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 Ϫ 1) ϫ 10 2432(n Ϫ 1) 2 6 ϩ 3 ϭ s , (2) 2 949 810 25 540 42 2 2 ϭ 6432.8 ϩϩ, (1) Nss(n ϩ 2) 6 Ϫ 7 146 Ϫ Ϫ2 41 Ϫ Ϫ2 where is the scattering cross section per molecule; Ns where ns is the refractive index of air and is the wave- is molecular density; the term (6 ϩ 3)/(6 Ϫ 7)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 is the depolarization factor or depolarization ratio, Hg (1013.25 mb), 15ЊC (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-
᭧ 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 , 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) ϭ (6 ϩ expressed as molecules per cubic centimeter, and that 3)/(6 Ϫ 7) ϭ 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 Ϫ 1)/ duced Rayleigh scattering values by 1.2%; however, it 2 (ns ϩ 2) is proportional to Ns, the resulting expression cannot be applied over the entire spectrum because for 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 ϩ 2 ഠ 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 ϩ 3)/(6 Ϫ 7) 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 Ϫ 1) ϫ 108 1 s F(N ) ϭ 1.034 ϩ 3.17 ϫ 10Ϫ4 , (5) 2 2 2 406 030 15 997 ϭ 8342.13 ϩϩ, (3) 130 Ϫ Ϫ2 38.9 Ϫ Ϫ2 and for the depolarization of O2 as 1 although the maximum deviation of ns from the 1953 F(O ) ϭ 1.096 ϩ 1.385 ϫ 10Ϫ3 formula was given as only 1.4 ϫ 10Ϫ8. EdleÂn (1953, 2 2 1966) also discussed the variation of refractive index 1 with temperature and pressure, and also with varying ϩ 1.448 ϫ 10Ϫ4 . (6) 4 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 ϭ 1.00, F(CO2) ϭ 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 Ϫ 1) ϫ 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 ϭ 8060.51 ϩϩ (4) 132.274 Ϫ Ϫ2 39.329 57 Ϫ Ϫ2 written as I() ϭ I () exp[Ϫ()/cos], (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() is the extraterrestrial ¯ux at wavelength , ginning of their paper, Peck and Reeder (1972) specify I() is the ¯ux reaching the ground, is the solar zenith standard air as having 330 ppm CO2. Also, they repeat angle, and () is the optical depth. Clear-sky mea- EdleÂn's (1966) formula, which had clearly de®ned stan- surements of I() as a function of , and plotted as dard air as having 300 ppm CO2, but state that it applies lnI() versus sec, should yield a straight line with slope to air having 330 ppm CO2. Here we will use the equa- Ϫ() and intercept I 0 (extrapolated back to sec ϭ 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 (), 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 () ϭ () ϩ () ϩ (), (8) R a g of this paper to survey all of the approximations in use where R() is the Rayleigh optical depth, a() is aero- by various authors nor is it to compare accuracies of sol optical depth, and g() is the optical depth due to the various methods; however, a few examples will absorption by gases such as O3,NO2, and H2O. In prin- serve to illustrate some of the dif®culties. The simplest ciple it is possible to measure () and then derive approach, taken by many authors, is to ®t an equation aerosol optical depth by subtracting estimates of R() of the form and g(). In practice, however, arriving at reasonable estimates of these quantities can be dif®cult, particularly () ϭ AϪB, (12) during fairly clean atmospheric conditions such as those found at Mauna Loa, Hawaii. At this point it should be where A and B are constants to be determined from a apparent that in order to isolate the individual compo- power-law ®t and the equation is normalized to 1013.25- nents of optical depth it is necessary to provide accurate mb pressure. An example was given by Dutton et al. estimates of Rayleigh optical depth. (1994), who performed such a ®t over the visible range Rayleigh optical depth is relatively easy to calculate and provided the equation once the scattering cross section per molecule has been p determined for a given wavelength and composition be- Ϫ4.05 R() ϭ 0.008 77 , (13) cause it depends only on the atmospheric pressure at p0 the site. That is, it is necessary to calculate only the total number of molecules per unit area in the column where p is the site pressure, p 0 is 1013.25 mb, and is above the site, and this depends only on the pressure, in micrometers. Clearly, one problem with this approx- as shown in the formula imation is that it cannot be extrapolated to other parts of the spectrum, particularly the UV, where the power- PA law exponent is signi®cantly