1854 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 On Rayleigh Optical Depth Calculations BARRY A. BODHAINE NOAA/Climate Monitoring and Diagnostics Laboratory, Boulder, Colorado NORMAN B. WOOD Cooperative Institute for Research in Environmental Sciences, NOAA/Climate Monitoring and Diagnostics Laboratory, Boulder, Colorado ELLSWORTH G. DUTTON NOAA/Climate Monitoring and Diagnostics Laboratory, Boulder, Colorado JAMES R. SLUSSER Natural Resource Ecology Laboratory, Colorado State University, Fort Collins, Colorado 21 January 1999 and 3 May 1999 ABSTRACT Many different techniques are used for the calculation of Rayleigh optical depth in the atmosphere. In some cases differences among these techniques can be important, especially in the UV region of the spectrum and under clean atmospheric conditions. The authors recommend that the calculation of Rayleigh optical depth be approached by going back to the ®rst principles of Rayleigh scattering theory rather than the variety of curve- ®tting techniques currently in use. A survey of the literature was conducted in order to determine the latest values of the physical constants necessary and to review the methods available for the calculation of Rayleigh optical depth. The recommended approach requires the accurate calculation of the refractive index of air based on the latest published measurements. Calculations estimating Rayleigh optical depth should be done as accurately as possible because the inaccuracies that arise can equal or even exceed other quantities being estimated, such as aerosol optical depth, particularly in the UV region of the spectrum. All of the calculations are simple enough to be done easily in a spreadsheet. 1. Introduction ppm CO2. It is an empirical relationship derived by ®tting the best available experimental data and is de- Modern Rayleigh scattering calculations have tradi- pendent on the composition of air, particularly CO and tionally been made by starting with those presented by 2 water vapor. Next, Penndorf (1957) calculated the Ray- Penndorf (1957). In Penndorf's paper, the refractive in- leigh scattering coef®cient for standard air using the dex of air was calculated using the equation of EdleÂn classic equation that is presented in many textbooks (1953): (e.g., van de Hulst 1957; McCartney 1976): 8 (ns 2 1) 3 10 24p32(n 2 1) 2 6 1 3r s 5 s , (2) 2 949 810 25 540 42 2 2 5 6432.8 11, (1) l Nss(n 1 2)12 6 2 7r 146 2 l22 41 2 l22 where s is the scattering cross section per molecule; Ns where ns is the refractive index of air and l is the wave- is molecular density; the term (6 1 3r)/(6 2 7r)is length of light in micrometers. This equation is for called the depolarization term, F(air), or the King factor; ``standard'' air, which is de®ned as dry air at 760 mm and r is the depolarization factor or depolarization ratio, Hg (1013.25 mb), 158C (288.15 K), and containing 300 which describes the effect of molecular anisotropy. The F(air) term is the least known for purposes of Rayleigh scattering calculations and is responsible for the most Corresponding author address: Barry A. Bodhaine, NOAA/ uncertainty. The depolarization term does not depend CMDL, R/E/CG1, 325 Broadway, Boulder, CO 80303. on temperature and pressure, but does depend on the E-mail: [email protected] gas mixture. Also, Ns depends on temperature and pres- q 1999 American Meteorological Society NOVEMBER 1999 NOTES AND CORRESPONDENCE 1855 sure, but does not depend on the gas mixture. The re- Possible errors in the depolarization term were con- sulting value of s, the scattering cross section per mol- sidered by Hoyt (1977), FroÈhlich and Shaw (1980), and ecule of the gas, calculated from Eq. (2), is independent Young (1980, 1981). The correction proposed by Young of temperature and pressure, but does depend on the (1981) had been accepted for modern Rayleigh scatter- composition of the gas. Note that Ns depends on Avo- ing calculations in atmospheric applications. In brief, gadro's number and the molar volume constant, and is Young (1981) suggested that the value F(air) 5 (6 1 expressed as molecules per cubic centimeter, and that 3r)/(6 2 7r) 5 1.0480 be used rather than the value values for ns and Ns must be expressed at the same 1.0608 used by Penndorf (1957). This effect alone re- 2 temperature and pressure. However, since (ns 2 1)/ duced Rayleigh scattering values by 1.2%; however, it 2 (ns 1 2) is proportional to Ns, the resulting expression cannot be applied over the entire spectrum because for s is independent of temperature and pressure (Mc- F(air) is dependent on wavelength. Furthermore, since Cartney 