Analytic spectral functions for atmospheric transmittancecalculations
Alex E. S. Green, John C. Wagner, and Ady Mann
Analytic formulas for the transition energies, intensities, and spectral absorption coefficients of the big (mostintense) band systems of H20 and CO2 are presented. Analytic spectral representations of the Chappuisband of 03, three bands of 02, and two bands of CH4 are also given. These inputs in conjunction with bandmodel transmission formulas can be used with spectral functions for the extraterrestrial solar irradiance andRayleigh and aerosol attenuation for engineering-type calculations of the direct solar spectral irradiancereaching the ground between 0.35 and 4.5 um. Several algorithms may then be used for estimating the diffuseirradiance.
1. Introduction
While the topic of atmospheric transmittance has avery long history, a tremendous increase of effort be-gan about thirty years ago following the launching ofSputnik 1. The most advanced methodology whichhas emerged is based on the Air Force GeophysicalLaboratory (AFGL) atmospheric absorption line pa-rameter compilation.1 This original compilation is adata bank of molecular absorption parameters, whichin 1982 contained 181,000 transitions of the seven mostactive IR terrestrial absorbers: H2 0, C0 2, 03, N2 0,CO, CH4, and 02 in their various isotopic forms. Inaddition, an AFGL trace gas compilations lists thecorresponding data on twenty-one additional molecu-lar species. A model and computer code FASCODE3
(fast atmospheric signature code) has been developedfor line-by-line calculations, which calls on these at-lases. The need for FASCODE3 in large measure stemsfrom advances in high resolution detection systemsand lasers.
For many applications moderate resolution trans-mittance calculations are adequate, and for such pur-poses a FORTRAN computer code LOWTRAN has beendeveloped through several versions.4 5 The LOWTRAN6 code5 calculates atmospheric transmittance andthermal radiances averaged over 20-cm-1 intervals in
The authors are with University of Florida, Gainesville, Florida32611.
5-cm- 1 steps from 350 to 40,000 cm-' (0.25-28.5 gm).LOWTRAN and components of it6 7 are under continueddevelopment, and a LOWTRAN 7 version should beavailable soon.8 A commercial version for a PC is nowavailable.
While LOWTRAN satisfies most of the needs of atmo-spheric scientists, solar and illumination engineers,photobiologists, and other applied scientists have aneed for even simpler lower resolution approximatecalculations which give the gross features of atmo-spheric transmittance and the solar irradiances reach-ing the ground. Several simpler numerical programsare also available 9-11 which can be utilized with a per-sonal computer for such engineering-type estimates oftransmittance and solar irradiance. In this work weattempt to simplify further these engineering-type cal-culations in the 0.35-4.5-Atm region. We do so bydeveloping simple analytic models of the spectral func-tions for the major atmospheric molecular absorbers,which at the same time point to the physical origins ofthe main spectral features.
For specificity we note that the downward spectralirradiance D(X,6) reaching the ground may be placed inthe form 2
D(X,O) = uH(X)T 1T2 T3T4 T5 T6 T7 , (1)
where 0 is the solar zenith angle, = cosO, H(X) is theextraterrestrial solar irradiance, T, T2, T3, T4 denoteoptical transmittances due to molecular scattering,aerosol scattering, ozone absorption (in the Chappuisband), and aerosol absorption, respectively. Here T5denotes H20 transmittance, T6 denotes C02 transmit-tance, and T7 denotes the transmittance of other mo-lecular gases which we here take as 02 and CH4. Thefirst four transmittances are customarily calculated
using the Lambert-Beer-Bouguer law of transmit-tance. The last three transmittances approximatelyobey band model transmittance formulas. We shallignore some small correction associated with molecu-lar band overlaps. The main new features of this workare approximate expressions for the spectral transmit-tance functions T5, T6, and T7 associated with the big(most intense) bands of H20, C02, CH4, and 02. Forpurposes of completeness, formulas approximatelyrepresenting H(X), T1, T2 , T3 , and T4 are first summa-rized.