different. To account for R() ϭ , (9) mga the fact that the exponent changes, some authors (e.g., FroÈhlich and Shaw 1980; Nicolet 1984) used equations where P is the pressure, A is Avogadro's number, m is a of the form the mean molecular weight of the air, and g is the ac- celeration of gravity. Note that ma depends on the com- molecules position of the air, whereas A and g are constants of () ϭ AϪ(BϩCϩDϪ1), (14) R cm2 nature. Although g may be considered a constant of nature, where the term molecules cmϪ2 is calculated from the it does vary signi®cantly with height and location on surface pressure, as explained above. This equation is the earth's surface and may be calculated according to likely to be more accurate over a greater range of the the formula (List 1968) spectrum. A slightly different approach was taken by g (cm sϪ2) Hansen and Travis (1974), who suggested the equation
ϭ g Ϫ (3.085 462 ϫ 10Ϫ4 ϩ 2.27 ϫ 10Ϫ7 cos2)z Ϫ4 Ϫ2 Ϫ4 0 R() ϭ 0.008 569 (1 ϩ 0.0113 ϩ 0.000 13 ), ϩ (7.254 ϫ 10Ϫ11 ϩ 1.0 ϫ 10Ϫ13 cos2)z2 (15)
Ϫ17 Ϫ20 3 Ϫ (1.517 ϫ 10 ϩ 6 ϫ 10 cos2)z , where R() is normalized to 1013.25 mb. As a ®nal (10) example Stephens (1994) suggested the equation
(Ϫ4.15ϩ0.2) (Ϫ0.1188zϪ0.001 16z2) where is the latitude, z is the height above sea level R() ϭ 0.0088 e , (16) in meters, and g 0 is the sea level acceleration of gravity given by where the expression is given in terms of altitude (km) above sea level using the standard atmosphere. The g0 ϭ 980.6160(1 Ϫ 0.002 637 3 cos2 point here is that all of these equations were useful for ϩ 0.000 005 9 cos2 2). (11) the particular authors over a limited wavelength range and at limited accuracy. Comparing these various equa- tions shows signi®cant differences, especially in the UV 3. Approximations for Rayleigh optical depth (Teillet 1990). More importantly, the differences among Many authors have simply taken Rayleigh scattering these equations can be signi®cantly greater than typical cross-section data from Penndorf (1957) over a partic- aerosol optical depths found in the atmosphere. In the ular wavelength interval of interest and applied a curve- case of clean conditions at Mauna Loa, it is possible ®tting routine to approximate the data for their own for aerosol optical depth to be calculated as negative purposes. Some, but not all, of these authors have ap- values because of these errors. NOVEMBER 1999 NOTES AND CORRESPONDENCE 1857
TABLE 1. Constituents and mean molecular weight of dry air. (%Vol ϫ MolWt) m ϭ . (17) Gas % volume Molecular wt % vol ϫ mol wt a (%Vol) N 78.084 28.013 2187.367 2 Note that the error arising from the fact that ⌺ (%Vol) O2 20.946 31.999 670.2511 Ar 0.934 39.948 37.311 43 is not exactly 100 is negligible. Assuming a simple lin- Ϫ3 Ne 1.80 ϫ 10 20.18 0.036 324 ear relationship between ma and CO2 concentration, ma He 5.20 ϫ 10Ϫ4 4.003 0.002 082 may be estimated from the equation ma ϭ 15.0556(CO2) Ϫ4 Kr 1.10 ϫ 10 83.8 0.009 218 Ϫ1 Ϫ5 ϩ 28.9595 gm mol , where CO2 concentration is ex- H2 5.80 ϫ 10 2.016 0.000 117 Xe 9.00 ϫ 10Ϫ6 131.29 0.001 182 pressed as parts per volume (use 0.000 36 for 360 ppm).
CO2 0.036 44.01 1.584 36 We recommend starting with Peck and Reeder's Ϫ1 Mean molecular weight with zero CO2 28.959 49 gm mol (1972) formula for the refractive index of dry air with Ϫ1 Mean molecular weight with 360 ppm CO2 28.964 91 gm mol 300 ppm CO2 concentration: 2 480 990 (n Ϫ 1) ϫ 108 ϭ 8060.51 ϩ 300 132.274 Ϫ Ϫ2 4. Suggested method to calculate Rayleigh optical 17 455.7 depth of air ϩ , (18) 39.329 57 Ϫ Ϫ2 Here we suggest a method for calculation of Rayleigh and scaling for the desired CO concentration using the optical depth that goes back to ®rst principles as sug- 2 formula gested by Penndorf (1957) rather than using curve-®t- ting techniques, although it is true that the refractive (n Ϫ 1) CO2 ϭ 1 ϩ 0.54(CO Ϫ 0.0003), (19) index of air is still derived from a curve ®t to experi- (n Ϫ 1) 2 mental data. We suggest using all of the latest values 300 of the physical constants of nature, and we suggest in- where the CO2 concentration is expressed as parts per cluding the variability in refractive index, and also the volume (EdleÂn 1966). Thus the refractive index for dry air with zero ppm CO is mean molecular weight of air, due to CO2 even though 2 these effects are in the range of 0.1%±0.01%. It should 2 480 588 (n Ϫ 1) ϫ 108 ϭ 8059.20 ϩ be noted that aerosol optical depths are often as low as 0 