The information content of the aerosol profile retrieval from the O2 A-band is primarily determined by the SNR and the spectral resolution of the instrument. To better understand the dependence on these two instrument parameters, we have carried out simulations for a range of SNRs and resolutions for a generic instrument based on the noise model of S-5 P.

For the simulations we have used a surface albedo of 0.5, at a solar zenith angle (SZA) of 30∘ and aerosol scenario A. We have kept the number of spectral pixels constant for all configurations which results in a high spectral oversampling for configurations with low spectral resolution; this has been done so as not to confuse loss of information with loss of spectral pixels .

The left panels of Fig.  show the error for the total AOD and the DoF as a function of the spectral resolution and continuum SNR for a retrieval without an intensity offset. Note that the DoFs are calculated with respect to the a priori covariance matrix described in Sect.  and thus they do not necessarily reflect the true DoFs with respect to the atmospheric variability. Instead, they provide a meaningful relative measure for characterizing the ability of different instrumental setups to provide information on the vertical aerosol distribution.

From Fig.  we find that a precise retrieval of total AOD does not necessarily require high spectral resolution but does require high SNR. For high values of SNR, the errors tend to be smaller for lower resolution. This is due to the SNR model limiting the usefulness of the higher resolution and therefore lower signals, particularly within the absorption lines and the fact that we have kept the number of spectral points constant for all values of resolution. However, for very high spectral resolution (0.03 nm), increasing resolution leads to smaller AOD errors regardless of SNR.

As expected, vertical information, shown by the DoFs, shows a clear dependence on the spectral resolution as the vertical information is inferred from the absorption lines of the O2 A-band. For low values of SNR (< 200), the noise is limiting the information that can be obtained from the absorption lines and the effect of spectral resolution becomes weak. Consequently, large values for DoF require very high spectral resolution combined with very high SNR.

The impact of including an intensity offset in the retrieval is shown on the right hand side of Fig. . The increase in the error for total AOD is typically small with an average value of 5 × 10-3. The largest increase is observed for low SNR, and the smallest effect is found for high-resolution and modest-to-high SNR. For DoF, we observe a small decrease of 0.05 on average. Again at low SNR the loss in information is more substantial and can be up to a 50 % decrease in DoFs for very low values of SNR.

Based on Fig. , we can make several tentative predictions for the performance of the four instruments described in Sect. . Firstly, we expect S-5 P to behave somewhat differently from the other three instruments due to it operating in the low-resolution and high-SNR regime that should result in low total AOD errors but also low DoFs. OCO-2 has high spectral resolution and high SNR, so it should achieve high DoFs together with small total AOD errors. GOSAT has very high spectral resolution but lower SNR such that the values for DoFs will be lower and for the total AOD error higher. CarbonSat has lower spectral resolution but higher SNR than GOSAT, which should lead to a similar performance. All four instruments should not be substantially affected by the inclusion of an intensity offset in the retrieval. It can be expected that DoFs will be around 4 to 5 and AOD errors might exceed a value of 0.05, especially for GOSAT and CarbonSat.

We now study the performance of the four instruments in Figs.  and for scenarios A and B as described in Table  using the spectral and noise properties of the instruments as given in Table .

The LAOD error, shown in the top row of Figs.  and , is typically between 0.02 and 0.04 at pressures of 200 to 800 mb. Layers represent roughly 26 mb each with typically smallest errors for OCO-2 and largest errors for S-5P. For scenario A, errors tend to increase significantly below 800 mb due to correlations between surface albedo and the aerosol residing near the surface. However, instruments with low spectral resolution (S-5 P) tend to behave differently from the instruments with higher spectral resolution, with the low-resolution instrument showing a weaker increase in error towards the surface, especially for large SZA. This is a result of saturation effects in the aerosol weighting functions near the line centres for low altitude for high spectral resolution.

For scenario B, such a clear increase in error with decreasing altitude is not observed as the bulk of the aerosol in this simulation is in the free troposphere, resulting in less saturation of the Jacobians and weaker correlations to the surface albedo.

