A shortcut from broadband to spectral aerosol optical depth

The concept behind the shortcut idea is a close correlation between column broadband aerosol optical depth (BAOD) and aerosol optical depth at 500 nm (AOD500). The method uses only two input parameters: (a) the Bouguer broadband coefficient of column transparency for optical mass m = 2 (solar elevation about 30°) and (b) integrated column precipitable water vapour which can be roughly estimated using surface water vapour pressure. In creating the method, a large database, including almost 20 000 complex, spectral and broadband direct solar beam observations at Tõravere, Estonia, during all seasons of a 8-year period, 2002–2009, was used. The AOD500 observations were performed by the NASA project AERONET and the broadband direct beam ones by the Estonian Meteorological and Hydrological Institute. Analysis of this database revealed a high correlation between BAOD and AOD500 which enabled transition from broadband to spectral AOD. Almost 82% of the observations in the database belonged to lower turbidities when AOD500 < 0.2. The root mean square deviation (RMSD) for AOD500 prediction in this range was 0.022. For AOD500 = 0.2–0.4, the RMSD was 0.035, for 0.4–0.6, the RMSD was 0.042. Relative RMSD for these ranges was about 22%, 12%, and 9%, respectively. For AOD500 > 0.6, relative RMSD remained 9%. For comparison, the same database was used to test Gueymard’s broadband parameterization based on his SMARTS2 classic model. The last one, apparently due to problems with circumsolar radiation, slightly but systematically underestimated the AOD500. However, there was a close correlation between our shortcut results and Gueymard’s broadband parameterization.


INTRODUCTION
* Spectral aerosol optical depth of the atmospheric column, or extinction coefficient due to aerosol particles, AODλ, is a central quantity in optics of atmospheric aerosols.It is usually measured with a multispectral instrument, sunphotometer, at a number of wavelengths, commonly including λ = 500 or 550 nm.There are two reasons for the AODλ use.First, this parameter informs about turbidity of the atmospheric column.Second, dependence of AODλ on wavelength λ is inversely related to size distribution of aerosol particles.
Due to high initial cost sunphotometers are expensive to obtain.Also their maintainance, because of regular servicing, is expensive − annual filter changing and recalibration at mountain locations with cloudless sky, at high Sun and very clear, stable atmosphere.For that reason, the net of sunphotometers was sparse until the end of the 1990s.The situation changed completely with the advent and expansion of the NASA huge AERONET project providing the scientific community with massive high-quality standardized global surface information on AODλ (Holben et al., 1998).It should be underlined that until now, the AERONET information is free to download.Like several other global atmospheric projects − NAAPS, HYSPLIT, MODIS, etc. − the free use of data is generously funded by the US government and tax-payers.
However, there are strong reasons to continue elaboration of alternative, mainly broadband evaluations for AODλ.An example of this kind of necessity is retrospective (or prospective) retrieval of AODλ for periods in the past (or future) decades when spectral measurements were not (or will not be) available.Another example is a quick AODλ estimation for correction of satellite remotely sensed data for regions where spectral solar observations are not available but the broadband ones are.
Sometimes, during a later inspection of time series of AODλ, the recorded data seem too large (overestimated) for a certain period.As usual in solar radiation observations, a doubt arises about an undesirable object (insect, spider's thread, trash, etc.) dwelling on or inside the instrument's tube.When there was only one sunphotometer operating but at the same time broadband direct beam was also recorded, an alternative AODλ estimation, through broadband approach, makes available reliability evaluations of spectral observations.And obviously, sites, which have enjoined a free use of AERONET observations without starting up their own solar spectral instrumentation, should be prepared to decrease in the number of simultaneously monitoring autonomous photometers or even general completion or commercialization of the AERONET project.
In Estonia, the AERONET CIMEL photometer began observations on 3 June 2002, at Tõravere (58°15′, 26°27′, 70 m ASL), on the territory of the Tartu-Tõravere Meteorological Station.The station is included into the Baseline Surface Radiation Network (Kallis et al., 2005).Simultaneous registration of both spectral and broadband irradiances provided the opportunity to create a joint, integrated database for AODλ and broadband parameters of atmospheric transparency (turbidity).For eight years, 2002-2009, our joint database includes 19 592 spectral-broadband solar direct irradiance and surface water vapour pressure observations.About 75% of observations were made in April, May, June, July, and August, 9% in September, 8% in March, 3.6% in October, and only 4.4% together in January, February, and November.Due to low Sun and calibrations no joint observations were made in December.
We consider a slant atmospheric column with optical mass 2 m = (solar elevation 30 ), ≈ ° consisting of three successive layers: (a) an ideal, clean, and dry atmosphere which includes O 3 and NO 2 ; (b) water vapour; and (c) aerosol particles.This consideration enables us to express extinction of the broadband direct beam using a product of individual transmittances of each layer and to calculate, as a residual term, broadband aerosol optical depth, BAOD2, at 2, m = for each of the 19 592 joint observations.The plot of the obtained values of BAOD2 against AOD500 revealed a high parabolic correlation enabling a "shortcut" from AOD500 to BAOD2.
To get an idea about the quality of the proposed broadband shortcut to AOD500, we applied Gueymard's (1998) parameterization, based on his known SMARTS2 model.Apparently due to a wider than traditional field of view (FOV 10 )  = ° of the Tõravere actinometer AT50, values of the broadband direct beam in our database are slightly overestimated in cases of larger aerosol turbidity.This explains slightly under-estimated AOD500 as predicted by Gueymard's parameterization.Despite this systematic difference, there was a very high consistency (coherence) between both predicted results, expressed by the coefficient of determination, 2 0.994.R =