Ϫ2 0.01 at Mauna Loa. Since Rayleigh optical depth is of 132.274 Ϫ the order of 1 at 300 nm, it is seen that a 0.1% error 17 452.9 ϩ , (20) in Rayleigh optical depth translates into a 10% error in 39.329 57 Ϫ Ϫ2 aerosol optical depth. Furthermore, it simply makes sense to perform the calculations as accurately as pos- and the refractive index for dry air with 360 ppm CO2 sible. is We should note that the effects of high concentrations 2 481 070 8 of water vapor on the refractive index of air may be of (n360 Ϫ 1) ϫ 10 ϭ 8060.77 ϩ 132.274 Ϫ Ϫ2 the same order as CO2 (EdleÂn 1953, 1966). However, for practical atmospheric situations the total water vapor 17 456.3 ϩ , (21) in the vertical column is small and does not signi®cantly 39.329 57 Ϫ Ϫ2 affect the above calculations. Furthermore, the water vapor in the atmosphere is usually con®ned to a thin where it must be emphasized that Eqs. (18)±(21) are layer near the surface, which signi®cantly complicates given for 288.15 K and 1013.25 mb, and in units of the calculation, whereas CO is generally well mixed micrometers. 2 2 throughout the atmosphere. We recommend that the scattering cross section (cm Ϫ1 To facilitate the following Rayleigh optical depth cal- molecule ) of air be calculated from the equation culations, the latest values of Avogadro's number 24 32(n Ϫ 1) 2 6 ϩ 3 (6.022 136 7 ϫ 1023 molecules molϪ1), and molar vol- ϭ , (22) 42 2 2 ume at 273.15 K and 1013.25 mb (22.4141 L molϪ1) Ns (n ϩ 2) 6 Ϫ 7 were taken from Cohen and Taylor (1995). In order to where n is the refractive index of air at the desired CO2 calculate the mean molecular weight of dry air with concentration, is expressed in units of centimeters, Ns 19 Ϫ3 various concentrations of CO2, the percent by volume ϭ 2.546 899 ϫ 10 molecules cm at 288.15 K and of the constituent gases in air were taken from Seinfeld 1013.25 mb, and the depolarization ratio is calculated and Pandis (1998), and the molecular weights of those as follows. Using the values for depolarization of the gases were taken from the Handbook of Physics and gases O2,N2, Ar, and CO2 provided by Bates (1984), Chemistry (CRC 1997). These results are shown in Table we recommend that the depolarization of dry air be
1. The mean molecular weights (ma) for dry air were calculated using Eqs. (5)±(6) and the following equation calculated from the formula to take into account the composition of air: 1858 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16
78.084F(N22 ) ϩ 20.946F(O ) ϩ 0.934 ϫ 1.00 ϩ C CO2 ϫ 1.15 F(air, CO2 ) ϭ , (23) 78.084 ϩ 20.946 ϩ 0.934 ϩ CCO2
whereCCO2 is the concentration of CO2 expressed in in the table. Next a least squares straight line was passed parts per volume by percent (e.g., use 0.036 for 360 through the resulting zc values up to 10 500 m, giving ppm). The results of Eq. (23) for standard air (300 ppm the following equation: CO ) are shown in Fig. 1. Note that the value of N in 2 s z ϭ 0.737 37z ϩ 5517.56, (26) Eq. (22) was calculated from Avogadro's number and c
the molar volume, and then scaled to 288.15 K accord- where z is the altitude of the observing site and zc is ing to the formula the effective mass-weighted altitude of the column. For
Ϫ3 example, an altitude of z ϭ 0 m yields an effective mass- Ns(molecules cm ) weighted column altitude of zc ϭ 5517.56 m to use in 23 1 6.022 136 7 ϫ 10 molecules molϪ 273.15 K the calculation of g. The resulting values or R should ϭ be considered the best currently available values for the 22.4141 L molϪ1 288.15 K most accurate estimates of optical depth. 1L ϫ . (24) 1000 cm3 5. Optical depths of the constituents of air Finally, we recommend that the Rayleigh optical depth As a sensitivity study, the contribution of CO2 to the be calculated from the formula Rayleigh optical depth of air may be estimated as a PA function of wavelength by using the above formulas R() ϭ , (25) expressed for CO2. Owens (1967) gives the refractive mga index of CO2 at 15ЊC and 1013.25 mb as
where P is the surface pressure of the measurement site 8 Ϫ2 Ϫ2 (nCO2 Ϫ 1) ϫ 10 ϭ 22 822.1 ϩ 117.8 (dyn cm ), A is Avogadro's number, and ma is the mean molecular weight of dry air calculated from the formula 2 406 030 15 997 m ϭ 15.0556(CO ) ϩ 28.9595, as in Eq. (17). The ϩϩ, (27) a 2 130 Ϫ Ϫ2 38.9 Ϫ Ϫ2 value for g needs to be representative of the mass- weighted column of air molecules above the site, and where is expressed in units of micrometers as before.
