Aerosol optical depths (AODs) used for the Edition 4.1 Clouds and the Earth's Radiant Energy System (CERES) Synoptic 1
Accurate estimates of the radiative effects of clouds and aerosols are essential for an understanding the radiative forcing to the Earth's climate system (Bauer and Menon, 2012; Boucher et al., 2013). In addition, through the reflection and absorption of solar radiation as well as the absorption and emission of terrestrial thermal radiation, clouds and aerosols affect the radiative heating of both the atmosphere and the surface, which in turn governs the atmospheric circulation and the hydrological cycle (e.g., Stephens et al., 2020; L'Ecuyer et al., 2015). Under the Earth Observing System (EOS) program, the National Aeronautics and Space Administration (NASA) has placed into orbit a series of satellites devoted to long-term observations of the climate state. Among these are Terra and Aqua, the flagship satellites of the EOS. Central to observation of climate evolution are the Moderate Resolution Imaging Spectroradiometer (MODIS) and the Clouds and the Earth's Radiant Energy System (CERES) instrument pairs that fly on both the Terra (March 2000–present) and Aqua (July 2002–present) platforms (Wielicki et al., 1996). Additional CERES instruments have been launched (October 2011) upon the Suomi National Polar-orbiting Partnership (NPP) satellite along with the MODIS successor, the Visible Infrared Imager Radiometer Suite (VIIRS), and on the NOAA-20 satellite (November 2017). In addition to observations from these satellites, the CERES mission also integrates observations from the Geostationary Operational Environmental Satellites (GOES) (West and East), as well as other geostationary satellites around the globe, for full diurnal coverage of clouds and radiation.
The CERES instruments measure broadband radiances over the solar spectrum (shortwave), the thermal infrared (longwave radiance is obtained from a total channel minus the shortwave channel), and the near-infrared atmospheric window, with frequent onboard calibration. CERES measurements, in conjunction with MODIS information, are used to infer broadband irradiances through empirical angular distribution models (ADMs). Geosynchronous satellite imagery observes the diurnal cycle of clouds, which is not fully sampled by the polar-orbiting satellites upon which CERES and MODIS reside.
While top-of-atmosphere (TOA) irradiances are derived from broadband radiances measured by CERES instruments (Loeb et al., 2005; Su et al., 2015a, b), surface and atmospheric irradiances are computed with a radiative transfer model. Inputs used for the computations include cloud properties derived from MODIS and geostationary satellites, aerosol optical depth (AOD) derived from MODIS radiances, and surface albedo derived from MODIS and CERES observations (Rutan et al., 2009). Temperature and humidity profiles are provided by a reanalysis product produced by the NASA Goddard Modeling and Assimilation Office (GMAO).
Irradiances at the surface produced by the CERES team have been compared with surface observations (Rutan et al., 2015; Kato et al., 2013, 2018). These comparisons are for all-sky conditions (i.e., including any clouds). Irradiances under clear-sky conditions are not explicitly separated from all-sky conditions in the evaluations. There are several factors that impede efforts at rigorous validation of clear-sky irradiances with surface observations: (1) a clear-sky condition at a given site does not persist over a long time period (e.g., a month or longer), (2) there are mismatches between clear-sky conditions determined by satellite- and ground-based instruments, and (3) the field-of-view size between CERES instruments and ground-based radiometers differ.
Despite difficulties in evaluating computed clear-sky irradiances, they play an important role in quantifying aerosol and cloud radiative effects (Loeb and Su, 2010; Soden and Chung, 2017). Therefore, the uncertainty in surface irradiances needs to be understood in order to assess the uncertainty in the aerosol and cloud radiative effect. This work is the first attempt by the CERES team to evaluate clear-sky surface irradiances provided by its data products. One of the essential variables in computing clear-sky irradiances is AOD. In this paper, we evaluate the AOD used for irradiance computations in the CERES project and analyze how the error propagates to clear-sky surface irradiances. Computations of surface irradiances provided by the Edition 4.1 SYN1deg data products use AOD derived by a chemical transport model (the Model for Atmospheric Transport and Chemistry – MATCH; Collins et al., 2001) that assimilates MODIS-derived AOD. In Sect. 2, we explain in the MATCH aerosol transport model and the assimilation of AOD with MODIS. We then compare MATCH AOD to the MODIS and Modern-Era Retrospective analysis for Research and Applications (Version 2; MERRA-2) aerosol products as well as to AOD from the AErosol RObotic NETwork (AERONET; Holben et al., 1998). Section 3 discusses differences found between the various estimates of AOD. Section 4 looks at clear-sky surface irradiance calculations from the SYN1deg product compared to observed values as well as the impact of AOD and particle size on the results. Conclusions are presented in Sect. 5.
