# Spectral aerosol optical depth prediction by some broadband models. Validation with AERONET observations/Aerosooli spektraalse optilise paksuse arvutusmudelite vordlus ja kontroll AERONET-i mootmistega.

Artiklis on omavaheliseks vordluseks valitud seitse lihtsat ja uks
keerukam aerosooli spektraalse optilise paksuse (AOD500) arvutamise
integraalset ehk laiaribalist mudelit. Lihtsad mudelid koosnevad vaid
monest valemist ja vajavad ainult kaht peamist sisendparameetrit,
milleks on Paikese integraalse otsekiirguse kiiritustihedus koos Paikese
korgusega voi atmosfaarisamba labipaistvuskoefitsient ning atmosfaari
veeaurusisaldus. Mudelites pole tehtud eeldusi aerosooliosakeste
fuusikaliste omaduste kohta, kuid kaks mudelit (M1 ja T1) lubavad
kolmanda sisendparameetrina varieerida Angstromi lainepikkuse eksponenti
[alpha], mis on seotud osakeste suurusjaotusega.

Ainus analuusitud keerukam mudel (G1) koosneb umbes 30 valemist ja lubab spetsifitseerida atmosfaari gaasilist koostist detailsemalt, naiteks osooni- ning lammastikdioksiidikoguste kaudu.

AOD500 arvutused toetusid Toraveres moodetud Paikese laiaribalisele otsekiirguse kiiritustihedusele, mille mooteandmeid voib kvaliteetseteks pidada, sest Tartu-Toravere ilmajaam kuulub rahvusvahelisse kiirgusmootmiste vorgustikku BSRN. Teine sisendsuurus, ohusamba veeaurusisaldus, ei olnud otseselt moodetud, vaid leitud kaudselt, veeaururohu kaudu. Koigi mudelite saadud AOD500 tulemusi vordlesime AOD500 tegelike vaartustega, moodetuna Toraveres AERONET Cimel-i fotomeetri poolt.

Mudelite voimekuse analuusiks koostasime ulatusliku andmebaasi, milles iga vaatlusrida uhendas nii ilmajaamas moodetud suurused (Paikese integraalne otsekiirgus, veeaururohk) kui ka AERONET-i fotomeetri samaaegselt moodetud suurused (AOD500 tegelik vaartus, Angstromi lainepikkuse eksponent, [alpha] (440-500-675-870)). Kokku sisaldas andmebaas 26 091 integraalset-spektraalset vaatlust, mis toimusid koikidel kalendrikuudel (valja arvatud detsember) kumne aasta jooksul, 2002-2011. Selline andmebaas voimaldas iga mudeli pohjalikku vordlust AERONET-i tegelike AOD500 mootmistega.

Suhteliselt puhta ohu korral, AOD500 < 0,6, andsid koik mudelid moistlikke tulemusi. Saastatuma ohu korral, mil AOD500 > 0,6, kaldusid statistiliselt korrigeerimata mudelid aerosooli optilist paksust alahindama. Alahindamist pohjustab Paikese oreooli nn parasiitkiirgus, mis suurendab aktinomeetri moodetud otsekiirgust ja loob mulje puhtamast ohust.

Testimistulemused tostsid esile kaks mudelit: T2 ja M2a. Tuleb siiski markida, et mudeli M2a valjundis on kunstlik piirang, puuduvad arvutatud vaartused vahemikus 0,4 < AOD500 < 0,477. Vaatlusalustest mudelitest kaks, M1 ja T1, voimaldasid sisendsuurusena muuta ka Angstromi lainepikkuse eksponenti a. Tavapraktikas pole avaartused a priori teada, koostatud andmebaasis aga sisaldus selline voimalus. Monevorra ootamatult ei parandanud a kaasamine AOD500 arvutustulemusi. Leidmaks pohjusi selle ebaonnestunud numbrilise eksperimendi kohta, analuusisime atmosfaarisamba optiliste ja niiskusparameetrite sesoonset muutlikkust Toraveres aastatel 2002-2011. Selgus, et Angstromi a korreleerub vaid osakeste nn peenmoodi fraktsiooni (FMF) panusega AOD500-sse. Kuu keskmiste loikes naitas FMF head korrelatsiooni Angstromi eksponendiga ja peenosakeste domineerimist suvekuudel.

Korvalproduktina AOD500 mudelite vordlemisele selgus, et atmosfaari labipaistvusel on juunis kevadsuvine maksimum, pohjuseks ilmselt ohusamba vaiksem aerosooliosakeste sisaldus. Esialgu oskame seda fakti pohjendada arvamusega, et juunis on Eestis valja kujunenud varske taimkate, mis takistab tolmu teket ja levikut. Samuti pole juunis veel levinud kohalikud raba- ja metsapolengud.

Uuringut toetasid kaks projekti: Euroopa regionaalarengu fondi rahastatav "Eesti kiirguskliima" ja Eesti Teadusnoukogu sihiteema TFP SF0180038s08.

1. M TRODUCTION

Aerosol optical depth, AOD[lambda], is a central parameter for the description of column aerosol content and column optical properties, including the rate of atmospheric turbidity. From the 1990s, after the start of the US NASA programmes AERONET, TerraMODIS, and AquaMODIS, this parameter became available and very popular (Holben et al., 1998; Toledano et al., 2007). However, in order to obtain a more capacious temporal and spatial overview of aerosol distribution, especially for retrospective (backward) extrapolation of AOD[lambda] time series to pre-1990 years when the photometric network was sparse, it is necessary to use alternative methods for calculation. There are also two other reasons for AOD[lambda] proxy, mainly broadband calculations: (1) quality inspection of recently measured AOD[lambda] time series, although by a modern solar photometer, because when the recorded values seem too large (overestimated for a certain period), a doubt always arises about an undesirable object (insect, spider's thread, trash, etc.) dwelling on or inside the instrument's tube; and (2) a quick AOD[lambda] estimation for correcting satellite remotely sensed data for regions or moments where/when spectral solar observations are not available but broadband ones are.

Several AOD[lambda] broadband models can be found from an extensive literature survey. The high-performance models are laborious for processing large amounts of data or they require either special or very accurate input quantities. However, there are other models which, especially for physical climatology, allow easy programming with no need of ancillary meteorological input data. Continuing activity in the development and modification of simple broadband models indicates that such approaches do correspond to practical needs.

In general, there are three main input parameters for simple AOD[lambda] models. One describes column attenuation of the broadband direct solar beam, e.g. broadband transmittance ([[tau].sub.m]), Bouguer coefficient of column transparency ([p.sub.m]), column broadband optical depth ([[delta].sub.m]), Linke turbidity factor ([T.sub.L,m]), which all are equal, linked by the well-known Bouguer-Lambert exponential law:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Here [S.sub.m] is the measured broadband irradiance at optical mass m; [S.sub.0] is the extraterrestrial broadband solar irradiance at the actual Sun-Earth distance, its average value, the "solar constant", is 1.367 kW [m.sup.-2] (Lenoble, 1993); and [[delta].sub.CDA,m] is the optical depth of an ideal, i.e. clean and dry atmosphere (CDA).

To eliminate the Forbes effect, inherent to column broadband optical characteristics, the generally accepted practice is to reduce them from the actual optical air mass m to a standard air mass, mainly to m = 2 (solar elevation h [approximately equal to] 30[degrees]). As to possible processes of reduction, it seems to us that quite successful methods have been developed in regard to the Bouguer coefficient [p.sub.m] (Ohvril et al., 1999, 2009). For this reason, [p.sub.2] is the input parameter of several models in this study.

A second input parameter to simple models is column humidity (water vapour content). Its unit, mass per unit area, is in practice usually given as the thickness of the layer of liquid water: 1 mm corresponds to 1 kg [m.sup.-2] and 1 cm to 1 g [cm.sup.-2].

A third parameter, the Angstrom wavelength exponent, [alpha], is closely linked to the size distribution of aerosol particles, provided that the size distribution, in part, follows a power law (Liou, 2002). However, the column aerosol particles are in permanent change, expressing deviations from the power law. Hence the wavelength exponent is actually a very unstable parameter, which has different values for different parts of the solar spectrum. As it is poorly correlated with AOD[lambda], its use in atmospheric optical models is questionable.

The present work aims at validation of seven simple broadband models and one more complicated model for AOD[lambda] calculations.

Five simple models (M1, M2, M2a, M2b, and M2c) were developed in Moscow during 1991-2013, and two in Tartu (T1, T2) during 2007-2012. The most complicated model (G1) is Gueymard's parameterization from 1998, based on his known SMARTS2 code, and it allows considering numerous column parameters. The set of the used models (including references) will be described in detail in Section 2.

All models were tested against AOD500 reference values obtained by an AERONET photometer at Toravere, Estonia, located in the territory of the Tartu-Toravere Meteorological Station during 10 years, 2002-2011. The station (58.26[degrees]N, 26.46[degrees]E, 70 m ASL), is included into the Baseline Surface Radiation Network (Kallis et al., 2005).

