Improved algorithm for calculations of Rayleigh-scattering optical depth in standard atmospheres

Improved algorithm for calculations ofRayleigh-scattering optical depth in standardatmospheres

Claudio Tomasi, Vito Vitale, Boyan Petkov, Angelo Lupi, and Alessandra Cacciari

Precise calculations of the total Rayleigh-scattering optical depth have been performed at 88 wavelengthsranging from 0.20 to 4.00 �m for the six well-known standard atmosphere models by integrating thevolume Rayleigh-scattering coefficient along the vertical atmospheric path from sea level to a 120-kmheight. The coefficient was determined by use of an improved algorithm based on the Ciddor algorithm[Appl. Opt. 35, 1566 (1996)], extended by us over the 0.20–0.23-�m wavelength range to evaluate themoist air refractive index as a function of wavelength, air pressure, temperature, water-vapor partialpressure, and CO2 volume concentration. The King depolarization factor was also defined taking intoaccount the moisture conditions of air. The results indicate that the influence of water vapor on Rayleighscattering cannot be neglected at tropospheric altitudes: for standard atmospheric conditions representedin terms of the U.S. Standard Atmosphere (1976) model, the relative variations produced by water vaporin the Rayleigh scattering parameters at a 0.50-�m wavelength turn out to be equal to �0.10% in themoist air refractivity at sea level (where the water-vapor partial pressure is equal to �7.8 hPa), �0.04%in the sea-level King factor, �0.24% in the sea-level Rayleigh-scattering cross section, and �0.06% in theRayleigh-scattering optical depth. © 2005 Optical Society of America

OCIS codes: 010.1310, 280.1310.

1. Introduction

The Rayleigh-scattering optical depth is used to mea-sure the overall attenuation of incoming direct solarirradiance along the vertical atmospheric path due tothe scattering effects produced by air molecules. Thisquantity is dimensionless since it is given by theintegral of the volume Rayleigh-scattering coefficientalong the vertical path of the atmosphere, includingall the molecular-scattering effects produced by theRayleigh component consisting of the molecular scat-tering that occurs “without change of frequency” andthe Raman component “as the Raman-scattered pho-tons have come from the incident beam, just as theunshifted photons have.”1 Thus the Rayleigh opticaldepth can be defined by an appropriate term compris-ing both Rayleigh scattering and vibrational Raman-

scattering effects, where the Rayleigh-scatteringcomponent consists of rotational Raman lines and thecentral Cabannes line, the latter being composed ofthe Brillouin doublet and the central Gross orLandau–Placzek line.2

Molecular scattering is certainly the strongestmechanism causing the atmospheric extinction of so-lar radiation in the cloudless atmosphere over theremote oceanic and land areas and, in particular, inthe Antarctic and Arctic high-latitude regions. Infact, calculations of Rayleigh-scattering optical depthperformed at various latitudes and seasons usuallyprovide values considerably higher than 1.00 atwavelengths shorter than 0.30 �m, equal to � 0.14 atthe 0.50-�m wavelength and lower than 0.01 atwavelengths longer than 1.00 �m.3–5 Conversely, themeasurements of aerosol optical depth (AOD) per-formed for clear-sky conditions in the remote regionsof Earth over the past 20 years have presented valuesmostly lower than 0.20 at ultraviolet (UV) wave-lengths, 0.10 at visible wavelengths, and 0.05 in thenear infrared (NIR).6–10 In particular, for very cleanair conditions of the atmosphere in Antarctica, theAOD assumes often smaller values than 0.05throughout the whole range from visible to NIRwavelengths. Thus the availability of precise values

The authors are with the Institute of Atmospheric Sciences andClimate, Consiglio Nazionale delle Ricerche, via Gobetti 101, Bo-logna I-40129, Italy. The e-mail address for C. Tomasi [email protected].

Received 21 July 2004; revised manuscript received 23 Decem-ber 2004; accepted 12 January 2005.

0003-6935/05/163320-22$15.00/0© 2005 Optical Society of America

3320 APPLIED OPTICS � Vol. 44, No. 16 � 1 June 2005

of Rayleigh-scattering optical depth within the widewavelength range from 0.20 to 4.00 �m is particu-larly important to obtain correct evaluations of theAOD from direct solar radiation extinction measure-ments and to realistically analyze the radiativetransfer processes that occur in the atmospherewithin the UV spectral interval. The need for thesecalculations is evidenced not only by the above com-parison between the values of aerosol and Rayleigh-scattering optical depths, but also by the fact thatRayleigh-scattering optical depth is considerablylarger at all the window wavelengths than those pro-duced in remote regions by ozone and water vapor. Infact, it is known that the total content of ozone in theatmospheric column seldom exceeds the value of0.35-cm STP11,12 in the polar regions and the otherremote regions, whereas precipitable water generallyassumes values lower than 1 g cm�2 for clear-sky con-ditions in the polar regions and of a few grams perinverse square centimeter in the mid- and low-latitude regions, also during the warmer period of theyear.13 Since the ozone absorption coefficient pre-sents values smaller than 0.15 �cm STP��1 at visibleand NIR wavelengths and the weak absorption coef-ficient of water vapor is considerably lower than0.01 g�1 cm2 at all the window wavelengths,14 the val-ues of ozone and water-vapor optical depth usuallyturn out to be both lower than 0.05 at all wave-lengths.

These evaluations clearly indicate that Rayleigh-scattering optical depth causes most of the solar ra-diation extinction occurring in the clear-skyatmosphere of remote planetary regions, yielding val-ues that are frequently at least 20 times greater thanthose of the AOD at wavelengths shorter than0.30 �m, from five to three times greater at a wave-length � � 0.50 �m, and of comparable magnitude inthe range of � � 1.00 �m. This implies that slighterrors of a few percent in calculations of the Rayleigh-scattering optical depth at UV and visible wave-lengths may often be comparable to the AOD valuesthemselves, suggesting that precise calculations ofRayleigh optical depth at these wavelengths can al-low the avoidance of important errors in the analysisof remote sensing measurements that aim to defineaerosol extinction parameters and particle radiativeproperties.15–19

Over the past 40 years, several attempts have beenmade to define realistic physical models of molecularscattering, which have led to more or less reliableevaluations of the Rayleigh optical depth at UV, vis-ible, and IR wavelengths.3–5,20–22 Such calculationswere performed with algorithms achieved throughvarious approximations generally made neglectingthe spectral dependence features of the depolariza-tion factor, underestimating the influence of watervapor and CO2 on the moist air refractive index, fol-lowing inaccurate procedures for integrating the vol-ume Rayleigh-scattering coefficient along the verticalpath of the atmosphere, and assuming that air com-position (and hence also the moisture characteristics)

does not change with height. As a result of theseapproximations, the evaluations of the Rayleigh-scattering parameters were found to exhibit discrep-ancies of several percent in the worse cases.

Most of the calculations have often been performedfor different thermodynamic characteristics of the at-mosphere, usually represented by means of standardatmosphere models. They are defined by the verticalprofiles of air pressure, air temperature, air density,and absolute humidity of the U.S. Standard Atmo-sphere model20,23–25 or supplementary standard at-mosphere models for different seasons and latitudesfrom 15 °N to 60 °N.24–26 Considerable changes in thevertical profiles of these thermodynamic parameterscan be observed when passing from one atmospheremodel to another of different latitudes or seasons;however, such variations usually cause only smallchanges in the Rayleigh-scattering optical depth, es-timated in general to be within �1%.4,27 As pointedout by Bréon,28 the discrepancies are, in practice, ofcomparable magnitude to those due to surface pres-sure changes.

Considering the high precision required byRayleigh-scattering calculations at different lati-tudes and the recent reduction of computation time,we decided to define an improved algorithm based onthe rigorous physical model of the total molecular-scattering processes occurring in the atmosphere, at-tempting to avoid approximations that are too roughfor determining the effects of pressure, temperature,water vapor, and CO2 on the dispersion of air. Thefinal purpose was to obtain a reliable algorithm forthe analysis of radiation measurements at UV, visi-ble, and IR wavelengths, where Rayleigh-scatteringeffects must be correctly accounted for.

2. Rayleigh-Scattering Model for Atmospheric Gases

As pointed out by Penndorf,3 the Rayleigh-scatteringoptical depth �� of the atmosphere is given at eachwavelength � by the integral

�(�) ��z0

z�

(�, z)dz, (1)

where ��, z� is the monochromatic volume coefficientfor the total molecular scattering in cloudless air ataltitude z, and z0 � 0 and z� are the sea level and thetop level of the atmosphere, respectively. The coeffi-cient ��, z� depends closely on the thermodynamicconditions of the atmosphere and hence varies withheight; it is given at each altitude by the product ofthe molecular number density N�z� of air by the totalRayleigh-scattering cross section ��, z� per mole-cule:

(�, z) � N(z)(�, z), (2)

where N�z� varies as a function of total air pressurep�z� and air temperature T�z� according to the follow-ing dependence form derived from the well-known

1 June 2005 � Vol. 44, No. 16 � APPLIED OPTICS 3321

equation of the state for moist air3:

N(z) � Ns

p(z)ps

Ts

T(z), (3)

with p�z� measured in hectopascals, T�z� in degreeskelvin, and NS � 2.546899 � 1019 cm�3 as calculatedby Bodhaine et al.5 from the Loschmidt constant (de-fined as the ratio of the Avogadro constant to themolar volume of an ideal gas) and normalized to stan-dard air pressure ps � 1013.25 hPa and standard airtemperature Ts � 288.15 ºK to yield the molecularnumber density for standard air conditions.

Several approximated and not fully reliable ana-lytical forms have been proposed in the literature torepresent the spectral behavior of the Rayleigh-scattering cross section per molecule s��, relative tostandard air conditions.3,21,22 According to van deHulst,29 McCartney,30 and Bodhaine et al.,5 the clas-sical equation used to calculate the Rayleigh-scattering cross section per molecule for incidentunpolarized (natural) radiation can be written in thefollowing form for nonstandard values of air pressure,temperature, and moisture parameters relative to acertain height z:

(�, z) �24�3[n(�, z)2 � 1]2

�4N(z)2[n(�, z)2 2]2 F(�, z), (4)

where n��, z� is the refractive index of air, calculatedat wavelength � and height z (and hence as a functionof air pressure p, air temperature T, water-vaporpartial pressure e, and CO2 volume concentration C);F��, z� is the King31 factor for the depolarization of airat wavelength � and height z (i.e., for parameters p,T, e, and C) used to take into account the effects dueto the anisotropic properties of air molecules sinceanisotropic molecules scatter more radiation at 90°scattering angles than isotropic molecules with thesame index of refraction1,2,32,33; and N�z� is the mo-lecular number density at height z, determined interms of Eq. (3).

Therefore air refractive index n��, z� must be cal-culated taking into account not only the spectral de-pendence features of the main atmospheric gases andthe variable features of their concentration parame-ters with height (including CO2 and water vapor), butalso the changes in the vertical profiles of air pressureand temperature. The factor F��, z� depends on thedepolarization ratio �n��, z� for randomly orientedmolecules illuminated by natural light according tothe classical dependence form defined by King31:

F(�, z) �6 3�n(�, z)6 � 7�n(�, z), (5)

where the depolarization ratio �n��, z� is expected todepend on both wavelength and height since the com-position of moist air can vary appreciably withheight, depending on the parameters p, T, e, and C,

and the dispersion of the depolarization ratios of themain anisotropic air molecules must be properly ac-counted for.

Eqs. (1)–(5) give form to a physical model of totalmolecular scattering by air. However, some physicalconcepts concerning this set of equations require fur-ther discussion to take into account realistically thatmolecular scattering varies not only as a function ofwavelength but also of the atmospheric conditionsbecause of the variability of refractive index n��, z�with wavelength and altitude and hence with totalair pressure, air temperature, and partial pressuresof the main atmospheric constituents (N2, O2, Ar,H2O, and CO2); and the dependence of the King31

factor on wavelength and composition parameters ofthe atmosphere. On this matter, a number of con-fused ideas have circulated in the literature over thepast decades, as pointed out by Young,1 Bates,34 andBodhaine et al.5 Therefore it is important to pay par-ticular attention to the influence of pressure, temper-ature, and atmospheric composition on the refractiveindex of moist air and the corresponding effects onthe King31 depolarization factor to determine themost appropriate dependence functions for use inEqs. (3)–(5).

