3.1 The effect of changes in input BRDF

Figure 1 shows the difference in the black sky surface albedo calculated with the Ross–Li BRDF using parameters of V2 and V3. The calculations are done for four AERONET sites with different surface types: GSFC (USA, MD) – urban/forest, Mezaira (United Arab Emirates) – desert, Mongu (Zambia) – treed savanna, and Kanpur (India) – semiarid/urban with strong seasonality. Figure 1 shows reasonably good agreement among average values of surface albedo for non-desert sites, with a maximum difference of ∼0.025. For desert, the agreement is good at 440 nm but decreases with increasing wavelength, reaching a maximum of ∼0.085 at 1020 nm. This discrepancy can be explained by the difference in the angular shape of the BRDFs in V2 and V3 as illustrated in Fig. 2a, which presents the difference in surface albedo as a function of SZA for Mezaira. Figure 2a shows that the surface albedo difference at 440 nm does not exhibit any substantial SZA dependence and therefore can be explained by the difference in the reflectance magnitude. For longer wavelengths, the SZA dependence of the surface albedo difference is much stronger, resulting in maximum variability from −0.05 to −0.16 at 1020 nm, and can be explained by the BRDF of V2 being more anisotropic than that of V3. For non-desert sites the difference in BRDF angular shape is less significant, as illustrated by Fig. 2b for the GSFC site, showing albedo differences within ±0.05, which is the reported uncertainty of the MODIS BRDF product (Wang et al., 2018). The large difference in surface albedo for desert sites has the potential to influence retrievals of SSA at longer wavelengths. Figure 3 illustrates the sensitivity of normalized sky radiances (NSRs) to the changes in BRDF parameters. The NSRs constitute the input to the inversion code and are defined as the measurements divided by extraterrestrial solar flux and multiplied by π. The calculations are made for two values of the SZA (50 and 75^∘) using dust aerosol parameters retrieved at the Mezaira site when the AOD at 440 nm is 0.65 and the Angström exponent (440–870 nm) is 0.25. As is seen from Fig. 3, the BRDF effect on NSR at 440 nm is moderate and decreasing with increasing SZA (∼2.0 % to ∼0.9 %). For longer wavelengths the relative difference of NSR is larger, reaching −8.8 % and −16 % at 1020 nm for SZA 50 and 75^∘, respectively.

Figure 1The V2 and V3 average differences and standard deviations in surface albedo calculated using BRDF parameters at four sites: Goddard Space Flight Center (GSFC, USA), Mezaira (UAE), Mongu (Zambia), and Kanpur (India).

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Figure 2Difference in surface albedo calculated with BRDF from V2 and V3 for (a) Mezaira (2004–2018) and (b) GSFC (1993–2018).

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3.2 Effects of changes in extraterrestrial solar flux and temperature correction

Figure 4 shows the relative difference between the extraterrestrial solar fluxes of V2 and V3 for the four standard sky wavelengths of the AERONET retrievals. The agreement is reasonable (differences of ∼1 % or less) at most of the wavelengths, except 675 nm, where the difference is anomalously large (∼3 %). Figure 5 shows the combined effects of both changes in solar flux and application of the temperature correction of sky radiances on NSR for two instruments, nos. 721 and 1026, which were deployed at the Mezaira site in 2013–2014 and 2017–2018, respectively. Each point corresponds to an individual observation taken at a specific value of the Sun photometer sensor head temperature. In addition to temperature, part of the difference is due to solar flux changes and accounting for NO₂ and water vapor absorption in V3. For example, at 675 nm, the ∼3 % bias is mostly due to differences in solar flux. The change in temperature dependence of the difference is due to the improved temperature correction in V3 (Giles et al., 2019). As is seen from Fig. 5a, the largest temperature correction effect is for 1020 nm, which is the most temperature-sensitive channel (Holben et al., 1998). At the same time for instrument 1026, the 1020 nm NSR difference is almost temperature-neutral. The reason is that instrument 1026 is the latest version – a T instrument equipped with a new model of filters (Iridian), which is much less temperature-sensitive than older filter models (Barr, Spetragon, POC) utilized in older versions of CIMEL instruments. Subtraction of the solar flux offset (V3 versus V2) in the case of the 1026 instrument gives a maximum NSR difference of ∼1.3 % at 440 nm and less than 1 % at 1020 nm. For instruments of older versions, the effect of temperature correction is expected to be larger than for those of Version-T; however, the specific numbers will vary from instrument to instrument.

Figure 3The relative percent difference in normalized sky radiances (NSRs) calculated with BRDF parameters from V2 and V3 for the Mezaira site.

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Figure 4The relative percent difference in spectral solar flux data used in V2 (ASTM E-490 https://www.nrel.gov/grid/solar-resource/spectra.html, last access: 8 June 2020) and V3 (Coddington et al., 2012) for four standard AERONET sky wavelengths (i.e., 440, 675, 870, and 1020 nm).

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3.3 Joint effects of temperature correction, solar flux and BRDF on retrievals of SSA: case study

The joint effect of the above factors on aerosol retrievals is illustrated by a case study of diurnal variability of dust SSA retrieved at the Hamim AERONET site on 25 August 2004 during the UAE field campaign (e.g., Reid et al., 2008b). This site is located in a desert region in the United Arab Emirates and is in proximity to several dust aerosol sources. This particular day was selected due to a combination of a very stable AOD level (0.486±0.016) and a very stable 440–870 nm Angström exponent (0.22±0.01), which makes an assumption about the stability of aerosol properties throughout the day reasonable. Also, a large temperature gradient (29.1–47.8 ^∘C), especially in the morning, makes an analysis of the diurnal variability of the temperature correction possible. Figure 6a shows the dependence of the difference in NSR on solar zenith angle, where negative values correspond to the morning and positive ones to the afternoon. In Fig. 6b the effect of solar flux is subtracted, so the variability is due to difference in temperature correction and BRDF only. Figure 6a shows that the total range of the variability of NSR difference is within $\sim \pm \mathrm{4}\mathit{\%}$ for 675 and 1020 nm. Comparison of Fig. 6a and b shows that the 675 nm channel is primarily affected by changes in solar flux, while the difference at 1020 nm is mostly due to the temperature correction. The diurnal variability of NSR difference at 1020 nm shows clear dependence on SZA, reaching a maximum absolute value in the early afternoon when the instrument sensor head temperature is maximal. The NSR differences at 440 and 870 nm are the least affected by solar flux values due to their similarity in both graphs of Fig. 6. In addition, comparison of Fig. 6a and b shows that the difference due to the temperature correction at 440 nm is partially compensated by the change in solar flux, while at 675 and 1020 nm both factors work in the same direction.