1976; Bucholtz 1995). Note that the usual ap- the depolarization has been measured for the constitu- 2 proximation ns 1 2 ø 3 was not included in Eq. (2) in ents of air (at least in a relative sense), it is possible in the interest of keeping all calculations as accurate as principle to estimate the depolarization of air as a func- possible. Results of such calculations were presented by tion of composition. Bates (1984) and Bucholtz (1995) Penndorf (1957) in his Table III. It is this table of values discussed the depolarization in detail. It appears that that has been used by many workers in the ®eld to currently the best estimates for (6 1 3r)/(6 2 7r) use estimate Rayleigh optical depths, usually by some the equations given by Bates (1984) for the depolariza- curve-®tting routine over a particular wavelength range tion of N2,O2, Ar, and CO2 as a function of wavelength. of interest. It is therefore possible to calculate the depolarization of Soon after Penndorf's paper was published, EdleÂn air as a function of CO2 concentration. (1966) presented a new formula for estimating the re- Bates (1984) gave a formula for the depolarization fractive index of standard air: of N2 as a function of wavelength as (n 2 1) 3 108 1 s F(N ) 5 1.034 1 3.17 3 1024 , (5) 2 2 2 406 030 15 997 l 5 8342.13 11, (3) 130 2 l22 38.9 2 l22 and for the depolarization of O2 as 1 although the maximum deviation of ns from the 1953 F(O ) 5 1.096 1 1.385 3 1023 formula was given as only 1.4 3 1028. EdleÂn (1953, 2 l2 1966) also discussed the variation of refractive index 1 with temperature and pressure, and also with varying 1 1.448 3 1024 . (6) l4 concentrations of CO2 and water vapor. In light of the EdleÂn (1966) revisions, Owens (1967) presented an in- Furthermore, Bates (1984) recommended that F(air) be depth treatment of the indexes of refractions of dry CO2- calculated using Eqs. (5) and (6), assuming that F(Ar) free air, pure CO , and pure water vapor, and provided 2 5 1.00, F(CO2) 5 1.15, and ignoring the other con- expressions for dependence on temperature, pressure, stituents of air. and composition. However, Owens' (1967) main interest was in temperature and pressure variations, and his anal- ysis does not signi®cantly impact our present work be- 2. Optical depth cause our calculations are performed at the temperature A quantity of fundamental importance in atmospheric and pressure of ``standard'' air. Peck and Reeder (1972) studies is the optical depth (or optical thickness). This further re®ned the currently available data for the re- quantity has been discussed by numerous authors (e.g., fractive index of air and suggested the formula Dutton et al. 1994; Stephens 1994) and is derived from the exponential law of attenuation variously known as (n 2 1) 3 108 s Bouguer's law, Lambert's law, or Beer's law. For pur- 2 480 990 17 455.7 poses of illustration only, Bouguer's law may be simply 5 8060.51 11 (4) 132.274 2 l22 39.329 57 2 l22 written as I(l) 5 I (l) exp[2t(l)/cosu], (7) for the most accuracy over a wide range of wavelengths. 0 Equation (4) is speci®ed for standard air but at the be- where I 0(l) is the extraterrestrial ¯ux at wavelength l, ginning of their paper, Peck and Reeder (1972) specify I(l) is the ¯ux reaching the ground, u is the solar zenith standard air as having 330 ppm CO2. Also, they repeat angle, and t(l) is the optical depth. Clear-sky mea- EdleÂn's (1966) formula, which had clearly de®ned stan- surements of I(l) as a function of u, and plotted as dard air as having 300 ppm CO2, but state that it applies lnI(l) versus secu, should yield a straight line with slope to air having 330 ppm CO2. Here we will use the equa- 2t(l) and intercept I 0 (extrapolated back to secu 5 0). tion of Peck and Reeder (1972) and assume that it holds An excellent example, along with a discussion of this for standard air having 300 ppm CO2. process, is shown by Stephens (1994) in his Fig. 6.1. 1856 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 An important point is that t(l), the total optical depth, plied Young's (1981) correction. Teillet (1990) com- may be composed of several components given by pared the formulations of several authors and found sig- ni®cant differences among them. It is not the purpose t(l) 5 t (l) 1 t (l) 1 t (l), (8) R a g of this paper to survey all of the approximations in use where t R(l) is the Rayleigh optical depth, t a(l) is aero- by various authors nor is it to compare accuracies of sol optical depth, and t g(l) is the optical depth due to the various methods; however, a few examples will absorption by gases such as O3,NO2, and H2O.