A THREE-LAYER STRUCTURE OF COLUMN BROADBAND TRANSMITTANCE
Spectral content of the direct solar beam reaching the Earth's surface is not equal during a day.Broadband characteristics, therefore, depend on solar elevation even in the case of stationary and azimuthally homogeneous atmosphere, causing the Forbes effect − virtual diurnal variation in atmospheric broadband optical characteristics (Ohvril et al., 1999).For this effect, even though spectral optical depth and related spectral optical parameters (transmittance, the Bouguer coefficient, etc.), are independent of optical mass , m their broadband counterparts are not.Although there are no good solutions for strict transformation of column broadband optical parameters from one solar elevation to another, multiannual pyrheliometric time series, recorded at various stations, mainly on the territory of the former USSR, have stimulated creation of corresponding semi-empirical methods.Historically, a destination optical mass, 1, m = was initially chosen (Kalitin, 1938).However, for control measurements the case where the Sun is at the zenith, is not available at most radiometric stations.The next integer number, 2 m = (solar elevation angle about 30 ), ° was further preferred as a standard one in interpretation of pyrheliometric observations.A set of simple formulas links different broadband optical parameters like the Bouguer coefficient of transparency, transmittance, optical depth, the Linke turbidity factor (Ohvril et al., 2009).Concerning aerosol optics, the same reference air mass is also recommended (Gueymard and Kambezidis, 1997).
We further limit the structure of extinction of the broadband direct solar beam by three processes or substances: (a) an ideal or clean and dry atmosphere (CDA), which includes Rayleigh scattering, absorption by ozone (O 3 ), and nitrogen dioxide (NO 2 ); (b) integrated column water vapour or precipitable water, ; W (c) atmospheric aerosol particles.
Denoting their transmittances by respectively, it is of considerable computational convenience to express the incident broadband beam irradiance at normal surface, "beam irradiance", , m S as a direct product of individual beam transmittances which is equal to a presumption of three successive extinction layers (Gueymard, 1998) where 0 S is the extraterrestrial broadband irradiance at the actual Sun-Earth distance, its average value, the "solar constant", is 1.367 kW m -2 (Lenoble, 1993).
On the other hand, using broadband total transmittance, τ , m and the Bouguer coefficient of transparency, m p (Kondratyev, 1969): BAODm is equal to the Unsworth-Monteith turbidity coefficient (Unsworth and Monteith, 1972;Gueymard, 1998).This parameter can be used to estimate aerosol contribution in attenuation of the broadband direct beam, in addition to a clean-wet atmosphere, consisting of CDA and water vapour only (Kambezidis et al., 1998).
We now continue with 2 m = and have for BAOD2: p from m S observations, which both give close results (Ohvril et al., 1999)

S d p
where d is the actual Earth-Sun distance in astronomical units.In Estonian actinometric practice the Bouguer coefficient of transparency, , m p is first calculated using Eq. ( 2).As a second step, 2 p is recalculated from m p using the formula that has been founded by Mürk and Ohvril (Myurk and Okhvril, 1990;Ohvril et al., 1999) Broadband coefficients of transparency, CDA,2 τ , of a CDA, depend, besides molecular (Rayleigh) scattering, on absorption by trace gases, mainly by O 3 and NO 2 .For that, column amounts of trace gases should be first estimated.

