This paper presents a weighted least squares approach to retrieve
aerosol layer height from top-of-atmosphere reflectance measurements
in the oxygen A band (758–770

Algorithms that estimate properties of atmospheric species from satellite measurements of top-of-atmosphere (TOA) radiance (including spectral signatures of gases) in planetary atmospheres typically employ an inverse method based on least squares. When retrieving terrestrial properties, this approach requires spectrally resolved measurements of the TOA Earth radiance, solar irradiance and a forward model as the minimal base ingredients with which the state vector parameters can be retrieved (which are also model parameters). The goal of the least squares approach is to minimize a cost function, which aims to reduce discrepancies between the forward model and the measurement by iteratively manipulating state vector parameters. Upon minimization, the iterative scheme converges to a solution that, in principle, best describes the forward model's representation of the measurement.

Many atmospheric retrieval algorithms employ a weighted least squares
estimation (WLSE) method modified to include a priori information on
the state vector. An example of such an inverse method set-up is
optimal estimation

Due to the large spectral variability in absorption within the oxygen A band, the measured TOA radiance and the measurement noise have a high dynamic range. The minimization of the propagation of measurement noise to the final retrieval solution should be a critical component of any retrieval algorithm. In WLSE, this is accomplished by the inverse measurement error covariance matrix, which ranks the measurement on each detector pixel using the information available on the measurement noise. Due to the extent of the dynamic range of the measurement noise in the oxygen A band, this ranking matrix becomes a primary controlling entity; if the measurement noise is very large, the inverse noise variance is very low, which results in a lower rank to the measured signal from that specific detector pixel.

Since the measured signal is scene dependent, the spectral rank of
each detector pixel is also scene dependent. This has special
consequences over bright surfaces, where the dynamic range of the
measured signal is much larger than over dark surfaces. Due to this,
photons at wavelengths where the oxygen A band has a lower absorption
cross section are less absorbed (subsequently travelling further into
the atmosphere) and have a much larger representation in the WLSE
method. A consequence of this, reported by

In order to account for unknown instrument and model errors,

The retrieval algorithm is described in Sect

The algorithm is comprised of a forward model and an inverse
method. The forward model uses a radiative transfer model described by

The atmospheric model describes the interaction of photons with
various components of the Earth's atmosphere that either absorb
photons or scatter them in different directions. The oxygen absorption
cross sections are derived from the NASA Jet Propulsion Laboratory
database, and first-order line mixing and collision-induced absorption
between

The radiative transfer calculations are done line by line within the
wavelength range of 758–770

The solar irradiance and Earth radiance are convolved with an
instrument spectral response function (ISRF)

OE is a maximum a posteriori (MAP) estimator designed to find a
solution for unknowns

Biases in retrieved

A spectrometer's detector pixel (in the spectral dimension) that
contains a higher concentration of oxygen absorption lines receives
less number of photons, in comparison to spectral points that contain
fewer or no absorption lines. As a result of this, the relative error
at these spectral points is larger, resulting in a lower
signal-to-noise ratio (SNR). The expression of noise in the

If the information on ALH is derived from absorption by oxygen, this
design of the

Input parameters for synthetic experiments.

The dynamic scaling method identifies favourable spectral points for
ALH retrieval by first identifying spectral points that are the least
favourable. The noise is increased at these unfavourable points while
keeping the noise at the other points unchanged. These favourable and
unfavourable spectral points are identified using a class of vectors
known as modifying vectors (with the symbol

To identify the unfavourable spectral points at which the measurement
noise is to be modified, a modifying vector

Spectral points with a

The reason for increasing the noise at specific unfavourable spectral
points is to increase the value of

The choice of modifying the SNR based on

To run a retrieval using the dynamic scaling method, the derivatives
of the reflectance with respect to

Examples of modifying vectors and

To demonstrate the dynamic scaling method, synthetic spectra are
generated for randomly varying values in

Results of the retrieval accuracy of

MODIS Terra images of the two test cases.

The synthetic spectra generated assume an aerosol layer thickness
(

In comparison with the formal approach
(Fig.

To generate errors in surface albedo, randomly varying relative
errors (with respect to the true surface albedo in the synthetic
spectra) ranging between

The analysis of retrieval biases from the synthetic sensitivity
analyses are very encouraging for the dynamic scaling method. The
method has shown significant improvements for

Results from processing 85 GOME-2A pixels over Russia on
8 August 2010 using the formal approach and the dynamic scaling
method. Empty GOME-2A pixels with a white border represent
non-convergences.