The Model for Atmospheric Transport and Chemistry (MATCH) is a transport model of intermediate complexity driven by off-line meteorological fields from the National Centers for Environmental Prediction (NCEP) reanalysis. It is run on a
Aerosol types and climatological sources.
Aerosol types included in MATCH are dust, sulfate, sea salt, soot, sulfates, carbon, and volcanic particles (Table 1). Model physics included in MATCH are parameterizations for convection and boundary layer processes that include prognostic cloud and precipitation schemes for aqueous chemistry and the scavenging of soluble species. MATCH also includes the ability to resolve the transport of aerosols via convection, boundary layer transport, and scavenging and deposition of soluble gases and aerosols. MATCH can simulate most cloud processes currently in use in a global climate model (GCM) (e.g., cloud fraction, cloud water and ice content, fraction of water converted to rain and snow, and evaporation of condensate and precipitate). It also includes vertical turbulent-eddy processes. These processes are then used for convective transport, wet scavenging, wet deposition, and dry deposition of the MATCH aerosols. These various parameterizations were originally developed for the NCAR Community Climate Model (CCM) and subsequently incorporated into the MATCH model. Descriptions of these parameterizations are given by Rasch et al. (1997, 2001), Collins et al. (2001), and additional papers described therein.
The MATCH aerosol suite includes a detailed mineral dust scheme in the Dust
Entrainment and Deposition model (Zender et al., 2003) and a diagnostic
parameterization for sea salt aerosol based on the 10 m wind speed (Blanchard and Woodcock, 1980). The sulfur cycle and the chemical reactions for sulfate aerosol creation rely on monthly climatological oxidant fields and emission inventories (Table 1) for sulfur oxides and oceanic dimethyl sulfide (photochemistry and nitrate aerosol are omitted). The reaction scheme is similar to that of the Model for Ozone and Related Chemical Tracers (MOZART; Emmons et al., 2010). Carbon aerosols (both organic compounds and soot) evolve with simple mean lifetime
The optical properties of the various aerosol types (e.g., mass extinction coefficient and single-scattering albedo), which are key parameters for aerosol assimilation, are drawn from the standard Optical Properties of Clouds and Aerosols (OPAC; Hess et al., 1998) database. However, scattering properties of maritime and dust aerosols used in the radiative transfer calculations in the SYN1deg product are not from MATCH. Instead, aerosol types from MATCH are mapped to a similar set of scattering properties (see Table 2) embedded in the Langley Fu–Liou radiative transfer (LFLRT) code (Fu and Liou, 1993; Fu et al., 1998; Rose et al., 2013). These include OPAC, as in MATCH, for all but the small and large dust particles. Dust scattering and absorption properties in the LFLRT code are from Sinyuk et al. (2003).
Mapping of MATCH aerosol types into radiative transfer code.
Figure 1 shows the single-scattering albedo (SSA) and asymmetry parameter (ASY)
for the seven constituents in the LFLRT code at 500
The single-scattering albedo (SSA) and asymmetry parameter (ASY) for the seven aerosol types available in the Langley Fu–Liou model SYN1deg calculations. Only those that vary with relative humidity are plotted; others are listed as constants. All values are for properties at 550
The difference in MATCH AOD due to the assimilation of MODIS AOD. Panel
One major advantage of the MATCH model is its ability to reliably assimilate
satellite-based retrievals of AOD to constrain the
climatologically forced aerosols generated within the chemical transport
portion of the code. Edition 4 MATCH algorithms ingest MODIS Collection 6.1
AOD (Remer et al., 2005), beginning in March 2000 from the Terra satellite
and June 2002 from both the Terra and Aqua satellites. MATCH assimilates MODIS AOD at the green wavelength of 550 nm, and it combines AOD derived by
the Dark Target (Levy et al., 2013) and Deep Blue algorithms (Hsu et al.,
2006). A global daily mean AOD in a
Climatological mean aerosol optical depth (AOD, i.e.,
The assimilation process begins by combining the Dark Target and Deep Blue AOD from MODIS (both Terra and Aqua when available) and creating daily averages. As MATCH progresses through time, the AODs at local solar noon are
assimilated by taking a 15
In this section, we compare AODs between MATCH and MERRA-2 (Randles et al., 2017) in which MODIS clear-sky radiances are assimilated. MERRA-2 also assimilates surface-observed AOD from AERONET; ship-based AOD observations; and Advanced Very High Resolution Radiometer (AVHRR) and Multi-angle Imaging SpectroRadiometer (MISR) retrievals for the years 2000–2002 and 2000–2014, respectively. We compare AODs in two different ways: first, MATCH and MERRA-2 AODs are compared with MODIS AODs, which tests the consistency of daily means when MODIS AOD is available (i.e., clear sky somewhere in the grid box at Terra and Aqua overpass time); second, MATCH and MERRA-2 AODs are compared under all-sky conditions, which is only possible with modeled AODs.