EVALUATION OF COLUMN NO 2 AMOUNTS
Vertical NO 2 profiles as described by standard atmospheres usually refer to very clean areas without intensive combustion processes and therefore the column amounts are low.The 1976 U.S. Standard Atmosphere − USSA 1976 (Thomas and Stamnes, 2002) gives for the tropospheric (0-11 km) NO 2 column amount only 0.386 × 10 15 molecules cm -2 .Normalizing this result with the Loschmidt constant (2.69 × 10 19 molecules cm -3 , 0 °C, 1 atm), we obtain for the tropospheric NO 2 layer column thickness 1.4 × 10 -5 cm, which we denote, following Gueymard's (1998) nomenclature, as 0.014 matm cm (≡ 0.014 DU).However, this tropospheric NO 2 content seems to be considerably underestimated compared to present-day European values.
Since the 1980s, the global mapping of column NO 2 has been possible with the help of satellites.Tropospheric NO 2 geography, with special focus on Central and Northern Europe and the Baltic Sea Region, is given by Ionov (2010).His review, for 2004-2009, shows low tropospheric NO 2 for Estonia, (0.8−1) × 10 15 molecules cm -2 , or 0.03-0.037matm cm.High tropospheric NO 2 values were found over St. Petersburg (up to 9 × 10 15 molecules cm -2 or 0.335 matm cm), and even higher values over Belgium, Netherlands, Germany, and Moscow (up to 18 × 10 15 molecules cm -2 or 0.669 matm cm).
An extended review on stratospheric NO 2 global distribution is given by Dirksen et al. (2011).For latitudes 25°-65° the NO 2 annual mean is about 3.2 × 10 15 molecules cm -2 or 0.12 matm cm.The thinnest stratospheric NO 2 layer is on the equator, 0.5 × 10 15 molecules cm -2 or 0.019 matm cm.The amplitude of the seasonal cycle increases with latitude, with the largest stratospheric NO 2 column over the South Pole in summer, 6 × 10 15 molecules cm -2 or 0.223 matm cm.

BROADBAND TRANSMITTANCES FOR SOME CDAs
In order to get a picture about variability of column transparency of ideal atmospheres with different O 3 and NO 2 contents, we made numerous runs of Gueymard's (1998) parameterization.Table 1 shows three typical of them.Row 1 corresponds to very low concentrations: 150 DU (0.15 atm cm) for O 3 and only 88 pptv (10 -12 ) for total NO 2 column amount.The ozone content of 350 DU (0.35 atm cm), in row 2, represents an annual mean for Estonia (Okulov, 2003;Okulov and Ohvril, 2010;Veismann and Eerme, 2011).The NO 2 amounts are also typical of this region.
The table exhibits a relatively small sensibility of transparency and transmittance of different CDAs (the last three columns) to changes in column concentrations of trace gases.Considering now that the relative error of the observed (by an actinometer) broadband direct beam is ± 4% which leads to ± 2% errors in coefficients 2 , p we can assume that CDA, 2 ln 0.1 p ≈ − and rewrite (8) in a simpler form:

