provides a little fewer than 1.3 million sets of retrievals for the whole 851 AERONET sites used.

The AERONET network has existed since 1993. Figure 3 shows the number of AERONET sites we used for this study since 1993. For the last 9 years, we used more than 350 sites, 250 sites, and 350 sites, respectively, for characterizing the size distribution, the refractive index, and the sphericity. The decrease observed in 2020 is because all data have not yet been validated.

A technical description of values for all aerosol microphysical properties for the 851 AERONET sites is presented Table 1, showing the percentile at 1 %, 5 %, 95 %, and 99 % and the median value for each of the properties. This gives a global overview of aerosol microphysical properties over land.

AERONET sites do not have the same number of observations (see Fig. 4). In the database we developed, one site may contain several thousands of selected retrievals for each aerosol microphysical property. For example, 8 sites provided more than 10 000 sets of retrievals for the size distribution, i.e., Sede Boker (Israel), Solar Village (Saudi Arabia – no longer in the network), GSFC (USA), Burjassot (Spain), El_Arenosillo (Spain), Carpentras (France – no longer in the network), Sevilleta (USA), and Granada (Spain). On the other hand, one site may contain fewer than 100 sets (this is the case for 138 sites). This means that one site may represent the equivalent of hundreds of other sites. To avoid the impact of those too well-represented sites, we show in Table 2 another way to present similar information as Table 1. By applying a single median value per AERONET site for each aerosol microphysical parameter retrieval, we have 651 values for each microphysical parameter (400 for the refractive index). Then, we derive a median value reported in Table 2. In this case, the median values do not change much (except for %Sph), but the range between both percentiles is reduced by 20 % to 50 %. With the assumption of a median value per site, Fig. 5 shows the frequency of τ440 and α440–870, while Figs. 6, 7, and 8 show the frequencies of each aerosol microphysical property from our selected dataset.

The results of the retrievals are evaluated using three performance metrics – accuracy (A), precision (P), and uncertainty (U):

The accuracy A represents the average bias of the estimates;2A=1N∑i=1NCi-Ri.

The relative uncertainty is defined here as U/V, where V can be the mean value of a specific site or of the whole set of a specific parameter.

Two measurements protocols are followed to acquire AERONET data. The aerosol optical thicknesses (AOT) is regularly measured every 15 min following a direct measurement of the sun when cloud-free. For the retrieval of the aerosol model microphysical properties, as specified above, the protocol required an almucantar measurement (Holben et al., 1998; Dubovik and King, 2000), which is performed early in the morning or late afternoon. The main issue is that this AERONET measurement might not be coincident with the Earth observation satellites' overpass times. Moreover, for various reasons (e.g., inhomogeneous sky, small clouds, calibration procedure) some measurements might be missing. We can obviously interpolate data between two available measurements, but we miss the variability in the considered aerosols. As an illustration, Fig. 9 shows an example of the impact of changing the aerosol model for size distribution from early morning (07:21:30 local time, LT) to late afternoon (16:28:45 LT). In this example, there is an increase in coarse aerosols between the morning and the evening, but we do not exactly know when that occurred.

In 2002, Dubovik et al. (2002a) suggested to determine each microphysical parameter with a direct regression (Eq. 5) using the aerosol optical thickness at 440 nm from the AERONET dataset. 5Aerosolmicrophysicalparameter=a+b⋅τ440 For each AERONET site, this approach has been used so far for the official validation of the MODIS and VIIRS surface reflectance products (Vermote et al., 2002, 2014), for the NASA HLS (Harmonized Landsat Sentinel-2) project (https://earthdata.nasa.gov/esds/harmonized-landsat-sentinel-2, last access: 1 March 2022; Claverie et al., 2018; Vermote et al., 2016), and for the CEOS ESA–NASA ACIX exercise (Doxani et al., 2018). Our objective here is to better account for the temporal and spatial variability in the aerosol microphysical parameters, which cannot be only related to the aerosol optical thickness itself. In an operational context, another possible and simple variable available for the aerosol description is the Ångström coefficient α (Ångström, 1929). Indeed, it is well accepted that this coefficient is related to the aerosol size (which is important in terms of light–matter interaction). If we take the example given in Fig. 9, we can see from Fig. 10 that the aerosol optical thickness does not change between the two almucantar procedures, while the Ångström coefficient does. The value of the latter decreases, indicating a bigger particle represented by a bigger coarse mode, which is consistent with Fig. 9. Another reason to select a multiplication of the optical thickness τ and the Ångström coefficient α is conceptual. The aerosol optical thickness τ is an extensive parameter, the Ångström coefficient α is an intensive parameter, and it is preferable to have a multiplication of a couple of intensive and extensive variables in physical parametrization as their multiplication remains an extensive parameter. Indeed, an intensive parameter can be used for identifying a sample, while an extensive parameter can be used for describing this sample.