Simultaneous registration of both spectral and broadband irradiances provided an opportunity to create a joint, integrated database for AOD[lambda] together with the Angstrom exponent and broadband parameters of atmospheric transparency and turbidity. The joint database includes 26 091 spectral-broadband solar direct irradiance and surface water vapour pressure observations covering all months except December.

Although the AERONET observations, besides AOD[lambda], give also values of precipitable water, W, we have considered that the input values for the models should be independent in regard to a reference instrument. For that, precipitable water was evaluated using surface water vapour pressure, [e.sub.0] (Okulov et al., 2002; Okulov and Ohvril, 2010).

2. A REVIEW OF THE USED BROADBAND MODELS

A quick review of the used broadband models together with their inputs and outputs can be obtained from Table 1.

2.1. Models Ml and M2

The models were developed in the Meteorological Observatory of the Moscow State University (Tarasova and Yarkho, 1991a, 1991b). Model M1 consists of 13 formulas; it is somewhat general and unique because it allows varying the Angstrom wavelength exponent, a. Model M2 is a simplified version of M1, where the Angstrom exponent has a fixed value, [alpha] = 1. Model M2 is expressed by only one formula

AOD550 = ln [S.sub.m] -(0.189[W.sup.-0.183] + [[0.880[W.sup.-0.009] - 1]/sin h])/ 0.813[W.sup.-0.002] -1 + [[0.435[W.sup.-0.0321] - 1]/sin h] (2)

where h is solar elevation and W zenith precipitable water in centimetres.

Model M2 was widely used for monitoring aerosol turbidity in the Russian territory (Abakumova and Gorbarenko, 2008). Routine registration of broadband direct solar irradiance has been performed in Moscow since 1955. Before the development of models M1 and M2 in 1991, there was one summer, in 1972, when the region around Moscow was affected by extensive forest and peat fires, but in that summer the capabilities to check broadband calculations of AOD[lambda] by reference spectral instruments were limited. A solar-sky AERONET Cimel photometer was deployed in Moscow for measuring AOD[lambda] as early as in 2001. The next, extremely dry summer of 2002 favoured huge fires around Moscow and the city was filled with smoke (the CIMEL photometer is located near the main building of the university, in the territory of the botanic garden). Prediction of AOD[lambda] under the conditions with a large amount of smoke aerosols (in summer 2002) revealed the fact that model M2 gives lower values at very turbid air in comparison to AERONET simultaneous observations.

The main reason for the underestimation of AOD[lambda] at high turbidities by the broadband models is that the models consider only a "true" narrow solar direct beam, exactly from the solar disc with its mean angular diameter of about 32 arc minutes. Actually the opening angles of older broadband instruments, but still in use for homogeneity of multidecadal time series, have considerably wider apertures reaching up to 10[degrees]. For example, the Kipp & Zonen Linke-Feussner actinometer has an aperture of 10.2[degrees] (Gueymard, 1998; Garg and Prakash, 2006). The AT-50 actinometers, continuously in use in actinometric networks in the territory of the former USSR, have the full field of view, FOV = 10[degrees].

Although the FOV of most current pyrheliometers is smaller (e.g. for the Eppley Laboratory Inc. normal incidence pyrheliometers (NIP) the FOV = 5.7[degrees]), the measured direct beam is anyway increased by undesirable diffuse irradiance intercepted by a broadband instrument (Gueymard, 1998; Carlund et al., 2003).

The magnitude of this increase is greater at low solar elevation and heavy aerosol loading, and also in cases of large aerosol particles such as maritime aerosol, biological aerosol, desert or ground dust, etc. In cases of small particles (e.g. almost pure molecular scattering in a clear atmosphere after a rain) circumsolar radiation is weaker. According to calculations made by Gueymard (1998) for FOV = 10.2[degrees], the circumsolar magnification factor can reach 35% in regard to the true direct irradiance. Increased artificial values of the observed broadband direct beam, [S.sub.m], lead to underestimation of the modelled AOD[lambda]. The artificial increase in readings of modern spectral photometric observations is decreased due to the smaller aperture (e.g. for the CIMEL-318 radiometer, the FOV = 1.2[degrees]).

2.2. Model M2a

Analysis of data from a smoky summer of 2002 when AERONET observations in Moscow were already available, led to a conclusion that a better match between predicted and observed values at AOD500* > 0.4 can be achieved by substitution of initially predicted AOD500 * with its increased counterpart, AOD500 (Chubarova, 2005):

AOD500 = 1.301X [(AOD500*).sup.1.095]. (3)

The correction from AOD500* = 0.4 towards bigger values increases the initially predicted AOD500* but also leads to an artificial discard of corrected aerosol optical depths in a range of 0.4 < AOD500 < 0.477, which is a secondary visual defect rather than a functional imperfection. Application of model M2 together with the correction for AOD500* > 0.4 is further denoted by M2a.

The third catastrophically dry and hot summer with flaming and smoldering wildfires around Moscow occurred in 2010. In contrast to previous smoky periods, in 1972 and 2002, the summer of 2010 was characterized by higher aerosol optical depths, reaching even a value of AOD500 = 4.6. In a review on radiation monitoring of all three smoky summers Chubarova et al. (2011a, 2011b) recommended implementation of correction by Eq. (3) from AOD500* = 0.5 onward, which means an absence of corrected AOD500 even in a larger range, 0.5-0.609.

2.3. Model M2b

However, in massive data processing, generation of a permanent empty zone for corrected AOD500 values is not desirable. On the other hand, for low AOD500* values correction by Eq. (3) is not significant. Moreover, below a certain value, AOD500* [less than or equal to] 0.062, the implementation of Eq. (3) will lead instead of AOD500* enlargement to its reduction. For example, inserting AOD500* = 0.025 into Eq. (3) one obtains AOD500 = 0.023. In such cases the correction factor, CF,

CF = AOD500/AOD500* (4)

obtains values <1.0. Use of Eq. (3) only for AOD500* [greater than or equal to] 0.063 (CF [greater than or equal to] 1) avoids appearance of an empty zone for corrected AOD500 and secures a smooth correction. The use of Eqs (2) and (3) together with a condition for AOD500* [greater than or equal to] 0.063 only is further denoted by M2b.

2.4. Model M2c

Recently Gofbarenko and Rublev (2013) reported about using a solar elevation-dependent correction of AOD500*. Assuming continental aerosols with a fixed Angstrom exponent, [alpha] = 1, they derived a correction algorithm:

AOD550 = (AOD550*)[0.9 + [(AOD550*)[0.7/5sin h+0125]/5], (5)

which they further applied for AOD550* > 0.5. Because AOD500* = 1.1 x AOD550*, the next algorithm for 500 nm can be obtained:

AOD500 =

(AOD500*/1.1)[0.9 + 0.2[(AOD500*/1.1).sup.[0.7/0.75sinh+0.125]]. (6)

However, the two last formulas reduce the first calculated AOD* for cleaner air also (i.e. in cases of low aerosol content) like the use of correction by Eq. (3). For example, in case of AOD500* = 0.36 and sin h = 0.5, Eq. (6) gives AOD500 = 0.34.

The minimal turbidity, min AOD500*, where correction (6) has a sense, depends on the solar elevation:

minAOD500* = [1.1x 0.5.sup.0.75sinh+0.125/0.7], (7)

e.g. for sin h = 0.3, 0.5, 0.7, minAOD500* = 0.78, 0.68, 0.58, respectively. These values are definitely higher than minAOD500* = 0.063 for correction with formula (3) for M2b. The use of Eqs (2) and (6) together with condition (7) is denoted as model M2c.

Figure 1 compares results of correction with Eqs (3) and (6), in regard to their common input counterpart (AOD500* [greater than or equal to] 0, including values even below the minAOD500*). Differences appear at very large turbidities and low sun. For AOD500* = 4 the correction by Eq. (3), published in 2005, gives AOD500 = 5.9, but the new one by Eq. (6) from 2013 gives AOD500 = 6.8, 8.5, 14.2 for sin h = 0.7, 0.5, 0.3, respectively.

2.5. Model G1

Apparently the most advanced broadband model was derived by Gueymard (1998), further denoted as G1. The model, originally for the prediction of AOD1000, contains about 30 formulas and allows varying several minor column gaseous components such as [O.sub.3] and tropospheric and stratospheric N[O.sub.2].

However, based on our evaluations and supported by Gueymard's error analysis (2013), the variability of the ozone amount can be considered a second-order input because of its small impact on the solar broadband direct beam. Besides, nitrogen dioxide would only be of concern over polluted areas. Therefore, in the extensive runs of Gueymard's model, we used the given fixed input values, typical for the Baltic Sea region: [O.sub.3] = 0.35 atm cm, N[O.sub.2](stratospheric) = 0.00012 atm cm, N[O.sub.2](tropospheric) = 0.00004 atm cm, p = 1013.25 hPa (Kannel et al., 2012).