Figure 3 shows differences in the climatological mean AOD between MERRA-2 and
MODIS (panel a) and MATCH and MODIS (panel b). To compute the monthly
mean AOD differences, both MERRA-2 and MATCH daily mean AODs are sampled when
daily mean MODIS AOD (MODIS products MOD08 and MYD08) from the same
While MATCH shows large positive differences over land, especially China and
Southeast Asia, Australia, the Amazon, and North Africa, MERRA-2 shows significant negative differences over the major rain forest regions of South America, Africa, and the tropical western Pacific. Both products are closer to MODIS AOD over ocean compared with
Scatterplot of daily
Figure 4 shows the difference in
We now consider the differences between the MATCH and MERRA-2
climatological AOD fields for all-sky and
The above results indicate that both MATCH
Figure 6 shows an hourly time series of AOD from MATCH, MERRA-2, and AERONET for January 2010 at the Beijing (China) AERONET site. Figure 6a shows the cloud fraction time series derived from MODIS and geostationary imagers (GEOS) from the SYN1deg Edition 4.1 product (Rutan et al., 2015), and Fig. 6b shows the AOD time series. Generally, both models produce a large variability in AOD at this site fairly well over the course of the month. While both MERRA-2 and MATCH AODs increase near times when the cloud fraction approaches 100 %, the increase in the MATCH AOD, which correlates relatively well with the increase in AERONET AOD, is larger than the increase in MERRA-2 AOD. Although the temporal correlation coefficient of the MATCH and AERONET AODs is smaller at this site during summer months than during winter months (not shown), a good temporal correlation between MATCH and AERONET AODs is consistent across most of the locations and times that we considered. To show this statistically, in the following, we extend this analysis to a number of AERONET sites grouped geographically based on general aerosol type.
Hourly time series of the
The location of AERONET sites and how they are grouped for calculations of the mean, bias, and RMS with respect to the MATCH and MERRA-2 optical depths found in Tables 3 and 4.
AODs from AERONET are nominally provided at eight spectral
channels every 20 min, given favorable conditions. We use two channels to derive observed AOD at 550 nm for comparison with the AOD provided by the MATCH model. Because the SYN1deg radiative transfer calculation is done hourly, we average any observations within a given hour period centered on the 30th minute for each site co-located within a SYN1deg grid box. The AERONET sites chosen are shown in Fig. 7, with a complete listing of all sites given in Appendix A. Although we examine 55 sites over more than 20 years, we aggregate the statistics within continental regions which naturally isolates them by general climatic conditions. Tables 3 and 4 show comparisons for each site grouping for clear-sky (less than 1 % cloud identified by MODIS and geostationary satellites in the SYN1deg grid box) conditions and for all-sky (any cloud condition within the SYN1deg grid box) conditions, respectively. Using clear-sky scenes identified by MODIS only gives the same statistical results with a lower number of samples. Statistics shown in Tables 3 and 4 are the average observed value, mean bias (MATCH
Hourly AERONET station statistics for MATCH and MERRA-2 for continental groups under clear-sky conditions
Hourly AERONET station statistics for MATCH and MERRA-2 for continental groups under all-sky conditions
The sign of the MATCH AODs compared to AERONET AODs for all-sky conditions is generally consistent with the sign of their clear-sky counterparts. The RMS difference under all-sky conditions is generally larger than the clear-sky RMS difference, while the correlation coefficient is nearly the same. The biases for MERRA-2 comparisons are generally comparable to MATCH, although the RMS values for MERRA-2 tend to be slightly smaller and correlations tend to be higher, due in part to the assimilation of AERONET into the MERRA-2 model.