We decided to select the Ångström coefficient for the 440 and 870 nm wavelengths, i.e., α440–870. Even if the Ångström coefficient has a dynamic behavior over the visible range, and it is not entirely constant, α440–870 is a good compromise between all values. At the end, we selected τ440 and α440–870 as variables of the regression. Within the AERONET network, these variables are available every 15 min under clear-sky conditions for all sites.

We can also use the water vapor content as it is a very important parameter in terms of the microphysical properties. Some aerosols are hydrophilic, and others are hydrophobic. Water vapor also modifies the size of the aerosol and its absorption capacity. We explored this option, but it did not improve the retrieval in terms of uncertainties. The aerosol optical thickness parameter already includes the effect of the water vapor over the aerosol size distribution, and it explains in part why there were no improvements.

One limited aspect of our approach is that these two parameters τ440 and α440–870 directly correspond to the aerosol scattering, and we may not properly characterize the aerosol absorption (Fraser and Kaufman, 1985; Vermote et al., 2007; Russell et al., 2010; Giles et al., 2012; Lenoble et al., 2013; Tsikerdekis et al., 2021). Therefore, the complexity of the radiative transfer through the atmosphere partially allows mitigation of this phenomenon. Indeed, coupling between the scattering and the absorption of light allows us to indirectly capture the aerosol absorption information.

With our AERONET database (over the 400 sites where we have all microphysical properties), we explored several mathematical formulations for a regression between an aerosol microphysical property, called AMP in the following equations, and the two variables τ440 and α440–870. We used a similar idea after Dubovik's law (Eq. 5). We first tested Eq. (5). Then, we tested a linear regression with the Ångström coefficient α440–870. 6AMPi=ai+bi⋅α440–870, where i represents one of the microphysical properties (e.g., Cvf, Cvc, %Cvf, %Cvc, rvf, rvc, σr, σc, nr440, nr650, nr850, nr1020, ni440, ni650, ni850, ni1020, %Sph).

Finally, we tested several mathematical formulations using our two predicted variables, and we found that each aerosol microphysical parameter, AMP, can be optimally described by 7AMPi=ai+bi⋅τ440cidi+ei⋅α440–870fi. In practice, to better use Eq. (7) and for the stability of retrievals, all six coefficients ai, bi, ci, di, ei, and fi are not derived with a single interaction. The aerosol microphysical parameters mainly depend on τ440 or on α440–870 (they rarely depend on both at the same level). Thus, to get a stable retrieval of the six coefficients, we used a so-called residue approach by checking which of the ai+bi⋅τ440ci or di+ei⋅α440–870fi is the most representative (i.e., with the best regression coefficient) regarding the behavior of the microphysical parameters. Following this procedure, we apply the first regression law ai+bi⋅τ440ci or di+ei⋅α440–870fi to derive (ai,bi,ci) or (di,ei,fi), respectively. Then, according to which one has the best correlation coefficient and using the remaining residue, we apply the second regression law di+ei⋅α440–870fi or ai+bi⋅τ440ci to derive the missing triplet of coefficients. Table 3 shows the percentage of occurrence for τ440 or α440–870 as the most representative variable for all microphysical parameters and for all available AERONET sites (see Fig. 2).

In Table 3, for 7 of the 11 parameters, α440–870 is more correlated with the microphysical parameter than τ440. This confirms that the use of α is pertinent to define these parameters. As expected, Cvf and Cvc are mostly driven by τ440 (Sinyuk et al., 2020), while %Cvf and %Cvc are driven by α440–870. Parameters Cvf and Cvc, which are extensive parameters, are directly related to the volume loading (mass) of the aerosol and, in the end, to the number of particles (accumulation of particles). Thus, it is not surprising that Cvf is more correlated to τ440 than Cvc. Indeed, we know that the fine mode optically reacts more efficiently in the visible light than the coarse mode in terms of extinction (Van der Hulst, 1981), considering that the number of particles present in the fine mode is usually much higher than the number of particles of the coarse mode. By the same reasoning, %Cvf and %Cvc, which are intensive parameters, are not sensitive to accumulation but rather to the spectral dependency of the aerosol extinction, meaning that %Cvf and %Cvc are more correlated to α440–870. In the AERONET processing, the complex refractive index is applied when the AOT is higher than 0.4 at 440 nm. This limits the variability in terms of AOT and probably artificially increases the occurrence for α440–870.