Concerning the Angstrom wavelength exponent, in accordance with Gueymard, it is in general not possible to know a priori whether the observed aerosol particles belong to the continental, maritime, or any other specific type. Therefore, a fixed conventional value of [alpha] = 1.3, representative of particles of rural-continental origin, was proposed by Gueymard.

The preliminary output for models M1 and M2 is originally AOD550, for G1 it is the Angstrom turbidity coefficient, [beta] = AOD1000. Transitions to AOD500 are easy applying the Angstrom exponential formula in a general form:

AOD([[lambda].sub.2]) = AOD([[lambda].sub.1]) AOD([[lambda].sub.1])[([[lambda].sub.2]/[[lambda].sub.1]).sup.-[alpha]]. (8)

2.6. Models T1 and T2

Two broadband models, T1 and T2, were developed at the University of Tartu. Model T1 is actually a modification of model M1 (Kannel et al., 2007), with some changes to consider the effects of circumsolar radiation (Kannel, 2007; Ohvril et al., 2009). The model is expressed by a single formula and has three input quantities: (1) Angstrom exponent, [alpha]; (2) coefficient of column broadband transparency, [p.sub.2] transformed to atmospheric mass, m = 2; (3) precipitable water, W (Kannel, 2007; Kannel et al., 2007; Ohvril et al., 2009).

Model T2 was derived using barely a statistical approach. In creating the method, a large database, including almost 20 000 complex, spectral, and broadband direct solar beam observations at Toravere, Estonia, during all seasons of an 8-year period 2002-2009, was used. Apparently, the model is local, and could be used only in conditions similar to Toravere. Monthly climatology of column optical parameters for Toravere will be given below in Section 6.

The model relies only on two input parameters: (1) coefficient of Bouguer column broadband transparency, [p.sub.2]; and (2) precipitable water, W. These parameters allow calculating for m = 2 a specific quantity, the column broadband aerosol optical depth (BAOD2). According to Kannel et al. (2012), the two optical depth parameters, AOD500 and BAOD2, are strongly correlated ([R.sup.2] = 0.96) through a second-degree polynomial:

AOD500 = 1.7[(BAOD2).sup.2] + 1.3(BAOD2), (9)

which enables an easy calculation of AOD500. Table 1 lists all considered broadband AOD[lambda] models with their input and output quantities.

3. OBSERVATIONAL DATA

We used two institutionally independent databases from the period 2002-2011.

1. Broadband direct solar irradiance and surface humidity measurements acquired at the Tartu-Toravere Meteorological Station, Estonia. The station is included into the Baseline Surface Radiation Network (Kallis et al., 2005). The data allowed calculation of the coefficient of column transparency ([p.sub.2]) and broadband aerosol optical depth (BAOD2) for each single observation. Note that the calculation of BAOD2 is not sensitive to uncertainties in precipitable water (W) estimations, thus W was estimated from the station surface water vapour pressure. Details of data processing are given in (Kannel et al., 2012). The quality of the W estimation is discussed in Section 6.

2. Spectral aerosol optical depth measurements by AERONET CIMEL photometers, which began regular observations in the territory of the meteorological station at Toravere on 3 June 2002.

The two databases provided the opportunity to create a joint, integrated database for AOD[lambda] and broadband parameters of atmospheric transparency (turbidity). Our joint database includes 26 091 spectral-broadband solar direct irradiance and surface water vapour pressure observations for ten years, 2002-2011. About 75% of the observations were made in April, May, June, July, and August; 9% in September; 8% in March; 4% in October; and only 4% together in January, February, and November. In December no joint observations were made due to the low Sun and photometer calibrations.

The abundance of data enabled a comprehensive comparison of each model against the AERONET AOD500 observations.

4. TEST OF MODELS

Test runs of the eight models contained 26 091 single calculations of AOD500 by each model, followed by comparison against the reference AOD500(AERONET) value. Two models, M1 and T1, allow varying the input value of the Angstrom exponent, [alpha]. For this reason additional runs were performed for these models with the variation of [alpha].

To obtain quantitative measures on the accuracy of the models, we use two linear regression parameters, slope and correlation [R.sup.2] (actually coefficient of determination), and the following three commonly used statistical parameters (Iqbal, 1983; Gueymard, 1993):

(1) the mean bias deviation (MBD), expressing the average deviation (difference) of the predicted values, [y.sub.i] = AOD500(Model), from the reference values, [x.sub.i] = AOD500(AERONET):

MBD = [1/N] [N.summation over (i=1)] [[y.sub.i] - [x.sub.i]]; (10)

(2) the root mean square deviation (RMSD), a measure of the variation of predicted values around the reference values:

RMSD = [1/N] [[[N.summation over (i=1)][([y.sub.i] - [x.sub.i]).sup.2]].sup.1/2] (11)

(3) the mean absolute relative deviation (MARD), also known as the mean absolute percentage deviation (MAPD), expressing the average value of relative deviations:

MARD = [1/N] [N.summation over (i=1)] [absolute value of ([[y.sub.i] - [x.sub.i]]/[x.sub.i])] (12)

The amount of column water vapour (W) is an input parameter to all considered AOD[lambda] models. It was estimated from surface conditions using correlation with vapour pressure, [e.sub.0] (Okulov et al., 2002). At the level of monthly means the used correlation performs quite well (Section 6), but for single observations the coefficient of determination, [R.sup.2] = 0.83, indicates a moderate scatter of W([e.sub.0]) around W (AERONET) (Kannel et al., 2012). Overestimation of W leads to underestimation of AOD[lambda], even to a physically unrealistic negative AOD[lambda]. For the sake of brevity, sensibility analysis of the considered models to possible W errors is not included in the present paper. We only give the number of predicted negative AOD500 values together with the number of corrections applied for each run of the model (Table 2).

Results of the eight main runs are presented in Fig. 2, where each panel also reviews performance statistics of the run: slope, correlation [R.sup.2], number of corrected values and predicted negative values. Table 2 lists also results of six additional runs: (1) two runs for both M1 and T1 with a different fixed wavelength exponent, and (2) a run for both M1 and T1 with an a priori known wavelength exponent. The last two runs are visualized in Fig. 3.

From Table 2, which includes six statistics, one can rank the accuracy of the performance of the different models (with different Angstrom [alpha], if applied) in regard to different statistics. In summary, giving a point for the best result in each "event" of the "hexathlon", an overall, combined ranking for T2 appears to score 4 points, and for M2a, 2 points. These two models can be recommended, because of their consistently high performance in all items, for the evaluation of AOD500 in conditions similar to those at Toravere. The two only models, M1 (row 4) and T1 (row 13), that consider a priori known Angstrom exponents did not stand out. We try to analyse this disappointing result in the next two sections.

5. FIXED VS INDIVIDUAL ANGSTROM EXPONENT

Usually broadband models for the calculation of AOD[lambda] do not enable changing the Angstrom wavelength exponent, [alpha]. In this sense model Ml and its derivate, T1, represent an exception. Nevertheless, it is meaningful that the authors of model Ml themselves have actually not used this opportunity; they first began to use M2 and then M2a, in both of which the exponent is fixed, [alpha] = 1. In model G1 the exponent is also fixed, [alpha] = 1.3, but it assumes validity of the Angstrom exponential formula (8). On the other hand, this means that the size distribution n(r) is partly given by the Junge power law (Liou, 2002):

n(r) = C(r) [r.sup.-([alpha]+3)] (13)

where C is a scaling factor proportional to particles column concentration and r is a particle radius. "Partly given" means that only the downgoing part of the size distribution (onward from the maximum) can be approximated by the power law.

Our joint 26 091 observation database is special, because each single observation contains also an Angstrom AERONET-evaluated exponent, [alpha](440-500-675-870), calculated as a best fit for the indicated four wavelengths. In this way the database enabled taking into consideration a priori known Angstrom exponents for predicting single AOD500 values.

However, runs of models M1 and T1 with an a priori known [alpha] (rows 4 and 13, Table 2) did not give the expected improvement of the predictions (Fig. 3).

So, a priori known Angstrom exponents did not improve AOD500 predictions. But should they? Consider the Angstrom formula is obeyed, then the exponent enables only transition from a known AOD[[lambda].sub.1] to any other AOD[[lambda].sub.2], not to start with the magnitude of AOD[[lambda].sub.1] itself. In terms of the Angstrom formula, prediction of AOD[[lambda].sub.1] can be done using a second parameter, the Angstrom turbidity coefficient, [beta].

In the next section we examine the background, mainly seasonal variability of column optical and humidity properties at Toravere, 2002-2011, and try to find regularities in the behaviour of the Angstrom exponent.

