We have developed a Bayesian aerosol retrieval (BAR) algorithm for the
retrieval of aerosol optical depth (AOD) over land from the Moderate Resolution
Imaging Spectroradiometer (MODIS). In the BAR algorithm, we simultaneously
retrieve all dark land pixels in a granule, utilize spatial correlation
models for the unknown aerosol parameters, use a statistical prior model for
the surface reflectance, and take into account the uncertainties due to fixed
aerosol models. The retrieved parameters are total AOD at 0.55

Atmospheric aerosols are small solid or liquid particles suspended
in the atmosphere.
They have a significant effect on the climate

The Moderate Resolution Imaging Spectroradiometer (MODIS) on board
NASA's Terra and Aqua satellites are among the oldest operating
instruments orbiting the Earth and collecting information on Earth's surface
and atmosphere.
Terra and Aqua are both polar-orbiting satellites with wide swaths
and they scan the entire surface of the Earth every 1–2 days.
The primary operational algorithm to retrieve aerosol properties,
such as the aerosol optical depth (AOD), is the Dark Target (DT),
which uses MODIS data measured over dark surfaces

Another widely used retrieval algorithm for MODIS is
Deep Blue (DB;

Both the DT and DB carry out the retrieval pixel by pixel.
This means every pixel is retrieved independently of each other.
This pixel-by-pixel approach makes the algorithm computationally efficient.
Often, however, aerosol properties have strong spatial correlations

In this work, we developed a Bayesian aerosol retrieval (BAR) algorithm
for MODIS aerosol retrieval over land.
The new algorithm is based on the DT algorithm and
the inversion part of the algorithm is reformulated as a statistical
(Bayesian) inverse problem

MODIS aerosol products retrieved using the DT are among the
most widely used aerosol products.
The MODIS C6 standard aerosol products
include the retrieved aerosol properties and measurement data
with spatial resolution of about 10

BAR is a retrieval algorithm that uses the same aerosol models and
preprocessing of the data, such as cloud-screening, as the DT.
Because the same preprocessing is used, the BAR algorithm retrieves the same
pixels as the operational DT algorithm.
In BAR, the inversion part of the DT algorithm is formulated in a statistical (Bayesian) framework.
In this statistical framework, the solution to the inverse retrieval problem is not a single value but
a posterior probability distribution model of the unknown parameters given the
measured MODIS TOA reflectances and prior information that we have on the unknowns.
As the complete statistical model of the problem is the posterior probability distribution,
it allows us to derive single point estimates that are referred to as the retrievals
and quantify the posterior uncertainties of the retrievals for each pixel.
The statistical framework also allows us, for example, to utilize information about the measurement
noise and use data from as many MODIS spectral bands as available for the retrieval.
The BAR algorithm is characterized by the following:

We use data from MODIS bands 3 (0.47

We retrieve the total AOD at 0.55

The surface reflectances at all bands are simultaneously retrieved with AOD and FMF. The surface reflectance relationships that are used in DT are not needed.

We simultaneously retrieve all unknown parameters in all dark land pixels of a granule.

We use prior probability density models for the values and the spatial correlation structure of the unknowns. The prior probability density models are used to encode the prior knowledge such as spatial correlation information, seasonal variability, or positivity constraints into the retrieval.

We utilize an approximation error model for the model uncertainties in the simulated TOA reflectances caused by the uncertainties in the aerosol models and radiative transfer simulations.

In the BAR AOD retrieval, statistical prior models for the retrieved parameters can be used.
We make the following modeling selections in the BAR:

To avoid negative AOD retrievals, we retrieve AOD in logarithmic scale

Instead of TOA reflectances

We model all unknown parameters in a granule by multivariate Gaussian prior models.
The prior models are fully described by their expected value vectors and covariance matrices:

AOD

FMF

surface reflectances

We model AOD, FMF, and surface reflectances at all bands as mutually uncorrelated variables.