The impact of including an intensity offset is fairly minor. A slight increase in errors can be observed that is most pronounced for CarbonSat and S-5 P using scenario A.

The middle panels of Figs.  and show the vertical resolution of the aerosol profile retrieval as a function of altitude, where we define the resolution by the reciprocal of the trace of the averaging kernels multiplied by the pressure grid and converted into kilometres as in .

On average the resolution is 4.5 km with the resolution becoming substantially poorer for high altitude. For scenario A, the vertical resolution increases from 2 km at 800 mb to 6–10 km at 200 mb pressure for all four instruments. For S-5 P the resolution remains similar near the surface, while for the other instruments the resolution decays rapidly at low altitude. For scenario B, the best resolution is observed near the surface with values approaching 1 km for all instruments and the average resolution improves to 4 km.

Similar to the LAOD errors, OCO-2 has typically the highest vertical resolution in the free troposphere. S-5 P typically outperforms all other instruments near the surface but often shows the worst vertical resolution near the free and upper troposphere.

The averaging kernels for the aerosol profile retrieval for OCO-2 and S-5 P are given in the bottom panels of both figures. As expected, the averaging kernels are relatively broad which reflects the limited vertical resolution. Between 200 and 600 mb, the peak roughly corresponds to the associated pressure height, while below and above we often find that averaging kernels overlap. Overall, the averaging kernels suggest that it may be possible to resolve 3–4 layers within the atmosphere. The figures also further demonstrate the better sensitivity nearer to the surface for S-5 P, particularly at higher SZA.

As the effect of an intensity offset is small in both scenario A and B, all further retrieval simulations include intensity offset in the retrieval as it will provide a more realistic assessment of aerosol retrievals over vegetated land.

For a more quantitative comparison, we have integrated the LAOD profile over the height range between 0 and 2 km (∼ 1000–750 mb) to represent roughly the range of the boundary layer, over 2 to 5 km (∼ 750–500 mb) to represent the lower free troposphere and over the full atmosphere to obtain total AOD. The AOD errors and DoFs for the three height ranges are given in Figs.  and for the scenarios A and B respectively. Scenarios labelled A + B, A + C or B + C include two aerosol layers and they will be discussed in Sect. 4.2.

For scenario A (Fig. ), we find that AOD errors for the 0–2 km range are consistently lowest for S-5 P with errors between 0.09 and 0.21. The superior retrieval performance of S-5 P is most pronounced for large SZAs, whereas for lower SZAs the errors for the four instruments become more similar, with errors for OCO-2 approaching those obtained for S-5 P. DoFs range from 0.62 (CarbonSat for SZA of 30∘ and albedo of 0.05) to 1.35 (S-5 P for SZA of 70∘ and albedo of 0.5) but are mostly slightly less than one, which means that AOD for this height range cannot be independently retrieved. For the 2–5 km range, we find the lowest errors and largest DoFs for OCO-2, with errors between 0.11 and 0.13 and DoFs close to 1. For S-5 P, errors are particularly large for the low-albedo cases. As expected, we find that results for CarbonSat and GOSAT are similar to each other. For total AOD, the retrievals for the four instruments compare in a similar manner as for the 0–2 km range, with errors ranging from 0.06 up to 0.29. This represents a significant reduction in error compared to the a priori uncertainty of 1 for all cases. Note that the errors for total AOD are often smaller than the errors for the 0–2 or 2–5 km height range, owing to the presence of negative correlations between layers. The DoFs for the full range are between 3.3 to 4.7, with the highest values for OCO-2 and the lowest values for S-5 P and CarbonSat.

As has been discussed above, the retrieval of aerosol vertical information depends on the surface albedo. At low albedo, aerosols act to shorten the path length and at high albedo they tend to lengthen the path length with a transition in-between where aerosols will have little to no impact on the radiance. We find that aerosol weighting functions become small for a surface albedo of ∼ 0.2 and retrieval errors increase accordingly. Due to the modulation of this coupling between aerosol scattering and surface with the spectrally variable O2 absorption, the weighting functions differ from zero and the increase in AOD error is modest.