3. Dependence of Air Refractive Index on Wavelengthand Atmospheric Parameters

In 1953, Edlén35 proposed a formula for the disper-sion of air that was based on the refractivity equa-tions derived 14 years before by Barrell and Sears36

from absolute measurements on dry and moist air.Examining the Edlén35 formula for the refractivity ofstandard air, where ns is the refractive index of stan-dard air at pressure ps and temperature Ts and nrefers to nonstandard conditions of air characterizedby total air pressure p and temperature T, Penndorf3

established that the ratio �n � 1��ns � 1� is propor-tional to the product of the ratio p�ps by a tempera-ture correction factor decreasing almost linearly withtemperature at all the visible and NIR wavelengths,with the slope coefficient assuming quite stable val-ues close to �3.6 � 10�3 �ºC��1 at all the wavelengthsfrom 0.40 to 1.00 �m. Nine years later, an improveddispersion formula for standard air was proposed byEdlén37 in the following form:

[ns(ps, Ts, �) � 1]108 � 8342.13 2,406,030

130 � ��2

15,997

38.9 � ��2, (6)

with � measured in micrometers. Equation (6) wasrecommended by Edlén37 for use with other formulasthat are suitable to define the linear dependence fea-tures of refractivity on air pressure and temperature;the linear effect of CO2 volume concentration on airrefractivity varying with wavelength; and the effectof water vapor at visible wavelengths, which dependson the parameters p, T, and e and varies appreciably


as a function of wavelength. In particular, consider-ing that water-vapor refractivity is �15% lower thanthat of air at the same air pressure and temperatureconditions, Edlén37 pointed out that “as uncontrolledair often holds one to two per cent by volume of watervapor, the refractivity may thus be changed by 40� 10�8 to 80 � 10�8, which is an order of magnitudemore than changes due to CO2. In the latter case, thechange in dispersion was found to be negligible, butthis is not so for moist air.”

Taking into account the Erickson38 equation for therelative refractivity of water vapor and the resultsprovided by Barrell and Sears,36 Owens39 analyzedthe density dependence of the refractive index andredefined the effects produced by CO2-free air, watervapor, and CO2, separately. Through a comparisonbetween his findings and the Edlén37 results, he de-termined the general formulas based on the Lorentz–Lorenz equation used to represent the air refractiveindex in terms of the mixture of the three above-mentioned gases. Furthermore, he defined the depen-dence formulas of dry CO2-free air refractive index onthe parameters p, T, and �; water-vapor refractiveindex on �, p, T, and e; and CO2 refractive-indexparameters on �, p, T, and C. On the basis of thisrefractivity model, he proposed the following simpli-fied formula for the refractive index of moist air, withC � 300 parts per million by volume (ppmv):

[n(p, T, e, �) � 1]108 � �2371.34 683,939.7

130 � ��2

4547.3

38.9 � ��2�Ds(p � e, T)

(6487.31 58.058 ��2

� 0.71150 ��4

0.08851 ��6)Dw(e, T), (7)

where the three-term sum given in the parenthesis ofthe first term describes the spectral dependence ofthe dry air refractive index, including the effects dueto 0.03% CO2 volume concentration; the functionDs�p�e, T� for dry air is given by a four-term formula,written as a function of dry air pressure (defined asthe difference p�e between total air pressure andwater-vapor partial pressure, both measured in hec-topascals) and air temperature T measured in de-grees kelvin; the four-term sum given in theparenthesis of the second term defines the spectraldependence of the water-vapor refractive index; andthe function Dw�e, T� for water vapor is given by aformula consisting of 12 terms depending on the pa-rameters T and e.

By analyzing a large set of experimental data, Peckand Reeder40 determined two formulas to representthe dependence form of the standard air refractiveindex ns�ps, Ts, �� as a function of �. The first has thefollowing form:

[ns(ps, Ts, �) � 1]108 �5,791,817

238.0185 � ��2

167,909

57.362 � ��2, (8)

with � measured in micrometers within the0.23–1.69-�m wavelength range and C � 300 ppmv,as pointed out by Bodhaine et al.,5 but not for C� 330 ppmv, as asserted by Peck and Reeder.40 Thecomparison of Eq. (8) with a set of 54 selected labo-ratory data throughout the above wavelength rangeyielded a value of the root-mean-square (rms) devia-tion equal to 2.1 � 10�9, which turns out to be moreaccurate than that found for Eq. (6) by more than 1order of magnitude.

The second Peck and Reeder40 formula was definedwithin the 0.185–1.690-�m wavelength range in thefollowing five-parameter form:

[ns(ps, Ts, �) � 1]108 � 8060.51 2,480,990

132.274 � ��2

17,455.7

39.32957 � ��2, (9)

with � measured in micrometers. Equation (9) wasevaluated to fit the above set of laboratory data, en-riched by a further six experimental values pertain-ing to the 0.185–0.230-�m wavelength range, with anoverall rms deviation equal to 1.0 � 10�8, and therms deviation values were correspondingly found tobe equal to 2.7 � 10�7 and 1.6 � 10�8 for Eqs. (8) and(6), respectively.

These findings indicate that Eq. (8) is suitable foruse to describe the spectral dependence features ofthe air refractive index within the 0.23–1.69-�mwavelength range. However, it does not appear to beappropriate for the definition of the spectral varia-tions of air refractivity from the 0.185- to 0.230-�mwavelength, where Eq. (9) was estimated to providemore realistic results. The upper panel of Fig. 1shows the spectral curves of standard air refractivity,as given by Eqs. (6), (8), and (9) in terms of the dif-ference ns�ps, Ts, �� 1 throughout the wavelengthrange from 0.185 to 0.250 �m, where the largest dis-crepancies were found to exist between the two for-mulas proposed by Peck and Reeder.40 Moreover, thelower panel of Fig. 1 presents the spectral curves ofthe ratios �1 and �2. The first ratio was calculated bydividing the air refractivity defined in Eq. (8) by thatgiven by Eq. (9), and the latter was determined bydividing the air refractivity in Eq. (8) by that in Eq.(6). As can be seen, Eq. (8) provides standard airrefractivity values lower than those of Eq. (6) by�0.09% at � � 0.200 �m and by slowly decreasingpercentages at the higher wavelengths, until reach-ing the same value of Eq. (6) at �0.236 �m and be-coming slightly higher at the 0.250-�m wavelengthby 10�3%. Equation (8) also yields appreciably lowervalues of air refractivity than those in Eq. (9) from


0.185 to 0.230 �m, with discrepancies of 9 � 10�2%at 0.200 �m and 8 � 10�4% at 0.230 �m. At longerwavelengths, the air refractivity values given by Eq.(8) approximate very well those given by Eq. (9), withdiscrepancies within � 10�3%. In particular, the ratio�1 presents values of 0.99538 at � � 0.185 �m,0.99912 at � � 0.200 �m, and 0.99999 at �� 0.230 �m. Such values suggest that Eq. (8) needsto be corrected at wavelengths shorter than 0.230 �mby taking into account the evaluations furnished byEq. (9), which was found to fit the laboratory mea-surements taken within the 0.185–0.230-�m wave-length range better than Eq. (8).40 We thereforecalculated the values of parameter �1 equal to theinverse of the ratio �1 at nine wavelengths from 0.185to 0.225 �m and then determined the best-fit curve of�1 as a function of wavelength within this narrowspectral range by imposing the normalization condi-tion that �1 is equal to unity at � � 0.230 �m. Theprocedure led us to determine the following spectral

function:

�1(�) � 1.000057551

�1.0011 � 11.6795� 34.2542�2

1 � 11.6702� 34.2347�2 , (10)

with � measured in micrometers within the0.185–0.230-�m wavelength range. Thus Eq. (10)provides the conversion function �1��, which can beused as a correction factor for Eq. (8) to obtain reliablevalues of standard air refractivity within the0.185–0.230-�m spectral interval, and Eq. (8) can bedirectly used at wavelengths from 0.23 to 4.00 �m, asproposed by Peck and Reeder.40 Beyond the 0.23-�mwavelength, Eq. (8) yields standard air refractivityvalues that slowly decrease from a value of 3.0147� 10�4 at the 0.25-�m wavelength to values of2.9155 � 10�4 at 0.30 �m, 2.7896 � 10�4 at0.50 �m, 2.7415 � 10�4 at 1.00 �m, and 2.7270� 10�4 at 4.00 �m, with relative discrepancies fromthe estimates given by Eq. (9) that are within � 1.4� 10�3% throughout the whole wavelength rangefrom 0.25 to 4.00 �m.

Alternative dispersion equations for dry air andwater vapor were subsequently proposed41–43 on thebasis of more precise measurements. According toOwens39 and using the spectral curve of Ar refractiveindex defined by Dalgarno and Kingston44 togetherwith that of CO2 given by Bideau-Mehu et al.,45

Bates34 determined the spectral dependence curve ofthe standard air refractive index by calculating theweighted sum of the contributions due to N2, O2, Ar,and CO2, hence neglecting the water-vapor influenceon the dispersion of air. Also, more precise measure-ments and careful studies were performed in the fol-lowing years.46–55 They clearly demonstrated theimportance of taking into account the water-vaporeffects on air dispersion to achieve more realisticevaluations of moist air refractivity at various atmo-spheric altitudes.

Taking into account the above results,36–55 Cid-dor56,57 developed a new set of equations based on themost recent equations for air density and air disper-sion at visible and NIR wavelengths. In particular, headopted the concepts pointed out in the Owens39

model, in which the refractivity of each component ofmoist air was determined at some standard condi-tions, together with the density of each componentcalculated for real conditions of air and relative to thecorresponding air density values estimated for stan-dard air conditions. The moist air refractivity at non-standard pressure and temperature conditions wascalculated as the sum of a first term giving the re-fractivity of dry air containing CO2 multiplied by itsdensity normalized to that of standard dry air condi-tions, where each molecule of CO2 was assumed toreplace a molecule of O2, and a second term giving therefractivity of pure water vapor multiplied by its den-sity normalized to that defined at standard conditions

Fig. 1. Upper panel: comparison between the spectral curves ofstandard air refractivity defined by the difference ns�ps, Ts, �� 1 within the 0.185–0.250-�m wavelength range obtained withthe Edlén37 formula in Eq. (6), the first Peck and Reeder40 formulain Eq. (8), and the second Peck and Reeder40 formula in Eq. (9).Lower panel: spectral curves within the same wavelength range ofthe spectral ratio �1�� between the standard air refractivity fromEq. (8) and the one from Eq. (9) (solid curve) and the spectral ratio�2�� between the standard air refractivity from Eq. (8) and the onefrom Eq. (6) (dashed curve).


as follows:

n(p, T, e, �, C) � 1 � (�a��axs)[naxs(�s, Ts, �, C) � 1] (�w��ws)[nws(e*, T*, �) � 1],

(11)

where �a is the dry air density at pressure p, temper-ature T, and CO2 volume concentration C; �axs is thedensity of dry air at standard air pressure ps

� 101, 325 Pa, standard air temperature Ts

� 288.15 K � 15 ºC, and a general value of CO2 vol-ume concentration C measured in ppmv; �w is thedensity of pure water vapor at nonstandard condi-tions, defined in terms of the parameters e and T; �ws

is the density of pure water vapor at temperatureT* � 293.15 K � 20 ºC and e* � 1333 Pa;n�p, T, e, �, C� is the refractive index of moist air attotal pressure p, temperature T, water-vapor partialpressure e, wavelength �, and CO2 volume concentra-tion C; naxs�ps, Ts, �, C� is the refractive index of dryair at standard total pressure ps, standard tempera-ture Ts, wavelength �, and CO2 volume concentrationC, with e � 0 Pa; and nws�e*, T*, �� is the spectralrefractive index of pure water vapor at temperatureT* � 293.15 K � 20 ºC and e* � 1333 Pa.