6. VARIABILITY OF THE USED COLUMN OPTICAL AND HUMIDITY PARAMETERS

Figure 4 provides monthly means of the Angstrom a and AOD500 and, in addition, seasonal variation of the fine mode fraction (FMF), which is one of the AERONET inversion products describing the contribution of fine particles to AOD500. The AERONET inversion code finds the minimum of the size distribution within the radius interval from 0.439 to 0.992 [micro]m. This minimum, approximately at 0.6 [micro]m radius, is used as a separation point between fine and coarse mode particles. Using that separation, the inversion code calculates the contribution of fine particles to the formation of AOD500 (AERONET Inversion Products, 2010; O'Neill et al., 2001). The AERONET term, fine particles, actually includes three traditional subregions of aerosol size distribution: the nucleation, Aitken, and accumulation mode.

As expected, the intra-annual evolution of the Angstrom exponent is consistent with monthly changes in the FMF. The higher values in summer (Jun-Jul-Aug) indicate domination of fine aerosol particles.

Somewhat surprising is a local minimum of AOD500 in June. For the cold season, we have no good explanations to the higher AOD500 in January and November compared to February and October; this is a topic of further studies.

The monthly means of column transparency ([p.sub.2]) for 2002-2011 (Fig. 5) are in opposite phase with AoD500, with an expected local maximum in June. Actually, higher column transparency in June was noticed already since 1994. This finding can be explained by a general cleaning of the European atmosphere as a part of the global brightening (Ohvril et al., 2009; Okulov and Ohvril, 2010). During June, the Estonian landscape is already totally covered with fresh vegetation, which restricts creation and vertical distribution of dust. The number of forest and bog fires is also low in June.

Column precipitable water (W) is usually evaluated from surface humidity and temperature, in our study:

W (mm) = 1.48[e.sub.0] + 0.40, (14)

where [e.sub.0] (mb) is the 12 UTC surface water vapour pressure. This parameterization was developed from clear-sky radio soundings in Tallinn (Okulov et al., 2002). For Toravere, Kannel et al. (2012) compared column humidity predictions, W ([e.sub.0]), with W(AERONET), where the latter were considered as reference. Use of almost 20 000 parallel observations from 2002-2009 showed that the prediction overestimates the reference as an average only by 3%. For monthly means of W over all considered years (2002-2011), approximation (14) gives values close to those obtained by the AERONET photometer (Fig. 6).

To model the extinction of broadband direct solar beam, for optical mass m = 2, we assume that the atmosphere consists of three layers or substances (Kannel et al., 2012): an ideal or clean and dry atmosphere (CDA), integrated column water vapour (W), and atmospheric aerosol particles.

Figure 7 shows monthly mean broadband transmittances for these three layers at Toravere, calculated from our database for a 10-year period, 2002-2011. The plots were prepared using only meteorological data, not the AERONET observations. Noticeable is the summer maximum of aerosol broadband transmittance in June, which apparently is the main reason of higher total column transmittance in this month. The lowest total transmittance occurs in July and August, caused by low transmittances of both column water vapour and aerosols.

Figure 8 shows the seasonal variation of total broadband optical depth, and its division into the main atmospheric constituents.

Unfortunately, the annual cycles of the reviewed column parameters (except the fine mode fraction) do not give us the expected relationships with the annual cycle of the Angstrom exponent (Fig. 4). Moreover, a plot of the Angstrom [alpha] against AOD500 revealed no correlation (Fig. 9). The [alpha] varies from 0.05 to 3.43 for cleaner air, when AOD500 < 0.2 encompassing the simple (nonclimatological) average of the exponent, [alpha] = 1.43. The standard deviation (one sigma uncertainty) of the exponent for the whole database [sigma] = 0.35.

Supposing normal distribution and coverage factor 3, the expanded 3[sigma]-uncertainty becomes 1.05. Compared to the average value of our database, [alpha] = 1.43, or the conventional value, [alpha] = 1.3, the irregularity of the Angstrom exponent is really noteworthy. For high turbidities, AOD500 > 0.85, the scatter of [alpha] is smaller, 1.2 < [alpha] < 1.9.

To conclude, Fig. 9 is a visual argument about the lack of relationship between the Angstrom [alpha] and AOD500. Atmospheric aerosol particles, apparently, represent a composite of several different types and sizes, which means that the relationship between ln(AOD[lambda]) and ln([lambda]/[[lambda].sub.0]) is more complicated than a linear one. The Angstrom exponent is not correlating with AOD[lambda] and, thus, including it into the AOD[lambda] broadband models does not improve the predictions.

7. CONCLUSIONS

Eight broadband models have been selected for investigation of their accuracies to predict aerosol spectral optical depth, AOD500. Seven of the considered models were simple ones, expressed with some formulas only. Such a simplistic approach is often sufficient for the interpretation of long-term changes in the column aerosol content, description of radiation regime, modelling of radiative transfer, validation of aerosol impact in climatological models, correction of satellite imagery, etc.

Broadband direct solar irradiance, as an input parameter to the models, was observed at the Tartu-Toravere Meteorological Station (Estonia), which is included in the Baseline Surface Radiation Network. Precipitable water (W) was derived from surface water vapour pressure. All models were tested against AOD500 reference values obtained by the AERONET Cimel photometer located at the same station.

Even a visual review (Fig. 2) demonstrates that for relatively clean air, AOD500 < 0.6, all considered broadband models give reasonable results. For more turbid air, AOD500 > 0.6, the models (if not corrected) tend to underestimate the aerosol optical depth. The underestimation is inherent only in physical (i.e. nonstatistical) models. The main reason for the underestimation of AOD[lambda] at higher turbidities is that models consider only a narrow solar direct beam, exactly from the solar disc with its mean angular diameter of about 32'.

Analysis of turbidity data from the smoky summers of 2002 and 2010 in Moscow led to a necessity to increase physically predicted AOD500 values using a statistical approach. Three different correction schemes were proposed.

Results of the test runs (Table 2) showed that two models, T2 and M2a, performed better compared to others. Model T2 scored best in three statistics without any negative AOD500 predictions. Model M2a was best in two statistics, but gave 45 negative predictions. It should be noted that model M2a has an artificial limitation: predicted values between 0.4 < AOD500 < 0.477 are not applicable. In summary of intercomparison, these two models can be recommended for proxy calculation of AOD500.

Two considered models, M1 and T1, enable input of an a priori known Angstrom wavelength exponent a for each single prediction, but with no improvement of performance. In order to find reasons for this failed numerical experiment, we analysed climatological variability of column optical and humidity parameters in terms of monthly means at Toravere during 2002-2011. Correlation with the coefficient [alpha] was established only with the fine mode fraction (FMF).

A study of the annual courses of column optical parameters revealed an interesting fact: a spring-summer maximum of atmospheric column transparency in June. This finding was supported by low values of both broadband aerosol optical depth and AOD500. Cleaner air in June can be explained by (1) fresh vegetation, which restricts generation of dust, and (2) low number of forest and bog fires in Estonia as well as in its surrounding areas.

doi: 10.3176/proc.2014.4.06

Received 4 November 2013, revised 3 February 2014, accepted 5 February 2014, available online 20 November 2014

ACKNOWLEDGEMENTS

This investigation was supported by the project "Estonian Radiation Climate" funded by the European Regional Development Fund and by the Estonian Research Council through project TFP SF0180038s08. The authors would like to thank anonymous referees for their commitment and relevant comments.

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Martin Kannel (a) *, Hanno Ohvril (a), Oleg Okulov (b), Kaidi Kattai (a), and Lennart Neiman (a)

(a) Laboratory of Atmospheric Physics, Institute of Physics, University of Tartu, Ulikooli 18, 50090 Tartu, Estonia

(b) Estonian Environment Agency, Mustamae tee 33, 10616 Tallinn, Estonia

* Corresponding author, martin.kannel@ut.ee

Ainus analuusitud keerukam mudel (G1) koosneb umbes 30 valemist ja lubab spetsifitseerida atmosfaari gaasilist koostist detailsemalt, naiteks osooni- ning lammastikdioksiidikoguste kaudu.

AOD500 arvutused toetusid Toraveres moodetud Paikese laiaribalisele otsekiirguse kiiritustihedusele, mille mooteandmeid voib kvaliteetseteks pidada, sest Tartu-Toravere ilmajaam kuulub rahvusvahelisse kiirgusmootmiste vorgustikku BSRN. Teine sisendsuurus, ohusamba veeaurusisaldus, ei olnud otseselt moodetud, vaid leitud kaudselt, veeaururohu kaudu. Koigi mudelite saadud AOD500 tulemusi vordlesime AOD500 tegelike vaartustega, moodetuna Toraveres AERONET Cimel-i fotomeetri poolt.

Mudelite voimekuse analuusiks koostasime ulatusliku andmebaasi, milles iga vaatlusrida uhendas nii ilmajaamas moodetud suurused (Paikese integraalne otsekiirgus, veeaururohk) kui ka AERONET-i fotomeetri samaaegselt moodetud suurused (AOD500 tegelik vaartus, Angstromi lainepikkuse eksponent, [alpha] (440-500-675-870)). Kokku sisaldas andmebaas 26 091 integraalset-spektraalset vaatlust, mis toimusid koikidel kalendrikuudel (valja arvatud detsember) kumne aasta jooksul, 2002-2011. Selline andmebaas voimaldas iga mudeli pohjalikku vordlust AERONET-i tegelike AOD500 mootmistega.