We model the observation noise and the approximation errors in TOA reflectances due to aerosol and radiative transfer
models as additive multivariate Gaussian random variable

In the BAR, we look for the maximum a posteriori (MAP) estimate for the unknown parameters.
The prior and likelihood models that are used in the construction of the posterior model are explained in more detail in Sect. 3.
With the models selected, the MAP estimate can be computed as

To quantify the uncertainties corresponding to the retrieved parameters we can compute
an approximation for the posterior covariance matrix as

Prior probability density models are used in the BAR retrieval to model information we have on unknown parameters prior to the retrieval. In the BAR, we use Gaussian prior models augmented with constraints that exclude non-physical solutions. For example, for the FMF the retrieval is restricted to an interval between 0 and 1. In practice, these constraints are implemented in the optimization algorithm. The multivariate Gaussian prior models are defined by their expected value vector and covariance matrix. In aerosol retrievals, the expected value vectors for aerosol parameters can be constructed, for example, by using values from aerosol climatologies. Covariance matrices encode information on the prior uncertainty of the parameters and correlations between different pixels.

In the BAR algorithm, the AOD is retrieved on a logarithmic scale to
avoid negative AOD retrievals and multivariate Gaussian distributions
are used as the prior models for the logarithm of the AOD.
The expected value vector for AOD is based on the MAC-V2 climatology by

The spatial correlations and variances in the logarithm of AOD are modeled
by using a covariance function that defines the AOD covariance matrix as

The covariance function parameters used in aerosol optical depth (AOD) and fine-mode fraction (FMF) prior models.

For the FMF, we use a similar Gaussian prior as for the AOD.
The prior expectation value for FMF is taken from the MAC-V2 climatology as for the AOD.
The FMF is modeled as a spatially correlated parameter and the same type of covariance
function as for the AOD is used to construct the prior covariance matrix

In the BAR algorithm, the surface reflectances at different wavelengths
are treated as unknown parameters and they are simultaneously retrieved
with AOD and FMF.
In the BAR algorithm, we use Gaussian prior models for the surface reflectances.
We model the surface reflectances at different bands as uncorrelated and the
surface reflectances at each band as spatially uncorrelated.
We note that this selection may not result in the best possible retrieval accuracy
but makes the processing of a large number of MODIS granules significantly faster
than with correlated models.
With these choices for the surface reflectance, the prior model becomes an
uncorrelated Gaussian density which is described by the expected surface reflectance
values and their variances at each pixel.
As expected values for the surface reflectance, we use the MODIS MCD43C3 albedo product blue-sky
albedos computed with the weighting coefficient 0.5 (50 % of the white-sky albedo
and 50 % of the black-sky albedo).
This selection to use the blue-sky albedo was done based on a test in which we carried out
retrievals with white-sky, black-sky, and blue-sky albedo-based prior models.
The differences between the different surface albedo types were small but the blue-sky albedo
resulted in the best results when compared with the collocated AERONET AOD values.
The daily MODIS albedo product is stored in 0.05

In the DT algorithm, the TOA reflectance

To make the retrieval algorithm computationally efficient, the values of

Before the DT retrieval is carried out, the LUTs are prepared for the retrieval. The LUT models are first interpolated to the fixed measurement geometry and then corrected for the surface elevation. In the retrieval, the LUT models are then evaluated by linearly interpolating the values as function of total AOD. In BAR, we use the same LUTs (for four different bands) as in the DT. While the DT algorithm uses piecewise linear interpolation, in BAR we use fifth-order polynomial interpolation of the LUTs in order to make the model differentiable with respect to the unknown AOD at all points. The differentiability is required as the retrieval is carried out by solving an optimization problem using gradient-based methods.

In the BAR algorithm, the random observation noise in MODIS observations, for example due to
measurement electronics in the instrument, is modeled by an additive noise
process:

In the statistical (Bayesian) retrieval framework, it is possible to model the
uncertainties and inaccuracies related to the physical models that are used in the retrieval (both aerosol and radiative transfer models).
The model uncertainties can be related, for example, to uncertainty in the values of the auxiliary
model parameters such as measurement geometry and fixed aerosol models.
In the field of statistical inverse problems, these model errors are often referred
to as approximation errors

In this study, the estimation of the mean

In BAR retrieval, we model the observation noise

To test the performance of the BAR algorithm, all MODIS daytime granules of the year 2015 are used.
We retrieve all granules from Terra and Aqua (MOD04_D3 and MYD04_D3) and compare the retrievals to AERONET observations
(version 3, level 1.5).
In the AERONET collocation we follow similar comparison protocol as in

To compare the overall performance and to make the comparison fair between different algorithms, we compare all pixels in which the retrieval was carried out regardless of the DT quality assurance (QA) information of the retrieval.