In Eq. (11) the parameters �a, �axs, �w, and �ws weredetermined by Ciddor56 using the equation for moistair density called the BIPM (Bureau Internationaldes Poids et Mesures) 1981�91 equation, as reportedby Giacomo43 and revised by Davis,53 that is,

� � (pMa�ZRT)[1 � Xw(1 � Mw�Ma)], (12)

where (1) Ma is the molar mass of dry air calculatedas a function of CO2 volume concentration C with thefollowing linear form:

Ma � 10�3<28.9635 12.011 � 10�6(C � C1)=,(13)

where Ma is measured in kilograms per mole and C inppmv, and the reference value C1 is equal to400 ppmv, according to the BIPM equation43,53; (2)Mw � 0.018015 kg�mol is the molar mass of watervapor; (3) T is the air temperature measured in de-grees kelvin; (4) R � 8.31451 J mol�1 K�1 is the gasconstant; (5) p is the total air pressure, measured inpascals; (6) Xw is the molar fraction of water vapor inmoist air, evaluated in terms of the quantityf h E�T��p, where (i) the enhancement factor f of wa-ter vapor in air is a function of p and T,43,53 calculatedwith the following form:

f(p, T) � 1.00062 3.14 � 10�8p 5.6 � 10�7(T

� 273.15)2, (14)

with p and T measured in pascals and degrees kelvin,respectively; (ii) h is the relative humidity of air; and(iii) the saturation vapor pressure E�T� of water va-

por in air is a function of T measured in degreeskelvin, given by the following equation:

E(T) � exp(1.2378847 � 10�5T2 � 1.9121316

� 10�2T 33.93711047 � 6343.1645�T);(15)

and (7) Z is the compressibility of moist air, calcu-lated as a function of the parameters p, T, and Xw bymeans of the following equation:

Z(p, T, Xw) � 1 � (p�T)[a0 a1(T � 273.15) a2(T

� 273.15)2 b0Xw b1Xw(T � 273.15)

c0Xw2 c1Xw

2(T � 273.15)

(p�T)2(d0 d1Xw2), (16)

with parameters p and T measured in pascals anddegrees kelvin, respectively, and a0 � 1.58123 �10�6 K Pa�1, a1 � �2.9331 � 10�8 Pa�1, a2 �1.1043 � 10�10 K�1 Pa�1, b0 � 5.707 � 10�6 K Pa�1,b1 � �2.051 � 10�8 Pa�1, c0 � 1.9898 � 10�4 KPa�1, c1 � �2.376 � 10�6 Pa�1, d0 � 1.83� 10�11 K2 Pa�2, and d1 � �7.65 � 10�9 K2 Pa�2.

In particular, the dry air parameters �a and �axs

were determined for the total dry air pressure givenby the difference p�e in Eq. (12) and for Xw � 0, andthe parameters �w and �ws were calculated by puttingthe product f e in place of p in Eq. (12).

To describe the spectral dependence features of airrefractivity, Ciddor56 preferred to use the first Peckand Reeder40 formula given in Eq. (8) over the0.23–1.69-�m wavelength range rather than the onedefined in Eq. (9) because the first was found to fit themeasurements in the IR better than Edlén’s37 equa-tion and Eq. (9). The Peck and Reeder40 formula inEq. (8) was amended by Ciddor56 taking into accountthe results achieved by Birch and Downs,54,55 in turnadjusted for the change of the CO2 volume concentra-tion from 300 to 450 ppmv, as assumed in the disper-sion equations for CO2 defined by Old et al.58 andSimmons.59

In this way, Ciddor56 obtained the following equa-tion for the dry air refractivity at temperature Ts

� 288.15 K � 15 ºC, pressure ps � 101,325 Pa, e� 0 Pa, and CO2 volume concentration C2� 450 ppmv, with wavelength � measured in mi-crometers:

[nas(ps, Ts, �, C2) � 1]108 �5,792,105

238.0185 � ��2

167,917

57.362 � ��2, (17)

and the ratio between the refractivity of standard dryair for unknown CO2 volume concentration C and forthe above-assumed value C2 is given by


naxs(ps, Ts, �, C) � 1nas(ps, Ts, �, C2) � 1 � 1 0.534 � 10�6(C � C2),

(18)

in substantial agreement with the Edlén37 results,showing that the variations of air refractivity withCO2 concentration are characterized by a linear de-pendence rate. In practice, Eq. (18) states that anincrease of standard dry air refractivity equal to 2.7� 10�3% is expected to occur in cases where C isassumed to increase from 330 to 380 ppmv, as ob-served over the past 30 years.

Moreover, according to Owens,39 Ciddor56 com-bined the absolute value of refractive index given byBarrell and Sears36 with the relative dispersion equa-tion proposed by Erickson38 and defined the water-vapor refractivity at temperature T* � 293.15 K� 20 ºC and partial pressure e* � 1333 Pa (theseconditions being those assumed by Barrell andSears36 used to define the standard conditions of purewater vapor) in terms of the following formula:

[nws(e*, T*, �) � 1]108 � 1.022 � (295.235

2.6422��2 � 0.032380��4

0.004028��6), (19)

which agrees very well with the experimental dataconsidered by Schiebener et al.51 Thus realistic eval-uations of moist air and dry air refractivity can beobtained with the set of Eqs. (10)–(19) at all the at-mospheric heights for each value of C and for eachtriplet of p, T, and e.

Figure 2 shows the spectral curves of moist airrefractivity calculated as the difference betweenn�p, T, e, �, C� and unity by Eqs. (10)–(19) within the0.2–4-�m wavelength range for the atmospheric con-ditions defined at three different levels by Model 6(U.S. Standard Atmosphere, 1976) proposed byAnderson et al.25 The three spectral curves were ob-tained for CO2 concentration C � 385 ppmv and thetriplets of parameters p, T, and e corresponding to thelevels z � 0 km, z � 10 km, and z � 20 km, i.e., p� 101, 300 Pa, T � 288.2 K, and e � 785.075 Pa atsea level; p � 26, 500 Pa, T � 223.3 K, and e� 1.855 Pa at altitude z � 10 km; and p � 5529 Pa,T � 216.7 K, and e � 2.16 � 102 Pa for z � 20 km. Ascan be seen, the moist air refractivity decreases ap-preciably as the height increases, assuming sea-levelvalues close to 2.82 � 10�4 at the visible and IRwavelengths; lower values by �66% at the 10-kmlevel; and values decreasing by a further 78%, pass-ing from the 10- to 20-km level. To give a measure ofthe dependence features of the moist air refractivityn�p, T, e, �, C� � 1 on parameters p, T, and e, calcu-lations of the partial variations of parameter n � 1were performed at the 0.23-, 0.50-, and 2.00-�mwavelengths when passing from case (a) to case (b)and from case (b) to case (c) in Fig. 2. The resultsshow that the decrease of �66% through the lowertropospheric layer of a 10-km depth is mostly due to

the decreasing pressure p, which causes by itself arelative decrement of �74%, only in part��8%� counterbalanced by the increase terms due tothe simultaneous lowering of parameters T and e.The greater part (99.8%) of this partial compensationis due to the temperature decrease whereas only theremainder (0.2%) is due to the decrease in the water-vapor parameter. In addition, the occurrence of a CO2volume concentration C � 385 ppmv in the tropo-sphere was estimated to cause a relative increase inmoist air refractivity of �0.02%. Similarly, passingfrom the 10- to 20-km level, we found that the partialvariation of pressure explains by itself a relative de-crease of more than 79%, which is compensated to avery limited extent ��1%� by the opposite effects dueto the decreasing temperature and only slightly�10�4%� by water vapor. These results indicate thatthe variations in moist air refractivity caused by wa-ter vapor are considerably smaller than those due topressure and temperature, but cannot be neglected attropospheric levels in calculations of Rayleigh-scattering parameters along the atmospheric verticalpath since water-vapor effects are higher than thoseattributed to CO2 by �1 order of magnitude.

4. Influence of Atmospheric Parameters on the KingFactor

The disparate contradictory estimates of the depolar-ization ratio �n for dry air proposed in the literatureare obtained from experimental measurements or

Fig. 2. Spectral curves of the moist air refractivity in the0.2–4.0-�m wavelength range, determined as the difference be-tween n�p, T, e, �, C� and unity with the Ciddor56 model defined byEqs. (10)–(19) for CO2 volume concentration C � 385 ppmv. Thethree curves were calculated for the triplets of parameters p, T,and e relative to the three following levels of Model 625: (a) z� 0 km, with p � 1013.00 hPa, T � 288.2 K, and e� 7.85075 hPa; (b) z � 10 km, with p � 265.00 hPa, T� 223.3 K, and e � 1.855 � 10�2 hPa; (c) z � 20 km, with p� 55.29 hPa, T � 216.7 K, and e � 2.16 � 10�4 hPa.


through calculations based on values of �n deter-mined separately for the main atmospheric gases.The King factor F defined in Eq. (5) is greater thanunity in all cases where anisotropic molecules arepresent and is equal to unity only in the ideal casewhere all the air molecules are isotropic. A number ofconfused ideas on this matter have been diffused bythe literature over recent decades, as pointed out byYoung.1,2,33 The determination of parameters �n andF is a crucial step. As discussed in depth by Young,1it can be an important source of error in the calcula-tions of the Rayleigh-scattering parameters ��, z�,��, z�, and ��. Various estimates of the mean val-ues of �n for standard air are available in the litera-ture, as shown in Table 1, together with thecorresponding values of the King factor F evaluatedin terms of Eq. (5) for standard conditions of air.

King31 proposed a value of F equal to 1.049 at allwavelengths, which corresponds to �n � 0.0284. Ex-amining a large set of mean values of depolarizationratio �n for incident unpolarized light, determined byvarious investigations for several atmospheric gases(N2, O2, Ar, CO2, N2O, NO, CO, H2, and He) and dryair, Penndorf3 found �n � 0.035 at temperature T� 15 ºC, from which the value F � 1.0608 can bedetermined in terms of Eq. (5). These values of �n andF were subsequently used by Elterman,20 whereasKasten61 proposed the pair of values �n � 0.0295 andF � 1.0509 in the same year. Examining the mea-surements of the depolarization ratio carried out inseveral laboratory experiments and neglecting thedepolarization effects produced by the Raman lines,Hoyt21 assumed the value �n � 0.0139, with F� 1.0235. Considering that air is a mixture of78.1% N2 molecules, 20.9% O2 molecules, and1.0% Ar molecules, Fröhlich and Shaw22 found thevalue �n � 0.0095 at the 0.5879-�m wavelength, from

which the value F � 1.016 was derived, i.e., a value4.2% lower than that calculated by Penndorf.3 Young1

severely criticized these estimates of parameters �n

and F, pointing out that it is of basic importance totake into account the partial effects of depolarizationproduced by CO2 molecules. Therefore, taking valuesof �n equal to 0.021 for N2, 0.058 for O2, and 0.078 forCO2, but omitting the depolarization effects causedby water vapor, Young1,2 determined the value ofdepolarization ratio �n � 0.0279 for dry air and theKing factor F � 1.048. This value turned out to beconsiderably higher than that obtained by Hoyt,21

3.1% higher than that determined by Fröhlich andShaw,22 and only 1.2% lower than that of Penndorf.3It is curious that the most accurate evaluations of Fcalculated 20 years ago in practice coincide with theone determined over 80 years ago by King,31 whobased his calculation on the depolarization measure-ments carried out by Cabannes.32

To define the most realistic value of F for its correctuse in the calculations of Rayleigh-scattering opticalparameters, Young33 stated that the King factor es-timates must include not only the anisotropy effectsproduced by the classical Rayleigh scattering, butalso those associated with rotational Raman scatter-ing, and the weak effects produced by the vibrationalRaman scattering can be in general neglected sincethey are estimated to be lower than 0.1%. Conse-quently, a very small underestimation error is madein all cases where one neglects the vibrational Ramanscattering in calculating the total molecular-scattering effects occurring in the atmosphere.