BROADBAND TRANSMITTANCE OF COLUMN WATER VAPOUR
In order to calculate the ability of solar radiation to pass the atmospheric water vapour, the amount of total column water vapour, , W should first be estimated.The number of measurement techniques for W observations has increased considerably since the 1990s and now includes ground-based and space-borne optical soundings, microwave radiometry, as well as propagation delay estimation using ground-based GPS data.In most countries, however, the classic balloon-borne radiosounding remains the main routine method for W monitoring (Jakobson et al., 2009), but the network of radiosonde stations is sparse and sondes are launched only 1−2 times per day.Therefore, especially for solar radiation and aerosol studies, correlation between W and surface meteorological parameters (mainly surface temperature and pressure of water vapour) is used.A short historical review of approximate calculations of W is listed by Okulov et al. (2002).For the Baltic region Jakobson et al. (2005) expressed seasonal means of W as linear functions of the geographical latitude degree.
Suppose that the amount of W is already known.To investigate how W affects broadband transmittance through a hypothetical atmosphere consisting of water vapour only, we have to use some kind of radiative transfer model.However, there are two major problems: (a) the complexity of the extraterrestrial solar spectrum and (b) the peculiarity of the water molecule, leading to an extremely complicated vibration-rotation absorption spectrum (Maurellis and Tennyson, 2003).
Nevertheless, during the last decades progress has been made in models calculating the water vapour attenuation of the broadband solar beam, leading to larger values of absorption.This statement follows, for example, from comparing parameterizations proposed by Zvereva (1968) and Gueymard (1995Gueymard ( , 1998)).
For the calculation of extinction of broadband solar beam irradiance, we recommend the parameterization developed by Gueymard (1995Gueymard ( , 1998)), based on his SMARTS2 model.He used a solar spectrum of 1881 wavelengths, at 1-nm intervals within the most the transmittance of water vapour can be expressed by a single formula: where W (cm) is precipitable water in the zenith direction.
Two results are evident: (a) approximation ( 12) provides an excellent agreement with Gueymard's complicated parameterization − visually the two lower curves in Fig. 1 are difficult to distinguish from each other and (b) Gueymard's method gives considerably stronger attenuation of direct solar beam irradiance than the model by Zvereva 30 years before.
Broadband optical depths of water vapour: are plotted in Fig. 2. As expected, Gueymard's parameterization, for 2, m = can be successfully replaced by ( 12), but Zvereva's model considerably underestimates the optical depth of column water vapour.
The curves of Gueymard and Ohvril et al., plotted  for any zenith precipitable water .W For example, a planetary mean column humidity, W = 2.5 cm (Peixoto, 1992) corresponds to transmittance 0.82 and optical depth 0.10.Typical summer precipitable water in the Baltic area, 2.0 cm, gives a transmittance of 0.83, and an optical depth of 0.094.During winter, at lower column humidity, transmittance and optical depth are more sensible to W changes (see Figs 1 and 2), but the average diurnal peak-to-peak (PtP) changes in W are usually low, e.g. in the Baltic region, PtP = 0.64 mm for summer and only 0.2 mm for winter.Of course, W can show fast variations, reaching up to 5 mm h -1 during several hours, but exclusively during changes in the synoptic situation and substitution of airmasses above the location of observation (Jakobson et al., 2009).We should not worry about these transitional cases, because, as a rule, cloudiness restricts observations of direct solar beam during synoptic changes.Substitution of ( 12) into (11) gives the final equation for BAOD2: where zenith W is in centimetres.This formula is one of the main results of the given work.Below it will be used for transition from BAOD2 to AOD500. Figure 3 gives a visual review about limits of the extent of BAOD2 as a function of 2 p and .W As input we have used five values for precipitable water.The cm, corresponds to a cold winter atmosphere, 1.0 W = cm is typical in Estonia in April and October (Okulov, 2003;Okulov and Ohvril, 2010) and it follows from (4) that condition ( 14) is equal to where 2,max p is, for given , W the maximal possible value of 2 p in a wet and clean atmosphere (Fig. 4).Insertion of aerosol particles into a wet and clean atmosphere would diminish values of column transparency, 2 .
p For a typical precipitable water summer value in Estonia, 2 W = cm, a maximal coefficient of column transparency is: 2,max 0.82, p ≈ for the highest Estonian value 4 W = cm, calculation gives 0.80.