Figure 5Effect of solar spectrum and temperature correction on relative percent difference in NSR for Mezaira (a) instrument 721 and (b) instrument 1026.

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Figure 6Effect of solar flux and temperature correction on relative percent difference in normalized sky radiances for Hamim: (a) joint effect of temperature correction and solar flux; (b) the effect of solar flux is subtracted.

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In analysis of the effect of the changes implemented in the V3 aerosol retrieval algorithm on SSA retrievals from almucantar scans, the following approach was used. It consists of generating three different input files for each set of measurements. The NSR and BRDF data of the first input file (1st) are identical to those of V2. The second input file (2nd) differs from the first one by replacing V2 NSRs with those of V3, and in the third input file (3rd) both NSR and BRDF parameters are replaced, utilizing only V3 inputs. These input files are then combined with SORD in different modes, scalar or vector, which allows for analysis of the effects of accounting for polarization on retrieval results. All the combinations are inverted, and retrieval results are compared to each other, allowing for analysis of sequential changes in retrieved SSA values as individual changes in input data take place. Figure 7a compares the diurnal variability of SSA retrieved by the V2 retrieval algorithm and that of V3 with the 1st file as an input and SORD in the scalar mode. The comparison shows a good agreement between two sets of retrievals, with a maximum difference of ∼0.0074 at SZA 19.5^∘. The diurnal variability of SSA retrieved by Version 3, V3, is also shown.

Figure 7Data from Hamim (UAE) on 25 August 2004 showing (a) diurnal variability of SSA at 440 nm retrieved from almucantars by the V2 and V3 algorithms and algorithm of V3 in scalar mode using inputs of V2; (b) the relative percent difference between measured and fitted NSRs at 440 nm for three different SZAs.

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Figure 8Hamim on 25 August 2004. The diurnal variability of Level 2 SSA retrievals at 440 nm for three combinations of SORD and inversion inputs: V2 (red), SORD scalar mode (blue), and SORD vector mode using V2 inputs (green).

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A distinct feature of Fig. 7a is a sharp decrease in the retrieved SSA for SZAs smaller than 50^∘. This partially can be explained by the decrease in the range of the scattering angles as SZA decreases, in which case the solution becomes less stable with respect to both random noise and biases. However, the systematic decrease in retrieved SSA with SZA implies the presence of a bias which itself is SZA dependent. The recent analysis of Sinyuk et al. (2015) showed that this bias could be related to the accuracy of the model of randomly oriented spheroids at small scattering angles. This model was originally proposed by Mishchenko et al. (1997) for modeling of light scattering by non-spherical dust aerosols, was further developed in Dubovik et al. (2002, 2006) and is currently employed by the AERONET aerosol retrieval algorithm. The model has proven to be successful in interpreting the results of laboratory measurements of the light scattered by non-spherical aerosols in the backscattering direction (Dubovik et al., 2006). However, careful inspection of the fitting of the laboratory measurements by the model shows that it underestimates measured intensity at small scattering angles in the range centered at ∼7⁰ (e.g., Lenoble et al., 2013, p. 49, Fig. 2.18, left upper panel). The influence of this bias on the accuracy of SSA retrievals depends on SZAs through the range of scattering angles, with the maximum scattering angle for almucantar geometry being double that of SZA. For large SZAs the effect of the bias is counterbalanced by the fit at the large scattering angles where the spheroidal model is accurate. In these cases, the bias is not or just partially fitted by the retrieval code and has little effect on the accuracy of SSA retrievals. However, with decreasing range of the scattering angles the contribution of the bias to the fitting of NSR increases, and with no measurements constraining the solution at larger scattering angles, it pushes the retrieved SSA lower. This reasoning is illustrated by Fig. 7b, which shows the relative difference between measured and fitted angular NSRs as a function of scattering angle for the three values of SZA. In particular, all curves exhibit a peak in the vicinity of ∼7⁰ where the fit underestimates the measurements. The magnitude of the peak decreases with decreasing SZA, corresponding to a more accurate fit of the bias. Also, a decrease in fitting accuracy for the smallest SZA at larger scattering angles (> 50^∘) can be seen.

Figure 8 shows the diurnal variability of SSA retrieved at 440 nm for SZAs greater than 50^∘, a limit which is a key quality control for Level 2 almucantars (Dubovik et al., 2000; Holben et al., 2006). The results are presented for two different combinations of SORD and inversion inputs listed in the figure captions. As is seen from the figure, the SSA retrievals by both algorithms under the same conditions (scalar RTC) exhibit noticeable SZA dependence. Switching to the vector mode in the V3 retrieval code reduces the SZA dependence of SSA retrievals in both morning and afternoon. This can be explained by the higher accuracy of vector RTC in situations with substantial contribution of molecular scattering. In this particular Hamim case, the Rayleigh optical depth at 440 nm comprises ∼32 % of the total optical depth, which together with strong polarization of molecular scattering makes accurate accounting for polarization important.

Figure 9Hamim on 25 August 2004. The diurnal variability of Level 2 SSA retrievals at 440 nm for three combinations of SORD and inversion inputs: V2 inputs (red), combined inputs with BRDF parameters of V2 and NSR of V3 (blue), and combined inputs with BRDF parameters of V3 and NSR of V3 (green) for (a) 440 nm and (b) 675 nm.

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Figure 10The same as Fig. 9 but for (a) 870 and (b) 1020 nm.

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Figure 9 shows Level 2 SSA retrievals by the V3 aerosol retrieval algorithm with RTC in vector mode for the three different types of input files described in the figure captions. Figure 8 shows a near-constant offset in the values of retrieved SSA in the morning versus the afternoon, which averages ∼0.014 for 440 nm. One of the possible explanations for this difference is that the dust in the morning and the afternoon was transported to Hamim from different regions. This assumption is supported by the analysis of GSFC back trajectories for the Hamim site (https://aeronet.gsfc.nasa.gov/cgi-bin/bamgomas_interactive, last access: 8 June 2020), which shows different back trajectories in the afternoon versus morning. Figure 9a shows that the contribution from the changes in NSRs dominates that of BRDFs for 440 nm SSA retrievals, while according to Fig. 9b, the contributions from changes in NSR and BRDF at 675 nm are comparable. This is consistent with the results in Figs. 2, 4 and 6, which show that the changes in the temperature correction dominate the 440 nm NSR, while the 675 nm NSRs are mostly affected by the changes in solar flux and BRDF. For SSA retrievals at 870 nm the contribution of the changes in BRDF is dominant, as shown in Fig. 10a, versus comparatively smaller effects for both solar flux and temperature correction. The SSA retrievals at 1020 nm, shown in Fig. 10b, are affected primarily by the changes in temperature correction and BRDF, acting significantly however in opposite directions.