For scenario B (Fig. ), the AOD errors for the 0–2 and the 2–5 km range tend to be much smaller for all instruments compared to scenario A, with the exception of the case of high albedo and low SZA. The retrieval performance for S-5 P varies largely with SZA, with lowest errors of around 0.05 for SZA of 70∘ and highest errors with values between 0.1 and 0.18 for SZA of 30∘. For the low SZA case, OCO-2 gives the smallest AOD errors for the 2–5 km height range, where the bulk of the aerosol resides in these simulations, with values ranging from 0.08 to 0.12.

The AOD errors will be a composite of the error due to the measurement noise and the so-called smoothing error which reflects the errors introduced by the limited vertical resolution of the observations and the smoothing of the a priori constraint . We have estimated the smoothing errors for the different instruments using the a priori covariance matrix to represent the atmospheric variability of an ensemble of scenes. This will certainly lead to an overestimation of true smoothing errors for most scenarios as the assumed uncertainty of the aerosol a priori covariance matrix is very large with a standard deviation of the total AOD of one (see Sect. ). For scenario A, the estimated smoothing errors are significant with values of about 35 to 55 % of the total AOD error for GOSAT, OCO-2 and CarbonSat and 30 to 40 % for S-5 P. The estimated smoothing errors significantly decrease for scenario B with values of 13 to 30 % for GOSAT, OCO-2 and CarbonSat and larger values of 32 to 42 % for S-5 P.

To illustrate the effect of the height of the aerosol layer on the aerosol retrieval, the error of the total AOD has been calculated as a function of centre layer height ranging from 1 to 7 km (∼ 850–400 mb) and the total AOD ranging from 0.1 to 0.9 for the case of SZA of 30∘ and albedo of 0.5. As shown by Fig. , a clear decrease in relative error with increasing total AOD and with increasing height can be observed. As the AOD increases to much higher values, the errors tend to then increase again. This is caused by two opposing effects: an increase in sensitivity of the aerosol retrieval with aerosol weighting functions showing larger values with increasing AOD and a decrease in SNR with increasing AOD due to the high surface albedo.

This figure further emphasises the difficulty in retrieving aerosol that is present near the surface. Errors can be around 100 % for typical values of background AOD (∼ 0.1). OCO-2 tends to perform significantly better than the other three instruments with errors in the range of 20–25 % if the centre height is 2 km or higher. With increased layer height, the observed AOD error from GOSAT and CarbonSat are also smaller than those from S-5 P. Note that for a scenario with large SZAs, the performance of S-5 P would improve considerably compared to the case shown in Fig. .

So far we have only included scenarios with a single aerosol layer in the atmosphere, but in reality multiple aerosol layers will often be present, such as an elevated dust layer present above boundary-layer aerosol. Furthermore, a high-altitude cirrus layer can be present above an aerosol layer, especially in the tropics. Here, we study the performance of the aerosol profile retrieval for the four instruments, for scenarios that combine an aerosol layer in the boundary layer (scenario A), an elevated layer in the free troposphere (scenario B) and a cirrus cloud layer (scenario C).

Figure shows the AOD error for the 0–2 and 2–5 km ranges, as well as the total AOD error for the combined scenarios A + B and A + C.

Compared to scenario A, the additional layer of scenario A + B leads to an increase in the AOD error of 0.05–0.1 for the 0–2 km range for GOSAT, OCO-2 and CarbonSat, except for the low-SZA and high-albedo case where little change is seen. A very significant increase in the AOD error for the 0–2 km range is also found for S-5 P for high albedo, whereas the opposite behaviour is found for low albedo with a large decrease in the AOD error by ∼ 50 %; this an be simply described by the SNR decreasing and increasing in each case respectively, whereas for the other instruments they are again hampered by Jacobian saturation.