COMPARING SPECTRAL AND BROADBAND OPTICAL DEPTHS
where 0 e (mb) is the 12 UTC ground water vapour pressure.This parameterization was developed according to clear sky radio-soundings in Tallinn (Okulov et al., 2002).Note that unlike , W ground water vapour pressure, 0 , e has a remarkable diurnal cycle.For that, in approximations like ( 16), the 0 e should correspond to a fixed time, e.g. to 12 GMT.This condition, obviously, leaves the estimated column humidity unchangeable during a day.In cases of relatively dry winter air ( 0.5cm) W < prediction of W by ( 16) has a root mean square deviation (RMSD) equal to 0.135 cm (Table 2, the 1st column).Compared to the mean value of this particular range, 0.25 W = mm, the deviation is 54%.For higher W amounts the relative RMSD is lower, reaching a final level of RMSD ≈ 11−12% for W(AERONET) > 2 cm.Note that uncertainty in W(AERONET) is typically less than 12% (Holben et al., 1998).
In Figure 6 two sets of broadband optical depth, BAOD2, calculated by ( 13) are compared.For both sets, the same input of observed column transparency, 2 , p is used but with two different values for precipitable water, W(AERONET) and 0 ( ), W e respectively.Both BAOD2s are strongly correlated, 2 0.993, R = having an excellent linear relationship, 0.997 .
This result confirms that BAOD2 is not sensitive to estimations of precipitable water.
The relationship between the spectral AOD and its broadband counterpart, BAOD2, is presented in Fig. 7.Here a large set (the same 19 592 joint observations from 2002−2009) of AOD500, observed by the AERONET sunphotometer (Level 2, Version 2) at Tõravere, are plotted against the BAOD2 calculated by ( 13).
Recall that the results plotted in Fig. 7 were developed for the given location but without any presumptions about the properties of aerosol particles in the column (the Ångström exponent, size distribution, absorption coefficient, etc.).The two optical depth parameters AOD500 and BAOD2 are strongly correlated 2 ( 0.96) R = through the following second-degree polynomial: where BAOD2 is calculated by ( 13) using observed coefficients of column broadband transparency, 2 , p and column humidity, 0 ( ), W e is estimated by a simple model ( 16).As expected, replacement of 0 ( ) W e by W(AERONET) caused only negligible changes in Fig. 7 (a respective figure with this test is not presented).
In Fig. 8, AOD500, predicted by ( 17), using BAOD2 as a function of 2 p and 0 ( ), W e is compared to the reference AOD500 as observed by AERONET at Tõravere.Here 81.8% of points belong to the range AOD500(AERONET) < 0.2 where the RMSD between predicted and reference values is 0.022.The last number apparently represents uncertainty of this quite rough broadband approach to predict AOD500 for a quite typical turbidity range at a particular geographical location − Tõravere.For comparison, the instrumental uncertainty in AOD500(AERONET) is 0.01 (Holben et al., 1998).The RMSD of prediction increases with increasing AOD500, and reaches 0.106 in very turbid cases when AOD500 = 1.0−1.3(Table 3).
Relative RMSD, with regard to the middle value of the first range, AOD500 = 0.1, is 22% and less than 12% for higher turbidities.This accuracy is not bad,  reminding us again that in creating this broadband approach, no presumptions were made about season, meteorological situation, microphysical parameters of aerosol particles, etc.For the entire observed range of optical depth, covering the AOD500 = 0−1.3,there is an excellent linear relationship ( 0.994 ) and close correlation 2 ( 0.958) R = between predicted and reference AERONET values.
Can we expect for better prediction of AOD500, using a more advanced complicated broadband approach?Apparently, one of the best broadband parameterizations, until now, is based on the SMARTS2 model which was developed by Gueymard (1995Gueymard ( , 1998)).It allows transition from broadband solar irradiance, S, to AOD1000 (the model's basic wavelength, λ = 1000 nm).As a parametric input, the model uses precipitable water W, columnar O 3 and NO 2 contents, and a fixed Ångström exponent, α = 1.3.Using Ångström's classic formula, AOD1000, by multiplication to (1000/500) 1.3 = 2.46, is easily recalculated to AOD500.Running Gueymard's model, we inserted from observations: broadband solar irradiance, S 2 (optical mass m = 2, average Sun-Earth distance) and zenith precipitable water, W. We assumed columnar O 3 and NO 2 contents as all-year climatological averages.
Figure 9 shows AOD500(Gueymard) as predicted by Gueymard's model, against AOD500(AERONET) as observed at Tõravere.For larger turbidities, AOD500 > 0.4, the figure shows increasing underestimation of AOD500 by Gueymard's model.This discrepancy can mainly be explained by a peculiarity of Savinov-Yanishevsky thermoelectric actinometers (operational pyrheliometers) AT50, used for observations of broadband direct beam at Tõravere.This cheap but reliable instrument has a 10° field of view (FOV).For comparison, the Eppley pyrheliometer's FOV is only 5.7°.In the case of turbid atmosphere, the bigger FOV leads to a "parasitic" circumsolar radiation, which increases the observed direct solar beam, S, and creates an illusion about better transparency and lower turbidity.Inserting the increased S into Gueymard's model, one apparently obtains slightly lower AOD values.
Disagreement between the AOD500, predicted by BAOD2, and by Gueymard's model, is clearly seen in Fig. 10 where the same inputs for both models are used.The AOD500, derived from BAOD2, predicts higher values than AOD500 (Gueymard), and better compensates the circumsolar radiation in the case of larger turbidities.Of course, following the suggestion proposed by Carlund et al. (2003), it is possible to derive respective correction for Gueymard's model in order to increase predicted AOD500 values in the case of pyrheliometers with a larger FOV.
What is remarkable in Fig. 10  where x = AOD500 (Gueymard).A very high coefficient of determination, 2 0.994, R = indicates, despite systematic discrepancy, a dense coherence of AOD500, predicted by our rough shortcut model and by an advanced Gueymard's model, respectively.