Figure 10 shows higher variability of SSA retrievals with SZA in the afternoon than in the morning for the inversions employing both NSR and BRDF of V3. In particular, retrieved SSA values in the near-infrared wavelengths in the morning are all close to 0.99, which is close to the theoretical limit for SSA of 1.0. This can be explained by low dust absorption at longer wavelengths (Dubovik et al., 2002) and constraints on the variability of retrieved parameters employed in the AERONET inversion algorithm (Dubovik and King, 2000). These constraints are based on a priori assumed ranges of variability of aerosol parameters and impose limits on the values of retrieved parameters. Theoretically, the highest retrieved SSA value is limited by 1, but because SSA is not retrieved directly (Dubovik and King, 2000), the limits on SSA retrievals are imposed through the constraints on the imaginary part of the refractive index (IRI). For low-absorbing aerosols the actual IRI values are very small, and therefore even a small bias in simulated NSRs can cause the retrieval values to hit the assumed limit. In the morning, the IRI retrievals assume the AERONET-constrained lower limit value of 0.0005, in which case the corresponding SSA retrievals are concentrated around 0.99. In the afternoon, the lower IRI limit is not reached due to the higher aerosol absorption.

Table 1Statistics, average values and standard deviations (in parentheses) of the difference in SSA retrievals from almucantar scans of V2 and V3 for the Mezaira site. The difference is defined as V3−V2.

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Table 2Statistics, average values and standard deviations (in parentheses) of the difference in the real part of refractive index (RRI) retrievals of V2 and V3 for the Mezaira site. The difference is defined as V3−V2.

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3.4 Comparison of aerosol parameters retrieved by V2 and V3 retrieval algorithms

This comparison of operational almucantar retrievals of V2 versus V3 is presented for four selected AERONET sites (Mezaira, GSFC, Mongu, and Kanpur), and the results are summarized in Tables 1 through 12. For SSA and the real part of the refractive index, the analysis is done for three bins of 440 nm AOD, which allows for extension of the analysis of the inversion comparisons to different levels of sensitivity relative to the combined effects of measurement noise and modeling biases. For the size distribution parameters two AOD bins (smaller and greater than 0.2 at 440 nm) are used due to the significantly higher stability and accuracy of the aerosol size distribution retrievals (Dubovik et al., 2000, 2002). The tables display statistics (mean value and standard deviation) of the difference between aerosol parameters retrieved by the V2 and V3 retrieval algorithms.

For Mezaira, the SSA 440 nm retrievals by V2 and V3 are not biased with respect to each other. However, SSAs retrieved by V3 are systematically higher than those of V2 at longer wavelengths, with the mean difference at 675 nm being the largest due to cumulative effects of both solar flux and BRDF changes. Also, the mean differences are smaller for bins corresponding to larger AOD. The mean difference of the retrievals of the real part of the refractive index shows similar patterns at 440 and 675 nm but further increases, reaching a maximum at 1020 nm. The agreement in size distribution parameters for both modes is reasonable bearing in mind the mean climatological values for VMR (volume median radius) and STD (standard deviation, in parentheses) for the Mezaira site, which are 0.15 µm (0.5) for the fine mode and 2.3 µm (0.62) for the coarse mode, respectively. This translates into an average relative difference of only ∼0.1 % for VMRC and ∼1.5 % for STDC for AOD smaller than 0.2 and into ∼0.8 % and ∼0.5 % for greater AOD. The corresponding numbers for the fine mode are also small: 5.4 % and 0.5 % for a smaller AOD bin and 4.9 % and 0.05 % for AOD larger than 0.2.

Table 3Statistics, average values and standard deviations (in parentheses) of the difference in volume median radius (VMR) in microns and width of particle size distribution (STD) retrievals of V2 and V3 and for the Mezaira site for almucantar retrievals with AOD (440) > 0.2. The difference is defined as V3−V2.

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Table 4Statistics, average values and standard deviations (in parentheses) of the difference in SSA retrievals of V2 and V3 for the GSFC site. The difference is defined as V3−V2.

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For the GSFC, Mongu and Kanpur sites the mean differences between V2 and V3 in SSA retrievals at the two longer wavelengths are much smaller than those for Mezaira due to smaller BRDF differences. However, the effect of the solar flux on SSA 675 nm retrievals is much weaker for GSFC than that for Mongu and Kanpur, which is explained by the difference in the magnitude of aerosol absorption. To gain a better understanding of the influence of the aerosol absorption on the uncertainty in retrieved SSA, Eq. (10) of Dubovik et al. (2000) can be used. Below this formula is rewritten to make the dependence on SSA explicit:

where ω₀ is SSA, τ_ext and τ_sca are extinction and scattering optical depth, respectively, and Δτ_sca and Δτ_ext are their uncertainties. Both uncertainties are always present: Δτ_ext is due to the accuracy of the AOD measurements and Δτ_sca may result from the modeling bias. For example, the changes in the magnitude of sky radiances due to the adjustment in solar flux value will be interpreted by the inversion algorithm as a change in scattering optical depth. Equation (1) shows that the contribution of the Δτ_sca to the SSA uncertainty depends on aerosol absorption: the smaller the aerosol absorption (larger ω₀), the smaller the Δτ_sca contribution and vice versa. Figure 11 shows the results of simulation of the SSA uncertainty using Eq. (1) for the different levels of aerosol absorption (SSAs of 0.99, 0.96, and 0.86) and gives a qualitative picture of the dependence of SSA uncertainty on aerosol absorption.

Figure 11The uncertainty in SSA simulated using Eq. (1) for the three values of SSA: 0.99 (red), 0.96 (blue) and 0.86 (green).

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The parameters of size distribution retrieved by V2 and V3 are in good agreement for all three sites, which can be seen from the comparison V3−V2 mean differences to the following mean climatological values of VMR and STD (in parentheses) from all almucantar retrievals at each site: 0.166 (0.456) and 2.921 µm (0.672) for GSFC fine and coarse modes, respectively, 0.146 (0.420) and 3.432 µm (0.655) for Mongu, and 0.172 (0.486) and 2.85 µm (0.626) for Kanpur. The comparison gives the following average relative differences for VMR and STD (in parentheses) for the smaller AOD bin: 0.9 % (4 %) and 0.9 % (1.9 %) for GSFC fine and coarse modes, respectively, 5.4 % (4.2 %) and 3.7 % (0.8 %) for Mongu, and 0.4 % (4 %) and 3.3 % (0.65 %) for Kanpur. For AOD greater than 0.2, the corresponding differences are the following: 3.7 % (0.2 %) and 2.1 % (1.3 %) for GSFC, 0.9 % (0.5 %) and 3.8 % (1.7 %) for Mongu, and 0.9 % (1.5 %) and 4.7 % (2.6 %) for Kanpur.