To study how the DT QA information affects the retrievals, we carry out another comparison in which we use the DT and BAR retrievals only at the pixels with DT QA flag 3.

The variables we compare are the AOD at 0.55

FMF in the DT algorithm is actually the weighting coefficient between the TOA reflectances corresponding to fine and coarse aerosol models and do not necessarily correspond to physical size distribution information;

in the DT aerosol models, the fine aerosol model includes a small amount of coarse particles in it and the coarse aerosol model includes a small amount of fine particles in it;

it is ambiguous to derive AERONET-based FMF as there are multiple size-distribution-related products that are based on slightly different algorithms and definitions;

it is possible to derive AE from MODIS retrieval using the aerosol models, retrieved total AOD, and FMF, and the AE is also available in the AERONET Direct Sun algorithm outputs.

The metrics we use to evaluate the retrieval algorithm performance and compare the MODIS and AERONET retrievals are correlation coefficient

Regions used in the evaluation of the algorithm: West North America (WNA), East North America (ENA), Central and South America (CSA), Europe (EUR), North Africa and Middle East (NAME), South Africa (SA), Northeast Asia (NEA), Southeast Asia (SEA), and Oceania (OCE). The red and blue dots show positions of all the AERONET stations used in the comparisons. The blue dots indicate stations classified as an urban station in the study.

Figure

Figure

The global performance of the algorithm was evaluated using all the daytime retrievals from
the year 2015.
Figure

The results show that the BAR AOD retrievals are significantly more accurate
than the corresponding DT or DB retrievals when compared to the AERONET AOD.
The fractions of retrievals inside the DT EE envelope (

Figure

The results for global AE retrievals for the DT and BAR algorithms
are shown in Fig.

We also evaluated the effect of using the
approximation error model and spatial correlation models
in the retrieval.
The retrievals were carried out in all granules in year 2015
with and without the approximation error model and with and without
the spatial correlation models for the AOD and FMF.
In the retrievals without spatial correlation models, we set the
off-diagonal elements of the prior covariance matrices as zeros
both for AOD and FMF.
The results are shown in Tables

Global statistics of AOD retrievals for Bayesian aerosol retrieval (BAR) run with different models. The models considered are the approximation error model and the spatial correlation model for AOD and FMF. X and – in the table indicate that the corresponding model was and was not included in the retrieval, respectively. All pixels were considered in the retrieval and each row correspond to data from 346 AERONET stations and 45 240 collocated observations.

Global statistics of Ångström exponent retrievals for Bayesian aerosol retrieval (BAR)
run with different models.
The models considered are the approximation error model and the spatial correlation
model for AOD and FMF.
X and – in the table indicate that the corresponding model was and was not included in the retrieval, respectively.
Only results with AERONET AOD

Similar figure as Fig.

The global and regional results of the DT and BAR AOD retrievals with respect to the AERONET
are shown in Table

Global and regional statistics of AOD retrievals for Dark Target (DT) and Bayesian aerosol retrieval (BAR) retrieval algorithms. All DT quality assurance classes are considered. Bolded numbers indicate the algorithm with better performance.

The global and regional results of the DT and BAR AE retrievals are shown in Table

Global and regional AOD accuracy comparisons between the BAR and DB retrievals are shown in Table

Global and regional statistics of Ångström exponent retrievals for Dark Target (DT) and Bayesian aerosol retrieval (BAR) algorithms. All DT QA flags are considered. Only retrievals with AERONET AOD larger than 0.2 were included. Bolded numbers indicate the algorithm with better performance.

Global and regional statistics of AOD retrievals for Deep Blue (DB) and Bayesian aerosol retrieval (BAR) algorithms. All pixels are considered. Bolded numbers indicate the algorithm with better performance.