All the estimates of parameters �n and F presentedin Table 1 were obtained as average values of the twoparameters within the visible wavelength range, de-termined on the basis of the composition percentagesof the dry air constituents, assuming that the depo-

Table 1. Mean Values of the Depolarization Ratio �n of Dry Air and the King Factor F Proposed in the Literaturea

Bibliographic Sources Year Depolarization Ratio �n King Factor F

Kingb 1923 0.0284 1.049Volkmannc 1935 0.0429 � 0.0033 1.075 � 0.006Penndorfd 1957 0.035 1.061Penndorfd (average) 1957 0.0375 � 0.0047 1.065 � 0.009Weber et al.e (upper limit) 1967 0.0139 � 0.0006 1.0235Eltermanf 1968 0.035 1.0608Kasteng 1968 0.0295 1.0509Hoyth 1977 0.0139 1.0235Fröhlich and Shawi 1980 0.0095 � 0.0040 1.0160 � 0.0068Youngj 1980 0.0279 1.0480Batesk 1984 0.0284 1.049

aReported with the accuracy defined by the various authors.bRef. 31.cRef. 60.dRef. 3.eRef. 62.fRef. 20.gRef. 61.hRef. 21.iRef. 22.jRef. 1.kRef. 34.


larization parameters of such constituents do not de-pend on wavelength. Bates34 was the first to take intoaccount the wavelength-dependence features of depo-larization effects, defined through both laboratorymeasurements and theoretical remarks. He calcu-lated the spectral values of the King factor as theweighted sum of the partial contributions due to themolecules of N2, O2, Ar, and CO2, considering thateach contribution varies as a function of wavelengthwith different spectral features and each weight isgiven by the volume concentration of the atmosphericconstituent. For this purpose, he defined the spectralfunctions relative to N2 and O2 using a two-term an-alytical form for the N2 molecules and a three-termform for the O2 molecules and neglected the spectralchanges of Ar and CO2 depolarization parameters,assuming these two constituents as neutral withwavelength. This approximation seems to be quitecorrect for practical purposes, the Ar and CO2 contri-butions being considerably less important than thoseof N2 and O2. The influence of water vapor on theanisotropic properties of air was instead neglected byBates,34 as previously done by Young33 and the otherinvestigators mentioned above. In fact, Bates definedthe following spectral curves of the partial King fac-tors, relative to the main four atmospheric constitu-ents of dry air:

F1(�) � 1.034 3.17

� 10�4 ��2 (for the N2 molecules), (20a)

F2(�) � 1.096 1.385 � 10�3��2 1.448

� 10�4 ��4 (for the O2 molecules), (20b)

F3 � 1.00 (for Ar), (20c)

F4 � 1.15 (for CO2). (20d)

Bucholtz4 used Eqs. (20a)–(20d) to calculate thedispersion of the King factor with wavelength. Hepointed out that the Penndorf3 value of F is moreappropriate to UV wavelengths near 0.26 �m,whereas the Young1 value appears to pertain betterto wavelengths beyond 0.50 �m. Using the values ofthe depolarization parameters determined by Bates34

for the four above atmospheric gases, together withthe corresponding spectral dependence functionsgiven in Eqs. (20a)–(20d), Bodhaine et al.5 recom-mended calculating the spectral curve of the Kingfactor for dry air as an average weighted value de-fined by

F(�) �

jcjFj(�)

j

cj, (21)

where the following assumptions can be made: the N2volume concentration c1 is equal to 0.78084 and F1��is given by Eq. (20a); the O2 concentration c2 is equalto 0.20946 and F2�� is given by Eq. (20b); the Arconcentration c3 is equal to 0.00934 and F3 � 1.00 atall wavelengths; and the CO2 concentration c4 isequal to 10�6 C (with C measured in ppmv) and F4� 1.15 at all wavelengths.

To define the atmospheric composition, Bates34

and Bodhaine et al.5 both neglected the dispersioneffects due to water vapor, even though it is oftenthe third most abundant constituent of air in theatmosphere. This choice could be quite realistic incold air masses where the air relative humidity h isrelatively low, since neglecting the water-vapor ef-fects on the depolarization ratio of air, as done byYoung,1 can cause only small underestimation er-rors in the King factor evaluation. However, inwarm and very humid air masses, such as thoseusually occurring in the tropical atmosphere,63

water-vapor effects cannot be neglected. Thus, toobtain uniform estimates of moist air refractive in-dex, we decided to take into account the effectscaused by the water anisotropic properties on theKing factor F for the various moisture conditionsoccurring along the vertical path. This was achievedby simply inserting a fifth term concerning watervapor in the sum of Eq. (21), giving the followinggeneral form:

which can be used at each altitude z, where water-vapor pressure e�z� and total air pressure p�z� aremeasured in pascals and CO2 concentration C inppmv. In Eq. (22), the partial King factors F1��,F2��, F3, and F4 are taken according to Eqs. (20a)–(20d), and the partial King factor F5 for water vaporis assumed to be equal to 1.001 at all wavelengths, asevaluated by Sioris et al.64 at the 0.5145-�m wave-length. This value of F5 is slightly greater than thatobtained from the estimate of water-vapor depolar-ization ratio �n � 3 � 10�4 determined by Murphy,65

who found no evidence of the existence of appreciablespectral dependence features of parameters F and �n

in the visible. Thus Eq. (22) can be reliably used todetermine the spectral curves of the King factor F��for moist air in the 0.2–4.0-�m wavelength range forany value of CO2 volume concentration C and anypair of parameters p and e. Three spectral curves of

F(�, z) �0.78084F1(�) 0.20946F2(�) 0.00934F3 10�6CF4 [e(z)�p(z)]F5

0.999640 10�6C e(z)�p(z), (22)


F�� are shown in the upper panel of Fig. 3, obtainedin terms of Eq. (22) for CO2 concentration C� 385 ppmv and the atmospheric parameters p and erelative to the heights of 0, 10, and 20 km in Model6,25 where the ratio e�p assumes values equal to7.75 � 10�3, 7.00 � 10�5, and 3.91 � 10�6, respec-tively. The results indicate that the King factor tendsto vary slightly with altitude as a result of the com-bined effects due to the changes in pressure and mois-ture conditions of air with height. For example, factorF at the 0.50-�m wavelength increases by 0.04%passing from sea level to 10 km and by 3 � 10�4%only when z rises from 10 to 20 km. Calculations ofthe partial variations in F�� were performed at var-ious wavelengths by modifying separately the param-eters p and e. The results indicate that the increase inF�� observed at all wavelengths when passing fromthe 0- to 10-km level is mostly due to the water-vaporpartial pressure decrease, which causes more rapideffects than those produced in the opposite sense by

the air pressure changes in Eq. (22). Similar resultswere also found passing from the 10- to 20-km level,showing that F�� increases at all wavelengths by�3 � 10�4%, the main contribution being producedby the decrease in partial pressure e, although to amore limited extent than in the previous case sincethe ratio e�p is subject to decrease only by 94.4%instead of 99.1% as in the previous case from the 0- to10-km level.

The dependence features of the King factor on themoisture conditions of air are better shown in thelower panel of Fig. 3, where the spectral curves of thedifference 1 � �� are presented, the latter termbeing the ratio between moist air and dry air Kingfactors determined for the same atmospheric condi-tions considered in the upper panel of Fig. 3. Theresults indicate that 1 � �� assumes sea-level val-ues decreasing appreciably with wavelength in theUV spectral range and more slowly at visible and IRwavelengths and become equal to 3.542 � 10�4 at a0.50-�m wavelength, 3.397 � 10�4 at 1.00 �m, and3.360 � 10�4 at 4.00 �m. Similar spectral depen-dence features are also shown by the difference 1� �� at the 10- and 20-km levels, although theypresent considerably smaller values than those ob-tained at sea level. In fact, the spectral values of 1� �� at the 10-km level turn out to be lower thanthe sea-level ones by �2 orders of magnitude, andthose determined at the 20-km level are nearly 2000times smaller than those evaluated at sea level. Thisconfirms that the variations in F�� that occur attropospheric levels are mainly caused by changes inthe moisture conditions of air, each variation fromone level to the upper level becoming gradually lessmarked since the vertical decrease of water-vaporpartial pressure is more rapid than air pressure be-yond the tropopause region. Therefore Eq. (22) ap-pears to be suitable for providing realisticcalculations of quantity F�� as it takes into properaccount the effects due to water vapor and not onlythose of air pressure and CO2. With regard to thelatter parameter, our calculations of the King factorperformed with Eq. (22) for Model 6,25 with and with-out CO2, indicate that the presence of CO2 with con-centration C � 385 ppmv causes an increase in F��at sea level of �4 � 10�3% at visible wavelengths,slowly increasing toward IR wavelengths.

The spectral values of the King factor F�� calcu-lated with Eq. (22) for the sea-level atmospheric con-ditions of Model 625 are given in Table 2 within the0.2–4.0-�m wavelength range, together with the cor-responding values of depolarization ratio �n�� deter-mined in terms of Eq. (5). As can be seen, continuousspectral features of parameters F�� and �n�� areobtained with Eq. (22) throughout the whole spectralinterval. A comparison is shown in Fig. 4 between ourvalues of F�� and those determined by Bucholtz4 andBodhaine et al.5 within the 0.2–1.0-�m wavelengthrange. The spectral patterns of the ratio �� betweenour values of F�� and those of Bucholtz4 presentdiscontinuous features, with changes within �0.12%

Fig. 3. Upper panel: spectral curves of the moist air King factorF��, obtained in terms of Eq. (22) within the 0.2–4.0-�m wave-length range, for CO2 concentration C � 385 ppmv and the tripletsof parameters p, T, and e relative to the three following levels ofModel 625: (a) z � 0 km, with p � 1013.00 hPa, T � 288.2 K, ande � 7.85075 hPa; (b) z � 10 km, with p � 265.00 hPa, T� 223.3 K, and e � 1.855 � 10�2 hPa; (c) z � 20 km, with p� 55.29 hPa, T � 216.7 K, and e � 2.16 � 10�4 hPa. Lower panel:spectral curves of the difference 1 � ��, where �� is the spectralratio between the moist air King factor given by Eq. (22) and thedry air King factor, as derived from Eq. (22), for CO2 concentrationC � 385 ppmv and the three triplets of parameters p, T, and erelative to the same levels considered in the upper panel.


throughout the wavelength range under consider-ation, these discrepancies being presumably due tothe worse accuracy adopted by Bucholtz4 to evaluatethe parameters F�� and �n�� according to Bates34

who defined three decimals only. Conversely, thespectral curve of ratio �� between our values of F��

and those of Bodhaine et al.5 exhibit continuous fea-tures, both spectral series of F�� being determinedwith the same accuracy (i.e., with five decimals). Thereliability of the above explanation of the discontin-uous spectral patterns of �� is confirmed by thecomparison made in Fig. 4 between the spectral pat-terns of �� and those of the ratio �� given by thepresent estimates of �n�� normalized to those of Bu-choltz.4 It appears with great evidence that propor-tional variations characterize the spectral patterns ofboth parameters �� and �� due to the lower pre-cision adopted by Bucholtz4 in estimating the param-eters F�� and �n��. The ratios �� and �� turn outto be lower than unity at the shorter UV wavelengthsand then to increase with wavelength, assuming val-ues ranging between 0.9985 and 1.0011 and between0.983 and 1.018, respectively, at wavelengths shorterthan 0.4 �m and between 0.9994 and 1.0003 and be-tween 0.989 and 1.009, respectively, at the longervisible and NIR wavelengths.