Suhteliselt puhta ohu korral, AOD500 < 0,6, andsid koik mudelid moistlikke tulemusi. Saastatuma ohu korral, mil AOD500 > 0,6, kaldusid statistiliselt korrigeerimata mudelid aerosooli optilist paksust alahindama. Alahindamist pohjustab Paikese oreooli nn parasiitkiirgus, mis suurendab aktinomeetri moodetud otsekiirgust ja loob mulje puhtamast ohust.

Testimistulemused tostsid esile kaks mudelit: T2 ja M2a. Tuleb siiski markida, et mudeli M2a valjundis on kunstlik piirang, puuduvad arvutatud vaartused vahemikus 0,4 < AOD500 < 0,477. Vaatlusalustest mudelitest kaks, M1 ja T1, voimaldasid sisendsuurusena muuta ka Angstromi lainepikkuse eksponenti a. Tavapraktikas pole avaartused a priori teada, koostatud andmebaasis aga sisaldus selline voimalus. Monevorra ootamatult ei parandanud a kaasamine AOD500 arvutustulemusi. Leidmaks pohjusi selle ebaonnestunud numbrilise eksperimendi kohta, analuusisime atmosfaarisamba optiliste ja niiskusparameetrite sesoonset muutlikkust Toraveres aastatel 2002-2011. Selgus, et Angstromi a korreleerub vaid osakeste nn peenmoodi fraktsiooni (FMF) panusega AOD500-sse. Kuu keskmiste loikes naitas FMF head korrelatsiooni Angstromi eksponendiga ja peenosakeste domineerimist suvekuudel.

Korvalproduktina AOD500 mudelite vordlemisele selgus, et atmosfaari labipaistvusel on juunis kevadsuvine maksimum, pohjuseks ilmselt ohusamba vaiksem aerosooliosakeste sisaldus. Esialgu oskame seda fakti pohjendada arvamusega, et juunis on Eestis valja kujunenud varske taimkate, mis takistab tolmu teket ja levikut. Samuti pole juunis veel levinud kohalikud raba- ja metsapolengud.

Uuringut toetasid kaks projekti: Euroopa regionaalarengu fondi rahastatav "Eesti kiirguskliima" ja Eesti Teadusnoukogu sihiteema TFP SF0180038s08.

1. M TRODUCTION

Aerosol optical depth, AOD[lambda], is a central parameter for the description of column aerosol content and column optical properties, including the rate of atmospheric turbidity. From the 1990s, after the start of the US NASA programmes AERONET, TerraMODIS, and AquaMODIS, this parameter became available and very popular (Holben et al., 1998; Toledano et al., 2007). However, in order to obtain a more capacious temporal and spatial overview of aerosol distribution, especially for retrospective (backward) extrapolation of AOD[lambda] time series to pre-1990 years when the photometric network was sparse, it is necessary to use alternative methods for calculation. There are also two other reasons for AOD[lambda] proxy, mainly broadband calculations: (1) quality inspection of recently measured AOD[lambda] time series, although by a modern solar photometer, because when the recorded values seem too large (overestimated for a certain period), a doubt always arises about an undesirable object (insect, spider's thread, trash, etc.) dwelling on or inside the instrument's tube; and (2) a quick AOD[lambda] estimation for correcting satellite remotely sensed data for regions or moments where/when spectral solar observations are not available but broadband ones are.

Several AOD[lambda] broadband models can be found from an extensive literature survey. The high-performance models are laborious for processing large amounts of data or they require either special or very accurate input quantities. However, there are other models which, especially for physical climatology, allow easy programming with no need of ancillary meteorological input data. Continuing activity in the development and modification of simple broadband models indicates that such approaches do correspond to practical needs.

In general, there are three main input parameters for simple AOD[lambda] models. One describes column attenuation of the broadband direct solar beam, e.g. broadband transmittance ([[tau].sub.m]), Bouguer coefficient of column transparency ([p.sub.m]), column broadband optical depth ([[delta].sub.m]), Linke turbidity factor ([T.sub.L,m]), which all are equal, linked by the well-known Bouguer-Lambert exponential law:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Here [S.sub.m] is the measured broadband irradiance at optical mass m; [S.sub.0] is the extraterrestrial broadband solar irradiance at the actual Sun-Earth distance, its average value, the "solar constant", is 1.367 kW [m.sup.-2] (Lenoble, 1993); and [[delta].sub.CDA,m] is the optical depth of an ideal, i.e. clean and dry atmosphere (CDA).

To eliminate the Forbes effect, inherent to column broadband optical characteristics, the generally accepted practice is to reduce them from the actual optical air mass m to a standard air mass, mainly to m = 2 (solar elevation h [approximately equal to] 30[degrees]). As to possible processes of reduction, it seems to us that quite successful methods have been developed in regard to the Bouguer coefficient [p.sub.m] (Ohvril et al., 1999, 2009). For this reason, [p.sub.2] is the input parameter of several models in this study.

A second input parameter to simple models is column humidity (water vapour content). Its unit, mass per unit area, is in practice usually given as the thickness of the layer of liquid water: 1 mm corresponds to 1 kg [m.sup.-2] and 1 cm to 1 g [cm.sup.-2].

A third parameter, the Angstrom wavelength exponent, [alpha], is closely linked to the size distribution of aerosol particles, provided that the size distribution, in part, follows a power law (Liou, 2002). However, the column aerosol particles are in permanent change, expressing deviations from the power law. Hence the wavelength exponent is actually a very unstable parameter, which has different values for different parts of the solar spectrum. As it is poorly correlated with AOD[lambda], its use in atmospheric optical models is questionable.

The present work aims at validation of seven simple broadband models and one more complicated model for AOD[lambda] calculations.

Five simple models (M1, M2, M2a, M2b, and M2c) were developed in Moscow during 1991-2013, and two in Tartu (T1, T2) during 2007-2012. The most complicated model (G1) is Gueymard's parameterization from 1998, based on his known SMARTS2 code, and it allows considering numerous column parameters. The set of the used models (including references) will be described in detail in Section 2.

All models were tested against AOD500 reference values obtained by an AERONET photometer at Toravere, Estonia, located in the territory of the Tartu-Toravere Meteorological Station during 10 years, 2002-2011. The station (58.26[degrees]N, 26.46[degrees]E, 70 m ASL), is included into the Baseline Surface Radiation Network (Kallis et al., 2005).

Simultaneous registration of both spectral and broadband irradiances provided an opportunity to create a joint, integrated database for AOD[lambda] together with the Angstrom exponent and broadband parameters of atmospheric transparency and turbidity. The joint database includes 26 091 spectral-broadband solar direct irradiance and surface water vapour pressure observations covering all months except December.

Although the AERONET observations, besides AOD[lambda], give also values of precipitable water, W, we have considered that the input values for the models should be independent in regard to a reference instrument. For that, precipitable water was evaluated using surface water vapour pressure, [e.sub.0] (Okulov et al., 2002; Okulov and Ohvril, 2010).

2. A REVIEW OF THE USED BROADBAND MODELS

A quick review of the used broadband models together with their inputs and outputs can be obtained from Table 1.

2.1. Models Ml and M2

The models were developed in the Meteorological Observatory of the Moscow State University (Tarasova and Yarkho, 1991a, 1991b). Model M1 consists of 13 formulas; it is somewhat general and unique because it allows varying the Angstrom wavelength exponent, a. Model M2 is a simplified version of M1, where the Angstrom exponent has a fixed value, [alpha] = 1. Model M2 is expressed by only one formula

AOD550 = ln [S.sub.m] -(0.189[W.sup.-0.183] + [[0.880[W.sup.-0.009] - 1]/sin h])/ 0.813[W.sup.-0.002] -1 + [[0.435[W.sup.-0.0321] - 1]/sin h] (2)

where h is solar elevation and W zenith precipitable water in centimetres.

Model M2 was widely used for monitoring aerosol turbidity in the Russian territory (Abakumova and Gorbarenko, 2008). Routine registration of broadband direct solar irradiance has been performed in Moscow since 1955. Before the development of models M1 and M2 in 1991, there was one summer, in 1972, when the region around Moscow was affected by extensive forest and peat fires, but in that summer the capabilities to check broadband calculations of AOD[lambda] by reference spectral instruments were limited. A solar-sky AERONET Cimel photometer was deployed in Moscow for measuring AOD[lambda] as early as in 2001. The next, extremely dry summer of 2002 favoured huge fires around Moscow and the city was filled with smoke (the CIMEL photometer is located near the main building of the university, in the territory of the botanic garden). Prediction of AOD[lambda] under the conditions with a large amount of smoke aerosols (in summer 2002) revealed the fact that model M2 gives lower values at very turbid air in comparison to AERONET simultaneous observations.