AOD retrievals over urban areas were evaluated by comparing the MODIS AOD retrievals over AERONET stations
that are located in urban areas.
We selected 17 AERONET stations for this comparison and the results are presented in Table

Statistics of AOD retrievals for Dark Target (DT) and Bayesian aerosol retrieval (BAR) algorithms over urban AERONET stations.
The location information for the AERONET sites can be found at the AERONET web page

The BAR algorithm provides approximate posterior uncertainties for retrieved quantities.
We evaluate the AOD posterior uncertainty estimates of the BAR algorithm
by comparing them to the discrepancies between the BAR retrievals and AERONET observations.
Table

Fraction of AERONET AODs inside

A new AOD retrieval algorithm, Bayesian aerosol retrieval (BAR), was developed.
The algorithm is based on the widely used MODIS DT algorithm.
In the BAR algorithm, the inverse retrieval problem is formulated in a statistical
(Bayesian) framework that allows systematic use of probabilistic
models for prior information and approximation errors related to
inaccuracies in the physical observation models and pixel-based
uncertainty quantification for the retrieved parameters.
In the BAR algorithm, the retrieved unknown parameters are the total
AOD at 0.550

The BAR algorithm was evaluated by retrieving all MODIS granules from the year 2015 and compared with AERONET AOD and AE. Results showed that by using the BAR algorithm the accuracy of the AOD retrievals was significantly improved when compared to both DT and DB retrievals. Globally, the fraction of AOD retrievals inside the DT EE envelope increased from 55 to 76 % when BAR was used instead of DT. Moreover, the median bias in AOD was improved, and globally the bias was 0.01 while the bias of the DT algorithm was 0.05. The AOD retrievals were improved in all studied regions and the largest improvement was found in North America. Oceania was the region with the smallest improvement. The AE retrievals were also improved in most of the regions when BAR was used instead of the DT algorithm, but the improvement was not as clear as for the AOD. The reason why the AE did not improve similarly as the AOD retrievals is a topic of future research.

The BAR algorithm gives approximate posterior uncertainties in the retrieved parameters for each pixel. We compared the AOD uncertainty estimates with absolute values of retrieval errors over AERONET stations. The results show that BAR is capable of producing feasible uncertainty estimates for AOD.

The average retrieval time with the BAR algorithm was less than 1 min per granule on a modern personal computer and therefore the computational costs of the algorithm allow the use of BAR for near-real-time processing of MODIS data. The BAR algorithm is not restricted to MODIS retrievals only and by writing the observation models for different instruments it is possible to extend the algorithm to be used for aerosol retrievals with other instruments as well. The results show that modeling and taking into account the spatial correlations of unknown parameters and model uncertainties in the retrieval may significantly improve the accuracy of the retrievals. The inversion framework is not restricted to aerosol retrieval only and could be used for other types of remote sensing applications, such as cloud and trace gas retrievals.

The first version of the BAR algorithm was constructed especially to evaluate the feasibility
and accuracy of the new modeling and inversion approach and many models and selections
can still be improved to make the algorithm better.
The planned improvements for the BAR algorithm in the future include the following:

The MAC-v2 climatology used for prior models was downloaded
from

Let

Applying the well-known Bayes theorem to the posterior distribution
(Eq.

We model the AOD, FMF, and surface reflections as uncorrelated and the noise term

Combining Eqs. (

In the BAR algorithm, we construct an approximation error model that describes the uncertainties and inaccuracies in the simulated TOA reflectances due to imperfect models and unknown aerosol and surface parameters. The construction of the model is based on simulated TOA reflectances that are compared with the reflectances measured by the MODIS instrument. In the construction of the approximation error model, the MODIS measurements are considered as the ground truth measurements.

Let

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

We thank the AERONET PIs and their staff for establishing and maintaining the AERONET sites used in this investigation. We thank NASA MODIS team to kindly make the MODIS data publicly available. Ville Kolehmainen acknowledges the Academy of Finland (Project 250215, Finnish Centre of Excellence in Inverse Problems Research). Edited by: Alexander Kokhanovsky Reviewed by: Adam Povey and two anonymous referees