The GOME-2 instrument is a part of an operational mission by the
European Organization for the Exploitation of Meteorological
Satellites (EUMETSAT) to monitor trace gases and aerosols in the
atmosphere. It is a spectrometer with an across-track scanning mirror
that projects the TOA Earth radiance and solar irradiance through a
prism on a grating to get information in the ultraviolet, visible and
the near-infrared regions of the electromagnetic spectrum. In the
oxygen A band, the spectral sampling interval is typically about
0.20

In this section, measured spectra from the GOME-2A instrument on board
the Metop-A satellite over Russian wildfires on 8 August 2010
(Fig.

Histograms of fitted aerosol optical thickness (

Auxiliary information required for these retrievals are meteorological
data, surface albedo, and a priori values for the optimal estimation
(Table

Input data and algorithm set-up for retrieving aerosol properties from GOME-2 measurements in the oxygen A band.

GOME-2A-derived aerosol layer heights co-located within
100

For validation, atmospheric lidar data from satellite and ground-based
instruments are chosen. For the 2010 Russian wildfires, the lidar-attenuated backscatter at 1064

Ceilometer stations in western Europe used for validating the retrieved

The wildfire plumes in and around Moscow on 8 August 2010 are
chosen as the test case for the dynamic scaling method. Anti-cyclonic
conditions on this day meant that the region of interest was
predominantly cloud-free. This case is the same as analysed in

On applying the formal ALH retrieval approach, 49 pixels converge and
36 pixels do not converge to a solution (Fig.

Retrieval results from GOME-2 experiments. Columns marked
with A, B, C and D are mean retrieved

Results from processing 206 GOME-2B pixels over western
Europe using the formal approach and the dynamic scaling
method. Empty GOME-2B pixels with a white border represent
non-convergences.

Applying the dynamic scaling method to the same scenario, we observe
an increase in the number of convergences to 78 pixels out of the 85
chosen (60 % increase compared to the formal approach), as shown
in Fig.

The October 2017 Portugal wildfires began in the third week of
October. On 16 October, the hurricane Ophelia made landfall over
Ireland as a midlatitude cyclone. Due to the cyclonic conditions the
forest fire aerosol plumes were pulled from Portugal into western
Europe along with Saharan desert dust

Out of the 206 pixels, 161 pixels converge to a solution from the
formal approach (Fig.

Comparing the retrieved

Validation of the retrieved aerosol layer height over western
Europe from ceilometers located in the Netherlands and Germany from the
CEILONET and DWD networks. The black lines represent averaged
ceilometer profiles of acquisitions 1 h before and after the
GOME-2B overpass over each location (600 profiles). The profiles are
uncalibrated raw attenuated backscatter

The LER of a scene tells us which surface is brighter. In this case,
the surface in the 2010 Russian fires was brighter than that in
the 2017 western Europe case. The values of the modifying vectors

A comparison of the calculated matrices in the dynamic
scaling method for all chosen GOME-2 pixels as a function of
wavelength calculated for

Inversion algorithms that retrieve aerosol properties from spectral
measurements in the oxygen A band (between 758 and 770

The optimal estimation framework, an application of the weighted least squares technique, is designed to rank data points (in this case, spectral points in the measured TOA radiance and solar irradiance) higher when the SNR is higher, in order to reduce the influence of measurement error in the final retrieved solution. In the oxygen A band, these spectral points coincide with weak oxygen absorption cross sections, since low absorption equates to a high number of photons that can traverse through the atmospheric medium. Over oceans, due to its low albedo the number of photons that travel back from the surface are few. The signal recorded by satellites from an ocean scene hence predominantly arises from scattering and absorption by atmospheric species (in this case, aerosols). Over land, however, the number of photons that travel back from the surface increase dramatically. Due to this, the optimal estimation framework ranks spectral points representing photons that have travelled back from the surface higher than those from aerosol layers. This is the primary error source when it comes to biases in aerosol retrievals from oxygen A band measurements over land.

This paper introduces the dynamic scaling method, which is designed to retrieve ALH over bright surfaces from oxygen A band measurements. The core principle of this proposed improvement is the wavelength-dependent modification of the measurement error covariance matrix by the subsequent wavelength-dependent modification of the signal-to-noise ratio of the measured spectrum, in order to reduce its preference towards photons that interact with the surface. The modification uses the scene-dependent Jacobian matrix, which makes it robust. The dynamic scaling method is compared with a formal optimal estimation approach by retrieving ALH and AOT from synthetically generated spectra with randomly varied model parameters and model errors (that is, the forward models for simulation and retrieval have different model parameters). The results from the synthetic experiments generally favour the dynamic scaling method, which shows a significant improvement in the accuracy of retrieved ALH in the presence of errors in the assumed aerosol geometric thickness and the surface albedo (up to 10 % relative errors) in the model.