The main reason for the underestimation of AOD[lambda] at high turbidities by the broadband models is that the models consider only a "true" narrow solar direct beam, exactly from the solar disc with its mean angular diameter of about 32 arc minutes. Actually the opening angles of older broadband instruments, but still in use for homogeneity of multidecadal time series, have considerably wider apertures reaching up to 10[degrees]. For example, the Kipp & Zonen Linke-Feussner actinometer has an aperture of 10.2[degrees] (Gueymard, 1998; Garg and Prakash, 2006). The AT-50 actinometers, continuously in use in actinometric networks in the territory of the former USSR, have the full field of view, FOV = 10[degrees].

Although the FOV of most current pyrheliometers is smaller (e.g. for the Eppley Laboratory Inc. normal incidence pyrheliometers (NIP) the FOV = 5.7[degrees]), the measured direct beam is anyway increased by undesirable diffuse irradiance intercepted by a broadband instrument (Gueymard, 1998; Carlund et al., 2003).

The magnitude of this increase is greater at low solar elevation and heavy aerosol loading, and also in cases of large aerosol particles such as maritime aerosol, biological aerosol, desert or ground dust, etc. In cases of small particles (e.g. almost pure molecular scattering in a clear atmosphere after a rain) circumsolar radiation is weaker. According to calculations made by Gueymard (1998) for FOV = 10.2[degrees], the circumsolar magnification factor can reach 35% in regard to the true direct irradiance. Increased artificial values of the observed broadband direct beam, [S.sub.m], lead to underestimation of the modelled AOD[lambda]. The artificial increase in readings of modern spectral photometric observations is decreased due to the smaller aperture (e.g. for the CIMEL-318 radiometer, the FOV = 1.2[degrees]).

2.2. Model M2a

Analysis of data from a smoky summer of 2002 when AERONET observations in Moscow were already available, led to a conclusion that a better match between predicted and observed values at AOD500* > 0.4 can be achieved by substitution of initially predicted AOD500 * with its increased counterpart, AOD500 (Chubarova, 2005):

AOD500 = 1.301X [(AOD500*).sup.1.095]. (3)

The correction from AOD500* = 0.4 towards bigger values increases the initially predicted AOD500* but also leads to an artificial discard of corrected aerosol optical depths in a range of 0.4 < AOD500 < 0.477, which is a secondary visual defect rather than a functional imperfection. Application of model M2 together with the correction for AOD500* > 0.4 is further denoted by M2a.

The third catastrophically dry and hot summer with flaming and smoldering wildfires around Moscow occurred in 2010. In contrast to previous smoky periods, in 1972 and 2002, the summer of 2010 was characterized by higher aerosol optical depths, reaching even a value of AOD500 = 4.6. In a review on radiation monitoring of all three smoky summers Chubarova et al. (2011a, 2011b) recommended implementation of correction by Eq. (3) from AOD500* = 0.5 onward, which means an absence of corrected AOD500 even in a larger range, 0.5-0.609.

2.3. Model M2b

However, in massive data processing, generation of a permanent empty zone for corrected AOD500 values is not desirable. On the other hand, for low AOD500* values correction by Eq. (3) is not significant. Moreover, below a certain value, AOD500* [less than or equal to] 0.062, the implementation of Eq. (3) will lead instead of AOD500* enlargement to its reduction. For example, inserting AOD500* = 0.025 into Eq. (3) one obtains AOD500 = 0.023. In such cases the correction factor, CF,

CF = AOD500/AOD500* (4)

obtains values <1.0. Use of Eq. (3) only for AOD500* [greater than or equal to] 0.063 (CF [greater than or equal to] 1) avoids appearance of an empty zone for corrected AOD500 and secures a smooth correction. The use of Eqs (2) and (3) together with a condition for AOD500* [greater than or equal to] 0.063 only is further denoted by M2b.

2.4. Model M2c

Recently Gofbarenko and Rublev (2013) reported about using a solar elevation-dependent correction of AOD500*. Assuming continental aerosols with a fixed Angstrom exponent, [alpha] = 1, they derived a correction algorithm:

AOD550 = (AOD550*)[0.9 + [(AOD550*)[0.7/5sin h+0125]/5], (5)

which they further applied for AOD550* > 0.5. Because AOD500* = 1.1 x AOD550*, the next algorithm for 500 nm can be obtained:

AOD500 =

(AOD500*/1.1)[0.9 + 0.2[(AOD500*/1.1).sup.[0.7/0.75sinh+0.125]]. (6)

However, the two last formulas reduce the first calculated AOD* for cleaner air also (i.e. in cases of low aerosol content) like the use of correction by Eq. (3). For example, in case of AOD500* = 0.36 and sin h = 0.5, Eq. (6) gives AOD500 = 0.34.

The minimal turbidity, min AOD500*, where correction (6) has a sense, depends on the solar elevation:

minAOD500* = [1.1x 0.5.sup.0.75sinh+0.125/0.7], (7)

e.g. for sin h = 0.3, 0.5, 0.7, minAOD500* = 0.78, 0.68, 0.58, respectively. These values are definitely higher than minAOD500* = 0.063 for correction with formula (3) for M2b. The use of Eqs (2) and (6) together with condition (7) is denoted as model M2c.

Figure 1 compares results of correction with Eqs (3) and (6), in regard to their common input counterpart (AOD500* [greater than or equal to] 0, including values even below the minAOD500*). Differences appear at very large turbidities and low sun. For AOD500* = 4 the correction by Eq. (3), published in 2005, gives AOD500 = 5.9, but the new one by Eq. (6) from 2013 gives AOD500 = 6.8, 8.5, 14.2 for sin h = 0.7, 0.5, 0.3, respectively.

2.5. Model G1

Apparently the most advanced broadband model was derived by Gueymard (1998), further denoted as G1. The model, originally for the prediction of AOD1000, contains about 30 formulas and allows varying several minor column gaseous components such as [O.sub.3] and tropospheric and stratospheric N[O.sub.2].

However, based on our evaluations and supported by Gueymard's error analysis (2013), the variability of the ozone amount can be considered a second-order input because of its small impact on the solar broadband direct beam. Besides, nitrogen dioxide would only be of concern over polluted areas. Therefore, in the extensive runs of Gueymard's model, we used the given fixed input values, typical for the Baltic Sea region: [O.sub.3] = 0.35 atm cm, N[O.sub.2](stratospheric) = 0.00012 atm cm, N[O.sub.2](tropospheric) = 0.00004 atm cm, p = 1013.25 hPa (Kannel et al., 2012).

Concerning the Angstrom wavelength exponent, in accordance with Gueymard, it is in general not possible to know a priori whether the observed aerosol particles belong to the continental, maritime, or any other specific type. Therefore, a fixed conventional value of [alpha] = 1.3, representative of particles of rural-continental origin, was proposed by Gueymard.

The preliminary output for models M1 and M2 is originally AOD550, for G1 it is the Angstrom turbidity coefficient, [beta] = AOD1000. Transitions to AOD500 are easy applying the Angstrom exponential formula in a general form:

AOD([[lambda].sub.2]) = AOD([[lambda].sub.1]) AOD([[lambda].sub.1])[([[lambda].sub.2]/[[lambda].sub.1]).sup.-[alpha]]. (8)

2.6. Models T1 and T2

Two broadband models, T1 and T2, were developed at the University of Tartu. Model T1 is actually a modification of model M1 (Kannel et al., 2007), with some changes to consider the effects of circumsolar radiation (Kannel, 2007; Ohvril et al., 2009). The model is expressed by a single formula and has three input quantities: (1) Angstrom exponent, [alpha]; (2) coefficient of column broadband transparency, [p.sub.2] transformed to atmospheric mass, m = 2; (3) precipitable water, W (Kannel, 2007; Kannel et al., 2007; Ohvril et al., 2009).

Model T2 was derived using barely a statistical approach. In creating the method, a large database, including almost 20 000 complex, spectral, and broadband direct solar beam observations at Toravere, Estonia, during all seasons of an 8-year period 2002-2009, was used. Apparently, the model is local, and could be used only in conditions similar to Toravere. Monthly climatology of column optical parameters for Toravere will be given below in Section 6.

The model relies only on two input parameters: (1) coefficient of Bouguer column broadband transparency, [p.sub.2]; and (2) precipitable water, W. These parameters allow calculating for m = 2 a specific quantity, the column broadband aerosol optical depth (BAOD2). According to Kannel et al. (2012), the two optical depth parameters, AOD500 and BAOD2, are strongly correlated ([R.sup.2] = 0.96) through a second-degree polynomial:

AOD500 = 1.7[(BAOD2).sup.2] + 1.3(BAOD2), (9)

which enables an easy calculation of AOD500. Table 1 lists all considered broadband AOD[lambda] models with their input and output quantities.

3. OBSERVATIONAL DATA

We used two institutionally independent databases from the period 2002-2011.