11. Extraterrestrial Solar Spectral Irradiance and Beer'sLaw Transmittances
Green and Chaill have developed an analyticrepresentation of the extraterrestrial solar spectral ir-radiance H(X), which approximately fits the 10-nmresolution version of the Neckel and Labs13 solar spec-trum as listed in Bird and Riordan's Table 2-1.10 Thefunction used is
[ K(q/X)o 1+ E (X ' (2)~exp(q/X)'o - i i Zi 2r Jj1
where K = 2300 W/m2 Aim, q = 0.7285 Aim, ao = 4.8, and1o = 2.0, and the parameters Aj, Xj, and or2 are given intabular form. 1 The left bracket is a pseudo-Planckdistribution which approximately represents the over-all envelope of H(X). The right braces are used to allowfor major Fraunhofer features in the solar spectrum.
The transmittance functions Ti for (1) molecularscattering, (2) aerosol scattering, (3) ozone absorptionin the Chappuis band, and (4) aerosol absorption canall be represented in the form
T = exp[-k(X)W,/i], (3)
where ki(X) is a spectral scattering or absorption coeffi-cient, Wi is an effective vertical path length, and tdenotes a generalized cosine function appropriate forthese species to allow approximately for the roundnessof the earth. These functions have the form
A = [(,U2 + t)I(l + t)]1/2, (4)
where the t terms are small characteristic numbers,which depend on the altitude distribution of the spe-cies. Approximate data and analytic representationsof ki(X) for i = 1-4 and values of Wi and t are availablein the literature. 411 1 4 15 In the case of aerosol scatter-ing and aerosol absorption, Shettle and Fenn4 14 haverepresented in tabular and graphical form the majordependences of k2 and k4 on wavelength and relativehumidity r for several classes of aerosol. Within theuncertainties of such descriptions of available data it ispossible to represent the Shettle-Fenn models analyti-cally.15 This should be useful, particularly if a sunphotometer is available to estimate W2 and W4.
The Chappuis band of ozone is a weak absorptionband in the visible region of the spectrum.5 Ignoringlocal fluctuations we can approximately fit the ozoneabsorption cross section data by using a mathematicalfunction of the form
0- - / \ 0
-1- I I I I I I I I -- 112 16 20 24
WAVENUMBER (1000 cm 1)
Fig. 1. Absorption cross section vs wavenumber for the Chappuisband of ozone illustrating the use of spliced derivatives of the logisticfunction (SDL). The solid curve represents experimental data, thedashed curve the SDL fit. The SDL parameters are 4.62 X 10-21
cm2 , w, = 16,811 cm-1 , A1 = 877 cm-1 , and Ar = 121 cm-'.
4 exp[(w - wp)/iA
= P 1 + exp[(w -w)/A]12(5)
Here wp locates the wavenumber of the peak of thisdistribution, yp determines the peak height, and Adetermines the width. The dependent variable in Eq.(5) is the derivative of a function called variously thelogistic, Verhulst, 16 Fermi-Dirac, 17 or Wood-Saxon 18
function. The integral may be written in the form
(6)IdY4YP { 1 + exp[(w-w )/A] }
The total area under this curve when w is integratedfrom -- to +- is 4Ayp.
In actuality one can achieve a substantial improve-ment in fit to the ozone data set and many other datasets by letting the width parameter to the left of thepeak A1 be different from the width parameter to theright of the peak Ar. Then we must append to thedefinition of the function, the condition A = A1 , w < wpand A = Ar, w > wp. In this case the total area underthe curve or band strength becomes
Sa = 2y,(A, + Ar). (7)
We shall refer to Eq. (5) as the derivative of the logisticfunction (DL). When we splice two DLs with differ-ent A terms, we will refer to it as a spliced derivative ofthe logistic function (SDL).
Figure 1 illustrates the nature of a SDL fit to Chap-puis band absorption cross-section data. The verticalozone thickness W3 is variable running as high as 0.4cm at high latitudes and as low as 0.24 cm toward theequator.