Comparing the results presented in Table 2 andFig. 4 at sea-level standard air conditions, we canstate that parameters F�� and �n�� at a 0.50-�mwavelength are subject to a decrease by 3.54� 10�2% and 0.734%, respectively, as a result of thewater-vapor effects represented in terms of thepresent algorithm (that is, for e � 7.85075 hPa inplace of dry air conditions). Moreover, the above cal-culations indicate that F�� and �n�� at a 0.50-�mwavelength are subject to an increase on the averageby 4 � 10�5% and 8 � 10�4%, respectively, for asea-level air pressure increase of 1 hPa and by 2� 10�4% and 5 � 10�3%, respectively, for a CO2

Table 2. Values of the King Factor F(�) Calculated with Eq. (22) and Depolarization Ratio �n(�)a

Wavelength��m�

King FactorF��

DepolarizationRatio �n�� 102

Wavelength��m�

King FactorF��

DepolarizationRatio �n�� 102

0.200 1.07851 4.465 0.380 1.05150 2.9830.205 1.07610 4.335 0.390 1.05117 2.9640.210 1.07393 4.218 0.400 1.05087 2.9470.215 1.07199 4.112 0.450 1.04973 2.8830.220 1.07023 4.016 0.500 1.04898 2.8410.225 1.06864 3.930 0.550 1.04845 2.8120.230 1.06720 3.851 0.600 1.04808 2.7910.240 1.06470 3.714 0.650 1.04779 2.7750.250 1.06260 3.598 0.700 1.04758 2.7630.260 1.06084 3.501 0.750 1.04741 2.7530.270 1.05934 3.419 0.800 1.04727 2.7450.280 1.05806 3.348 0.850 1.04716 2.7390.290 1.05696 3.287 0.900 1.04707 2.7340.300 1.05601 3.234 0.950 1.04699 2.7300.310 1.05517 3.187 1.000 1.04693 2.7260.320 1.05444 3.147 1.500 1.04661 2.7080.330 1.05380 3.111 2.000 1.04650 2.7020.340 1.05323 3.079 2.500 1.04645 2.6990.350 1.05272 3.051 3.000 1.04642 2.6980.360 1.05227 3.026 3.500 1.04641 2.6970.370 1.05186 3.003 4.000 1.04640 2.696

aThe values are determined in terms of Eq. (5) for the sea-level atmospheric conditions given in Table 3 for Model 625 with the sameaccuracy adopted by Bodhaine5 and Bucholtz4, respectively.

Fig. 4. Spectral curves of ratios �� (solid curve), �� (dottedcurve), and �� (dashed curve) within the 0.2–1.0-�m wavelengthrange. The values of �� were obtained by dividing the presentvalues of the King factor F�� (see Table 2) by the original onesdefined by Bucholtz4 (see Table 2, Ref. 4) at the same wavelengthsand for standard air conditions at sea level. The values of �� weredetermined by dividing our values of F�� given in Table 2 by theoriginal ones defined by Bodhaine et al.5 (see Table 2, Ref. 5) at thesame wavelengths and for standard air conditions at sea level. Thevalues of spectral ratio �� were calculated by dividing the presentestimates of the depolarization ratio �n�� defined in Table 2 by thecorresponding values of �n�� proposed by Bucholtz4 (see Table 3,Ref. 4) at the same wavelengths and for standard air conditions atsea level.


volume concentration increase from 360 to 385ppmv.

5. Calculation of Rayleigh-Scattering Parameters inStandard Atmospheres

On the basis of the results provided by severalinvestigations,1–5,33,37,39,40 we defined in Sections 3and 4 a realistic physical model for representing therefractive index of moist air in terms of Eqs. (10)–(19)based on the Ciddor56,57 formulas and used togetherwith the King depolarization factor defined by meansof Eqs. (20)–(22). With the overall set of formulas, theRayleigh-scattering optical parameters can be calcu-lated realistically within the 0.2–4.0-�m wavelengthrange for all meteorological conditions occurring inthe atmosphere. The physical model proposed in Eq.(22) to calculate the King factor at all the above wave-lengths and different atmospheric heights wasadopted by us according to the criteria delineated byBates34 and Bodhaine et al.5 Therefore the presentmodel accounts for the presence of water vapor, fol-lowing homogeneous criteria in determining bothmoist air refractivity parameters and King factorproperties.

Equations (10)–(22) can be conveniently used tocalculate with good precision the moist air refractiveindex and the King factor used in Eq. (4), since bothquantities are of crucial importance to determine theRayleigh-scattering cross section per molecule at allwavelengths and altitudes. Using Eqs. (2)–(4) we de-fined the vertical profiles of the Rayleigh-scatteringcross section �� per molecule at various wave-lengths and those of the volume Rayleigh-scatteringcoefficient ��, z�. Using the vertical curves of ��, z�,we can calculate the spectral values of Rayleigh-scattering optical depth �� as integrals in terms ofEq. (1) for each set of vertical profiles of parameters p,T, e, and C. Thus it is important to define the verticalprofiles of parameters p, T, e, and C from sea level toa high enough level of the atmosphere, as done byBucholtz.4 To better steer the choice of the most suit-able top level of the atmosphere, in all cases whereaccurate calculations of atmospheric optical depth orother columnar radiative parameters are to be car-ried out,20 we took into account that volumeRayleigh-scattering coefficient ��, z� is closely re-lated to the molecular number density N�z�, as sug-gested by Eq. (2); and N�z� decreases rapidly withheight, as it is proportional to the total air pressurep�z� and the inverse of air temperature T�z�. Conse-quently, at altitudes of around 95 km, ��, z� as-sumes spectral values that are 106 times smaller thanthe sea-level ones, this parameter tending to decreasealmost regularly with height through the mesosphereand the upper atmospheric region by 1 order of mag-nitude every 15 km on the average. On the basis ofthese remarks, we decided to set the top level of theatmosphere for the present calculations at a 120-kmheight, as done by Anderson et al.25 in the U.S. Stan-dard Atmospheres.

Tab

le3.

Val

ues

of

Sea

-Lev

elM

eteo

rolo

gic

alan

dR

ayle

igh-

Sca

tter

ing

Par

amet

ers

atS

eaLe

veli

nth

eS

ixS

tand

ard

Atm

osp

here

sa

Sea

-Lev

elP

aram

eter

s

Sta

ndar

dA

tmos

pher

es

Mod

el1

(Tro

pica

l,15

°NA

nnua

lA

vera

ge)

Mod

el2

(Mid

-Lat

itud

eS

umm

er,4

5°N

,Jul

y)

Mod

el3

(Mid

-Lat

itud

eW

inte

r,45

°N,J

anua

ry)

Mod

el4

(Sub

-Arc

tic

Sum

mer

,60

°N,J

uly)

Mod

el5

(Sub

-Arc

tic

Win

ter,

60°N

,Jan

uary

)

Mod

el6

(U.S

.Sta

ndar

d,19

76)

Tot

alai

rpr

essu

rep 0

�hP

a�10

13.0

010

13.0

010

18.0

010

10.0

010

13.0

010

13.0

0A

irte

mpe

ratu

reT

0�K

�29

9.70

294.

2027

2.20

287.

2025

7.20

288.

20R

elat

ive

hum

idit

yf 0

�%�

75.6

76.4

77.1

75.0

80.7

45.9

Wat

er-v

apor

part

ial

pres

sure

e 0�h

Pa�

26.2

367

19.0

444

4.39

776

12.0

190

1.42

833

7.85

075

Wat

er-v

apor

mix

ing

rati

oe 0

�p0

2.59

0�

10�

21.

880

�10

�2

4.32

0�

10�

31.

190

�10

�2

1.41

0�

10�

37.

750

�10

�3

CO

2vo

lum

eco

ncen

trat

ion

C0

�ppm

v�38

237

939

038

239

038

5A

bsol

ute

hum

idit

y�

0�g

m�

3 �18

.969

314

.026

63.

5008

9.06

801.

2033

5.90

26M

oist

air

refr

acti

vity

n0

�0.5

0�

m��

12.

6720

�10

�4

2.72

46�

10�

42.

9658

�10

�4

2.78

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Thus we decided to use the six atmospheric modelsproposed by Anderson et al.,25 (see Table 3) i.e.:

1. Model 1, Tropic (15 °N annual average);2. Model 2, Mid-latitude Summer (45 °N, July);3. Model 3, Mid-latitude Winter (45 °N, January);4. Model 4, Sub-Arctic Summer (60 °N, July);5. Model 5, Sub-Arctic Winter (60 °N, January);

and6. Model 6, U.S. Standard Atmosphere (1976).

The models appear suitable for use in making re-alistic calculations of Rayleigh-scattering parametersby means of Eqs. (10)–(22) since they provide thevertical profiles of air pressure, temperature, density,and mixing ratios of water vapor, O3, O2, CO2, andother greenhouse gases, defining their values at 50levels from sea level to a 120-km height in increasingsteps of 1 km from sea level to 25 km, 2.5 km from 25to 50 km, and 5 km from 50 to 120 km. They werederived by Anderson et al.25 from the vertical profilesof air pressure, temperature, density, and moistureparameters determined by the working group of theU.S. Committee on Extension of the Standard Atmo-sphere23,26 for the U.S. Standard Atmosphere (1962)and its supplementary models. Such models describethe average variations of the meteorological param-eters of the atmosphere with latitude and season.Therefore their use allows us to evaluate the averagevariations of the Rayleigh-scattering coefficient thatare associated with the average variations in thethermal and moisture conditions of the atmospheredue to latitude and season changes.

The CO2 volume concentration C was assumed byAnderson et al.25 to be equal to 330 ppmv in all themodels and at all altitudes from sea level to 75 kmand to decrease at higher altitudes to reach the valueof 35 ppmv at a 120-km height. The value of330 ppmv is not updated to the current measure-ments of this concentration parameter since the nu-merous ground-based stations of the GlobalAtmosphere Watch network provided yearly meanvalues of C in 2001 that are mostly higher than370 ppmv. Thus we examined the monthly mean val-ues of C given at several stations by the World DataCenter for Greenhouse Gases (WDCGG) to calculatethe most suitable value of C for each of the above sixatmospheric models.25 The procedure followed in thepresent analysis is described in Appendix A, wherevalues of ground-level concentration C are given forthe year 2001, which differ slightly from one model toanother by some ppmv only since it is well knownthat this concentration parameter is subject to varyappreciably with latitude and season. Consideringthat the data require updating at the five-year periodfrom 2005 to 2010, we assumed that C increases onaverage by �1.5 ppmv�yr and determined the valuesof sea-level CO2 volume concentration C0 given inTable 3 by adding an increase of �10 ppmv to theabove average values found in 2001. These sea-levelvalues were assumed to be stable within the altituderange from 0 to 75 km and to decrease as a function

of height from 75 to 120 km in all the six standardatmosphere models, following the same decreasingpercentage rate established by Anderson et al.25

Moreover, to divide the atmosphere into more lay-ers than those fixed by Anderson et al.25 in an attemptto achieve more precise calculations of parameters�� and ��, z� in Eq. (2), we applied some interpo-lation procedures in height to the original data.25 Thevalues of the parameters p, T, and e�p (that providethe water-vapor mixing ratio measured in terms ofvolume parts25) were calculated at levels increasingin steps of 0.25 km from sea level to 20 km, 0.5 kmfrom 20 to 40 km, and 1 km from 40 to 120 km. Moreprecisely, pressure p and water-vapor mixing ratioe�p were determined with an exponential interpola-tion procedure in height between the values given atthe 50 levels defined by Anderson et al.,25 whereas airtemperature was calculated through linear interpo-lation in height. The above procedure was found toprovide interpolated values of p, T, and e, which turnout to be not far from the hydrostatic equilibriumconditions defined by the Anderson et al.25 profiles,with relative differences no greater than �0.03% upto the 25-km level. In this way, we obtained the val-ues of the three above parameters at 201 levels fromsea level to 120 km.

The vertical profiles of air pressure p�z� are verysimilar for all models until heights of around 40 km,revealing some small differences above the 80-kmlevel, especially between Models 3 and 4. Markeddiscrepancies exist among the six vertical profiles ofT�z� in the tropospheric region, with the highest val-ues in Model 1 and the lowest in Model 5. A well-pronounced minimum of T�z� characterizes thethermal structure of the tropopause in Model 1,whereas the lowest stratospheric values are those inModel 5 and the highest in Model 4. Through themesosphere and beyond the mesopause level, thecoldest air conditions pertain to Models 2 and 4,whereas Models 3 and 5 exhibit the warmest air con-ditions. The vertical profiles of partial pressure e�z�differ considerably in the troposphere, with the high-est values in Model 1 and the lowest in Model 5,presenting less marked discrepancies at the upperaltitudes until reaching (at the 120-km level) valuessmaller than those at sea level by more than 12 or-ders of magnitude. The vertical profiles of CO2 con-centration C do not differ appreciably since the valuesof C0 given in Table 3 vary from 379 to 390 ppmv, thatis, within �1.6%.