1. Broadband direct solar irradiance and surface humidity measurements acquired at the Tartu-Toravere Meteorological Station, Estonia. The station is included into the Baseline Surface Radiation Network (Kallis et al., 2005). The data allowed calculation of the coefficient of column transparency ([p.sub.2]) and broadband aerosol optical depth (BAOD2) for each single observation. Note that the calculation of BAOD2 is not sensitive to uncertainties in precipitable water (W) estimations, thus W was estimated from the station surface water vapour pressure. Details of data processing are given in (Kannel et al., 2012). The quality of the W estimation is discussed in Section 6.

2. Spectral aerosol optical depth measurements by AERONET CIMEL photometers, which began regular observations in the territory of the meteorological station at Toravere on 3 June 2002.

The two databases provided the opportunity to create a joint, integrated database for AOD[lambda] and broadband parameters of atmospheric transparency (turbidity). Our joint database includes 26 091 spectral-broadband solar direct irradiance and surface water vapour pressure observations for ten years, 2002-2011. About 75% of the observations were made in April, May, June, July, and August; 9% in September; 8% in March; 4% in October; and only 4% together in January, February, and November. In December no joint observations were made due to the low Sun and photometer calibrations.

The abundance of data enabled a comprehensive comparison of each model against the AERONET AOD500 observations.

4. TEST OF MODELS

Test runs of the eight models contained 26 091 single calculations of AOD500 by each model, followed by comparison against the reference AOD500(AERONET) value. Two models, M1 and T1, allow varying the input value of the Angstrom exponent, [alpha]. For this reason additional runs were performed for these models with the variation of [alpha].

To obtain quantitative measures on the accuracy of the models, we use two linear regression parameters, slope and correlation [R.sup.2] (actually coefficient of determination), and the following three commonly used statistical parameters (Iqbal, 1983; Gueymard, 1993):

(1) the mean bias deviation (MBD), expressing the average deviation (difference) of the predicted values, [y.sub.i] = AOD500(Model), from the reference values, [x.sub.i] = AOD500(AERONET):

MBD = [1/N] [N.summation over (i=1)] [[y.sub.i] - [x.sub.i]]; (10)

(2) the root mean square deviation (RMSD), a measure of the variation of predicted values around the reference values:

RMSD = [1/N] [[[N.summation over (i=1)][([y.sub.i] - [x.sub.i]).sup.2]].sup.1/2] (11)

(3) the mean absolute relative deviation (MARD), also known as the mean absolute percentage deviation (MAPD), expressing the average value of relative deviations:

MARD = [1/N] [N.summation over (i=1)] [absolute value of ([[y.sub.i] - [x.sub.i]]/[x.sub.i])] (12)

The amount of column water vapour (W) is an input parameter to all considered AOD[lambda] models. It was estimated from surface conditions using correlation with vapour pressure, [e.sub.0] (Okulov et al., 2002). At the level of monthly means the used correlation performs quite well (Section 6), but for single observations the coefficient of determination, [R.sup.2] = 0.83, indicates a moderate scatter of W([e.sub.0]) around W (AERONET) (Kannel et al., 2012). Overestimation of W leads to underestimation of AOD[lambda], even to a physically unrealistic negative AOD[lambda]. For the sake of brevity, sensibility analysis of the considered models to possible W errors is not included in the present paper. We only give the number of predicted negative AOD500 values together with the number of corrections applied for each run of the model (Table 2).

Results of the eight main runs are presented in Fig. 2, where each panel also reviews performance statistics of the run: slope, correlation [R.sup.2], number of corrected values and predicted negative values. Table 2 lists also results of six additional runs: (1) two runs for both M1 and T1 with a different fixed wavelength exponent, and (2) a run for both M1 and T1 with an a priori known wavelength exponent. The last two runs are visualized in Fig. 3.

From Table 2, which includes six statistics, one can rank the accuracy of the performance of the different models (with different Angstrom [alpha], if applied) in regard to different statistics. In summary, giving a point for the best result in each "event" of the "hexathlon", an overall, combined ranking for T2 appears to score 4 points, and for M2a, 2 points. These two models can be recommended, because of their consistently high performance in all items, for the evaluation of AOD500 in conditions similar to those at Toravere. The two only models, M1 (row 4) and T1 (row 13), that consider a priori known Angstrom exponents did not stand out. We try to analyse this disappointing result in the next two sections.

5. FIXED VS INDIVIDUAL ANGSTROM EXPONENT

Usually broadband models for the calculation of AOD[lambda] do not enable changing the Angstrom wavelength exponent, [alpha]. In this sense model Ml and its derivate, T1, represent an exception. Nevertheless, it is meaningful that the authors of model Ml themselves have actually not used this opportunity; they first began to use M2 and then M2a, in both of which the exponent is fixed, [alpha] = 1. In model G1 the exponent is also fixed, [alpha] = 1.3, but it assumes validity of the Angstrom exponential formula (8). On the other hand, this means that the size distribution n(r) is partly given by the Junge power law (Liou, 2002):

n(r) = C(r) [r.sup.-([alpha]+3)] (13)

where C is a scaling factor proportional to particles column concentration and r is a particle radius. "Partly given" means that only the downgoing part of the size distribution (onward from the maximum) can be approximated by the power law.

Our joint 26 091 observation database is special, because each single observation contains also an Angstrom AERONET-evaluated exponent, [alpha](440-500-675-870), calculated as a best fit for the indicated four wavelengths. In this way the database enabled taking into consideration a priori known Angstrom exponents for predicting single AOD500 values.

However, runs of models M1 and T1 with an a priori known [alpha] (rows 4 and 13, Table 2) did not give the expected improvement of the predictions (Fig. 3).

So, a priori known Angstrom exponents did not improve AOD500 predictions. But should they? Consider the Angstrom formula is obeyed, then the exponent enables only transition from a known AOD[[lambda].sub.1] to any other AOD[[lambda].sub.2], not to start with the magnitude of AOD[[lambda].sub.1] itself. In terms of the Angstrom formula, prediction of AOD[[lambda].sub.1] can be done using a second parameter, the Angstrom turbidity coefficient, [beta].

In the next section we examine the background, mainly seasonal variability of column optical and humidity properties at Toravere, 2002-2011, and try to find regularities in the behaviour of the Angstrom exponent.

6. VARIABILITY OF THE USED COLUMN OPTICAL AND HUMIDITY PARAMETERS

Figure 4 provides monthly means of the Angstrom a and AOD500 and, in addition, seasonal variation of the fine mode fraction (FMF), which is one of the AERONET inversion products describing the contribution of fine particles to AOD500. The AERONET inversion code finds the minimum of the size distribution within the radius interval from 0.439 to 0.992 [micro]m. This minimum, approximately at 0.6 [micro]m radius, is used as a separation point between fine and coarse mode particles. Using that separation, the inversion code calculates the contribution of fine particles to the formation of AOD500 (AERONET Inversion Products, 2010; O'Neill et al., 2001). The AERONET term, fine particles, actually includes three traditional subregions of aerosol size distribution: the nucleation, Aitken, and accumulation mode.

As expected, the intra-annual evolution of the Angstrom exponent is consistent with monthly changes in the FMF. The higher values in summer (Jun-Jul-Aug) indicate domination of fine aerosol particles.

Somewhat surprising is a local minimum of AOD500 in June. For the cold season, we have no good explanations to the higher AOD500 in January and November compared to February and October; this is a topic of further studies.

The monthly means of column transparency ([p.sub.2]) for 2002-2011 (Fig. 5) are in opposite phase with AoD500, with an expected local maximum in June. Actually, higher column transparency in June was noticed already since 1994. This finding can be explained by a general cleaning of the European atmosphere as a part of the global brightening (Ohvril et al., 2009; Okulov and Ohvril, 2010). During June, the Estonian landscape is already totally covered with fresh vegetation, which restricts creation and vertical distribution of dust. The number of forest and bog fires is also low in June.

Column precipitable water (W) is usually evaluated from surface humidity and temperature, in our study:

W (mm) = 1.48[e.sub.0] + 0.40, (14)

where [e.sub.0] (mb) is the 12 UTC surface water vapour pressure. This parameterization was developed from clear-sky radio soundings in Tallinn (Okulov et al., 2002). For Toravere, Kannel et al. (2012) compared column humidity predictions, W ([e.sub.0]), with W(AERONET), where the latter were considered as reference. Use of almost 20 000 parallel observations from 2002-2009 showed that the prediction overestimates the reference as an average only by 3%. For monthly means of W over all considered years (2002-2011), approximation (14) gives values close to those obtained by the AERONET photometer (Fig. 6).

To model the extinction of broadband direct solar beam, for optical mass m = 2, we assume that the atmosphere consists of three layers or substances (Kannel et al., 2012): an ideal or clean and dry atmosphere (CDA), integrated column water vapour (W), and atmospheric aerosol particles.

Figure 7 shows monthly mean broadband transmittances for these three layers at Toravere, calculated from our database for a 10-year period, 2002-2011. The plots were prepared using only meteorological data, not the AERONET observations. Noticeable is the summer maximum of aerosol broadband transmittance in June, which apparently is the main reason of higher total column transmittance in this month. The lowest total transmittance occurs in July and August, caused by low transmittances of both column water vapour and aerosols.