1ll. Water Vapor Absorption
There is a vast literature on the IR spectra of atmo-spheric molecules in general and H20 in particular,and we can only cite a few standard works. 19-22 In thecase of H2 0 one can group vibrational bands into clus-ters of bands at approximately the same wavenumberif they have the same polyad number23 24:
where v, V2, and 3 are the well-known vibrationalquantum numbers. Green and Mann25 have calledattention to the fact that within each polyad there isone strongest, most intense, or most IR active (big)band which dominates the other polyad members bylarge factors. Apart from the fundamental (0,1,0) N =1 band, these most intense bands correspond to a fam-ily of transitions in which v2 = 0 and V3 = 1 and a secondfamily corresponding to v2 = 1 and V3 = 1. Table I liststheir assembly of the big band centers w, and big bandstrengths Sn based on many tabulations, particularlythe recent one of Yamanouchi and Tanaka.26 To ahigh level of accuracy (0.2%) these two families ofband centers satisfy
W = W1V1 + W2V2 + W3V3 + X11Vl, (9)
where w, = 3562, w2 = 1562, W3 = 3760, and xli = -70.9(all in cm'1). The approximate relationships w, - 2w2- W3 make the polyad number N a useful gross struc-ture quantum number. To within experimental errors(-30%) the two families of band strengths Sn satisfy
log10 Sn = SO + SIVI + S2V2, (10)
where so = 5.21, si = -1.13, and s2 = -1.25.In this work we show that by using SDL functions in
conjunction with the big band of H20 and CO2 and anumber of secondary bands, it is possible to representapproximately k5(X) in analytical form while identify-ing the physical origin of the main spectral feature.We achieve this big picture of water vapor by using thefollowing approximations:
(1) We smooth over the internal structure, i.e., thePQR structure within a single band.
-1IU
EI _)i'-3
-J
-7 1 i i i I I i 1. 1 I -70 2000 1000 6000 8000 10000
WAVENUMBER (cm 1)
Fig. 2. Fits of SDL functions with parameters given in Table I tothe N = 1-10 big bands of H2 0.
(2) We ignore very weak bands within a polyad butinclude a small number of secondary bands.
(3) We ignore all but the predominant isotopic formH106.
Green and Chaill have already shown that by includ-ing these big bands and a small set of secondary bands,it is possible using DL functions to obtain a goodanalytical representation of the numerical water vaporspectral absorption coefficients listed by Bird andRiordan.10 They in effect used
Note: The band centers w, and band strengths S for the big bands of H20 are based on data cited in Refs.25 and 26.wp, Ip, Al, and Ar yield approximate fits to the numerical H2 0 spectral function. 7
(8) 1
conversion factor, and Fn denotes a near unity bandstrength renormalization factor used with a few bandsto improve agreement with long slant path atmospher-ic observations.
To emulate the results expected from the LOWTRAN
code we utilize the spectral functions for water vaportabulated by Pierluissi and Maragoudakis (PM).7Figure 2 shows these numerical functions for the N = 0to 10 big bands of H20. Also shown by the dashedcurves are SDL functional representations, which, asseen, approximate the 20-cm-l resolution data. Thecolumns in Table I headed by wp, Ip, Al, and Ar are theSDL parameters for the analytic fits in Fig. 2. Thequantity Sa = 2Ip(Ail + Ar). The transmittance func-tion used by Pierluissi et al.6 7 is
T= exp -(CWp)AP (13)
where
W = (P/P0 )OP(To/T)MPU, (14)
U = 0.7732 X 10- 4MpaZ, (15)
in which P (atm), T (K), M (ppmv), and Pa (g/m3) areaverage values of pressure, temperature, mixing ratio,and air density, respectively. Uand Ware the absorb-er amount and equivalent absorber amount, and Z(km) is the path length. The last three colums ofTable I give the Pierluissi transmittance formula pa-rameters Ap, Np, and Mp for the designated bands. Ineffect we represent the spectral function of Pierlussi etal. by
IV. Carbon Dioxide
The systematics of the big bands of CO2 share anumber of features with those of H20. This is sodespite the fact that the linear coefficients in the preci-sion formula for band centers wl - 2w2 * W3. Theclose degeneracy of w1 with 2w2 leads to Fermi reso-nances which cause an interesting mixing of states.The AFGL compilations use the notation v1,v2,l,v3,r,where is the angular momentum about the linear axisand r is a ranking number for states with the same vi,which goes from + 1,v,...J,. Rothman andYoung2 7 have given a list of -200 band centers andband intensities for all the natural isotopic forms ofCO2. The brightest bands in this list, belonging to themajor natural isotope of CO2, i.e., C12016, have thequantum numbers v2 = 0, 1 = 0, and V3 = 1. TheseviOOlr bands may be viewed as the big CO2 polyads,since for a given vi the bands of various ranks groupstogether in transition energies. Among the bandswithin a given polyad the bands with ranks near thecenter r, = (v1/2) + 1 are the brightest. Table II givesthe quantum numbers vi, r, and V3 and the band cen-ters w, and logloS based on the brightest bands select-ed from the Rothman and Young tabulation.