Results for all points across all sites and times are shown in Fig. 8. The color density plots are on a log scale and indicate that the vast majority of observations have an AOD of less than 1 for both the clear- and all-sky conditions observed within the SYN1deg grid box. Biases are less than 10 % of the mean value, but the RMS is large relative to the mean observed value. The overall correlation is approximately 0.8. The clear-sky hours (where SYN1deg estimated less than 1 % cloud in the grid box based on MODIS and GEOS observations) represent a little more than 10 % of the overall points. When MATCH AOD is compared to MERRA-2 AOD (not shown), MATCH is biased approximately 10 % higher.
All-sky
In this section, we investigate the reason for the AOD differences shown in the previous section. In addition, we estimate the effect of the AOD differences on surface irradiances when MATCH AODs are used for surface irradiance computations.
Generally, cloud contamination in MODIS AODs is caused by unresolved sub-pixel-scale clouds (Kaufman et al., 2005; Martins et al., 2002). Therefore, the difference shown over convective regions seems to be caused by the uncertainty due to 3D radiative effects that impact retrieved AODs by unknown amounts (Wen et al., 2007), by errors in estimating the fraction of hygroscopic aerosols, or by errors in estimating water uptake by hygroscopic aerosols (Su et al., 2008; Marshak et al., 2021). Larger AODs are screened out in the MOD08 data product, whereas the CERES team uses all retrieved AODs regardless of the QAC score, likely increasing MATCH AOD overall. The comparison with AERONET AODs is not decisive to determine how to screen MODIS AODs because MATCH AODs are positively biased and MERRA-2 AODs are negatively biased for the Brazil group. The result underscores the difficulty involved in deriving accurate AODs, which appear to involve requirements in addition to the identification of clear-sky scenes. Levy et al. (2013) list the factors lowering the QAC score as (1) pixels are thrown out due to cloud masking, (2) the retrieval solution does not fit the observation well, and (3) the solution is not physically plausible given the observed situation. Therefore, even though the difficulty in identifying clear-sky scenes is driven by cloud contamination by trade cumulus (Loeb et al., 2018), the difficulty in deriving AODs exists over convective regions (Varnai et al., 2017) as well.
Larger positive biases of MATCH AODs compared with AERONET AODs exist over Africa (Tables 3, 4). For North Africa, the bias is known to be caused by excessive dust generated by the MATCH algorithm. Even though modeled aerosols are not often used over North Africa owing to the abundance of clear-sky conditions, the dust problem leads to a larger positive AOD bias. In addition, MATCH uses fixed aerosol sources in time. Therefore, it tends to miss large aerosol events, such as forest fires, until clear-sky conditions occur, allowing observations of the event by MODIS. This leads to a larger RMS difference and lower correlation coefficient with AERONET AODs compared with those from MERRA-2 versus AERONET.
Because MODIS AODs are not generally available under overcast conditions, the
reliance on modeled AOD increases as the cloud fraction over a
AOD and precipitable water (PW) as a function of cloud fraction over the
We consider the impact of MATCH aerosols on computed surface irradiances by
comparing calculated hourly mean surface downward irradiances from the Edition 4.1 SYN1deg-1Hour product to observations of downward irradiance. In a
The location of surface observations of downwelling shortwave irradiance used to compare the SYN1deg Edition 4.1 calculations to observations for all available hours (from March 2000 through December 2019) for which the SYN1deg cloud analysis determines the hour and grid box to be 100 % clear sky.