The errors for the 2–5 km range are reduced for all instruments to values between 0.07–0.12, except for the low-SZA and high-albedo case.

Adding a high-altitude cirrus cloud (scenario C) has less impact on the AOD retrieval than adding the free tropospheric aerosol layer (scenario B). For the scenario A + C, the error for the 0–2 km range increases by up to 0.05 for GOSAT, OCO-2 and CarbonSat. For S-5 P, we find that the AOD error for the 0–2 km range is slightly decreased for low albedo and is almost unchanged for high albedo, which might reflect a weaker sensitivity of S-5 P to the upper atmosphere.

Figure demonstrates the impact of additional aerosol layers for scenario B, noting that scenario B + A is the same as scenario A + B shown in Fig. . Comparing scenario B + A to scenario B we see that the additional aerosol layer has far less impact than in the previous case, where an additional aerosol layer was added to scenario A.

The most noticeable variations for the 2–5 km range occur for the high-albedo cases where the error reductions, and error increases in the case of S-5 P, are between 0.001 and 0.025. In contrast, the total AOD error varies substantially in comparison to the single aerosol for the scenario B + A case (0.05–0.2), producing a greater relative change than in the previous example. The addition of an extra layer does not significantly effect the DoFs within the 2–5 km range, where values are consistently 1, though a drop is noticeable for the 0–2 km and total column.

For scenario B + C we see very little increase in errors for all instruments and typically a minor reduction in the 0–2 km range and total error, particularly at high albedo and SZA, whereas the 2–5 km range does not show the same degree of variability. S-5 P shows the greatest reduction of the four, particularly at low albedo and high SZA. The DoFs mirror this behaviour by only very slight fluctuations across all ranges, instruments and scenes. The impact of a cirrus layer is clearly weaker when the aerosol is higher in the atmosphere and thus more disentangled from the surface.

An alternative method for the aerosol retrieval is to retrieve a small number of parameters of an assumed profile shape instead of retrieving a full extinction profile. This means that we replace the soft constraint on the aerosol profile imposed by the a priori covariance matrix with a hard constraint given by an assumed profile shape. We assume a Gaussian function for the vertical distribution of aerosol optical depth that is defined by the total AOD, centre height and width so that the 39 parameters related to the aerosol extinction profile are replaced with two retrieved parameters only (note that width is effectively not retrieved). Accordingly, the retrieval precision for the two parameters will be higher compared to the 39 parameters of the full profile retrieval.

We have calculated the errors on the retrieved AOD and centre height for the four instruments for the same scenarios as before.

Comparing the AOD error of the parameter retrieval (Table ) to the total AOD error from the profile retrieval (Figs.  and ), we find that errors are much reduced in almost all cases for scenario B, whereas for scenario A the errors are comparable for high albedo but much increased for scenarios with low albedo, with errors up to 0.57. This is not surprising, as the extinction that is retrieved as a profile is collapsed into the total AOD, which corresponds to a single parameter, whereas for the parameterised retrieval two parameters are obtained. Interestingly, the AOD error for the parameterised retrieval for the boundary-layer scenario (scenario A) with low albedo even exceeds the AOD error for the 0–2 km range from the profile retrieval. Similar to the profile retrieval, the AOD errors for scenario A are smallest for S-5 P, except for low SZA and high albedo. AOD errors for scenario A tend to be large, and only for S-5 P (for large SZA) and OCO-2 (for low SZA and high albedo) do errors approach a value of 0.05 or less. AOD errors for scenario B are well below 0.05 for GOSAT and OCO-2 as well as for CarbonSat except for low SZA and high albedo, where for S-5 P errors tend to be much higher.

The error for the centre height tends to be correlated with the AOD error and are typically small, with values around a few hundred metres. For scenario A and low albedo, the errors can exceed 1 km, with the largest errors found for CarbonSat and GOSAT.