The second family of bands characterized by v1003ris included because it has one member (00031) whichexceeds in strength the brightest members of the 3001rpolyad and is in the same wavelength region.
The band centers of these two families are fit towithin 1% by the formula
Wr = W1V1 + W3V3 + xv1r,
C= 4Ip exp[(w - wp)/]al 1 + exp[(w -Wp)/A]2
bands
A =AL W<Wp,
AR W > P(16)
(17)
where w = 1430, 3 = 2301, and x = -39.3 (all incmi'). The band strengths are reasonably well corre-lated by
Table II. Quantum Numbers and Parameters for the Big Bands of CO2
A (Um) VI r V3 w, loglos Wp Ip Al Ar loglos Ap Np MP
Fig. 3. Fits of SDL functions with parameters given in Table II tothe big bands of CO2.
log10 SO = SO + s1V1 + S3V3 + q[r - (v1 /2 + 1)12, (18)
where so = 7.855, si = -1.668, S3 = -2.117, and q =-0.461.
To simulate the LOWTRAN spectral results we fitSDL functions to the numerical spectral function ofPierluissi and Maragoudakis. Table II also gives theSDL spectral band parameters along with the parame-ters of the transmittance formula. Figure 3 illustratesthis representation of the analytic and the numericalspectral functions. It should be clear that the analyticrepresentation of the CO2 spectral function will repro-duce nicely the gross structure of IR transmittanceswhile identifying the fundamental quantum transi-tions involved.
V. Methane and Oxygen
Pierluissi and Tsai6 have presented composite trans-mittance spectra from 0 to 10,000 cm-' for uniformlymixed gases along a vertical path from sea level to 100km. Apart from the CO2 bands one observes absorp-tion patterns due to vibration rotation bands of meth-ane near 3000 and 4300 cm-'. In addition there aresignificant absorption features due to magnetic dipoleelectronic transitions in 02 in the visible near 14,500cm' (0.69 ,m) in the near IR around 13,000 cm' (0.76/um) and the IR around 7900 cm-1 (1.27 gm). We havefit the PM spectral parameter C' = loglo(C) for CH4and 02 and have tabulated the results in Table III.
6 8 10 12 11 16WAVENUMBER (1000 cm 1)
Fig. 4. Fits of SDL function to absorption coefficients of CH 4 and02 normalized to the ppmv of CO2.
Also shown are the quantum designations of the majortransitions in these neighborhoods.21
To visualize the comparative effects of CH4 and O2with CO2 (see Fig. 3) we have multiplied the CH 4attenuation coefficients by the ratio (ppmv CH4/ppmvC02) and the 02 attenuating coefficient by (ppmv 02/ppmv C0 2). The results shown in Fig. 4 again showthat the analytic representations provide reasonablespectral functions. We have not included N20 or COin our study, since their effects are either outside ourrange of interest or overwhelmed by CO2 absorptionbands.
VI. Discussions and Conclusions
We have concentrated our attention on H2 0, CO2,CH4, and O2, which have the most observable effects inthe visible and IR spectral region of interest (0.35-4.5Mm) for solar irradiance calculations. It should bestraightforward to assign SDL parameters to representthe major features of the other species whose spectralfunctions have been tabulated by Pierluissi and Mara-goudakis (NH 3 , CO, NO, NO2 , N 20, 03, and SO 2).