We begin with a simple sensitivity calculation of AOD on surface downward
shortwave irradiance (DSI). Figure 12 shows a series of radiative transfer
calculations using the online Langley Fu–Liou radiative transfer code (
Calculated DSI error at the surface computed with the LFLRT model due to the error in AODs. AOD is assumed to be 0.09. The respective light and dark orange shading indicate positive and negative errors (in W m
Figure 13 shows hourly comparisons of computed clear-sky downward shortwave
irradiance compared to observations for the groups of sites shown in Fig. 11. In general, the calculated irradiance is larger than observed. We find that in every grouping, SYN1deg calculations tend to be too transmissive, overestimating DSI by between 3 W m
Comparisons of DSI at the surface from the SYN1deg Edition 4.1 calculations (
Clear-sky scenes used for Fig. 13 are those identified by MODIS and geostationary satellites over the 1
Bias (RMS) of the clear-sky surface shortwave calculation compared with observation
In this section, we consider the implications of errors in AOD and aerosol type on longwave LFLRT calculations as found in the SYN1deg product. Figure 14 shows SYN1deg surface downward longwave irradiance (DLI) calculations compared to surface observations similar to those shown in Fig. 13. Except for the polar region, where DLI is very sensitive to near-surface air temperature, the bias and standard deviations of the DLI are smaller than the shortwave equivalents in terms of both watts per square meter (W m
Comparisons of longwave downward irradiance at the surface from the SYN1deg Edition 4.1 calculations (
The effect of dust particle size on surface irradiance calculations
DLI is, thus, more sensitive to aerosol type in certain regions of the globe
where there is substantial dust. To see the potential impact on DLI, Fig. 15 shows calculated longwave downward radiative forcing (clear minus pristine calculations) at 57 AERONET sites across the 20 years of SYN1deg data under
consideration. The northwestern African sites (where dust is found seasonally) are shown as red boxes, and one clearly sees larger longwave forcing at these sites. Given the importance of particle size for the longwave effect, we check MATCH particle size against AERONET fine-/coarse-mode retrievals for several of the
African AERONET sites. Figure 16 plots canonical mean observations of fine-
and coarse-mode AOD from three AERONET sites along with groupings of AOD species from the MATCH model output. To undertake a comparison with the AERONET fine-mode
observations, we plot the sum of the MATCH AOD due to organic carbon (OC),
black carbon (BC), and sulfate (
The clear-sky direct radiative effect (clear minus pristine) in downward longwave irradiance averaged from 2000 through 2020, when AERONET observations are available. Boxes indicate the average, and the vertical bar is
Canonical monthly means across 20 years (2000–2020) showing AERONET fine-mode
Figure 16 indicates that resultant fine-mode and coarse-mode comparisons are encouraging, but the agreement is site dependent. In general, MATCH is capturing seasonal changes in fine-mode and coarse-mode particles at these sites, but the magnitude of the AOD values is biased.
CERES instruments observe TOA irradiances, which can be used to assess the
bias in computed irradiance. Global annual mean clear-sky TOA irradiances
derived from CERES observation averaged over 20 years from March 2000 through February 2020 are 53 W m
Radiative flux, aerosol optical depth (AOD), precipitable water, and surface albedo change to match observed TOA radiative fluxes.
To analyze how the EBAF tuning process changes surface irradiance, AOD, and
precipitable water, we computed the mean change separated by surface group
shown in Fig. 11. Generally, AOD increases and precipitable water decreases
to increase reflected shortwave flux, which in turn decreases surface downward shortwave irradiance over these regions (Table 7). For the midlatitude group, on average, AOD is increased by 0.02, precipitable water
is decreased by 0.06 cm, and surface albedo is increased by 0.03. These
adjustments reduce the diurnally averaged downward shortwave irradiance at
the surface by 2 W m
The adjustment made to match TOA shortwave irradiance in the EBAF product
is within the uncertainty of MODIS-derived AOD of
While we cannot identify the cause of the discrepancy between AOD comparison and downward shortwave irradiance comparison with surface observations, potential issues are as follows: (1) the aerosol type and optical properties used in irradiance computations and (2) the bias in downward shortwave irradiance measured by pyranometer, especially diffuse irradiance at smaller solar zenith angles. Because of the temperature gradient within a pyranometer, the downward shortwave irradiance measured by this instrument tends to be biased low under clear-sky condition (Haeffelin et al., 2001). Note that a study by Ham et al. (2020) indicated that the bias in the diurnally averaged surface downward shortwave irradiance computed by a four-stream model should be smaller than 1 %.