should be calculated from Eqs. (10)±(11), modi®ed by Next the scattering cross section of a CO2 molecule can using a value of zc determined from the U.S. Standard be calculated from Eq. (22), where Ns ϭ 2.546 899 ϫ Atmosphere, as provided by List (1968). To determine 1019 molecules cmϪ3 at 288.15 K and 1013.25 mb as
zc we used List's (1968, p. 267) table of the density of before, and the King factor F(CO2) taken to be 1.15, air as a function of altitude and calculated a mass- as suggested by Bates (1984). Finally (CO2, ) can be weighted mean above each altitude value, using an av- calculated using Eq. (25), where m ϭ 44.01 (the mo-
erage altitude and average density for each layer listed lecular weight of CO2), and multiplying by 0.000 36 (for a CO2 concentration of 360 ppm) to estimate the number of CO2 molecules. Note that (H2O, ) for 44 kg mϪ2 column water vapor was calculated in a similar
manner using the refractive index of H2O given by Har- vey et al. (1998) and a depolarization ratio of 0.17 for
H2O given by Marshall and Smith (1990). The results of these calculations are shown in Table 2, where the
change of optical depth ⌬(H2O) is given for the case where dry air molecules are replaced by H2O molecules for 44 kg mϪ2 column water vapor. Thus for a Rayleigh
optical depth of 1.224 for air at 300 nm, a CO2 con-
TABLE 2. Optical depths of the constituents of air (standard pressure 1013.25 mb and altitude 0 m).
(nm) (N2 ϩ O2 ϩ Ar) (CO2) (H2O) ⌬ (H2O) 300 1.2232 0.001 031 0.008 654 0.000 189 5 340 0.7163 0.000 605 8 0.004 928 Ϫ0.000 029 01 370 0.5015 0.000 424 8 0.003 514 0.000 043 45 FIG. 1. Depolarization factor for dry air with 300 ppm CO2. NOVEMBER 1999 NOTES AND CORRESPONDENCE 1859 centration of 360 ppm would contribute an optical depth about 0.001.
6. Some example calculations Using the above equations we now present example calculations to show new values for the scattering cross section (as a function of wavelength) of dry air con- taining 360 ppm CO2, similar to the presentations of FIG. 2. Percent error for Eq. (29) ®t to the scattering cross-section Penndorf (1957) and Bucholtz (1995). In addition we data in Table 3. present new values for Rayleigh optical depth for dry air containing 360 ppm CO2 at sea level, 1013.25 mb, and a latitude of 45Њ; and at Mauna Loa Observatory ®t in this case because the general form of the data being (MLO) (altitude 3400 m, pressure 680 mb, and a latitude ®t by the equation is also a ratio of polynomials. For of 19.533Њ). The results of these calculations are shown the scattering cross-section data in Table 3 the best-®t in Table 3. equation is For those readers who wish to use curve-®tting tech- (ϫ10Ϫ28 cm 2 ) niques, we have investigated several different equations similar to those used by other authors, as discussed ear- 1.045 599 6 Ϫ 341.290 61Ϫ22Ϫ 0.902 308 50 ϭ . lier in this paper. We found that the accuracies of those 1 ϩ 0.002 705 988 9Ϫ22Ϫ 85.968 563 equations were not suf®cient for our purposes, and therefore we looked for a more accurate approach. We (29) ®nd that the equation Equation (29) is accurate to better than 0.01% over the a ϩ bϪ22ϩ c 250±850-nm range, and still better than 0.05% out to y ϭ (28) Ϫ22 1000 nm when ®tting the scattering cross-section data 1 ϩ d ϩ e in Table 3. In fact, this equation is accurate to better gives excellent accuracy. This ®ve-parameter equation than 0.002% over the range 250±550 nm (see Fig. 2). falls in the class of ``ratio of polynomials'' commonly Best ®ts for the other two examples provided in Table used in curve-®tting applications. It gives an excellent 3 are