In the case of H20 the big bands give the overallpattern quite well. However, considerable improve-ments in the band wings are achieved by includinganalytic representations of a number of secondarybands. One could achieve even more precise represen-tations near the band peaks by using two or three SDLfunctions for each single band to match the internal(PQR) structure. It would then be very useful torelate the parameters of these branches either usingfundamental molecular physics or empirical analyses.For lower resolution engineering calculations smooth-ing over these fine structures should be adequate.Then the band centers wp and intensities Ip could be
Table I. Quantum Numers and Parameters for the Big Bands of CH4 and 02
X (,um) V1 V2 V3 V4 W, Wp Ip A1 A, logloS A, Np Mp
Fig. 5. Absorption coefficients vs wavelength for water vapor at 300K and for liquid water.
obtained from formulas for H20 such as Eqs. (9) and(10) and for CO2 from Eqs. (17) and (18) rather than alook-up table. If then a rule is developed for theapparent bandwidth parameters the required data us-ing the quantum mechanical engineering approachwould be compacted to one set per species rather thanone set per band. The LOWTRAN 6 code with its filterfunction could be used to generate the necessary datafor such an engineering representation.
Our focus has been on transmittance along slantpaths from the sun to the ground for the purpose ofspectral irradiance estimation. Equation (13) hasbeen used earlier2 8
31 for calculating transmittance
along finite paths at low resolution for optical thick-nesses leading to significant departures of transmit-tance from unity. We might achieve improved accura-cy by using the transmittance formula 3 1
T = exp -ki(A) Wi/i
[1 + Kiki(X)Wj/gj]1 -A
where Ki is a dimensionless constant for each band.This is a more general relationship which contains theMayer-Goody statistical model as a special case (whenA = '/2). It properly goes over to the thin atmospherelimit T = exp(-kiW/lui) and contains Eq. (13) foroptically thicker paths. It may be possible to readjustthe parameters of Pierluissi et al. and our analyticrepresentations of their logio(C) function to preservethe correspondence with the LOWTRAN 6 results yetalso accommodate the thin atmosphere limit.
With H(X) and all the spectral inputs for T, to T7 inanalytic form direct spectral irradiance calculationsare straightforward and could be accomplished with apersonal computer. Several algorithms would then beavailable9-11 for approximately estimating the diffusespectral irradiance and hence the global spectral irra-diance.
The effects of relative humidity on aerosol scatter-ing and absorption have been considered by Shettleand Fenn.14 In a study of the IR absorption of liquidwater, Green and Mann2 5 have noted a close corre-spondence between the big bands of water vapor andthe big bands of liquid water. This is illustrated in
Fig. 5 using tabulations of Ludwig et al. 32 and Williamset al.33 34 These curves indicate a complex set of over-laps between water vapor bands at 300 K and the liquidwater bands which has not been fully considered instudies of the influence of atmospheric transmissionon scattering by water droplets and vice versa.
It should be obvious from Figs. 2-4 that our analyticrepresentation of the H20, C02, CH4, and 02 spectralfunctions of Pierluissi et al.6 7 portrays reasonably ac-curately the main spectral features of the major IRabsorbing species in the earth's atmosphere. We havealso identified these features with specific sets of quan-tum numbers and molecular transitions which formorderly arrays. It must be conceded that the rapiddevelopment of computing power discounts the com-putational advantages of simple analytic models forengineering-type calculations. Indeed a commercialversion of LOWTRAN is now available for a PC. Never-theless the simplified representations should fosterphysical understanding of the gross features of atmo-spheric radiative transfer, and such physical under-standing can still be as useful as greater computerpower in attaining new knowledge.
The authors would like to thank Jerry Schwartz andShun-Tie Chai for help in the preparation of thismanuscript. This work was supported in part by theOffice of Health and Environmental Research of theU.S. Department of Energy, Corning Glass Works, andthe Florida Natural Gas Association.
Ady Mann is on leave from the Physics Departmentof the Technion-Israel Institute of Technology.
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