The mean difference in the retrieved real part of the refractive index is almost spectrally constant for the Mongu site and decreases with increasing wavelength for GSFC and Kanpur. However, in all three cases, the standard deviations of the retrieval differences are significantly smaller than the ranges of operational almucantar retrieval variability for cases when AOD (440) > 0.4 in both V2 and V3.

Table 5Statistics, average values and standard deviations (in parentheses) of the difference in the RRI retrievals of V2 and V3 for the GSFC site. The difference is defined as V3−V2.

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Table 6Statistics, average values and standard deviations (in parentheses) of the difference in volume median radius (VMR) and width of particle size distribution (STD) retrievals of V2 and V3 for the GSFC site. The difference is defined as V3−V2.

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4.1 HYB scan extension of SSA retrievals to smaller SZAs

Due to the increased range of scattering angles for the HYB geometry, the corresponding SSA retrievals are expected to exhibit less SZA variability than those of ALM, and to a much lower SZA. To test this assumption, the dependences of SSA retrieved from HYB scans on SZA were generated for three AERONET sites by aggregating retrievals in five SZA bins (each bin is 10^∘ wide, centered at 30, 40, 50, 60 and 70^∘ SZA) and calculating the mean value and the standard deviation for each bin. Figure 13 shows the dependencies of 440 nm SSA retrievals on SZA for the Mezaira, Mongu Inn, and Kanpur sites. These sites have the longest record of HYB-type observations starting from the fall of 2014. Figure 13a shows the mean SSA values for SZA 50^∘ and larger are nearly constant but decrease by ∼0.007 for smaller SZAs. The magnitude of the decrease, however, is much smaller than that observed in the Hamim case study from almucantar retrievals where SSA drops by ∼0.045 between 50 and 30^∘ (see Fig. 7a). Figure 13c shows a similar SZA dependence for Kanpur, with a mean SSA variability of ∼0.003 for SZAs larger than 40^∘ and a decrease by ∼0.015 between the 30 and 40^∘ angular bins. As was discussed in the analysis of the Hamim almucantar inversions, the reason for the decrease in retrieved SSA with decreasing SZA can be attributed to the bias in the modeling of light scattering by non-spherical dust aerosols when the scattering angle range is limited, which therefore can be expected to influence SSA retrievals at dust-dominated locations and smaller SZAs. In this regard, both the Mezaira and Kanpur sites are affected by dust aerosols: Mezaira is a desert site dominated by coarse-mode dust and Kanpur has a significant seasonal dust presence. The aerosol loading at Mongu Inn, on the other hand, is dominated by spherical fine-mode aerosols from seasonal biomass burning, and therefore no SZA-dependent bias in SSA retrievals is expected. This is illustrated in Fig. 13b, which shows that for Mongu Inn, the SSA does not exhibit any significant SZA dependence.

Table 11Statistics, average values and standard deviations (in parentheses) of the difference in the RRI retrievals of V2 and V3 for the Kanpur site. The difference is defined as V3−V2.

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Table 12Statistics, average values and standard deviations (in parentheses) of the difference in volume median radius (VMR) and width of particle size distribution (STD) retrievals of V2 and V3 for the Kanpur site. The difference is defined as V3−V2.

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Figure 14Dependence of 675 nm SSA hybrid scan retrievals on SZA for three AERONET sites: (a) Mezaira, (b) Mongu Inn, (c) Kanpur.

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The dependencies of SSA retrievals on SZA for HYB scans for longer wavelengths are shown in Figs. 14 through 16. The general feature of the results for Mezaira and Mongu Inn is the absence of any clear SZA trend for SSA retrievals. For Kanpur the SZA dependence at 440 nm (Fig. 13c) is gradually reversed as the wavelength increases, reaching the maximum 30^∘ to 40^∘ SZA difference of ∼0.018 at 1020 nm. The stability of dust SSA retrievals at Mezaira at longer wavelengths can be explained by the strong BRDF effect in V3 for barren desert sites which counterbalances the effect of modeling bias. Additional analysis revealed that the SSA trend reversal for Kanpur can be explained by changes in relative contributions of fine and coarse aerosol components to the total AOD with SZA bins. For example, at 1020 nm, the 30^∘ SZA bin is dominated by coarse-mode AOD, which is ∼0.3, while fine-mode AOD is only ∼0.085. At the same time, at the 40^∘ SZA bin the coarse- and fine-mode contributions are comparable, ∼0.18 (coarse) and ∼0.16 (fine), resulting in SSA at 30^∘ being larger than that at 40^∘ (e.g., Eck et al., 2010).

Table 13Hybrid sky scan with azimuth (AZ) and view zenith angle (VZA) for three selected solar zenith angles at 30, 60, and 75^∘. NA – not available.

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Figure 15Dependence of 870 nm SSA hybrid scan retrievals on SZA for three AERONET sites: (a) Mezaira, (b) Mongu Inn, (c) Kanpur.

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Figure 16Dependence of 1020 nm SSA hybrid scan retrievals on SZA for three AERONET sites: (a) Mezaira, (b) Mongu Inn, (c) Kanpur.

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4.2 Hybrid versus almucantar scan – comparison of SSA retrievals

Typically, HYB and ALM scans are taken ∼10–12 min apart for the SZA range of 50^∘ to 75^∘; therefore, the corresponding retrieval results are expected to be in relatively good agreement. Figures 17 and 18 show the statistics of the difference in SSA retrieved from HYB and ALM scans as a function of SZA. The statistics were generated using the data from all AERONET sites for which the HYB inversions were available by aggregating SSA retrievals in five SZA bins. Each bin is 1^∘ wide, centered at 50.5^∘ (387), 54.5^∘ (160), 59.5^∘ (187), 64.5^∘ (121) and 73.5^∘ (296) SZA, where the number of inversions corresponding to each SZA bin are shown in parentheses. In general, the agreement is good for all wavelengths, with the mean values of the differences below 0.005. At the same time the variability is increasing with increasing wavelength due to the predominant contribution of fine-mode aerosols to the generated statistics (aerosol loading of ∼80 % sites is dominated by fine-mode aerosols) and therefore much smaller AOD at the longest wavelengths, which results in less sensitivity to aerosol absorption. Also, there is a trend of decreasing difference with increase in SZA due to increasing similarity of both scan geometries, which is very clearly visible at 440 nm. At longer wavelengths SZA dependencies exhibit a maximum at ∼55⁰ due to a sharp decrease in the mean difference at the smallest SZA.