Figure 8 shows the seasonal variation of total broadband optical depth, and its division into the main atmospheric constituents.

Unfortunately, the annual cycles of the reviewed column parameters (except the fine mode fraction) do not give us the expected relationships with the annual cycle of the Angstrom exponent (Fig. 4). Moreover, a plot of the Angstrom [alpha] against AOD500 revealed no correlation (Fig. 9). The [alpha] varies from 0.05 to 3.43 for cleaner air, when AOD500 < 0.2 encompassing the simple (nonclimatological) average of the exponent, [alpha] = 1.43. The standard deviation (one sigma uncertainty) of the exponent for the whole database [sigma] = 0.35.

Supposing normal distribution and coverage factor 3, the expanded 3[sigma]-uncertainty becomes 1.05. Compared to the average value of our database, [alpha] = 1.43, or the conventional value, [alpha] = 1.3, the irregularity of the Angstrom exponent is really noteworthy. For high turbidities, AOD500 > 0.85, the scatter of [alpha] is smaller, 1.2 < [alpha] < 1.9.

To conclude, Fig. 9 is a visual argument about the lack of relationship between the Angstrom [alpha] and AOD500. Atmospheric aerosol particles, apparently, represent a composite of several different types and sizes, which means that the relationship between ln(AOD[lambda]) and ln([lambda]/[[lambda].sub.0]) is more complicated than a linear one. The Angstrom exponent is not correlating with AOD[lambda] and, thus, including it into the AOD[lambda] broadband models does not improve the predictions.

7. CONCLUSIONS

Eight broadband models have been selected for investigation of their accuracies to predict aerosol spectral optical depth, AOD500. Seven of the considered models were simple ones, expressed with some formulas only. Such a simplistic approach is often sufficient for the interpretation of long-term changes in the column aerosol content, description of radiation regime, modelling of radiative transfer, validation of aerosol impact in climatological models, correction of satellite imagery, etc.

Broadband direct solar irradiance, as an input parameter to the models, was observed at the Tartu-Toravere Meteorological Station (Estonia), which is included in the Baseline Surface Radiation Network. Precipitable water (W) was derived from surface water vapour pressure. All models were tested against AOD500 reference values obtained by the AERONET Cimel photometer located at the same station.

Even a visual review (Fig. 2) demonstrates that for relatively clean air, AOD500 < 0.6, all considered broadband models give reasonable results. For more turbid air, AOD500 > 0.6, the models (if not corrected) tend to underestimate the aerosol optical depth. The underestimation is inherent only in physical (i.e. nonstatistical) models. The main reason for the underestimation of AOD[lambda] at higher turbidities is that models consider only a narrow solar direct beam, exactly from the solar disc with its mean angular diameter of about 32'.

Analysis of turbidity data from the smoky summers of 2002 and 2010 in Moscow led to a necessity to increase physically predicted AOD500 values using a statistical approach. Three different correction schemes were proposed.

Results of the test runs (Table 2) showed that two models, T2 and M2a, performed better compared to others. Model T2 scored best in three statistics without any negative AOD500 predictions. Model M2a was best in two statistics, but gave 45 negative predictions. It should be noted that model M2a has an artificial limitation: predicted values between 0.4 < AOD500 < 0.477 are not applicable. In summary of intercomparison, these two models can be recommended for proxy calculation of AOD500.

Two considered models, M1 and T1, enable input of an a priori known Angstrom wavelength exponent a for each single prediction, but with no improvement of performance. In order to find reasons for this failed numerical experiment, we analysed climatological variability of column optical and humidity parameters in terms of monthly means at Toravere during 2002-2011. Correlation with the coefficient [alpha] was established only with the fine mode fraction (FMF).

A study of the annual courses of column optical parameters revealed an interesting fact: a spring-summer maximum of atmospheric column transparency in June. This finding was supported by low values of both broadband aerosol optical depth and AOD500. Cleaner air in June can be explained by (1) fresh vegetation, which restricts generation of dust, and (2) low number of forest and bog fires in Estonia as well as in its surrounding areas.

doi: 10.3176/proc.2014.4.06

Received 4 November 2013, revised 3 February 2014, accepted 5 February 2014, available online 20 November 2014

ACKNOWLEDGEMENTS

This investigation was supported by the project "Estonian Radiation Climate" funded by the European Regional Development Fund and by the Estonian Research Council through project TFP SF0180038s08. The authors would like to thank anonymous referees for their commitment and relevant comments.

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Martin Kannel (a) *, Hanno Ohvril (a), Oleg Okulov (b), Kaidi Kattai (a), and Lennart Neiman (a)

(a) Laboratory of Atmospheric Physics, Institute of Physics, University of Tartu, Ulikooli 18, 50090 Tartu, Estonia

(b) Estonian Environment Agency, Mustamae tee 33, 10616 Tallinn, Estonia

* Corresponding author, martin.kannel@ut.ee

Table 1. List of the considered broadband AOD[lambda] models. Possible inputs: h -solar elevation; [S.sub.m] -broadband direct solar irradiance; [p.sub.2] -broadband (integral) Bouguer coefficient of column transparency for optical air mass m = 2; W -precipitable water; [alpha] -Angstrom wavelength exponent; [O.sub.3] -column ozone content; N[O.sub.2] -column nitrogen dioxide content Model Reference Input Number of formulas M1 Tarasova and h, [S.sub.m], W, 13 Yarkho, 1991a, [alpha], [O.sub.3] = 1991b 300 DU M2 Abakumova and h, [S.sub.m], W, 1 Gorbarenko, 2008 [alpha] = 1, [O.sub.3] = 300 DU M2a Chubarova, 2005 h, [S.sub.m], W, 2 [alpha] = 1, [O.sub.3] = 300 DU M2b Chubarova, 2005 h, [S.sub.m], W, 2 [alpha] = 1, [O.sub.3] = 300 DU M2c Gorbarenko and h, [S.sub.m], W, 2 Rublev, 2013 [alpha] = 1, [O.sub.3] = 300 DU G1 Gueymard, 1998 h, [S.sub.m], W, > 30 [alpha] = 1.3, [O.sub.3], N[O.sub.2] T1 Kannel, 2007; [p.sub.2], W, 1 Kannel et [alpha], [O.sub.3] = al., 2007 300 DU T2 Kannel et [p.sub.2], W 2 al., 2012 Model Output Correction of prediction M1 AOD550 Not used M2 AOD550 Not used M2a AOD500 For AOD500 > 0.4 M2b AOD500 For AOD500 [greater than or equal to] 0.063 M2c AOD550 Depends on solar elevation G1 AOD1000 Not used T1 AOD500 Not used T2 AOD500 Not used Table 2. Inputs of the Angstrom exponent and performance statistics for AOD500 predictions by different models against AERONET observations during 2002-2011. In each run, the total number of predictions was 26 091. Numbers in bold indicate the best result in the column. Negative predictions are physically unrealistic Row Model, Angstrom Number of figure [alpha] corrections as input 1 M1 1.2 not used 2 M1, Fig. 2 1.3 not used 3 M1 1.4 not used 4 M1, Fig. 3 individual not used 5 M2, Fig. 2 1.0 not used 6 M2a, Fig. 2 1.0 822 7 M2b, Fig. 2 1.0 21 733 8 M2c, Fig. 2 1.0 213 9 G1, Fig. 2 1.3 not used 10 T1 1.2 not used 11 T1, Fig. 2 1.3 not used 12 T1 1.4 not used 13 T1, Fig. 3 individual not used 14 T2, Fig. 2 not used not used Row Performance statistics Slope [R.sup.2] Negative predictions 1 0.935 0.933 107 2 0.974 0.936 129 3 1.016 0.938 143 4 1.055 0.945 159 5 0.912 0.923 45 6 0.995 0.947 45 7 1.060 0.947 45 8 0.917 0.927 45 9 0.972 0.929 1 10 0.915 0.934 0 11 0.962 0.936 1 12 1.009 0.938 1 13 1.088 0.941 2 14 1.013 0.957 0 Row Performance statistics MBD RMSD MARD 1 -0.002 0.032 0.203 2 0.003 0.030 0.214 3 0.009 0.031 0.233 4 0.013 0.032 0.247 5 -0.004 0.034 0.203 6 0.001 0.029 0.201 7 0.011 0.033 0.230 8 -0.004 0.033 0.203 9 0.009 0.029 0.221 10 -0.003 0.032 0.197 11 0.004 0.029 0.211 12 0.011 0.030 0.232 13 0.017 0.037 0.253 14 0.005 0.026 0.188

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Title Annotation: | PHYSICS |
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Author: | Kannel, Martin; Ohvril, Hanno; Okulov, Oleg; Kattai, Kaidi; Neiman, Lennart |

Publication: | Proceedings of the Estonian Academy of Sciences |

Article Type: | Report |

Geographic Code: | 4EXES |

Date: | Dec 1, 2014 |

Words: | 7001 |

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