We applied our approach for the three mathematical formulations given by Eqs. (5), (6), and (7) over the whole selected dataset and present the results in Table 4.

In terms of accuracy A (Eq. 2), results show very low values. Except for Cvf, Cvc, and %Cvf, which present an accuracy up to 2 %, accuracies of all other microphysical parameters are below 0.1 %. For uncertainty U (Eq. 4), the third mathematical formulation gives the overall best results (Table 4). As expected, τ440 better represents Cvf, while in contrast α440–870 better represents the %Cvf. Finally, including both variables, we get a non-negligible improvement for both volume concentrations (absolute and relative). For the other microphysical properties, we do not observe much of an improvement, but Eq. (7) gives consistently better results. One point to be noted is that all microphysical properties provided by the AERONET network have lower uncertainties than those presented in Table 4 (Dubovik et al., 2000; Sinyuk et al., 2020).

As pointed out, %Cvf and %Cvc globally present a better uncertainty than for Cvf and Cvc, but for exactly 20 % of the sites the volume concentration of the fine-mode Cvf is more accurate than the relative volume concentration %Cvf (Fig. 11). We are unable to find a clear reason to explain that. The only tiny explanation is that aerosols over these sites present a tendency described by (1) lower concentrations than the average (both fine and coarse modes), meaning relatively low optical thickness; (2) a relatively lower Ångström coefficient; and (3) a relatively lower absorption. Nevertheless, according to the radiative transfer theory used to define the optical properties (phase matrix, scattering and absorption coefficients), the phase matrix is normalized at the end. Thus, either the couple of volume concentrations (Cvf, Cvc) or the couple of relative volume concentrations (%Cvf, %Cvc) can (it should be a couple) be used depending on the uncertainty for one AERONET site. It should be noted that, in all cases, the uncertainty U of %Cvc, U%Cvc, is always lower than that of Cvc, UCvc.

Table 5 shows the new uncertainties U of %Cvf, U%Cvf, and the new uncertainties U of Cvf, UCvf, when we only select sites for which U%Cvf>UCvf (80 % of cases) or UCvf>U%Cvf (20 % of cases), respectively. The improvement is visible if we use both %Cvf and Cvf according to the lowest uncertainties.

As pointed out previously, we have 50 % of sites without any refractive indexes. One solution to improve the number of sites is to define mean parameters (a, b, c, d, e, f) for nr and ni by kind of environment (e.g., urban, urban coastal, forest, non-forest land, desert). In that context, we undertook a preliminary study which included all data independently of the site to retrieve mean parameters. It gave a relative uncertainty U of 3.0 % for nr with no change compared to Tables 4 and 5. In contrast, for ni, it showed a relative uncertainty of 52 % for ni, which is about 40 % higher than those shown in Tables 4 and 5, but this study includes all data without distinguishing the environment. If we are able to specifically define the environment of the missing sites, we should get a relative uncertainty closer to 37.5 % (as indicated in Tables 4 and 5) rather than 52 %. In both cases, the uncertainty remains acceptable.

To expand on Table 4, Fig. 12 give the A, P, and U of the retrieved microphysical properties over the whole dataset versus τ440 and α440–870. The interesting point of these figures is the dependency of uncertainties with τ440 and α440–870. Indeed, except for Cvf and Cvc, uncertainties are quite stable with the aerosol optical thickness. In contrast, most uncertainties present variation with the Ångström coefficient. This confirms the importance of considering α440–870 in the regression. Another point is the correlation between Tables 4 and 5 and Fig. 12. When the variability in the uncertainty with α440–870 is important (Fig. 12), the variability in the microphysical properties is more important as well (Tables 4 and 5). It should be noted that for %Cvf and for Cvf, the A, P, and U are for selected sites only (see Table 5).