Figure 17Difference in SSA retrieved from HYB and ALM scans. (a) 440; (b) 675 nm.

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Figure 18Difference in SSA retrieved from HYB and ALM scans. (a) 870; (b) 1020 nm.

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5.1 Approach for individual inversions

The assumption made in this analysis is that the perturbations in measurements and auxiliary inputs are due to uncertainties (biases, systematic errors) in AOD, radiometric calibration of sky radiances, solar spectral irradiance, and surface reflectance. Sun photometer pointing bias is not included in the list of uncertainties because this error is quite small ($\sim \mathrm{0.1}{}^{\circ }$), and furthermore averaging the left and right parts of almucantar scans before use as input to the retrieval code reduces this uncertainty significantly. Each bias assumes three values: positive value, negative value, and zero (non-biased), and the values and the signs of biases of the same type are assumed to be spectrally independent. This means, for example, that AODs at different wavelengths are biased in the same direction with the same assumed bias value. The same is true for sky radiances, solar irradiance spectrum, and surface reflectance. The radiometric calibration and solar spectrum irradiance uncertainties are combined in one bias because both affect the magnitude of sky radiances. As a result, there are three distinct biases, each assuming three values, which makes 27 distinct combinations in perturbing measurements in inversion inputs. The following values for 1 sigma biases of the input data are assumed. AOD: $\pm \mathrm{0.01}/m$ (Eck et al., 1999b), where m is optical air mass, radiometric calibration bias: ±3 %, solar irradiance spectrum: ±2 %, and surface reflectance: ±0.05. The combined calibration and solar irradiance bias assume the ±5 % values, which overestimate the sum of individual biases in cases when they compensate each other. In the Ross–Li BRDF model, only the first of the three BRDF parameters is perturbed. This is equivalent to an assumption that the angular shape of BRDF is unchanged and only the surface albedo is biased. For each set of Sun photometer observations an array of 27 inputs corresponding to 27 combinations of biases is produced. All of these inputs are then separately inverted, and the following statistics of the inversion results are generated: average value, standard deviation, and minimal and maximal values. These statistics describe the variability in retrieved aerosol parameters due to the assumed uncertainties in inversion inputs for each individual inversion. From these statistics only the standard deviation (which we label U27) is used as a proxy for the estimated uncertainty as an indicator of a spread of the retrievals corresponding to the different combinations of input uncertainties.

5.2 Lookup table (LUT) approach

Because operationally running 27 inversions instead of 1 substantially increases the computational time, a LUT approach was designed to speed up the determination of U27 for all new observations when they are reprocessed to Level 2 with final calibrations applied. A climatological LUT is generated from the entire AERONET almucantar and hybrid scan database by binning U27s in AOD (440 nm), AE (440–870 nm), and SSA (440, 675, 870, 1020 nm). A total of ∼1 million sky scans that passed Level 2 quality controls were inverted 27 times in order to produce the U27 values for all parameters. Using this LUT approach, U27 for each individual new Level 2 retrieval can be obtained by interpolation using the corresponding measured and inverted combination of AOD, AE, and SSA. The analysis of U27 for dust aerosol individual retrievals of SSA at longer wavelengths revealed physically nonrealistic small values of U27 related to IRI hitting its lower limit (boundary) constraint of the inversion, as was discussed in the analysis of the results presented in Fig. 10. Therefore, a second LUT was generated from the subset of U27s for only individual inversions when no boundary hits were detected for any of the retrieved parameters. The second LUT is used for obtaining the unbiased estimates of uncertainty in situations when the effect of boundary hitting by retrievals on U27 is substantial (typically for SSA of dust aerosol, which is already very weakly absorbing).

The following binning intervals were used in generating the LUT. AOD: less than 0.05, 0.05–0.1, 0.1–0.2, 0.2–3.4 (in 0.2 increment bins). AE (440–870 nm): less than 0.3, 0.3–2.4 (in 0.3 increment bins). SSA: 440, 675 nm: less than 0.7, 0.7–0.8, 0.8–0.85, 0.85–0.9, 0.9–0.95, 0.95–0.975, 0.975–1.0. SSA: 870–1020 nm: two additional bins: less than 0.6, and 0.6–0.7. The U27s are provided for the following parameters: VMR of size distribution for both modes, width of size distribution for both modes, IRI at four standard wavelengths, SSA at four standard wavelengths, and asymmetry parameter at four standard wavelengths. The uncertainty estimates for the real part of the refractive index (RRI) are not provided for the reason that will be discussed later in Sect. 5.4.2.

All the parameters use AOD and AE in the LUT; however, some absorption-dependent parameters also utilize the binning of SSA, which produces two types of LUTs. The first LUT is two-dimensional with binning in AOD and AE. This LUT is used to generate U27 for VMR, width of PSD, and asymmetry parameter. The second LUT is binned by AOD, AE, and SSA and used to obtain U27 for IRI and SSA in order to account for the effect of different absorption levels by aerosols with overlapping AE ranges, e.g., pollution and smoke aerosols (both of which exhibit a wide range of absorption: Dubovik et al., 2002; Giles et al., 2012). The two-dimensional (2D) LUT can be viewed as a rectangular grid, each node of which corresponds to an AOD, AE pair. Each inversion has its own AOD and AE and thus represents a point in the AOD, AE 2D space. If a point is inside of that grid, four surrounding nodes are used for 2D interpolation. Similarly, the three-dimensional (3D) LUT can be viewed as a parallelepiped lattice, each node of which corresponds to an AOD, AE, SSA triad. Each inversion has its own AOD, AE and SSA and thus represents a point in the AOD, AE, SSA 3D space. If a point is inside of the lattice, we use the eight surrounding nodes for 3D interpolation. If one or more points that are required for interpolation are missing, the values of the closest nodal points are used. The 2D interpolation is done on the original LUT (no restriction on boundary hits of the a priori constraints). The 3D interpolation uses the no hits LUT (no constraint limits reached for any retrieved parameters) only for the following combination of AE, SSA bins: AE – less than 0.3 and SSA – 0.975–1.0, conditions for which most of the boundary hits of the IRI parameter occur.

Figure 19Comparison of SSA U27 estimated for individual inversions and by interpolation over LUT for the Mezaira site: (a) 440; (b) 675 nm.