We evaluated MATCH AODs used to produce the CERES SYN1deg product. AODs derived from Terra and Aqua by the Dark Target and Deep Blue algorithms were merged to produce daily gridded AODs. Daily gridded AODs were used for assimilation by MATCH at local solar noon. As a consequence, monthly mean AODs under clear-sky conditions identified by MODIS closely agree with those derived from MATCH, although MATCH uses climatological aerosol sources. Because AODs are not screened by QAC, MATCH AODs are larger over convective regions (e.g., the Amazon, central Africa, and Southeast Asia) for both clear-sky and all-sky conditions.
MATCH AODs under all-sky conditions are larger than those under clear-sky conditions. Time series of AERONET AODs indicate that AODs generally increase with cloud fraction, which is consistent with, primarily, water uptake by hygroscopic aerosols (Varnai et al., 2017). In addition, surface observations at the ARM SGP site suggest larger AODs and larger precipitable water values under all-sky conditions than those under clear-sky conditions. AOD biases from AERONET AODS in MATCH are comparable to biases of MERRA-2 AOD from AERONET AODS for both all-sky and clear-sky conditions. However, MERRA-2, which uses AERONET AODs to train the algorithm, has better temporal correlation with AERONET AODs than MATCH AODs.
Once MATCH AODs are used for surface irradiance computations, downward shortwave irradiances are positively biased by 1 % to 2 % compared with
those observed at surface sites. TOA reflected clear-sky shortwave irradiances are negatively biased compared with those derived from
CERES observations. Increasing AODs by
A great deal of the data used in this study were collected by dedicated site
scientists measuring critical climate variables around the world. The tables
included in this Appendix outline the sites, the in situ measurements taken and their locations, and the dates of available data. Table A1 lists the locations of the AERONET sites, our source for observed AOD, which can
be found online at
Sources of surface-observed downwelling irradiance are outlined in Table A2
(land) and Table A3 (buoys). For land, we utilize data from the Baseline Surface Radiation Network (BSRN; Driemel et al., 2018; Ohmura et al., 1998); the US Department of Energy's Atmospheric Radiation Measurement (ARM) program; and the NOAA SURFRAD network, available from the NOAA Air Resources Laboratory/Surface Radiation Research Branch (Augustine et al., 2000). Buoy observations come from two sources through four separate projects. The Upper Ocean Processes Group at the Woods Hole Oceanographic Institute have maintained the Stratus, North Tropical Atlantic Site (NTAS), and Hawaii Ocean Time-series (HOTS) buoys for more than a decade, providing valuable time series of radiation observations in climatically important regions of the ocean. These data can be retrieved from:
AERONET observation sites.
Continued.
Surface irradiance validation sites (land).
The abbreviations used in the table are as follows: BSRN – Baseline Surface Radiation Network (
Surface observation sites for ocean buoy locations.
UOP:
The underlying code to process both MATCH and SYN1deg are publicly available upon request from the authors (Seiji Kato – SYN1deg surface and atmospheric radiation budget code; David Fillmore – MATCH code). However, each code requires significant data inputs. These data are provided from various institutions through data sharing agreements via the NASA Langley Atmospheric Science Data Center (ASDC). These data agreements do not extend beyond the ASDC, although they might be replicated as the data are also publicly available.
All surface observation and SYN1deg data are publicly available. Websites from which the surface observations can be accessed are listed in Appendix A. SYN1deg data may be accessed at
DF was one of the original developers of the MATCH model. DF, DR, and SK wrote significant parts of this paper. DR and FR provided the statistical analysis and plots for the paper. FR and TC developed and implemented significant portions of the SYN1deg code, and DF, FR, and TC developed and implemented the MATCH processing code. DR, SK, DF, and FR reviewed and edited the manuscript.
The contact author has declared that none of the authors has any competing interests.
Publisher'note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was funded by the NASA CERES project. The products and the validation could not have been accomplished without the help of the CERES TISA team. These data were obtained from the NASA Langley Research Center EOSDIS Distributed Active Archive Center. We also wish to acknowledge the hard work put in by the many dedicated scientists maintaining surface instrumentation in many diverse climates to obtain high-quality observations of downwelling shortwave and longwave surface flux; these groups are noted in Appendix A. We would also like to thank the anonymous reviewers for their in-depth reading and assessment of the paper which led to significant improvements of the manuscript.
This research has been supported by the National Aeronautics and Space Administration (CERES project).
This paper was edited by Nikos Hatzianastassiou and reviewed by three anonymous referees.