1.045 599 6 Ϫ 341.290 61Ϫ22Ϫ 0.902 308 50 (sea level, 45ЊN) ϭ 0.002 152 0 and (30) R 1 ϩ 0.002 705 988 9Ϫ22Ϫ 85.968 563 1.045 599 6 Ϫ 341.290 61Ϫ22Ϫ 0.902 308 50 (MLO, 3.4 km, 680 mb) ϭ 0.001 448 4 . (31) R 1 ϩ 0.002 705 988 9Ϫ22Ϫ 85.968 563
It should be noted that only the leading coef®cients in used to calculate the scattering cross section per mol- Eqs. (30)±(31) are different from those in Eq. (29). ecule of air, taking into account the concentration of
These numerical values represent the values of the PA/ CO2. In most cases the effects of water vapor may be mag terms (molecules in the column) given in Eq. (25) neglected. The recommendations of Bates (1984) were for the two cases (taking into account the 10Ϫ28 factor used for the depolarization of air as a function of wave- that was removed from the scattering cross section data length. Next the Rayleigh optical depth should be cal- to facilitate curve ®tting calculations). culated using the atmospheric pressure at the site of interest. Note the importance of taking into account var- iations of g. We do not necessarily recommend the use 7. Conclusions of curve-®tting techniques to generate an equation for We have presented the latest values of the physical estimating Rayleigh optical depth because the inaccu- constants necessary for the calculation of Rayleigh op- racies that arise can equal or even exceed other quan- tical depth. For the most accurate calculation of this tities being estimated, such as aerosol optical depth. quantity it is recommended that users go directly to ®rst Furthermore, all of the above calculations are simple principles and that Peck and Reeder's (1972) formula enough to be done in a spreadsheet if desired, or can be used to estimate the refractive index of standard air. easily be programmed in virtually any computer lan- Next, we recommend that Penndorf's (1957) method be guage. However, for those who wish to use a simple 1860 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16
TABLE 3. Scattering cross section (per molecule) and Rayleigh op- TABLE 3. (Continued ) tical depth (R) for dry air containing 360 ppm CO2. Rayleigh optical depths are given for a location at sea level, 1013.25 mb, 45Њ latitude, R R and at MLO at altitude 3400 m, pressure 680 mb, and latitude 19.533Њ. WV (sea level, (MLO, (m) (cm2) 45ЊN) 680 mb) King factor
R R WV (sea level, (MLO, 0.560 4.1908EϪ27 9.0188EϪ02 6.0698EϪ02 1.04873 0.565 4.0416EϪ27 8.6977EϪ02 5.8537EϪ02 1.04869 (m) (cm2) 45ЊN) 680 mb) King factor 0.570 3.8990EϪ27 8.3908EϪ02 5.6472EϪ02 1.04865 0.250 1.2610EϪ25 2.7137Eϩ00 1.8264Eϩ00 1.06308 0.575 3.7627EϪ27 8.0974EϪ02 5.4497EϪ02 1.04861 0.255 1.1537EϪ25 2.4828Eϩ00 1.6710Eϩ00 1.06215 0.580 3.6323EϪ27 7.8168EϪ02 5.2608EϪ02 1.04858 0.260 1.0579EϪ25 2.2766Eϩ00 1.5322Eϩ00 1.06130 0.585 3.5075EϪ27 7.5483EϪ02 5.0801EϪ02 1.04854 0.265 9.7206EϪ26 2.0919Eϩ00 1.4079Eϩ00 1.06052 0.590 3.3880EϪ27 7.2912EϪ02 4.9071EϪ02 1.04851 0.270 8.9496EϪ26 1.9260Eϩ00 1.2962Eϩ00 1.05979 0.595 3.2736EϪ27 7.0450EϪ02 4.7414EϪ02 1.04847 0.275 8.2552EϪ26 1.7766Eϩ00 1.1956Eϩ00 1.05912 0.600 3.1640EϪ27 6.8092EϪ02 4.5827EϪ02 1.04844 0.280 7.6282EϪ26 1.6416Eϩ00 1.1048Eϩ00 1.05850 0.605 3.0590EϪ27 6.5831EϪ02 4.4305EϪ02 1.04841 0.285 7.0608EϪ26 1.5195Eϩ00 1.0227Eϩ00 1.05793 0.610 2.9583EϪ27 6.3664EϪ02 4.2847EϪ02 1.04837 