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Figure 19 shows a comparison of U27 estimated for SSA retrievals by the two methods for Mezaira for SSA at 440 and 675 nm. Figure 19a shows good agreement between them at 440 nm, although the dependence of LUT-interpolated U27 on AOD exhibits less scattering than that for individual inversions, which can be explained by the additional LUT averaging. Figure 19b at 675 nm demonstrates the effect of the boundary hits by IRI retrievals on U27 obtained for individual inversions, in which case the variability among the 27 individual retrievals is reduced due to the lower limit constraint for IRI, resulting in an upper limit for SSA. Because of this artificial variability reduction, the corresponding standard deviations are underestimated in some of the individual retrievals. It is also seen from Fig. 19b that combining the original LUT with the one with no hits for the lowest AE and the highest SSA bins corrects these unphysically small U27 values, especially for low levels of AOD.

5.3 Quality control

For V3 retrieval parameters, we define the Level 2 criteria for estimated retrieval uncertainties in a manner very similar to that of Dubovik et al. (2000) and Holben et al. (2006), thus limiting the quality-assured U27 retrievals to those within AOD/SZA boundaries close to or equal to those defined in the previous V2. The following Level 2 criteria are used and U27 estimates of uncertainty are provided for these parameters.

• SSA, IRI: for ALM 440 nm AOD larger than 0.4 and SZA larger than 50^∘. For HYB 440 nm AOD larger than 0.4 and SZA larger than 25^∘. This ensures that for both ALM and HYB scans the minimum angle of the scattering angle range is 100^∘ for measured directional sky radiances.

• Volume median radius, width of particle size distribution, asymmetry parameter: for ALM 440 nm AOD larger than 0.02 and SZA larger than 50^∘. For HYB 440 nm AOD larger than 0.02 and SZA larger than 25^∘.

5.4 Analysis

The approach utilizing LUT interpolation was investigated by generating U27 for aerosol retrievals at four selected AERONET sites: GSFC, Mezaira, Kanpur, and Mongu, for which the results are presented below.

Figure 20Uncertainties for SSA retrievals at the GSFC site: (a) 440, (b) 675, (c) 870, and (d) 1020 nm.

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5.4.1 SSA

Figure 20 shows the uncertainty estimates for SSA retrieved at the GSFC site. As is seen, at all the wavelengths the U27 exhibits power-like functional dependencies on AOD, with smaller uncertainties corresponding to the larger optical depth and vice versa. The scattering of the data increased with increasing wavelength, which can be explained by the decrease in AODs at longer wavelengths due to the large AE, a 1.61-yearly climatological value. In addition, the data displayed in Fig. 20 are comprised of retrievals for the range of SZA greater than 50^∘ (upper limit ∼77^∘), which can contribute to additional U27 scattering due to SZA variability within narrow AOD bins. The solid blue lines in Fig. 20 indicate the upper boundary of Level 2 SSA retrieval uncertainty established in Dubovik et al. (2000) for AOD (440 nm) greater than 0.2 for the water-soluble aerosol type (weakly absorbing) which dominates GSFC aerosol loading especially at the highest AOD levels. Close inspection of Fig. 20a shows that the crossing point between the blue line (Dubovik et al., 2000, estimate) and U27 AOD dependence is close to an AOD (440) of 0.3, thus increasing the AOD by 0.1 in V3 for the same estimate of SSA uncertainty (0.03) that was assumed in V2 (for water-soluble aerosol). Similar analysis of Fig. 20b though Fig. 20d shows that the AOD (440 nm) corresponding to the uncertainty of 0.03 increases with wavelength: ∼0.4, 0.5, and 0.6 for 675, 870, and 1020 nm, respectively.

Figure 21Uncertainties for SSA retrievals at the Mezaira site: (a) 440, (b) 675, (c) 870, and (d) 1020 nm.

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Figure 21 shows the uncertainty estimates for SSA retrieved at the Mezaira site. The uncertainties exhibit some minor scattering at 440 nm and much larger variability at longer wavelengths. The main reason for the larger scattering of U27 values at longer wavelengths is the close proximity of the original IRI retrievals to the lower limit set by the inversion code even after the direct boundary hits were filtered out, resulting in small uncertainty values near the limit. In this case the different degree of proximity to the lower limit results in a different degree of reduction of U27, thus producing the resulting distribution of U27 values for each narrow AOD bin. The closeness of the IRI retrievals to the inversion limit at a longer wavelength in V3 is partially due to the BRDF input in V3 as discussed previously. The U27 exhibits smaller scattering for AODs larger than ∼0.9, which can be explained by the higher sensitivity to aerosol absorption in this AOD range as well as by the smaller number of the occurrences of high-AOD events. Additionally, as AOD increases, the aerosol signal increases relative to the fixed uncertainties of data and auxiliary inputs. Figure 21a shows the 0.03 uncertainty level corresponds to the 440 nm AOD value of ∼0.45, which is close to the 0.5 value by Dubovik et al. (2002) which gave the same AOD value for both dust and biomass burning aerosols. At longer wavelengths, the intersection point of the 0.03 line with U27 AOD dependence is harder to define due to a fair amount of scattering; however, intersection with the upper boundary of U27 AOD dependence is in the vicinity of 0.45 as well. However, it may be somewhat less than AOD (440) =0.45 due to the fact that absorption by dust is very weak at the longer wavelengths, therefore resulting in greater sensitivity to absorption, while it is much stronger at 440 nm, where absorption occurs from the iron oxides in dust.

Table 14Average SSA uncertainties estimated at GSFC.

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Table 15Average SSA uncertainties estimated at Mezaira.

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Figure 22SSA uncertainties generated by averaging over AOD bins with subsequent interpolation to the regularly spaced AOD grid: (a) GSFC, (b) Mezaira, (c) Mongu, (d) Kanpur.