The Rayleigh-scattering cross section �� per mol-ecule was calculated in terms of Eqs. (4) and (5) at 88selected wavelengths from 0.2 to 4.0 �m for all the201 above levels of the six atmosphere models.25 Thevertical profile of �0.50 �m, z� obtained for Model625 is shown in the left panel of Fig. 5, together withthe vertical curves of �0.50 �m, z� calculated for theother five models25 and normalized to the values ofthis quantity found for Model 6.25 The results clearlyindicate that the values of �0.50 �m� at troposphericlevels lower than 5 km differ little from one model toanother, the highest values being obtained for Model


5 and the lowest for Model 1, with relative discrep-ancies at sea level within �0.6%. At upper altitudes,the values of ��, z� become gradually closer at all thewavelengths under consideration, until presentingdiscrepancies smaller than 0.06% at the 8-km heightand even more limited differences at the strato-spheric levels. The spectral values of �� and theirwavelength-dependence features can be better exam-ined in Table 4, where the values of this quantity aregiven for the sea-level conditions of Model 6 at 88selected wavelengths. The results indicate that ��follows the wavelength-dependence form predicted bythe Rayleigh-scattering theory,65 defined in terms ofthe inverse of the wavelength power with a spectralexponent, which first decreases quickly from thebest-fit value of 4.759 � 0.039 within the0.20–0.25-�m spectral range to 4.091 � 0.001 withinthe 0.50–0.55-�m interval and then more slowly atupper wavelengths, reaching a value of 4.002 withinthe 3.5–4.0-�m spectral range. The sea-level values

of �� given in Table 4 within the 0.2–4.0-�m wave-length range are compared in Fig. 6 with those cal-culated by Bucholtz4 at the same wavelengths forstandard air conditions, i.e., for air pressure and tem-perature conditions very close to those considered inTable 4 at sea level for Model 6.25 The spectral valuesof the ratio �� between the present values of ��and those determined by Bucholtz4 were found to bemostly lower than 1, as they vary mainly between0.9980 and 0.9990 at the visible wavelengths andbetween 0.9975 and 0.9985 at IR wavelengths. Thesefindings indicate that the Bucholtz4 evaluations of�� are mostly overestimated by very small percent-ages varying between 0.1% and more than 0.2% withrespect to the present ones, these overestimationsbeing mainly caused by use of appreciably highervalues of factor F��, as shown in Fig. 4, and to alesser extent by the neglect of water-vapor effects.

Using Eqs. (2) and (3), we calculated the volumeRayleigh-scattering coefficient �� at the same 88wavelengths selected above for all the above-fixedlevels of the six atmospheric models.25 The verticalprofile of �0.50 �m, z� obtained for Model 6 is shownin the right panel of Fig. 5, together with those of�0.50 �m, z� calculated for the other five atmo-sphere models25 and normalized to the values of thisquantity determined for Model 6.25 The results showthat the parameter �0.50 �m, z� decreases by morethan 7 orders of magnitude from sea level to a 120-kmheight and that the various vertical profiles presentdiscrepancies within �20% throughout the whole tro-posphere and low stratosphere and more marked dif-ferences at altitudes ranging between 40 and 80 kmand above the 100-km level, the lowest values in thestratosphere and mesosphere being those given byModel 5 and the highest by those of Model 4.

The values of �� calculated for the sea-level con-ditions of Model 6 are given in Table 4 at the 88wavelengths selected above. Compared with thoseproposed by Bucholtz4 for standard air conditions,they turn out to be generally lower by 0.2–0.3%, ascan be seen in Fig. 6, where the spectral curve of theratio �� is shown in the 0.2–4.0-�m spectral range,this quantity being obtained by dividing at eachwavelength the value of �� given in Table 4 by thatdetermined by Bucholtz4 for standard air conditions.Figure 6 shows that ratio �� increases at UV wave-lengths reaching a maximum close to unity and thendecreases rapidly throughout the visible range toreach IR values smaller than 0.999 at 0.40 �m, whichcontinue to diminish slowly as the wavelength in-creases. As pointed out above for the ratio �� rela-tive to the Rayleigh-scattering cross-section, theseresults are mostly due to use of overestimated valuesof the King factor.

6. Calculation of the Rayleigh-Scattering Optical Depth

The above considerations indicate that the presentalgorithms proposed for the calculation of the refrac-tive index of moist air and depolarization propertiesof moist air provide evaluations of the Rayleigh-scattering cross section and volume coefficient that

Fig. 5. Vertical profile of Rayleigh-scattering cross section permolecule �0.50 �m� (solid curve in the left panel) and verticalprofile of volume Rayleigh-scattering coefficient �0.50 �m� (solidcurve in the right panel), both determined for Model 6.25 The othercurves represent the vertical profiles of parameters �0.50 �m�(left panel) and �0.50 �m� (right panel) calculated for the otherfive atmosphere models25 (labeled My with y � 1, 2, 3, 4, and 5 asgiven in Table 3) and normalized to those calculated for Model 6,25

their scales defined at the top of the figure: open squares refer toModel 1, open triangles (up) to Model 2, open diamonds to Model 3,open circles to Model 4, and inverted open triangles to Model 5.


are appreciably lower than those estimated by Bu-choltz4 for standard air conditions, presumably be-cause of the better approximations used to determinethe spectral values of the King factor and the morerealistic representation of the water-vapor and CO2effects on the moist air refractivity characteristics.On the basis of the previous remarks, a computa-tional program was written to determine the spectralvalues of optical depth �� in terms of Eq. (1), wherethe coefficient ��, z� is defined by Eq. (2); N�z� isassumed to vary with height according to Eq. (3); and

the cross section ��, z� is calculated in terms of Eq.(4), which includes the effects due to spectral varia-tions in the King factor. In Eq. (4) the refractive indexof moist air n��, z� is a function of parameters p, T, e,C, and �, according to the dependence features de-scribed by Eqs. (10)–(19), whereas the King factorF�� is evaluated by means of Eq. (22) based on Eqs.(20) and (21). This computational program, based onuse of an overall number of 17 equations, was appliedto the six atmospheric models25 to calculate the val-ues of the monochromatic volume coefficient ��, z� at

Table 4. Values of the Rayleigh-Scattering Cross Section �0(�) and Volume Rayleigh-Scattering Coefficient �0(�) Calculated at the 88 WavelengthsSelected from 0.20 to 4.00 �m with the Present Rayleigh-Scattering Model for Sea-Level Atmospheric Conditions Defined in Model 6a

Wavelength� ��m� 0�� cm�2� 0�� km�1�

Wavelength� ��m� 0�� cm�2� 0�� km�1�

0.20 3.6064 (�10�25) 9.1813 (�10�1) 0.64 2.4280 (�10�27) 6.1814 (�10�3)0.21 2.8349 7.2172 0.65 2.2799 5.80440.22 2.2686 5.7755 0.66 2.1430 5.45570.23 1.8422 4.6899 0.67 2.0162 5.13300.24 1.5146 3.8559 0.68 1.8987 4.83380.25 1.2586 3.2042 0.69 1.7896 4.55610.26 1.0558 2.6879 0.70 1.6883 4.29820.27 8.9311 (�10�26) 2.2737 0.71 1.5941 4.05830.28 7.6120 1.9379 0.72 1.5064 3.83490.29 6.5320 1.6629 0.73 1.4246 3.62680.30 5.6399 1.4358 0.74 1.3483 3.43250.31 4.8974 1.2468 0.75 1.2771 3.25120.32 4.2748 1.0883 0.76 1.2105 3.08170.33 3.7493 9.5451 (�10�2) 0.77 1.1482 2.92310.34 3.3031 8.4091 0.78 1.0899 2.77460.35 2.9221 7.4391 0.79 1.0352 2.63550.36 2.5950 6.6064 0.80 9.8394 (�10�28) 2.50490.37 2.3129 5.8882 0.82 8.9059 2.26730.38 2.0684 5.2658 0.84 8.0807 2.05720.39 1.8557 4.7243 0.86 7.3490 1.87090.40 1.6698 4.2511 0.88 6.6985 1.70530.41 1.5069 3.8362 0.90 6.1184 1.55760.42 1.3634 3.4711 0.92 5.5999 1.42560.43 1.2368 3.1487 0.94 5.1352 1.30730.44 1.1246 2.8631 0.96 4.7179 1.20110.45 1.0249 2.6093 0.98 4.3421 1.10540.46 9.3614 (�10�27) 2.3833 1.00 4.0030 1.01910.47 8.5681 2.1813 1.10 2.7284 6.9461 (�10�4)0.48 7.8575 2.0004 1.20 1.9234 4.89670.49 7.2194 1.8379 1.30 1.3947 3.55080.50 6.6451 1.6917 1.40 1.0359 2.63730.51 6.1271 1.5599 1.50 7.8549 (�10�29) 1.99970.52 5.6588 1.4406 1.60 6.0639 1.54380.53 5.2346 1.3326 1.70 4.7556 1.21070.54 4.8496 1.2346 1.80 3.7820 9.6282 (�10�5)0.55 4.4995 1.1455 1.90 3.0453 7.75280.56 4.1805 1.0643 2.00 2.4796 6.31270.57 3.8894 9.9017 (�10�3) 2.20 1.6927 4.30940.58 3.6233 9.2243 2.40 1.1947 3.04160.59 3.3797 8.6040 2.60 8.6714 (�10�30) 2.20760.60 3.1562 8.0352 2.80 6.4453 1.64090.61 2.9510 7.5127 3.00 4.8900 1.24490.62 2.7622 7.0322 3.50 2.6386 6.7173 (�10�6)0.63 2.5884 6.5896 4.00 1.5463 3.9367

aAir pressure p0 � 1013.00 hPa, temperature T0 � 288.20 K, water-vapor partial pressure e0 � 7.85075 hPa, and CO2 volumeconcentration C � 385 ppmv.


the 201 fixed levels from a 0 to 120-km height. Weavoided using the trapezoidal rule for the integrationof the volume coefficient ��, z� from one level to thesubsequent level, as done by Bucholtz.4 Instead, weassumed that this parameter decreases in an expo-nential fashion with height through all the atmo-spheric regions, following the typical trend ofmolecular number density N�z� (Ref. 3) and hencepresenting a value of scale height varying slowly withaltitude. Thus, considering that in Eq. (2) the crosssection ��, z� remains in practice nearly constantwith height, as shown in Fig. 5, we first calculated foreach ith layer (among the 200 atmospheric layersdefined above) the exponent following an exponentialinterpolation procedure between the values of ��, z�at the lower and upper levels zi and zi 1, respectively.We then calculated the scale height relative to the ithlayer by taking into account that the layer depth isgiven by the difference zi 1 � zi. Subsequently, theintegrals of coefficient ��, z� were calculated for eachof the 200 layers defined above, and optical depth ��was calculated by summing all these 200 integrals.The values of �� obtained at the 88 selected wave-lengths are presented in Table 5 for the six atmo-spheric models.25 It is worth pointing out that thesevalues were obtained for slightly different values ofthe sea-level air pressure p0 equal to 1013.00 hPa inModels 1, 2, 5, and 6; 1010.00 hPa in Model 4; and1018.00 hPa in Model 3, as given in Table 3. Anotherimportant point is that optical depth �� decreases asa function of wavelength following the dependenceform predicted by the Rayleigh-scattering theory,65

being proportional to the inverse of the wavelengthpower, where wavelength � is measured in microme-ters and the exponent decreases gradually withwavelength. In the case of Model 6, for example, theexponent assumes a best-fit value equal to 4.755

� 0.038 within the 0.20–0.25-�m spectral range, de-creasing rapidly to the value of 4.092 � 0.001 withinthe 0.50–0.55-�m range and more slowly to 4.002within the 3.5–4.0-�m wavelength range.

The values of �� presented in Table 5 differ ap-preciably from one model to another, showing thatvariations in air pressure, temperature, and humid-ity can produce appreciable changes in the Rayleigh-scattering optical depth of the atmosphere. Withrespect to Model 6, Model 1 presents higher values of�� by 0.4–0.3% as � increases from 0.2 to 4.0 �m;Model 2 presents higher values by 0.17–0.15%;Model 3 presents higher values by �0.51–0.56%;Model 4 presents lower values by � 0.35% at all wave-lengths; and Model 5 presents slightly differing val-ues within �0.05%.