0.290 6.5462EϪ26 1.4088Eϩ00 9.4813EϪ01 1.05739 0.615 2.8618EϪ27 6.1586EϪ02 4.1448EϪ02 1.04834 0.295 6.0785EϪ26 1.3081Eϩ00 8.8038EϪ01 1.05689 0.620 2.7691EϪ27 5.9592EϪ02 4.0107EϪ02 1.04831 0.300 5.6525EϪ26 1.2164Eϩ00 8.1868EϪ01 1.05643 0.625 2.6802EϪ27 5.7679EϪ02 3.8819EϪ02 1.04829 0.305 5.2638EϪ26 1.1328Eϩ00 7.6238EϪ01 1.05599 0.630 2.5948EϪ27 5.5842EϪ02 3.7582EϪ02 1.04826 0.310 4.9084EϪ26 1.0563Eϩ00 7.1092EϪ01 1.05559 0.635 2.5128EϪ27 5.4078EϪ02 3.6395EϪ02 1.04823 0.315 4.5830EϪ26 9.8629EϪ01 6.6379EϪ01 1.05521 0.640 2.4341EϪ27 5.2383EϪ02 3.5254EϪ02 1.04820 0.320 4.2846EϪ26 9.2206EϪ01 6.2056EϪ01 1.05485 0.645 2.3584EϪ27 5.0754EϪ02 3.4158EϪ02 1.04818 0.325 4.0103EϪ26 8.6304EϪ01 5.8084EϪ01 1.05452 0.650 2.2856EϪ27 4.9188EϪ02 3.3104EϪ02 1.04815 0.330 3.7580EϪ26 8.0873EϪ01 5.4429EϪ01 1.05421 0.655 2.2157EϪ27 4.7682EϪ02 3.2091EϪ02 1.04813 0.335 3.5254EϪ26 7.5868EϪ01 5.1060EϪ01 1.05391 0.660 2.1484EϪ27 4.6234EϪ02 3.1116EϪ02 1.04810 0.340 3.3108EϪ26 7.1249EϪ01 4.7952EϪ01 1.05363 0.665 2.0836EϪ27 4.4840EϪ02 3.0178EϪ02 1.04808 0.345 3.1124EϪ26 6.6981EϪ01 4.5079EϪ01 1.05337 0.670 2.0213EϪ27 4.3498EϪ02 2.9275EϪ02 1.04806 0.350 2.9289EϪ26 6.3031EϪ01 4.2421EϪ01 1.05312 0.675 1.9612EϪ27 4.2207EϪ02 2.8406EϪ02 1.04804 0.355 2.7589EϪ26 5.9372EϪ01 3.9958EϪ01 1.05289 0.680 1.9035EϪ27 4.0963EϪ02 2.7569EϪ02 1.04802 0.360 2.6011EϪ26 5.5977EϪ01 3.7673EϪ01 1.05266 0.685 1.8478EϪ27 3.9765EϪ02 2.6762EϪ02 1.04799 0.365 2.4546EϪ26 5.2824EϪ01 3.5551EϪ01 1.05246 0.690 1.7941EϪ27 3.8610EϪ02 2.5985EϪ02 1.04797 0.370 2.3183EϪ26 4.9892EϪ01 3.3578EϪ01 1.05226 0.695 1.7424EϪ27 3.7497EϪ02 2.5236EϪ02 1.04795 0.375 2.1915EϪ26 4.7162EϪ01 3.1741EϪ01 1.05207 0.700 1.6926EϪ27 3.6424EϪ02 2.4514EϪ02 1.04793 0.380 2.0733EϪ26 4.4619EϪ01 3.0029EϪ01 1.05189 0.705 1.6445EϪ27 3.5390EϪ02 2.3818EϪ02 1.04792 0.385 1.9631EϪ26 4.2246EϪ01 2.8432EϪ01 1.05172 0.710 1.5981EϪ27 3.4392EϪ02 2.3146EϪ02 1.04790 0.390 1.8601EϪ26 4.0030EϪ01 2.6941EϪ01 1.05156 0.715 1.5533EϪ27 3.3428EϪ02 2.2498EϪ02 1.04788 0.395 1.7639EϪ26 3.7959EϪ01 2.5547EϪ01 1.05140 0.720 1.5101EϪ27 3.2499EϪ02 2.1872EϪ02 1.04786 0.400 1.6738EϪ26 3.6022EϪ01 2.4243EϪ01 1.05126 0.725 1.4684EϪ27 3.1601EϪ02 2.1268EϪ02 1.04784 0.405 1.5895EϪ26 3.4207EϪ01 2.3022EϪ01 1.05112 0.730 1.4282EϪ27 3.0735EϪ02 2.0685EϪ02 1.04783 0.410 1.5105EϪ26 3.2506EϪ01 2.1877EϪ01 1.05098 0.735 1.3893EϪ27 2.9898EϪ02 2.0122EϪ02 1.04781 0.415 1.4363EϪ26 3.0910EϪ01 2.0803EϪ01 1.05085 0.740 1.3517EϪ27 2.9089EϪ02 1.9577EϪ02 1.04779 0.420 1.3667EϪ26 2.9412EϪ01 1.9795EϪ01 1.05073 0.745 1.3154EϪ27 2.8308EϪ02 1.9051EϪ02 1.04778 0.425 1.3013EϪ26 2.8004EϪ01 1.8847EϪ01 1.05062 0.750 1.2803EϪ27 2.7552EϪ02 1.8543EϪ02 1.04776 0.430 1.2398EϪ26 2.6680EϪ01 1.7956EϪ01 1.05051 0.755 1.2463EϪ27 2.6822EϪ02 1.8052EϪ02 1.04775 0.435 1.1819EϪ26 2.5434EϪ01 1.7118EϪ01 1.05040 0.760 1.2135EϪ27 2.6116EϪ02 1.7576EϪ02 1.04773 0.440 1.1273EϪ26 2.4261EϪ01 1.6328EϪ01 1.05030 0.765 1.1818EϪ27 2.5433EϪ02 1.7117EϪ02 1.04772 0.445 1.0759EϪ26 2.3154EϪ01 1.5583EϪ01 1.05020 0.770 1.1511EϪ27 2.4772EϪ02 1.6672EϪ02 1.04770 0.450 1.0274EϪ26 2.2111EϪ01 1.4881EϪ01 1.05011 0.775 1.1214EϪ27 2.4132EϪ02 1.6241EϪ02 1.04769 0.455 9.8165EϪ27 2.1126EϪ01 1.4218EϪ01 1.05002 0.780 1.0926EϪ27 2.3514EϪ02 1.5825EϪ02 1.04768 0.460 9.3841EϪ27 2.0195EϪ01 1.3592EϪ01 1.04993 0.785 1.0648EϪ27 2.2914EϪ02 1.5422EϪ02 1.04766 0.465 8.9755EϪ27 1.9316EϪ01 1.3000EϪ01 1.04985 0.790 1.0378EϪ27 2.2334EϪ02 1.5031EϪ02 1.04765 0.470 8.5889EϪ27 1.8484EϪ01 1.2440EϪ01 1.04977 0.795 1.0117EϪ27 2.1772EϪ02 1.4653EϪ02 