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The dependence of estimated retrieval uncertainties on AOD can be made more explicit by averaging the scattering of individual data points over narrow AOD bins. Following this reasoning, the U27 SSA data for each of the four representative AERONET sites were binned in AOD, with the number of bins selected in such a way as to cover the widest AOD range possible and at the same time keep the number of data points in each bin high enough for representative statistics. Then the average values for each AOD bin were calculated and the final U27 dependencies were obtained by interpolating the binned data to a regularly spaced AOD grid. Figure 22a through Fig. 22d show the resulting AOD dependencies of SSA uncertainties for each of four selected AERONET sites. The corresponding numerical data are summarized in Tables 14 through 17 for AOD (440 nm) less than 0.7. Figure 22 shows that as expected AOD averaging resulted in rather smooth AOD dependencies of U27. The graphs of Fig. 22 provide a clear visualization of the spectral dependence of SSA uncertainties by considering intersections of corresponding curves with the 0.03 uncertainty level. For fine-mode aerosols dominating both the GSFC and Mongu sites, the values of AOD corresponding to 0.03 SSA uncertainty are increasing with the wavelength, as can be seen from Fig. 22a and c. In the GSFC case, for example, the 0.03 accuracy for 1020 nm SSA is reached at AOD of ∼0.6 as compared to AOD of ∼0.3 for 440 nm SSA. For coarse-mode aerosols such as mineral dust (Mezaira site), the spectral variability of SSA uncertainty is much weaker, as is seen from the graphs of Fig. 22b. The decrease in U27 of ∼0.0085 for longer wavelengths relative to 440 nm can be explained in part by the underestimation of uncertainty due to the proximity of IRI to the inversion limit, as was previously discussed. In addition, the lower SSA uncertainties can be related to the lower dust absorption in this spectral range, as was discussed in relation to Eq. (1). The aerosol loading over Kanpur is representative of the mixture of fine-mode pollution and coarse-mode dust with dust aerosols dominating during April through June. The effect of different components of the mixture on AOD dependence of U27 is clearly seen in Fig. 22d. For example, for the AOD larger than ∼0.8 the contribution of fine-mode aerosols is dominant, resulting in clear separation of U27 values at different wavelengths. For the range of smaller AOD the dust contribution is stronger, as can be concluded from the much weaker spectral dependence of estimated SSA uncertainties. The analysis is supplemented by Table 18, which contains multiyear (1995–2019) averages of aerosol and surface characteristics at four selected AERONET sites. The following abbreviations are used in the Table 14 header: FMF stands for the fine-mode fraction of AOD, AE is the Angström exponent estimated for the (440–879) wavelength range, and SA is a surface albedo.

Table 16Average SSA uncertainties estimated at Mongu.

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Table 17Average SSA uncertainties estimated at Kanpur.

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Figure 23Uncertainties of the imaginary part of refractive index estimates: (a) GSFC and (b) Mezaira.

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5.4.2 Refractive index

The behaviors of the U27 uncertainty estimates for the IRI are very similar to that of SSA. Figure 23 shows examples of uncertainties for 440 nm IRI estimated at GSFC and those of 1020 nm IRI estimated at Mezaira. There is a clear similarity between AOD dependences of IRI uncertainties of Fig. 23 and AOD dependencies of uncertainties of the corresponding SSA; see Figs. 20a and 21d. The difference in the average magnitude of U27 estimates of GSFC and Mezaira is due to the very low dust absorption at 1020 nm as well as the effect of the lower boundary retrieval constraint for IRI.

Figure 24Uncertainties of the real part of refractive index estimated at 440 nm: (a) GSFC, (b) Mongu, (c) Mezaira.

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Figure 24 shows the uncertainties (by the U27 methodology) of the RRI at 440 nm estimated for the GSFC, Mongu, and Mezaira sites. The values of these uncertainties at the GSFC and Mongu sites are below 0.02 for AOD greater than 0.4 and further decrease with AOD, reaching a minimum value of ∼0.012, while the U27 estimated at Mezaira levels out at ∼0.021 for AOD greater than ∼0.30. The average U27s were calculated for the range of AOD larger than 0.5 and compared to the values reported in Dubovik et al. (2000) for different aerosol types: 0.02 (Dubovik: 0.04) for dust at Mezaira, 0.015 (Dubovik: 0.04) for biomass burning at Mongu, and 0.0175 (Dubovik: 0.025) for water soluble at GSFC. The difference in U27 estimates and uncertainties of Dubovik et al. (2000) can be explained by the different way of accounting for instrument-pointing bias. As was discussed before, V3 uncertainty estimation does not account for pointing because of its much-reduced effect due to the averaging of the left and right parts of the ALM or HYB scans. On the other hand, a pointing bias of 0.5^∘ was assumed in the sensitivity analysis of Dubovik et al. (2000). This assumed value of the pointing bias is in excess of a more realistic value of 0.1^∘, which was estimated by comparing the two sides of the ALM scans. Thus, an additional unrealistically large source of uncertainty was introduced in Dubovik et al. (2000), with potentially large effects on the RRI retrievals. Further analysis showed that U27 estimates are substantially lower than observed variability (standard deviations) of RRI retrievals: ∼0.05 at Mezaira, ∼0.06 at Mongu, and ∼0.049 at GSFC. In contrast, the U27 estimated for SSA and observed variability of SSA retrievals are more consistent.

The potential instability in the retrievals of RRI is further illustrated by the fact that the upper and lower limits imposed on RRI variability (1.33 and 1.60, respectively) are often reached especially for low AOD. The percentage values of RRI retrievals reaching 1.33 and 1.6 (in parentheses) were estimated for all three sites for two bins in 440 nm AOD: less than 0.2 and greater than 0.4. The estimates for smaller AOD bin show that RRI retrievals hit the 1.6 boundary more often than that of 1.33: 1.6 % (13.7 %) for Mezaira, 3.1 % (11.1 %) for GSFC, and 2.2 % (18.5 %) for Mongu. The corresponding estimates for larger AOD demonstrate a significant reduction in the number of boundary hits, especially that of 1.6: 0.6 % (6.8 %) for Mezaira, 2.5 % (0.9 %) for GSFC, and 0.4 % (4.3 %) for Mongu.

Table 18Multiyear (1995–2019) averages of aerosol and surface characteristics at four AERONET sites selected for analysis of uncertainties in retrieved aerosol parameters. The abbreviations FMF, AE and SA stand for the fine-mode fraction of AOD, Angström exponent estimated for the (440–879) wavelength range, and surface albedo, respectively.

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Figure 25Analysis of correlation between the real part of the refractive index and fine-mode concentration at the GSFC site.

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The U27s estimated for RRI at longer wavelengths are very similar to those at 440 nm. The similarity between estimated uncertainties for RRI at different wavelengths can be explained by the fact that spectrally RRI retrievals are not independent but are related through the smoothness constraints (e.g., Dubovik and King, 2000).