The values of �� given in Table 5 are comparedwith those obtained by Bucholtz4 in Fig. 7, in terms ofthe ratio �1�� of the present values of �� to those ofBucholtz, where the latter values are normalized tothe values of sea-level pressure p0 adopted in Table 3for the six models.25 The spectral curves of �1��present values mostly smaller than unity, indicatingthat the Bucholtz4 procedure provides slightly over-estimated values of Rayleigh-scattering optical depthwith respect to the present evaluations. More pre-cisely, the ratio �1�� pertinent to the six atmosphericmodels25 assumes values ranging between 0.997 and1.000 at UV and visible wavelengths and slightlylower values ranging between 0.997 and 0.999 at IRwavelengths longer than 1 �m. In particular, wefound values ranging between 0.998 and 0.999 forModels 3 and 5, values lower than 0.998 for Models 4and 6, and values close to 0.997 for Models 1 and 2.Calculations of the ratio �2�� between the presentvalues of optical depth �� calculated for Model 625

and those determined by Bodhaine et al.5 for thesame atmospheric model, and hence for the samesea-level air pressure conditions, are also shown inFig. 7. The values of �2�� turn out to be slightly lowerthan unity, showing that the procedure followed byBodhaine et al.5 provides values of �� very similar tothose obtained with the present algorithm, the smalldiscrepancies being mainly due to the water-vaporeffects accounted for in the present calculations and,to a considerably lesser extent, to use of CO2 volumeconcentration values differing by 25 ppmv on the av-erage from those used by Bodhaine et al.

7. Conclusions

The algorithm used to calculate the Rayleigh-scattering parameters consists of the classical Eqs.(1)–(4) representing the total molecular scattering ofair, where the refractive index of moist air is deter-mined in terms of Eqs. (10)–(22) based on the Cid-dor56,57 physical model, which takes into account thewater-vapor and CO2 effects, and the King factor for-mula, which considers the effects produced by themain constituents of air, including also those due towater vapor and CO2. Therefore the algorithm de-fined in the present paper is improved with respectto the algorithms previously proposed in the

Fig. 6. Spectral curves of the ratios �� (solid curve) and ��(dashed curve) within the 0.2–4.0-�m wavelength range. The ratio�� yields the present values of the Rayleigh-scattering cross sec-tion per molecule �� (given in Table 4 for the sea-level atmo-spheric conditions of Model 625) divided by those determined byBucholtz4 at the same wavelengths for standard air conditions.The ratio �� was determined by dividing the values of volumeRayleigh-scattering coefficient 0�� (given in Table 4 for the sea-level atmospheric conditions of Model 6) by those determined byBucholtz at the same wavelengths for standard air conditions.


Table 5. Values of the Rayleigh-Scattering Optical Depth �(�) Obtained for the Six Standard Atmosphere Modelsa

Wavelength� ��m�

Model 1,Tropical(15 °N,

Ann. Aver.)

Model 2,Mid-Latitude

Summer(45 °N, July)

Model 3,Mid-Latitude

Winter(45 °N, Jan.)

Model 4,Sub-ArcticSummer

(60 °N, July)

Model 5,Sub-Arctic

Winter(60 °N, Jan.)

Model 6,U.S. Standard

(1976)

0.20 7.7985 7.7816 7.8081 7.7407 7.7640 7.76610.21 6.1305 6.1178 6.1396 6.0860 6.1051 6.10630.22 4.9062 4.8963 4.9143 4.8711 4.8868 4.88750.23 3.9841 3.9763 3.9913 3.9560 3.9690 3.96940.24 3.2757 3.2694 3.2819 3.2527 3.2636 3.26380.25 2.7221 2.7170 2.7275 2.7032 2.7124 2.71250.26 2.2835 2.2792 2.2881 2.2677 2.2755 2.27550.27 1.9317 1.9281 1.9357 1.9184 1.9250 1.92500.28 1.6464 1.6434 1.6499 1.6351 1.6407 1.64070.29 1.4128 1.4102 1.4158 1.4031 1.4080 1.40800.30 1.2199 1.2177 1.2225 1.2115 1.2158 1.21570.31 1.0593 1.0574 1.0616 1.0520 1.0557 1.05570.32 9.2461 (�10�1) 9.2294 (�10�1) 9.2665 (�10�1) 9.1831 (�10�1) 9.2153 (�10�1) 9.2150 (�10�1)0.33 8.1096 8.0949 8.1275 8.0544 8.0827 8.08240.34 7.1445 7.1316 7.1604 7.0959 7.1209 7.12060.35 6.3203 6.3090 6.3345 6.2774 6.2996 6.29930.36 5.6129 5.6028 5.6255 5.5748 5.5945 5.59420.37 5.0027 4.9937 5.0140 4.9688 4.9863 4.98610.38 4.4739 4.4659 4.4840 4.4436 4.4593 4.45910.39 4.0138 4.0067 4.0229 3.9866 4.0008 4.00060.40 3.6118 3.6054 3.6201 3.5874 3.6001 3.59990.41 3.2593 3.2535 3.2668 3.2373 3.2488 3.24860.42 2.9491 2.9439 2.9558 2.9292 2.9396 2.93940.43 2.6752 2.6704 2.6813 2.6571 2.6665 2.66640.44 2.4325 2.4282 2.4381 2.4161 2.4247 2.42450.45 2.2170 2.2130 2.2221 2.2020 2.2098 2.20970.46 2.0249 2.0213 2.0295 2.0112 2.0184 2.01820.47 1.8533 1.8500 1.8576 1.8408 1.8473 1.84720.48 1.6996 1.6966 1.7035 1.6881 1.6941 1.69400.49 1.5616 1.5588 1.5652 1.5510 1.5566 1.55650.50 1.4373 1.4348 1.4407 1.4277 1.4328 1.43270.51 1.3253 1.3230 1.3284 1.3164 1.3211 1.32100.52 1.2240 1.2218 1.2269 1.2158 1.2201 1.22000.53 1.1323 1.1303 1.1349 1.1246 1.1286 1.12860.54 1.0490 1.0471 1.0514 1.0419 1.0456 1.04560.55 9.7324 (�10�2) 9.7153 (�10�2) 9.7551 (�10�2) 9.6669 (�10�2) 9.7014 (�10�2) 9.7007 (�10�2)0.56 9.0425 9.0266 9.0636 8.9816 9.0137 9.01310.57 8.4129 8.3980 8.4325 8.3562 8.3861 8.38550.58 7.8373 7.8235 7.8556 7.7845 7.8124 7.81180.59 7.3103 7.2974 7.3274 7.2611 7.2871 7.28650.60 6.8270 6.8150 6.8429 6.7810 6.8053 6.80480.61 6.3831 6.3719 6.3980 6.3401 6.3629 6.36240.62 5.9748 5.9643 5.9888 5.9346 5.9559 5.95540.63 5.5987 5.5889 5.6119 5.5611 5.5810 5.58060.64 5.2519 5.2427 5.2642 5.2166 5.2353 5.23490.65 4.9316 4.9229 4.9432 4.8984 4.9160 4.91560.66 4.6354 4.6273 4.6463 4.6042 4.6207 4.62040.67 4.3612 4.3535 4.3714 4.3318 4.3474 4.34700.68 4.1070 4.0998 4.1166 4.0794 4.0940 4.09370.69 3.8711 3.8643 3.8802 3.8451 3.8589 3.85850.70 3.6519 3.6455 3.6605 3.6274 3.6404 3.64010.71 3.4481 3.4420 3.4562 3.4249 3.4372 3.43690.72 3.2583 3.2526 3.2660 3.2364 3.2480 3.24780.73 3.0814 3.0760 3.0887 3.0607 3.0717 3.07150.74 2.9164 2.9113 2.9233 2.8968 2.9072 2.90700.75 2.7623 2.7575 2.7689 2.7438 2.7537 2.75340.76 2.6183 2.6137 2.6245 2.6007 2.6101 2.60990.77 2.4836 2.4792 2.4895 2.4669 2.4758 2.47560.78 2.3574 2.3533 2.3630 2.3416 2.3500 2.3498


literature3–5 since we used the well-known first Peckand Reeder40 formula to define the spectral depen-dence features of the air refractive index within the0.23–1.69-�m wavelength range, together with a newformula derived from the second Peck and Reeder40

formula to define the spectral variations of air refrac-tivity in the 0.185–0.230-�m wavelength range. Thisset of equations accounts more satisfactorily for thewater-vapor effects on air refractivity and for theirspectral dependence features, showing that the moistair refractivity at visible wavelengths increases by�0.1% in cases in which the water-vapor partial pres-sure is made to decrease by �75% at standard sea-level pressure and temperature conditions like thosedefined in Model 6.25

Use of Eq. (22) to determine the spectral depen-dence curve of the King factor for moist air in the0.2–4.0-�m wavelength range appears also to be re-alistic, this equation being obtained by adding thewater-vapor effects to those of N2, O2, Ar, and CO2molecules represented in the Bates34 formula. Thisnew algorithm allows us to show that the King factoris subject to decrease by �0.04% only at visible wave-lengths as a result of the water-vapor effects occur-ring for standard conditions of the atmosphere.

Table 5. Continued

Wavelength� ��m�

Model 1,Tropical (15

°N, Ann.Aver.)

Model 2, Mid-Latitude

Summer (45°N, July)

Model 3, Mid-Latitude

Winter (45 °N,Jan.)

Model 4, Sub-Arctic Summer(60 °N, July)

Model 5, Sub-Arctic Winter(60 °N, Jan.)

Model 6, U.S.Standard

(1976)

0.79 2.2392 2.2353 2.2445 2.2242 2.2322 2.23200.80 2.1283 2.1246 2.1333 2.1140 2.1216 2.12140.82 1.9264 1.9230 1.9309 1.9134 1.9203 1.92020.84 1.7479 1.7448 1.7520 1.7362 1.7424 1.74230.86 1.5896 1.5868 1.5934 1.5790 1.5846 1.58450.88 1.4489 1.4464 1.4523 1.4392 1.4444 1.44420.90 1.3234 1.3211 1.3266 1.3146 1.3193 1.31920.92 1.2113 1.2092 1.2142 1.2032 1.2075 1.20740.94 1.1108 1.1088 1.1134 1.1033 1.1073 1.10720.96 1.0205 1.0187 1.0229 1.0136 1.0173 1.01720.98 9.3921 (�10�3) 9.3757 (�10�3) 9.4144 (�10�3) 9.3291 (�10�3) 9.3627 (�10�3) 9.3619 (�10�3)1.00 8.6587 8.6436 8.6793 8.6006 8.6316 8.63091.10 5.9017 5.8914 5.9158 5.8622 5.8833 5.88281.20 4.1605 4.1532 4.1704 4.1326 4.1475 4.14711.30 3.0169 3.0117 3.0241 2.9967 3.0075 3.00721.40 2.2408 2.2369 2.2462 2.2258 2.2338 2.23361.50 1.6991 1.6961 1.7031 1.6877 1.6938 1.69361.60 1.3116 1.3094 1.3148 1.3029 1.3076 1.30741.70 1.0287 1.0269 1.0311 1.0218 1.0255 1.02541.80 8.1806 (�10�4) 8.1664 (�10�4) 8.2002 (�10�4) 8.1258 (�10�4) 8.1552 (�10�4) 8.1544 (�10�4)1.90 6.5872 6.5757 6.6030 6.5431 6.5667 6.56612.00 5.3636 5.3542 5.3764 5.3277 5.3469 5.34642.20 3.6615 3.6551 3.6703 3.6370 3.6501 3.64982.40 2.5843 2.5798 2.5905 2.5670 2.5762 2.57602.60 1.8757 1.8724 1.8802 1.8631 1.8698 1.86972.80 1.3942 1.3917 1.3975 1.3848 1.3898 1.38973.00 1.0577 1.0559 1.0603 1.0507 1.0545 1.05443.50 5.7074 (�10�5) 5.6975 (�10�5) 5.7211 (�10�5) 5.6692 (�10�5) 5.6897 (�10�5) 5.6891 (�10�5)4.00 3.3448 3.3390 3.3528 3.3224 3.3344 3.3341

aRef. 25.