Figure shows the AOD error for the four instruments as a function of aerosol centre height and AOD for a SZA of 30∘and albedo of 0.5 similar to the profile retrieval in Fig. . As discussed, the structure is caused by increasing information content and decreasing SNR with increasing AOD and with decreasing sensitivity to the 3–4 km when the aerosol is above or below this height.

For this high-albedo case shown in the figure, we find that the AOD error from the parameterised retrieval yields significantly smaller errors compared to the profile retrieval, especially if the aerosol centre height is 2 km or higher. AOD errors for the parameterised retrieval remain small even for low AODs with errors less than 10 % for OCO-2 for aerosols with centre heights above 1–2 km. In particular for S-5 P, the AOD errors in the parameterised and profile retrievals converge if the aerosols are near the surface.

The parameterised retrieval assumes a certain shape of the aerosol profile and any deviations from the assumed profile shape can result in biases in the retrieved AOD and centre height. To assess this effect, we have simulated scenarios that include two aerosol layers or one aerosol layer and a cirrus cloud and then retrieved them with the parameterised retrieval for a single aerosol layer using the full end-to-end retrieval. The a priori information for the retrieval has been taken from the lower aerosol layer of each simulated scenario.

Figure shows the retrieved AODs and centre heights for simulated scenarios A + B, A + C and B + C.

The total AOD for Scenario A + B is 0.6 with layers at 1 and 3 km, which appears to be well reproduced by all instruments for the high-albedo and low-SZA case. The retrieved height is around 2 km, representing the mean of the two aerosol layers in the simulation. For all other cases, OCO-2, GOSAT and CarbonSat appear mostly sensitive to the upper aerosol layer, and thus the total atmospheric AOD is underestimated. S-5 P shows some sensitivity to the lower layer and retrieved AOD values are consistently larger and retrieved heights lower compared to the other three instruments.

For the scenario A + C that combines boundary layer aerosol with a cirrus cloud, we find that retrieved AOD and height vary substantially from case to case and for the different instruments. Again, for the high-albedo and low-SZA case, the retrieved AOD reproduces approximately the true AOD of 0.4 and the retrieved height represents roughly a weighted average value of the two layers in the simulation. For all other cases, a centre height and AOD resembling those of cirrus layer are observed for instruments with high spectral resolution (OCO-2 and GOSAT), while for S-5 P and CarbonSat the retrieved height and AOD are more variable, with sensitivity to both layers.

For scenario B + C, we find that results for all four instruments are very similar to each other with a tendency to overestimate the total AOD for high albedo and low SZA and to underestimate AOD for all cases. The retrieved height tends to represent or somewhat overestimate the average height except for the high-albedo and low-SZA case, where the retrieved height is the height of the lower layer.

For the aerosol retrieval, we have imposed only a very weak constraint on surface albedo with an a priori uncertainty of 1. As already discussed in the previous sections, the errors for the AOD retrieval are especially large when the aerosol is near the surface due to strong correlations between AOD near the surface and surface albedo. Imposing a much tighter constraint on surface albedo will help to reduce these correlations, which will then lead to an improved precision of the AOD retrieval.

To investigate the potential benefit of albedo constraints, a series of retrieval simulations with varying levels of albedo constraint have been carried out.

The inferred errors of the total AOD for scenario A are shown in Fig. . As expected, we find that the error for the AOD retrieval decreases with increasing constraint on the surface albedo, but significant improvements in the errors are only obtained once the assumed albedo a priori uncertainty falls below a certain threshold. In the high-albedo case for OCO-2, CarbonSat and GOSAT, this threshold is around 0.1 (or 20 %) for a SZA of 70∘, though it reduces to 0.01 (or 2 %) for a SZA of 30∘. Again, we find that S-5 P behaves differently and a stronger constraint on the surface albedo is needed for the high-albedo cases. For the low-albedo case, the threshold is between 0.01 and 0.03 (20 to 60 %).