The apparent underestimation of RRI uncertainties by U27 can be related to additional factors other than uncertainties in inversion inputs which can influence retrieval variability. Figure 25a shows the dependence of original retrievals of RRI at 440 nm on the concentration volume of the fine mode (CVF) derived from the retrieved particle size distribution for the GSFC site data (nearly all fine-mode-dominated). For this plot the data were sampled into five AOD bins, which resulted in separation of the RRI dependence on CVF into five distinct branches with a rather strong correlation between RRI and CVF. Each branch is well approximated by linear dependence, with regression equations and correlation coefficients displayed in the figure. As is seen, the absolute values of the slopes of linear regressions decrease as AOD increases. Simultaneously, the range of the variability of CVF increases and that of RRI decreases as AOD increases. The strong variability of RRI retrievals for narrow AOD bins can be explained by the lack of sufficient measurement sensitivity to this parameter to provide stable retrievals under somewhat noisy measurement conditions at times. In particular, AOD measurements do not constrain RRI individually but in conjunction with CVF, as it follows from results of Fig. 25a. As RRI is changing, it affects extinction optical depth through the change in aerosol scattering. However, to keep modeled extinction optical depth consistent with the AOD measurements, CVF is adjusted by the inversion code to compensate for the change caused by RRI. Thus, decreases in RRI are compensated by increases in CVF and vice versa in such a way as to reproduce the total aerosol scattering. Therefore, accurate retrievals of SSA are possible despite the variability of individual RRI and CVF retrievals, since total aerosol scattering is accurately described by a combination of these two parameters. The effect of the correlated behavior of RRI and CVF retrievals can be seen in Fig. 25b, where it manifests itself in the scattering of CVF retrievals for the narrow AOD bins. Similar analysis was done for the Mongu site, which is also dominated by fine-mode aerosol. As in the GSFC case the RRI versus CVF anti-correlation was observed, which allows one to extend the above conclusion to all fine-mode-dominated aerosols.

In the case of dust aerosols, AOD measurements do not provide enough constraints for RRI retrievals due to the very small sensitivity of AOD to variability in RRI for these coarse-mode-size particles. This is a consequence of a well-known fact that the extinction efficiency factor of large particles tends to a constant limit as particle size increases (e.g., Bohren and Huffman, 1998). In both cases, fine and coarse aerosols, constraints provided by the measurements of sky radiance intensities are not sufficient to provide stable RRI retrievals under conditions of typical measurement noise. The addition of polarization measurements can increase the sensitivity to RRI, as demonstrated in Fedarenka et al. (2016). The U27s for individual inversions are estimated under the specific realization of measurement noise which equally affects the original retrieval as well as retrievals obtained by inverting perturbed input files. Thus, the U27 estimates are not sensitive to the variability of retrievals caused by slightly noisy measurement conditions and will underestimate the retrieval uncertainty for RRI due to a lack of sufficient measurement constraints. For this reason, the U27 estimates for the RRI are not reported in the AERONET database.

Figure 26Uncertainties for (a) volume median radius of fine mode and (b) width of size distribution for fine mode estimated at the GSFC site.

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Figure 27Uncertainties for (a) volume median radius of fine mode and (b) width of size distribution for fine mode estimated at the Mongu site.

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5.4.3 Size distribution

Figure 26a and b show estimated uncertainties (U27) for volume median radius for the fine mode (VMRF) and standard deviation (width) of the size distribution for the fine mode (STDF) plotted as a function of AOD for the GSFC site. Uncertainties for both parameters decrease as AOD increases, reaching ∼0.004 and ∼0.009 for VMRF and STDF, respectively, for high AOD (> 0.4 at 440 nm). Comparison of the U27 values from all L2 retrievals averaged over the entire AOD range to the variability of retrievals (in parentheses) shows that the U27 values are smaller than those of the standard deviation of retrieval variability: 0.008 (0.03) for VMRF and 0.02 (0.06) for STDF. This difference can be explained by the actual variability of VMRF and STDF due to the different levels of humidification growth, cloud processing and/or particle aging (Eck et al., 2012). Averaging over an AOD range greater than 0.4 results in decreasing U27 values but does not produce significant changes in the standard deviation of retrievals: 0.0043 (0.032) for VMRF and 0.009 (0.056) for STDF. The VMRF and STDF uncertainties (U27) estimated at the Mongu site are shown in Fig. 27a and b, respectively, and exhibit similar behavior with AOD to the uncertainties estimated at the GSFC site. Similarly, the U27 values averaged over the entire AOD range are significantly smaller than the standard deviations of retrieval variability: 0.0058 (0.0207) for VMRF and 0.0138 (0.0616) for STDF. Averaging over AOD greater than 0.4 reduces both U27 and standard deviation values: 0.0039 (0.0118) for VMRF and 0.0094 (0.0416) for STDF. Larger values of standard deviation at larger AOD can be explained by natural variability in aerosol size. For the Mongu site the average value of the Angström exponent (440–870 nm) is ∼1.67 over the entire AOD range, with a standard deviation of ∼0.24. We conclude that variability in aerosol size can contribute to the observed variability of aerosol retrievals (Eck et al., 2001, 2013).

Figure 28Uncertainties for (a) volume median radius of coarse mode and (b) width of size distribution for coarse mode estimated at the Mezaira site.

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Figure 29Uncertainties for (a) volume median radius of fine mode and (b) width of size distribution for fine mode estimated at the Kanpur site.

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Figure 30Uncertainties for (a) volume median radius of coarse mode and (b) width of size distribution for coarse mode estimated at the Kanpur site.

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Uncertainties (U27) in the size distribution parameters for the Mezaira site are presented in Fig. 28 for volume median radius and size distribution standard deviation for coarse mode (VMRC and STDC) since this is a desert-dust-dominated location. The VMRC dependence on AOD follows the pattern similar to that of fine-mode aerosols, with U27 values averaged over the entire AOD range being significantly smaller than standard deviations of L2 retrieval variability: 0.073 (0.356) and 0.068 (0.356) for averaging over AODs larger than 0.4. The ratio of VMRC U27 to the standard deviation of VMRC retrievals averaged over 440 nm AOD greater than 0.4 is ∼3.2 %. A significant amount of scattering can be seen from Fig. 28b for STDC uncertainties for AOD smaller than 0.5. This can be attributed to the presence of a significant fraction of smaller particles for AOD smaller than 0.5, for which the coarse mode is not well defined. This also partly explains the observed larger variability in retrieved size distribution parameters at low AOD.

For the Kanpur site, the U27 estimates of uncertainty for size distribution parameters are presented for both fine and coarse modes in Figs. 29 and 30, respectively. As is shown in Fig. 29, the U27 exhibits two populations, with one displaying a strong increase in U27 values between AOD values of 0.5 and 1.0. This increase can be explained by the presence of dust aerosols with a strong coarse mode for which therefore the fine-mode AOD is relatively low, thereby providing low information content for the fine mode in the retrieval which results in greater sensitivity to the uncertainties in the inversion inputs. Similarly, the U27 increase for a coarse-mode radius shown in Fig. 30 for AOD larger than 1 is related to the domination of fine-mode aerosol with relatively low coarse-mode AOD and therefore weak signal in the measurements attributed to the coarse mode. The U27 values averaged over the entire AOD range and for AOD larger than 0.4 are very close to each other and have the following values for the average of the entire AOD range, along with standard deviations of L2 retrieval variability: 0.0063 (0.0465) for VMRF, 0.013 (0.066) for STDF, 0.0753 (0.405) for VMRC, and 0.0187 (0.0538) for STDC.