TEST AGAINST AERONET OBSERVATIONS
Performance of the proposed model, expressed by Eq. ( 17) and based on data from Tõravere, 2002−2009, was first tested against fresh reference AOD500 AERONET-Tõravere observations from 2010.Single values and daily averages were compared (Fig. 11).
Figure 11 shows a close consistency between predicted and reference data.For single values the determination coefficient was 2 0.97, R ≈ and for daily means even higher, 2 0.98.R ≈ The slope of the regression, 1.08 , y x = seems somewhat high.However, the slope varies between 0.93 and 1.08 in prediction for single years during 2002−2009.Apparently, uncertainty for the slope is about 8%.

TESTS AGAINST TERRAMODIS AND AQUAMODIS OBSERVATIONS
Both Terra-and AquaMODIS allow download of daily means of AOD550 and Ångström exponents for 1 × 1 deg pixels on the underlying surface.Using the Ångström formula, AOD550 can be recalculated to AOD500.In this work we considered Tõravere in the centre of the 1 × 1 deg area.During 2010, there were 94 days when data from all three AOD instruments as well as from a broadband actinometer were available.Results of AOD500 comparisons are plotted in four panels of Fig. 12.The left panels represent tests of AquaMODIS and TerraMODIS against daily means of AERONET as a reference instrument.Although the uncertainty for the slope is about 6% only, the consistency of data in both cases is low, 2 0.775 R = and 0.702, respectively.In the right panels the AOD500 daily means from BAOD are tested against TerraMODIS and AquaMODIS, respectively.As expected, the uncertainty for the slope is greater, about 10%, and the consistency is lower than comparing against AERONET data, 2 0.749 R = and 0.683, respectively.But, what is special in all panels of Fig. 12, is a possibility of getting negative AOD values by both satellites.Apparently, this kind of mistake happens due to overestimation of column precipitable water, W.