Overall, we find that constraining the surface albedo will help to reduce the errors of the AOD retrieval, but the required level of a priori knowledge of the surface albedo is very variable and can be very high for some scenarios.

In the retrieval simulations so far, we have assumed that the surface pressure is sufficiently well known and would not need to be retrieved. To assess the impact of a potential bias in surface pressure on the retrieved aerosol parameters, we have simulated spectra for scenario A with a surface albedo of 0.5 and a SZA of 30∘. The spectra have then been retrieved using the full end-to-end retrieval, with the parameterised aerosol retrieval as described in Sect. , but with a systematic bias in surface pressure of 2 mb. The resulting biases in AOD and height are shown in Table .

Overall, we find that instruments with higher spectral resolution coupled with SNR are less sensitive to biases in surface pressure. A 2 mb bias in surface pressure results in a 0.1 (30 %) bias in retrieved AOD for S-5 P, CarbonSat and GOSAT and of 0.07 (23 %) for OCO-2. The bias in height is typically less. For a 2 mb surface pressure bias, a 200 m or more bias in retrieved height is obtained for S-5 P, CarbonSat and GOSAT, whilst a bias of only 70 m is obtained for OCO-2.

An additive temperature-scaling factor is a retrieved parameter, but the retrieval of aerosol is still subject to potential errors caused by a systematic difference in the shape of the atmospheric temperature profile. To assess this effect, a perturbation of 1 K has been added to the bottom kilometre of the assumed atmospheric profile before performing the retrieval. The inferred biases in AOD and height for the same scenario as above are given in Table . Overall, we find that biases are small: the largest bias for S-5 P is 0.02 for AOD and of 74 m for height. Similarly to surface pressure, we find that instruments with high resolution are less affected and biases are an order of magnitude smaller compared to S-5 P.

All retrieval simulations so far have assumed that the aerosol mixture is known. Usually this is not the case and wrong assumptions on the aerosol mixture can result in systematic errors in the retrieved aerosol parameters. Again, we have simulated spectra for scenario A for a surface albedo of 0.5 and a SZA of 30∘. We have used the full end-to-end retrieval to retrieve the simulated spectra generated using the dusty maritime and course dust mixture (mixture 2b) and carried out the retrieval for each of the other 12 aerosol mixtures from .

These 12 mixtures typically represent smaller particles with Angstrom coefficients as high as 1.16 for the continental carbonaceous and black-carbon mixture (mixture 5a). The single scattering albedos range from 0.856 for the carbonaceous and continental black-carbon mixture (mixture 5b) to 0.978 for the carbonaceous and dusty maritime mixture (mixture 1a).

The results are shown in the bottom half of Table . The biases observed for the AOD retrieval are all negative, which means that the AOD has been underestimated when using a mixture different to mixture 2b used in the simulation. AOD biases found for OCO-2, GOSAT and CarbonSat are very similar; the largest biases of up to ∼ 45 % are found for mixtures with the largest Angstrom coefficient, reflecting smaller particles (carbonaceous and black-carbon continental mixtures, mixtures 5a–5c; carbonaceous and dusty continental, mixture 4a; carbonaceous and black-carbon maritime, mixtures 4a–4b; carbonaceous and dusty maritime, mixture 1a). AOD biases observed for S-5 P are smaller at less than 30 % but with the same trend regarding the Angstrom coefficient.

For the high-resolution instruments OCO-2, GOSAT and CarbonSat we find an overestimation of the aerosol layer height of a few hundred metres with a tendency for mixtures with small particles to result in larger biases. However, the trend with Angstrom coefficient (or particles size) is much weaker compared to the AOD biases, and we find the largest biases for aerosol mixtures with the lowest value of the scattering phase function in the direction of the scattering angle (carbonaceous and black-carbon maritime, mixture 3a) or the lowest value of the single scattering albedo (carbonaceous and black-carbon continental, mixture 5b). Again the biases for height are significantly smaller for S-5 P compared to the instruments with high resolution and biases not exceeding 100 m.