Fig. 7. Spectral curves of the ratio �1�� between optical depth�� calculated with the present method for the six U.S. StandardAtmosphere models25 and optical depth �� calculated by Bu-choltz4 for the 1962 U.S. Standard Atmosphere model23 and thefive supplementary models26 and normalized by us to the sea-levelair pressure conditions given in Table 3 for the six atmospheremodels: Model 1 (filled squares), Model 2 (open squares), Model 3(filled circles), Model 4 (open circles), Model 5 (filled triangles), andModel 6 (open triangles). The filled diamonds represent the ratio�2�� between optical depth �� calculated with the presentmethod for Model 6 and optical depth �� obtained for the sameatmospheric model calculated with the method proposed byBodhaine et al.5


Calculations of the moist air refractivity and theKing factor performed at different altitudes indicatethat the present algorithm can be properly used toevaluate the variations of Rayleigh-scattering crosssection and volume coefficient with height, as it alsotakes into account the vertical changes in the airpressure, temperature, and moisture conditions of airthat can occur over small altitude scales. The calcu-lations of the cross section �0.50 �m� performed at201 levels from sea level to 120 km show that themost marked differences between the values of ��from one to another of the six atmosphere models canbe found within the lower tropospheric region of a fewkilometers in depth, where the largest discrepanciesin air temperature and moisture conditions can beobserved, leading to appreciable changes in bothmoist air refractivity and the King factor. Corre-spondingly, variations of a few percent in the volumecoefficient ��, z� were found from one model to an-other at tropospheric levels, whereas larger discrep-ancies took place at stratospheric and mesosphericaltitudes, as clearly shown in Fig. 5. These remarksoffer evidence of the fact that the small differencesfound between the values of Rayleigh-scattering op-tical depth �� obtained in Table 5 for the six atmo-sphere models are mainly due to the differentfeatures of air temperature and water-vapor partialpressure characterizing the six models in the lowerpart of the troposphere. In particular, �� was foundto decrease because of the presence of water vapor bypercentages that vary as a function of wavelengthand water-vapor vertical content. For example, inModel 1,25 where precipitable water exceeds 4 g cm�2,the decrease of �� due to water vapor was evaluatedas equal to �0.06% at wavelength � � 0.20 �m,�0.19% at � � 0.36 �m, �0.20% at � � 0.50 �m, and�0.21% at wavelengths increasing from 1 to 4 �m.These rates decrease gradually as precipitable waterW diminishes. For Model 6, with W � 1.42 g cm�2, wefound relative variations of �0.20% at � � 0.20 �m,�0.06% at both � � 0.36 �m and � � 0.50 �m,and �0.07% at � � 1.0 �m; and for Model 5, withW � 0.42 g cm�2, we obtained relative variationsof � 6.00 � 10�3% at � � 0.20 �m, �0.02% at �� 0.36 �m, �0.01% at � � 0.50 �m, and �0.02% at� � 1.0 �m. These variations in �� turn out to be ofcomparable magnitude but of opposite sign withthose produced by CO2, which were estimated to varybetween 0.04% and 0.05% for all the six atmo-sphere models throughout the whole spectral rangefrom 0.2 to 4.0 �m.

Calculations of optical depth ��0.5 �m, z� havebeen also performed by integrating ��0.50 �m� fromsea level to height z made to vary from 0 to 120 km.The results show that ��0.50 �m, 10 km� is equal to73.8% of the overall value ��0.50 �m, 120 km�;��0.50 �m, 20 km� to 94.5%; ��0.50 �m, 30 km� to98.8%; ��0.50 �m, 50 km� to 99.8%; ��0.50 �m,75 km� to 99.9%; and ��0.50 �m, 100 km� to 100%.From these evaluations, it can be concluded that thevalues of Rayleigh-scattering optical depth �� can

be correctly calculated at all the wavelengths follow-ing the present procedure based on the improved al-gorithm described by Eqs. (10)–(22) for verticalprofiles of parameters p, T, and e defined by one of thestandard atmosphere models25 over the altituderange from sea level to 100 km and for any triplet ofvertical profiles of p, T, and e determined from aradiosounding measurement, provided that thesevertical profiles (usually limited to the troposphereand the first kilometers of the stratosphere) are com-pleted with those defined at the upper levels by themost realistic atmosphere model among those consid-ered in the present study.25

Appendix A: Evaluations of the Average CO2 VolumeConcentration Values for an Updated Use of the SixStandard Atmosphere Models

To determine updated values of the CO2 volumeconcentration C in the six standard atmosphere mod-els, we examined the monthly mean ground-level val-ues of C measured at a large number of the GlobalAtmosphere Watch stations during 2001, this yearbeing the last in which the data of the major part ofstations belonging to the global network of the WD-CGG were available. Thirty-six stations were se-lected by us for use in the six standard atmospheremodels as follows:

1. Model 1 (Tropical, 15 °N annual average). Thefirst eight stations listed in Table 6 were considered,as they are at latitudes varying between 5 °N and 26°N, with annual average ground-level values of Cranging between 371.1 and 373.5 ppmv, from which ayearly mean value of 371.8 � 1.0 ppmv was obtained.Considering that an average increase of CO2 volumeconcentration equal to 1.5 ppmv�yr was observedover the past years, the sea-level value C0 was as-sumed to be equal to 382 ppmv in Model 1,25 thisvalue referring to the year 2008.

2. Model 2 (Mid-Latitude Summer, 45 °N, July);Model 3 (Mid-Latitude Winter, 45 °N, January); andModel 6 [U.S. Standard Atmosphere (1976), 45 °Nannual average]. For these three mid-latitude atmo-sphere models, the 18 stations listed in Table 6 (withnumbers from 9 to 26) were considered at latitudesvarying between 37 °N and 53 °N, with nine in Eu-rope, four in Asia, and five in North America. Themonthly mean ground-level values of C determinedin July are suitable for Model 2. They were found torange between 361.9 and 386.2 ppmv in 2001, givingan average value of 368.5 � 5.7 ppmv. Considering amean increase of 1.5 ppmv�yr for the following years,we assumed C0 � 379 ppmv in Model 2 for the year2008. The monthly mean ground-level values of Ccalculated in January, and hence suitable for Model3, were found to vary between 372.6 and 389.7 ppmvin 2001, giving an average value of 379.2� 5.7 ppmv. Considering the above mean increaseof 1.5 ppmv�yr, we assumed that C0 � 390 ppmv inModel 3, relative to 2008. The yearly mean ground-level values of C suitable for Model 6 were found to


range between 371.2 and 385.0 ppmv in 2001, givingan average value of 374. � 3.6 ppmv. For an averageincrease of 1.5 ppmv�yr, the value C0 � 385 ppmvwas assumed in Model 6 for the year 2008.

3. Model 4 (Sub-Arctic Summer, 60 °N, July) andModel 5 (Sub-Arctic Winter, 60 °N, January). For thetwo Sub-Arctic models, we considered the data takenat the ten stations listed in Table 6, with numbersfrom 27 to 36 and latitudes from 53 °N to 67 °N. Themonthly mean ground-level values of C determinedin July are suitable for use in Model 4. They werefound to range between 364.7 and 388.5 ppmv in2001, giving an average value of 371.2 � 7.9 ppmv.For a mean yearly increase of 1.5 ppmv, we assumedthat the value C0 � 382 ppmv in Model 4, relative to2008. The monthly mean ground-level values of Ccalculated in January, and hence suitable for Model

5, were found to vary between 373.9 and 389.1 ppmvin 2001, with an average value of 379.9 � 5.0 ppmv.For the mean yearly increase adopted above, we as-sumed that C0 � 390 ppmv in Model 5, relative to2008.

This research was supported by the ProgrammaNazionale di Ricerche in Antartide and developed asa part of subproject 2003�6.7 “Characterization ofaerosol induced climatic effects in polar regions: anassimilation and analysis of multispectral sun-photometer data from the POLAR-AOD network.”The authors gratefully acknowledge the Interna-tional Center for Theoretical Physics, Trieste, Italy,for its support of the participation of B. Petkov in theframework of the Programme for Training and Re-search in Italian Laboratories.

Table 6. WDCGG Stations, Their Latitude, Longitude, and Height and the Yearly Average and Monthly Mean Ground-Level Values of CO2 VolumeConcentration C Measured in 2001

WDCGG Station Latitude Longitude

Height(m

AMSL)a

Ground-Level CO2 VolumeConcentration �ppmv�

YearlyAverage

JanuaryAverage

JulyAverage

1. Mauna Loa, Hawaii 19° 32= N 155° 35= W 3397 371.07 370.46 371.512. Assekrem, Algeria 23° 16= N 5° 38= E 2728 371.37 371.93 370.293. Guam 13° 26= N 144° 47= E 2 371.02 370.09 370.454. Ragged Point, Barbados 13° 10= N 59° 26= W 45 371.15 371.36 370.365. Key Biscayne, Florida 25° 40= N 80° 12= W 3 373.20 375.78 371.626. Cape Kumukahi, Hawaii 19° 31= N 154° 49= W 3 371.43 371.72 370.767. Minamitorishima, Japan 24° 18= N 153° 58= E 8 371.77 372.52 370.128. Yonagunijima, Japan 24° 28= N 123° 01= E 30 373.47 374.64 371.739. Terceira Island, Portugal 38° 46= N 27° 23= W 40 371.53 372.58 370.20

10. Monte Cimone, Italy 44° 11= N 10° 42= E 2165 371.83 374.80 367.9011. Waldhof, Germany 52° 48= N 10° 46= E 74 385.00 389.71 386.1812. Zugspitze, Germany 47° 25= N 10° 59= E 2960 381.21 386.06 376.5413. Brotjacklriegel, Germany 48° 49= N 13° 13= E 1016 374.91 380.21 365.7614. Deuselbach, Germany 49° 46= N 7° 03= E 480 375.64 382.96 369.6815. Schauinsland, Germany 47° 55= N 7° 55= E 1205 372.40 376.87 364.6816. Hegyhatsal, Hungary 46° 57= N 16° 39= E 344 376.62 392.31 361.8817. Black Sea, Romania 44° 10= N 28° 41= E 3 377.52 384.35 368.5118. Issyk-Kul, Kyrgyzstan 42° 37= N 76° 59= E 1640 372.90 376.40 366.5019. Plateau Assy, Kazakhstan 43° 15= N 77° 53= E 2519 371.23 375.12 364.4120. Sary Taukum, Kazakhstan 44° 27= N 77° 34= E 412 373.76 378.50 364.3021. Ryori, Japan 39° 02= N 141° 50= E 230 373.41 376.29 366.4622. Wendover, Utah 39° 54= N 113° 43= W 1320 372.23 375.82 369.0823. Niwot Ridge, Colorado 40° 03= N 105° 35= W 3475 372.36 372.87 373.4124. Park Falls, Wisconsin 45° 56= N 90° 16= W 472 374.33 379.78 364.8325. Shemya Island, Alaska 52° 43= N 174° 06= E 40 372.43 375.90 363.7526. Estevan Point, Canada 49° 23= N 126° 32= W 39 372.26 374.82 368.3027. Heimaey, Iceland 63° 24= N 20° 17= W 100 371.80 374.12 366.2628. Mace Head, Ireland 53° 20= N 9° 54= W 25 371.71 373.85 366.8329. Shetland, U.K. 60° 05= N 1° 15= W 30 370.97 374.23 366.3730. Westerland, Germany 54° 56= N 8° 19= E 40 377.84 386.48 369.8531. Ocean Station M, Norway 66° 00= N 2° 00= E 7 371.78 376.27 366.6232. Cold Bay, Alaska 55° 12= N 162° 43= W 25 372.12 375.62 364.7333. Kollumerwaard, The Netherlands 53° 20= N 6° 17= E 0 378.00 384.10 379.7034. Baltic Sea, Poland 55° 30= N 16° 40= E 7 374.05 379.12 366.1335. Neuglobsow, Germany 53° 10= N 13° 02= E 65 388.50 389.10 388.4636. Zingst, Germany 54° 26= N 12° 44= E 1 381.21 386.06 376.54

aAMSL, above mean sea level.


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Improved algorithm for calculations of Rayleigh-scattering optical depth in standard atmospheres

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