1.04764 0.475 8.2231EϪ27 1.7696EϪ01 1.1910EϪ01 1.04969 0.800 9.8642EϪ28 2.1228EϪ02 1.4287EϪ02 1.04762 0.480 7.8766EϪ27 1.6951EϪ01 1.1408EϪ01 1.04962 0.805 9.6191EϪ28 2.0701EϪ02 1.3932EϪ02 1.04761 0.485 7.5483EϪ27 1.6244EϪ01 1.0933EϪ01 1.04954 0.810 9.3817EϪ28 2.0190EϪ02 1.3588EϪ02 1.04760 0.490 7.2370EϪ27 1.5574EϪ01 1.0482EϪ01 1.04948 0.815 9.1515EϪ28 1.9694EϪ02 1.3255EϪ02 1.04759 0.495 6.9417EϪ27 1.4939EϪ01 1.0054EϪ01 1.04941 0.820 8.9283EϪ28 1.9214EϪ02 1.2931EϪ02 1.04758 0.500 6.6614EϪ27 1.4336EϪ01 9.6480EϪ02 1.04935 0.825 8.7120EϪ28 1.8749EϪ02 1.2618EϪ02 1.04757 0.505 6.3951EϪ27 1.3763EϪ01 9.2624EϪ02 1.04929 0.830 8.5021EϪ28 1.8297EϪ02 1.2314EϪ02 1.04756 0.510 6.1421EϪ27 1.3218EϪ01 8.8959EϪ02 1.04923 0.835 8.2986EϪ28 1.7859EϪ02 1.2019EϪ02 1.04754 0.515 5.9015EϪ27 1.2700EϪ01 8.5475EϪ02 1.04917 0.840 8.1011EϪ28 1.7434EϪ02 1.1733EϪ02 1.04753 0.520 5.6727EϪ27 1.2208EϪ01 8.2161EϪ02 1.04911 0.845 7.9095EϪ28 1.7022EϪ02 1.1456EϪ02 1.04752 0.525 5.4549EϪ27 1.1739EϪ01 7.9006EϪ02 1.04906 0.850 7.7235EϪ28 1.6621EϪ02 1.1186EϪ02 1.04751 0.530 5.2475EϪ27 1.1293EϪ01 7.6002EϪ02 1.04901 0.855 7.5429EϪ28 1.6233EϪ02 1.0925EϪ02 1.04750 0.535 5.0499EϪ27 1.0868EϪ01 7.3140EϪ02 1.04896 0.860 7.3676EϪ28 1.5855EϪ02 1.0671EϪ02 1.04749 0.540 4.8615EϪ27 1.0462EϪ01 7.0412EϪ02 1.04891 0.865 7.1974EϪ28 1.5489EϪ02 1.0424EϪ02 1.04748 0.545 4.6819EϪ27 1.0076EϪ01 6.7811EϪ02 1.04886 0.870 7.0321EϪ28 1.5133EϪ02 1.0185EϪ02 1.04747 0.550 4.5105EϪ27 9.7069EϪ02 6.5329EϪ02 1.04882 0.875 6.8715EϪ28 1.4788EϪ02 9.9523EϪ03 1.04746 0.555 4.3470EϪ27 9.3549EϪ02 6.2960EϪ02 1.04878 0.880 6.7154EϪ28 1.4452EϪ02 9.7263EϪ03 1.04746 NOVEMBER 1999 NOTES AND CORRESPONDENCE 1861
TABLE 3. (Continued ) Mean molecular weight of dry air (360 ppm CO2) ϭ 28.9649 gm molϪ1 R R WV (sea level, (MLO, Acceleration of gravity (sea level and 45Њ latitude) 2 Ϫ2 (m) (cm ) 45ЊN) 680 mb) King factor g 0(45Њ) ϭ 980.6160 cm s 0.885 6.5638EϪ28 1.4126EϪ02 9.5067EϪ03 1.04745 Mass-weighted air column altitude zc ϭ 0.737 37z ϩ 0.900 6.1339EϪ28 1.3200EϪ02 8.8841EϪ03 1.04742 5517.56 0.905 5.9985EϪ28 1.2909EϪ02 8.6880EϪ03 1.04741 0.910 5.8668EϪ28 1.2626EϪ02 8.4972EϪ03 1.04740 0.915 5.7387EϪ28 1.2350EϪ02 8.3117EϪ03 1.04740 REFERENCES 0.920 5.6141EϪ28 1.2082EϪ02 8.1312EϪ03 1.04739 0.925 5.4929EϪ28 1.1821EϪ02 7.9556EϪ03 1.04738 0.930 5.3749EϪ28 1.1567EϪ02 7.7848EϪ03 1.04737 Bates, D. R., 1984: Rayleigh scattering by air. Planet. Space Sci., 32, 0.935 5.2601EϪ28 1.1320EϪ02 7.6184EϪ03 1.04737 785±790. 0.940 5.1483EϪ28 1.1079EϪ02 7.4566EϪ03 1.04736 Bucholtz, A., 1995: Rayleigh-scattering calculations for the terrestrial 0.945 5.0395EϪ28 1.0845EϪ02 7.2989EϪ03 1.04735 atmosphere. Appl. Opt., 34, 2765±2773. 0.950 4.9335EϪ28 1.0617EϪ02 7.1455EϪ03 1.04734 Cohen, E. R., and B. N. Taylor, 1995: The fundamental physical 0.955 4.8303EϪ28 1.0395EϪ02 6.9960EϪ03 1.04734 constants. Phys. Today, 48, 9±16. 0.960 4.7298EϪ28 1.0179EϪ02 6.8505EϪ03 1.04733 CRC, 1997: Handbook of Chemistry and Physics. D. R. Lide and H. 0.965 4.6319EϪ28 9.9682EϪ03 6.7087EϪ03 1.04732 P. R. Frederikse, Eds., CRC Press, 2447 pp. 0.970 4.5366EϪ28 9.7629EϪ03 6.5706EϪ03 1.04732 Dutton, E. G., P. Reddy, S. Ryan, and J. J. DeLuisi, 1994: Features 0.975 4.4437EϪ28 9.5630EϪ03 6.4360EϪ03 1.04731 and effects of aerosol optical depth observed at Mauna Loa, 0.980 4.3531EϪ28 9.3681EϪ03 6.3049EϪ03 1.04730 Hawaii: 1982±1992. J. Geophys. Res., 99, 8295±8306. 0.985 4.2649EϪ28 9.1782EϪ03 6.1770EϪ03 1.04730 EdleÂn, B., 1953: The dispersion of standard air. J. Opt. Soc. Amer., 0.990 4.1788EϪ28 8.9930EϪ03 6.0524EϪ03 1.04729 43, 339±344. 