CONCLUSIONS AND DISCUSSION
The following conclusions can be drawn from this study.(A) For prediction of AOD500, a simple "short-cut" broadband model is proposed.As an input, the model uses two quantities: (a) column broadband Bouguer coefficient of transparency, p 2 , which corresponds to optical mass, m = 2; and (b) column precipitable water, W. According to the input, a broadband aerosol optical depth, BAOD2, for the same solar elevation is calculated.Transition from BAOD2 to AOD500 is implemented using an experimental parabolic relationship (17).The "shortcut" model is based on a large database, 19 592 joint broadband (provided by the EMHI − Estonian Meteorological and Hydrological Institute), and spectral (provided by the AERONET) observations during all seasons of 2002−2009.In the database, 81.8% of observations corresponded to aerosol optical depth, AOD500 < 0.2.In this range of aerosol turbidity, the RMSD of prediction was 0.022.For comparison, the instrumental uncertainty in AOD500(AERONET) is 0.01 (Holben et al., 1998).Absolute deviations of prediction increased towards increase in AOD500 and were 0.035 for the next range of aerosol turbidity, 0.2−0.4(13.8% of observations).For very turbid cases, when AOD500 = 1.0−1.3, the RMSD reached 0.11.(B) Performance of the model was tested using fresh data − 2679 joint, broadband (provided by the EMHI) and spectral (AERONET) single observations during all seasons of 2010 at Tõravere.In terms of regression, the uncertainty in prediction (against AERONET reference values) was about 8%.The consistency between predictions and the reference values was surprisingly good, 2 R = 0.976 for single observations and 2 0.986 R = for daily means.
The model was also verified using named 94 daily average values during 2010, when the AOD data from TerraMODIS and AquaMODIS were available besides the AERONET and broadband actinometer observations.The values of AOD500 obtained by all four methods were roughly comparable but consistency of satellite data against the AERONET and broadband model was low.It should be noted that in several occasions the AOD from satellites is negative, apparently due to overestimation of column precipitable water, W. (C) One of the inputs of the model, column precipitable water, W, was calculated using its correlation with surface water vapour pressure, e 0 .It appeared that neither BAOD2 nor AOD500 was sensible to the W input. Replacement by W(e 0 ) to a more correct W(AERONET) did not improve predictions.(D) For an additional examination of our AOD500 predictions, Gueymard's broadband model (1998) was used.Our predictions systematically exceeded Gueymard's ones, a linear regression, y = 1.031x, was applicable for AOD500 < 0.5.For larger turbidities difference between predictions started to increase.Apparently, this discrepancy is not a shortcoming of Gueymard's model but due to a large (10°) field of view (FOV) of the AT50 actinometer used at Tõravere for observations of the broadband direct solar beam.A larger FOV than that of conventional pyrheliometers (5.7°) enables more circumsolar radiation enter an instrument, which increases the observed direct solar beam and creates an impression about better transparency and lower turbidity.When problems with circumsolar radiation which can be corrected with adjusting constants were ignored, both broadband models (our model and Gueymard's parameterization) demonstrated a surprising consistence of prediction between each other (Fig. 10), with a very high coefficient of parabolic regression, 2 0.994.R = The listed results lead to a speculation about the ability of a broadband approach to predict spectral AODλ.Climatology of AODλ, as well as of any other aerosol optical parameter, emphasizes its high variability (Olmo et al., 2001;Smirnov et al., 2002;Aaltonen et al., 2006;Toledano et al., 2007).Apparently, this variability can be only in part observed and predicted with broadband methods.Perhaps, using additional information on the column aerosol content (like the origin, seasonality and diurnal trend of particles, their size distribution, etc.), particular models can be created and the quality of AODλ prediction improved.
On the other hand, the introduced simple broadband "shortcut" can be recommended for estimation of column AOD500 using data on broadband direct irradiance archived in the past, and explain climatological trends related to volcanic activity, anthropogenic pollution, etc.
In radiometric stations, where spectral and broadband solar direct beams are currently observed, the method can also be used for quality control of automated AOD500 observations.
Obviously, for different geographical locations, adjusting constants in formulas ( 16) and ( 17) are not identical with the given ones, developed for Tõravere, Estonia.

Fig. 3 .
Fig.3.Broadband aerosol optical depth as plotted against the broadband column transparency coefficient.Calculations are implemented for m = 2 and a set of values of zenith precipitable water, from W = 0.1 to 6.0 cm.

Fig. 4 .
Fig. 4. Maximal possible broadband Bouguer coefficients of transparency, p 2,max , for a clean and wet atmosphere.

Fig. 11 .
Fig. 11.AOD500 as predicted by broadband aerosol optical depth, BAOD2 at m = 2, plotted against AOD500 from the AERONET Tõravere 2010 observations.The left panel represents 2679 single observations, the right panel 94 daily means.

Fig. 12 .
Fig.12.The left panels: AOD500 daily means from the MODIS satellites, plotted against the reference, AERONET Tõravere observations.The right panels: AOD500 daily means as predicted by broadband aerosol optical depth, BAOD2, plotted against AOD500 observed by the MODIS satellites.Note that in several occasions the satellite AOD500 values are negative.

Table 2 .
Root mean square deviations (RMSD) of W prediction for different W ranges