The use of α440–870 mostly improves the retrieval of both %Cvf and %Cvc (Tables 4 and 5). Figure 13 shows the comparison between uncertainties in %Cvf and %Cvc using Eq. (5) or (7) versus the mean value of %Cvf and %Cvc for each AERONET site (one dot represents one AERONET site). For %Cvf, we only consider sites where U%Cvf<UCvf. These figures highlight the improvement of retrievals (about 1/3 less). We can also point out that relative uncertainties are lower for high and low values of %Cvf.

Figure 14 gives the relative uncertainty for the other microphysical properties site by site, but only using Eq. (7) (for Cvf, we only consider sites where UCvf<U%Cvf). Again, except for the volume concentration Cvf and Cvc, we can notice the “arch” effect generating a lower relative uncertainty for lower values and for higher values of the considered properties. It is not shown here, but this arch effect is even more important with absolute uncertainties. At the end, we are able to characterize the uncertainties for each aerosol microphysical property and for each AERONET site.

As previously mentioned, this work is meant to support atmospheric correction validation over land. Thus, one question is how the uncertainty in the retrieved aerosol microphysical property affects the surface reflectance product validation. To address this issue, we decided to define, for each aerosol microphysical property, the impact of its uncertainty (Table 5) on the atmospheric correction and the determination of the surface reflectance over land. For that purpose, we defined a synthetic database of TOA reflectances for each AERONET site and for each specific satellite band. To generate this database, we used the 6S code (Vermote et al., 1997; Kotchenova et al., 2006, 2008; Kotchenova and Vermote, 2007) with the following inputs: (1) a set of 80 viewing conditions (solar angle, view angle, azimuth angle) describing all satellite angular configurations possible, (2) a set of different atmospheres (pressure, temperature, water vapor), (3) a set of surface reflectances (from 0 to 0.6 depending on the wavelength), and (4) a set of 40 aerosol microphysical properties with associated τ440 and α440–870 picked up in the real AERONET database. Then, we applied the atmospheric scheme developed for the land surface reflectance code (LaSRC) algorithm for MODIS, VIIRS, Landsat 8, and Sentinel-2 (Vermote et al., 2002, 2014, 2016; Claverie et al., 2018; Doxani et al., 2018). First, using each set of inputs, we computed the TOA reflectance. Then, inducing 20 cases of random uncertainties for each aerosol microphysical property, we applied an atmospheric correction to get the surface reflectance ρsurf to be compared to the one used as input. Table 6 gives the uncertainties we get for the MODIS red channel (band 1, 620–670 nm). For example, %Cvf is generated with an uncertainty of 22.0 %. This uncertainty generates, once we proceed with an atmospheric correction scheme, an uncertainty of 0.00014 in the surface reflectance (in reflectance units). The main relative uncertainty appears for the uncertainty Uni440 of the imaginary part of the refractive index (relies on the aerosol absorption), 1.0 × 10-3 in terms of surface reflectance, followed by the uncertainty in the radius of the fine mode. In a decreasing order of magnitude, Urvf and Unr440 appear around 1/3 lower. Then, another step below, UCvf and U%Cvf appear.

Many atmospheric correction schemes use a blue channel to retrieve the aerosol properties, so it is interesting to assess the impact of the aerosol model with the atmospheric reflectance in the blue channel. Figure 15 shows, for an example with the MODIS blue channel (band 3), the dependency between the uncertainties in ρsurf in the red channel and the atmospheric reflectance in the blue channel. This uncertainty is always low, below 0.005, for a range of reasonable atmospheric reflectance values. This figure also shows that this aerosol reflectance in the blue channel is almost linearly correlated to the uncertainties in the surface reflectance in the red channel. This means that a QA flag can be directly defined using the atmospheric reflectance in the blue channel rather than the optical thickness (Vermote et al., 2002, 2014).

Finally, Fig. 16 represents, in the end, the impact of the aerosol model uncertainties retrieved using Eq. (7) on the surface reflectance retrieval ρsurf in the MODIS red spectral band. Uncertainties, shown for two ranges of aerosol optical thicknesses at 550 nm (τ550; less than 0.25 and less than 0.50), are clearly always below the MODIS specification required for the surface reflectance (0.005 + 0.05∗ρsurf). For ρsurf ranged between 0.10 and 0.40, the uncertainty in ρsurf is relatively between 1 % and 2 %.

This confirms that our aerosol model description for the AERONET sites can be used with good confidence for the satellite atmospheric correction.