0.995 4.0950EϪ28 8.8125EϪ03 5.9310EϪ03 1.04728 , 1966: The refractive index of air. Metrologia, 2, 71±80. 1.000 4.0132EϪ28 8.6366EϪ03 5.8125EϪ03 1.04728 FroÈhlich, C., and G. E. Shaw, 1980: New determination of Rayleigh scattering in the terrestrial atmosphere. Appl. Opt., 19, 1773± 1775. Hansen, J. E., and L. D. Travis, 1974: Light scattering in planetary equation and are satis®ed with less accuracy, the tech- atmospheres. Space Sci. Rev., 16, 527±610. niques used to produce Eqs. (29)±(31) may be of in- Harvey, A. H., J. S. Gallagher, and J. M. H. Levelt Sengers, 1998: terest. As more accurate estimates of the various pa- Revised formulation for the refractive index of water and steam as a function of wavelength, temperature and density. J. Phys. rameters discussed above become available, the equa- Chem. Ref. Data, 27, 761±774. tions of interest may easily be modi®ed. Hoyt, D. V., 1977: A redetermination of the Rayleigh optical depth In some calculations of optical depth it may be desired and its application to selected solar radiation problems. J. Appl. to take into account the vertical distribution of the com- Meteor., 16, 432±436. position of air, particularly CO . In this case a layer- List, R. J., 1968: Smithsonian Meteorological Tables. Smithsonian, 2 527 pp. by-layer calculation may be done using the estimated Marshall, B. R., and R. C. Smith, 1990: Raman scattering and in- composition for each layer, and then the total optical water ocean optical properties. Appl. Opt., 29, 71±84. depth may be estimated by summing the optical depths McCartney, E. J., 1976: Optics of the Atmosphere. Wiley, 408 pp. for all of the layers. Nicolet, M., 1984: On the molecular scattering in the terrestrial at- mosphere: An empirical formula for its calculation in the hom- osphere. Planet. Space Sci., 32, 1467±1468. Acknowledgments. We thank Gail Anderson for her Owens, J. C., 1967: Optical refractive index of air: Dependence on helpful comments concerning curve-®tting techniques. pressure, temperature and composition. Appl. Opt., 6, 51±59. Peck, E. R., and K. Reeder, 1972: Dispersion of air. J. Opt. Soc. Amer., 62, 958±962. APPENDIX Penndorf, R., 1957: Tables of the refractive index for standard air and the Rayleigh scattering coef®cient for the spectral region Summary of Constants between 0.2 and 20.0 and their application to atmospheric Values for the constants of nature that have been used optics. J. Opt. Soc. Amer., 47, 176±182. Seinfeld, J. H., and S. N. Pandis, 1998: Atmospheric Chemistry and in this paper are listed below. Physics, from Air Pollution to Climate Change. Wiley, 1326 pp. Avogadro's number ϭ 6.022 136 7 ϫ 1023 molecules Stephens, G. L., 1994: Remote Sensing of the Lower Atmosphere. Ϫ1 Oxford University Press, 523 pp. mol Teillet, P. M., 1990: Rayleigh optical depth comparisons from various Molar volume at 273.15 K and 1013.25 mb ϭ sources. Appl. Opt., 29, 1897±1900. 22.4141 L molϪ1 van de Hulst, H. C., 1957: Light Scattering by Small Particles. Wiley, Molecular density of a gas at 288.15 K and 1013.25 470 pp. 19 Ϫ3 Young, A. T., 1980: Revised depolarization corrections for atmo- mb ϭ 2.546 899 ϫ 10 molecules cm spheric extinction. Appl. Opt., 19, 3427±3428. Mean molecular weight of dry air (zero CO2) ϭ , 1981: On the Rayleigh-scattering optical depth of the atmo- 28.9595 gm molϪ1 sphere. J. Appl. Meteor., 20, 328±330.