The dynamic scaling method is also demonstrated for real spectra by using GOME-2A and GOME-2B oxygen A band measurements of two separate wildfire incidences in Europe, one being the 8 August 2010 Russian wildfires and the other being the more recent 17 October 2017 Portugal wildfires. In the case of the 2010 Russian wildfires, the formal optimal estimation retrieval approach produces few convergences and misses out the primary biomass burning aerosol plume (as observed from a MODIS Terra image). The fitted AOT are unrealistically high and spatially inconsistent with the aerosol plume observed by MODIS Terra. Co-located CALIOP lidar profiles show that the retrieved ALH is biased low in the atmosphere, closer to the surface. The dynamic scaling method, on the other hand, increases the number of converged pixels by 60 % in comparison to the formal approach. The fitted AOT is still too high, but the spatial distribution of the AOT compared to same observed in the MODIS Terra image is consistent. The retrieved ALHs are also more realistic, as they are positioned close to the centroid of the CALIOP-backscatter-profile-describing aerosols. For the Portugal wildfire plume on 17 October 2017 over western Europe, the dynamic scaling method does not increase the number of convergences significantly. The dynamic scaling method retrieves ALHs that are only slightly higher and fits AOTs at values are slightly lower in comparison to those from the formal approach. The retrieved heights from both methods are compared to lidar profiles from the EUMETNET ACL network of ceilometers. The comparison shows that both methods retrieved heights that are within the profiles that could be associated with aerosol layers. Analysing a radiosonde profile of the relative humidity and calculated back trajectories, it is observed that the ceilometer profiles miss higher aerosol layers due to attenuation of the signal at lower atmospheric levels. This explains why the retrieved heights from the dynamic scaling method are slightly higher than the same from the formal approach.

In general, the dynamic scaling method improves the number of converged pixels. Between the two discussed cases, the dynamic scaling method provides a better improvement in the 2010 Russian wildfires case. This is primarily because the method is scene dependent. An important driver that determines the improvement of retrievals is the level at which the surface influences the TOA reflectance, which is jointly influenced by two parameters – the surface albedo and the AOT. The average surface albedo of the scene for the 2010 Russian wildfires was observed to be brighter than the same for the 2017 Portugal wildfires. This is a possible explanation for the differences in the performance of the dynamic scaling method for the two cases.

The fitted AOT is systematically lower for the dynamic scaling method in comparison to the formal approach. A part of this can be attributed to the reduction in the influence of spectral points in the measurement with a larger influence from the surface albedo. While this is expected, the method does not necessarily make the fitted AOT more realistic. It may well be the influence of assumptions in aerosol properties such as aerosol single-scattering albedo and the phase function. It could, however, also be that the method does not fully remove the influence of surface in the measured top-of-atmosphere reflectance signal. In any case, the dynamic scaling method improves the representation of the fitted AOT of the MODIS Terra observed smoke plume.

The dynamic scaling method is designed to modify the signal-to-noise ratio to an extent that is necessary and sufficient to reduce the influence that photons travelling from the surface back to the detector have on the weighted least squares estimate of aerosol properties. Using the Jacobian to dictate the preference of weight least squares for spectral points in the measurement makes the dynamic scaling method a robust, generally applicable retrieval set-up. Results from this paper are applicable to other algorithms using weighted least squares techniques to retrieve atmospheric properties from measurements of top-of-atmosphere reflectance in the oxygen A band over bright surfaces.

All GOME-2 data used for processing in this paper are freely available from the EUMETSAT data centre:

The authors declare that they have no conflict of interest.

This research is partly funded by the European Space Agency (ESA) within the EU Copernicus programme under the project name “Sentinel-4 Level-2 Processor Component Development”, number AO/1-7845/14/NL/MP. We acknowledge EUMETSAT for providing the GOME-2 L1b data. We thank Ina Mattis from the DWD and Marijn de Haij from the KNMI for providing us with valuable ceilometer profiles for validating satellite retrievals. We would also like to thank Marc Allaart from KNMI for providing the radiosonde profiles and Rinus Scheele from the KNMI for calculating the back trajectories. Edited by: Alexander Kokhanovsky Reviewed by: two anonymous referees