Aerosol-type retrieval and uncertainty quantification from OMI data

We discuss uncertainty quantification for aerosoltype selection in satellite-based atmospheric aerosol retrieval. The retrieval procedure uses precalculated aerosol microphysical models stored in look-up tables (LUTs) and top-of-atmosphere (TOA) spectral reflectance measurements to solve the aerosol characteristics. The forward model approximations cause systematic differences between the modelled and observed reflectance. Acknowledging this model discrepancy as a source of uncertainty allows us to produce more realistic uncertainty estimates and assists the selection of the most appropriate LUTs for each individual retrieval. This paper focuses on the aerosol microphysical model selection and characterisation of uncertainty in the retrieved aerosol type and aerosol optical depth (AOD). The concept of model evidence is used as a tool for model comparison. The method is based on Bayesian inference approach, in which all uncertainties are described as a posterior probability distribution. When there is no single best-matching aerosol microphysical model, we use a statistical technique based on Bayesian model averaging to combine AOD posterior probability densities of the best-fitting models to obtain an averaged AOD estimate. We also determine the shared evidence of the best-matching models of a certain main aerosol type in order to quantify how plausible it is that it represents the underlying atmospheric aerosol conditions. The developed method is applied to Ozone Monitoring Instrument (OMI) measurements using a multiwavelength approach for retrieving the aerosol type and AOD estimate with uncertainty quantification for cloud-free over-land pixels. Several larger pixel set areas were studied in order to investigate the robustness of the developed method. We evaluated the retrieved AOD by comparison with ground-based measurements at example sites. We found that the uncertainty of AOD expressed by posterior probability distribution reflects the difficulty in model selection. The posterior probability distribution can provide a comprehensive characterisation of the uncertainty in this kind of problem for aerosoltype selection. As a result, the proposed method can account for the model error and also include the model selection uncertainty in the total uncertainty budget.


Supplement
Computational implementation of the method This supplementary material presents a pseudo-code for implementation of the used method introduced earlier in paper Määttä et al. (2014) step-bystep for one Ozone Monitoring Instrument (OMI) pixel. The method is based on Bayesian inference approach.
• Averaged posterior distribution p avg (τ | R obs ) given as a discrete set of values for τ in the range of [0,τ max ] and stored in a table • Point estimate for AOD at 500 nm determined as maximum a posteriori (MAP) estimate, i.e. mode of the averaged posterior distribution We use a symbol τ for AOD in the formulas. The modeled reflectance R mod (τ, λ) depends on τ and is calculated by interpolation between nodal values of LUT while fitted to the measured reflectance R obs in order to find τ that minimizes Here is the residual of model fit. This is done for each aerosol microphysical model in turn. In the formula σ 2 obs (λ) are the measurement error variances and C is non-diagonal covariance matrix for model discrepancy (i.e. forward modelling uncertainty). In our experiment we calculated the elements of the covariance matrix C for wavelength pair λ i and λ j as

SUPPLEMENT
where parameter l is a correlation length, parameter σ 2 0 is non-spectral (i.e. non-spatial) diagonal variance and σ 2 1 is spectral (i.e. spatial) variance. We like to note that our used parameter values are specific for this study with OMI data and have been empirically evaluated. These parameter values were estimated from an ensemble of the residuals, i.e. the differences between the observed and modeled reflectances, as described in the paper Määttä et al. (2014). Here we used l = 90 nm and for σ 2 0 and σ 2 1 we used values of 1% and 2% of the observed reflectance, respectively. By Bayes' formula the posterior distribution for τ within the model m and given the observed reflectance R obs is In this case we have one unknown τ (i.e. AOD at 500 nm) and the full posterior distribution is calculated as described below.
The posterior is evaluated at a dense grid, e.g. at 200 points, of τ values, basically in the range of [0, τ max ]. The maximum allowed τ max is determined by the model LUT.
We calculate the likelihood as where χ 2 mod (τ ) is calculated from Eq. 1 for the set of τ values in the range of [0, τ max ]. The constant c ensures that the probability distribution is properly defined and it is the same for all the models m.
We assume that a prior distribution p(τ |m) for τ within aerosol microphysical model m follows a log-normal distribution This confirms that p(τ |m) can take only positive real values and ensures that AOD is positive. We set mean value τ 0 = 2 for the log-normal distribution.

SUPPLEMENT
Consequently, we have now calculated all the elements of the posterior distribution for τ (Eq. 3).
In our study we call p( R obs |m) as the model evidence that is used to make the model selection. We select models with the highest evidence value until the cumulative sum of the selected models' evidences pass the value of 0.8 or the number of chosen models is 10.
Next we calculate relative evidence for model m i with respect to the other models selected above (max 10) by These relative evidence values are used to compare models among the set of selected best fitting models.
The averaged posterior distribution over the selected best models m i is calculated as where n is the number of models.
We do this test only for the best model.
As a summary, we do the following for model selection, calculation of posterior distributions and getting MAP estimate of AOD: