THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aerospace Engineering High Altitude Engineering Laboratory Final Report THEORETICAL INVESTIGATIONS OF CARBON DIOXIDE RADIATIVE TRANSFER S. Roland Dravson and Charles Young ORA Project 07349 under contract with: U. S. DEPARTMENT OF COMMERCE WEATHER BUREAU CONTRACT NO. Cwb-11106 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1966

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TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii ABSTRACT ix 1. INTRODUCTION 1 2. CALCULATION OF ROTATIONAL LINE POSITIONS AND INTENSITIES 2 2.1 Calculation of Rotational Line Positions 2 2.1.1 The carbon dioxide molecule 2 2.1.2 Rotational line positions 7 2.2. Calculation of Rotational Line Intensities 14 3. DIRECT INTEGRATION TECHNIQUES 20 3.1 Introduction 20 3.2 Homogeneous Paths 22 3.3 Slant Paths 24 3.4 Programming Techniques 26 4. HOMOGENEOUS PATH TRANSMISSIVITIES 32 4.1 Calculation of Rotational Line Positions and Intensities 32 4.2 Half-Widths and Line Shapes 36 4.3 Comparison of Calculated and Experimental Transmissivities 37 5. SLANT PATH TRANSMISSIVITY 45 6. CONCLUSIONS 52 ACKNOWLEDGMENTS 55 REFERENCES 54 iii

LIST OF TABLES Table Page 2.1 Vibrational energy levels for carbon dioxide in the 12- to 18-micron region 8 2.2 Rotational constants for carbon dioxide in the 12- to 18 -micron region 11 4.1 Band intensities used in calculating rotational line intensities 33 4.2 Integrated intensity of the 15-micron CO2 bands 34 4.3 Comparison of calculated and experimental equivalent widths 38 5.1 Vertical path transmissivity from 10.0 mb for SIRS 48 5.2 Vertical path transmissivity from 10.0 mb for SIRS 48 5.3 Vertical components of radiance at 10 mb for U. S. Weather Bureau Texas balloon flight 51 V

LIST OF FIGURES Figure Page 2.1. Vibrations of the carbon dioxide molecule and changes in electric moment. 3 4.1. Comparison of calculated absorption with experimental values of Burch et al. (l962). 39 4.2. Comparison of calculated and experimental absorption at 0.39 mm Hg. 43 5.1. Temperature structure for balloon flight. 46 5.2. Comparison of slant path transmission for U. S. Weather Bureau balloon flight, Palestine, Texas, September, 1964. 47 vii

Abstract Rotational line positions and intensities for the 15-micron carbon dioxide bands have been calculated, taking into account the important carbon dioxide isotopes. This information has been used to calculate transmissivities for both homogeneous and atmospheric paths. In both cases the results were obtained by direct integration with respect to frequency. The homogeneous path calculations were compared with laboratory measurements and conclusions drawn concerning band strengths. The atmospheric path calculations were compared with measurements made by the SIRS instrument. ix

1. Introduction The need for accurate transmissivity calculations for atmospheric paths in the 15-micron spectral region, where most of the absorption is due to the carbon dioxide molecule, has become very important in recent years due to the interest in atmospheric probing using infrared radiation intercepted by satellites and for radiative transport calculations in planetary atmospheres.-t In order to calculate atmospheric transmissivities using the direct integration technique it is necessary to have available good estimates of certain basic molecular parameters for carbon dioxide; unfortunately a number of these parameters have contradictory values assigned to them. One of the main purposes of this research is to decide which values should be used and also to develop methods for calculating transmissivities for homogeneous and atmospheric paths by direct integration with respect to frequency.

2. Calculation of Rotational Line Positions and Intensities 2. 1 Calculation of Rotational Line Positions 2. 1. 1 The Carbon Dioxide Molecule Since carbon dioxide is a linear symmetric molecule, the theoretical evaluation of certain basic parameters is somewhat simplified. The infrared optical activity of a molecule depends on the change in its electric moment. Figure 2. 1 shows the modes of vibration of the carbon dioxide molecule. In mode vl there is no change in the electric moment and consequently no vibration band corresponding to this mode. The v1 fundamental vibration may, however, be studied using the Raman effect. There are also no pure rotation lines for carbon dioxide since on rotation there will be no change in the dipole moment. The v2 vibration is the degenerate representation of two equal frequencies and is centered about 667. 4 cm (approximately 15- microns). This mode is sometimes called the bending mode. Since the v2 band has a strong Q-branch we know that the change in electric moment is perpendicular to the axis of symmetry. The fundamental associated with the v3 mode is centered about 2349. 2 cm (approximately 4. 3 microns). The Q-branch associated with this band is very weak indicating that the change in electric moment is parallel to the axis of symmetry. This mode is sometimes called the valence mode. The above discussion applies to the symmetric isotopes such as 12 16 18 12C 16, 13 160; if we consider isotopes such as C 0 0 or C 02 1 C 0 2 12C 160 170 then we have destroyed some of the symmetry and we will see later how this influences our calculations. 2

Change in Electric Moment Oxygen Carbon Oxygen Mode None.ii oin -o 0 o out Q0 -o in o'' /I V2 II 3 Figure 2.1. Vibrations of the carbon and changes in electric moment. dioxide molecule

The unperturbed energy levels may be calculated using the following formula (Courtoy, 1959), 3 G(v, v2'v3':) = i (v.i+ d./2) i= I 3 +v ^ Z d. d. + X xij (v + ) (v. + ) i= 1 ji 2 3 2 3 +g22 2+ 1 Z Yijk (vi ( + (vk+ k) i= 1 jji kj 2 2 2 (2. 1. 1) where v. are the vibrational quantum numbers, I is a quantum number 1 associated with the degeneracy of the v2 - mode andgivesthe angular momentum (in units of h/2rr ) of the molecule about the symmetry axis of the degenerate vibration, d. is a degeneracy index associated with the v. - mode (for CO2 we have, d1 = d = 1 and d2 = 2) and the u. are the vibrational frequencies (in cm-l). The vibrational constants i., xij, Yijk and g22 are generally determined experimentally. Formula (2. 1. 1) enables the energy levels to be calculated with respect to the minimum energy level. Each vibration band is composed of a large number of spectral lines produced by the rotation of the molecule. The rotational energy levels are given by (Courtoy, 1959). F(J) = BJ(J + 1) - DJ2 (J + 1)2 (2. 1. 2) 4

where B is the rotational constant and is related to the moment of inertia of the molecule, the term involving D takes into ac.count the centrifugal stretching of the molecule and J is the rotational quantum number. Courtoy (1959) gives a formula for B in terms of certain constants which are experimentally determined. 3 3 B = B - v a.v. v. v. v1v2v3 000 1 1 ij 1 J i= 1 i= 1 j i (2. 1. 3) where B is the rotational constant corresponding to level of lowest 00ooo0 energy, viz. the level (o, o, o: o) The total energy for the molecule is thus given by T (v1, v2, v3: S: J) = G (vl, v2, v3: ) + F (J)2. 1. 4) We note that for degenerate vibrational levels (7', A... ) J must be larger or equal to & viz. J =., + 1,.... (2. 1. 5) The rotational levels of linear molecules are positive or negative depending on whether the total eigenfunction does not change sign or changes sign for an inversion. In the case of C02 the ground state is a state and the even rotational levels are positive with the odd levels negative. Also for C02 the positive rotational levels are symmetric and the negative antisymmetric with respect to an interchange of all pairs of identical nuclei. The ratio of the statistical weights of the symmetrical and antisymmetrical rotational levels is determined by the nuclear spins. For the symmetric C02 isotopes, such as 12C 1602 13C 02, the antisymmetric levels have C2 2st o es s uc as 5

12 16 180 weight zero while for the antisymmetric isotopes, such as 1C 10 10, 12 16 17 12C 160, the symmetric and antisymmetric levels have equal weight i. e. there is no distinction between symmetric and antisymmetric rotational levels. Certain secondary effects must be considered, they are centrifugal stretching, Coriolis interaction, & -type doubling and Fermi resonances. We noted previously the influence that centrifugal stretching has on the rotational energy levels. The Coriolis vibration - rotation interaction is due to the rotation of the molecule and leads to a coupling between rotation and vibration. It also influences the intensity distribution of the rotational lines in a particular band. If we determine our rotational constants experimentally then we automatically take it into account. The 7, A,... vibrational levels of linear polyatomic molecules are doubly degenerate due to the equivalence of the two directions of the angular momentum. The interaction between the vibration and rotation slightly splits the degenerate levels giving. -type doubling. It is thus necessary to consider different rotational constants for the split levels. In a molecule, such as CO2, it often happens that two vibrational levels belonging to different vibrations may have almost the same energy. The two levels are in "resonance" and we obtain a perturbation of the energy levels. This is known as Fermi resonance. For further discussion of the above topics Herzberg (1945) and Allen and Cross (1963) should be consulted. In the next section we discuss the formulae and constants used to calculate the rotational line positions. 6

2. 1. 2 Rotational Line Positions Equation (2. 1. 4) was used to calculate the rotational energy levels. To use this equation is was necessary to have available the vibrational energy levels and rotational constants. Tables 2. 1 and 2. 2 list the values used and their source. Fermi resonances, Q -type doubling and centrifugal stretching have been taken into account. The rotational constants have the superscripts c and d, using Courtoy's (1959) notation, corresponding to levels with even J positive or odd J positive, respectively. 12 16 13 16 ^ 12 The following isotopes have been considered 1C 60, C 102 C 160 80, 12C 160 70, 13C 160 180. Not all the bands for these molecules have been considered, only the stronger weighted by the isotopic abundance (ref. Section 2. 2). 12 The possibility that the traditional assignments of v1 and 2 v2 in 12C 161 2 02 should be reversed has been discussed by Amat and Pimbert (1965). 2 They concluded that more study was needed. However, very recently Gordon and McCubbin (1966) have confirmed that the assignments of v1 and 2 v2 should be reversed. Most of our calculations were completed when this information became available and we felt that we should continue our calculations rather than recalculate our line positions based on this information. The corrections involved are probably small. 7

TABLE 2. 1. Vibrational energy levels for carbon dioxide in the 12 -18-micron region (units cm ) 2. la: Isotope 12 16 C 02 LEVEL Lower Upper G Lower G Upper 0 - 000:0 010:1 010:1 010:1 020:0 020:0 020:2 020:2 020:2 100:0 100:0 030:3 030:3 030:1 030:1 010:1 020:0 100:0 020:2 030:1 110:1 030:1 110:1 030:3 030:1 110:1 040:2 120:2 120:2 120:0 - 667. 379(1) 667. 379(1) 667. 379() 1285. 412(1) 1285.412(1) 1335. 12 9( 1335. 129(1) 1335. 129(1) 1388. 187(1) 1388. 187( 2003. 28(2) 2003.28(2) 1932. 466(1) 1932. 466(1) 667. 379(1 1285. 412(1) 1388. 187() 1335. 129(1) 1932. 466(1) 2076. 859(1) 1932. 466(1) 2076. 859(1) 2003. 28(2) 1932. 466(1) 2076. 859(1) 2584. 9(2) 2760. 75(2) 2760. 75(2) 2670.83(3) 667.379 618. 033 720. 808 667. 750 647. 054 791.447 597. 337 741. 730 668. 151 544.279 688. 672 581. 62 757.47 828.284 738. 364 I I _ __ I I__, 8

2. Ib: Isotope 13C16 02 LEVEL Lower Upper G Lower G Upper v 0 000:0 010:1 010:1 010:1 020:0 020:2 100:0 010:1 020:0 100:0 020:2 030:1 030:3 110:1 648. 52(1) 648. 52(1) 648. 52(1) 1265. 81(1) 1297. 40(1) 1370. 05(1) 648. 52(1) 1265. 81(1) 1370. 05(1) 1297.40(1) 1896. 54(1) 1946. 69(2) 2037. 11(1) 648.52 617.29 721.53 648.88 630. 73 649.29 667.06 2.lc: Isotope 1C 1O 18 LEVEL G G Lower Upper Lower Upper o 000:0 010:1 662.29(1) 662.29 010:1 020:0 662.29(1) 1365.84(1) 703.55 010:1 100:0 662.29(1) 1259.43(1) 597.14 010:1 020:2 662.29(1) 1325.01(2) 662.72 020:0 030:1 1365.84(1) 2049.25(2) 683.41 (1) (2) (3) Gordon and McCubbin (1965) Courtoy (1959) Stull, Wyatt and Plass (1962) 9

2. Id: Isotope 2C 160 170 LEVEL Lower Upper G Lower G Upper v 0 000:0 010:1 664. 72(3) 664. 72 010:1 020:2 664. 72(3) 1329. 79(3) 665.07 2. le: Isotope 13C 160 180 LEVEL G G Lower Upper Lower Upper v 0

TABLE 2.2. Rotational constants for carbon dioxide in the 12 to 18-micron region (units cm ) 2.2a: Isotope 1C 16 02 Level BC Bd B DC Dd 000:0 010:1 020:0 100:0 020:2 030:1 110:1 030:3 120:0 040:2 120:2 0.39021 0.390635 0.390476 0. 390201 0.391657 0.390756 0.390372 0. 39236 0.388525 0.39187 0.39152 0.39021 0.391245 0.390476 0.390201 0. 391657 0.391675 0.391326 0.39236 0.388525 0. 39187 0.39152 13.5 13.8 16.1 12.8 13. 9 14 x 12 x 13 x 13 x 13x 12.2 x 10-8 x 10 xl-8 x 108 x 10-8 -8 x -8 10-8 o-8 10 -8 10 -8io 13. 13. 16. 12. 13. 14. 10. 13 13 13 5 x 10-8 7 x 108 1 xl0-8 8 x 108 7 x 10-8 6xl0-8 6 x 0-8 x 10 8 l-8 x 108 x -8 x 10 (3) (1) (1) (1) (1) (1) (1) (2) (3) (3) (2) 12.2 x 10 - 2.2b: Isotope 13C 160 C 02 Level Bc Bd DC Dd -8 13. 7 x 10~8)-8 000:0 0. 39025 0. 39025 13. 7 x 108 13. 7 x10 (3) 010:1 0.39064 0.39126 13. 7 x 10-8 13. 7 x 10-8 (3) 020:0 0.390935 0.390935 15. 9 x 10-8 15. 9 x 10-8 (3) -8 -8 100:0 0.389745 0.389745 12.2 x 10 12.2 x 10 (3) 020:2 0.39165 0.39165 13.3 x 10-8 13. 3 x 10-8 (3) 030:1 0. 39090 0. 39216 13 x 108 13 x 108 (3), (4) 110:1 0. 39023 0. 39084 13 x 10 13 x 108 (3), (4) 030:3 0. 39096 0. 39096 13 x10-8 13 x 108 (3), (4) 11

2.2c: Isotope 12C 160 180 Level BC Bd DC Dd 000:0 010:1 020:0 100:0 020:2 030:1 0. 36820 0. 36857 0. 36851 0. 36811 0. 36948 0. 36924 0.3682 0 0. 36 912 0. 36851 0. 36811 0. 36 948 0. 37008 11.5x-8 11.5 x 108 11.5 x 10o8 11 x lo 8 11 x 10 8 11 x 108 11 x 10 8 II. 5 11. 5 11 x 11 x 11 x 11 x x 10 8 -8 10-8 -8 10 o-8 -8 -8 (3) (5) (5) (5) (5) (5) 2.2d: Isotope 12C 160 170 Level Bc Bd DC Dd -8 -8 000:0 0. 37922 0. 37922 12.5 x 10 12. 5 x 10 (6) 010:1 0. 37826 0. 37884 12.5 x 10-8 12.5 x 108 (7) 020:2 0.37923 0.37923 12.5 0-8 12.5x10-8 (7) 020:2 0. 37923 0. 37923 12. 5 x 10 12.5 x 10 (7) 2.2e: Isotope 13C160 18 Level BC Bd DC Dd 12

(1). These values were obtained from the values given by Gordon and McCubbin (1965) using Courtoy's (1959) values for the 000:0 leven as a base value. (2). Benedict (1957). (3). Courtoy (1959). 3 2 (4). The D's were calculated using D = 4 Be /v2 (5). Calculated using Courtoy's (1959) constants, Fermi resonances were taken into account in the calculations. (6). The rotational constants were obtained by linear interpolation using the other isotopes as base values. (7). Calculated using interpolated parameters. 13

2. 2 Calculation of Rotational Line Intensities The intensity of an individual rotational line is given by (cf. Penner, 1959). S (n", n', n:.-":J"-n, n, n':Q':J') 3 T(ni/'1 nI" n" '11":J") 8wr v N (T) (exp T(n, n2 n::J (2. 2. (2. 2. 1) 3hc QQJ k T k T X gj, (Aj,,,,)2 2 (1-exp ( -hc )) where ' refers to the upper level, " to the lower level and v is the -l frequency (cm ) corresponding to the indicated change in quantum numbers. N(T) is the number of molecules per unit volume per unit pressure at temperature T. gj,,, is the statistical weight of the upper level given by gJi, = 2J' + 1 for ' = 0 gji, = 2(2J' + 1) for 4' 0,3 is an empirical factor corresponding to the matrix elements. QJ is the complete rotational partition function. Qv is the complete vibrational partition function. (A, f,, ) are the amplitude factors listed in Table 3. T(n', n, n": ":J") is the energy of the lower level. T1 n2 n3 14

TABLE 3. Amplitude Factors (Penner, 1959: Dennison, 1931) (Aj )2it 1/2 AJ = o, A^ = -1 (AJ" 0)2 (AJ"-1 1) J" -1 2(2J" +1) AJ = +i,A = -1 (J"-1 0)2 J"' +1 2(2J" -1) J = -1, a = -1 AJ" Q"l 2 - (J" ~ t") (J"T " + 1) t, ' 1 4J" (2J" + 1) J(AJ_ " _ )2 - (J" +~ ") (J" +~ " -1) J -1 e4 + 1 4J" (2J" + 1) "R o, AI = ~1 AJ = 0 al" o,^,A = ~ 1 AJ = +1 A(J"-1 " )2 AAijlffl"T I )2 J" a"+ 1~ (J"~ l") (J"" ~ + 1) 4J'1 (2J' -1) f" o0, A^ J = -1 = ~1 15

The intensity of a band is given by (cf. Penner, 1959). S (n;l, n2, n3:.- n, n n3:') 1' 2' 3*~n~n2 2 3 4.r 3 v N (T) 3hc Qv -G (nl, n2, n3"") 1 2' 3 g.I" (exp ) (2. 2. 2) kT X (1-exp -hc v ( kT ) where v is the frequency corresponding to the band center G(n', n, n" Q) is the vibrational energy 2, n2, n3' and g --- 1 for Q" = 0 2 for Q" + 0 gge I We calculate 3 2 from (2. 2. 2) and substitute into (2. 2. 1) to give S(n"1, " " n:Qj1 -—, n ' n Q 1' ' -n i. n2 3 2J' ) 2gj, (AJ'el' )2 2gj, l, (Aj,, ~,, ) hcv expF(J", ") ) S(T) (1-exp (-kT )) exp (- k kT -- kT (2. 2.3) -hcv ge, QJ vo (1-exp ( k)) where S(T) is the band intensity and F(J,' 1") is the rotational energy level.

Now, in addition, we have S(T) N(T) Q (To) (l-exp (-hc vo(kT)) ____ V 0__v 0__.......... (2.2.4) S(To) N(T ) Qv(T) (1-exp (-hc vo(kTo)) Xexp (-G (1/kT- 1/kTo)) If we keep the pressure constant then N(T) T 0 N(To) T 0 and consequently S(T) T Qv (To) (1-exp (-hc vo/kT)) (2. 2.5) S(To) T (T) (1-exp (-hc v /kT )) Therefore S(n!', n3, n3 n I, n n: 1' 2' 3 2 3 = 2gj,,I (AJ,,,)2 v S(To) To Qv(To)(1-exp (-hc v/kT)) (2. 2. 6) g. QJ(T) v o T Qv(T) (1-exp (-he v /kTo) (.2. X exp (-F(J", ")/kT) The above discussion is applicable to the symmetric molecules such as 2C 02, C 02. For non-symmetric molecules such as 12C 160 180 it is necessary to take g., as 2 for all values of." since the symmetric and anti-symmetric levels are all occupied. 17

1o 1C6 For 1C 602 the partition functions were calculated by interpolation on the values given by Gray and Selvidge (1965). In the case of the remaining isotopes the partition functions were calculated using the harmonic oscillator partition functions (Herzberg, 1945). 3 Q = 71 i- = 1 (1-exp (-he v./kT))di 1 (2.2.7) 0 where d. are the degrees of degeneracy of the vibrations v i, and 1 kT Qj = — + hc B e 1 1 3 15 he B 4 he B e + e )2 kT 315 kT 1 he B + 1 hc e 3 315 kT (2. 2.8) A program was written which computed the rotational line intensities for the various bands in the 12 to 18 micron region. The isotopes 12 16 13C 160 12 16 180 12C160170 13C160180 were considered. The band intensities for the various isotopes were weighted by their abundance, which can be calculated using the isotopic abundances of carbon and oxygen. They are: 12 16 C 02 13 16 02 12 160180 12 160 170 13 160 180 C O18O 1.00 1. 12 x 102 4.0 x 133 8. 0x 10 4.5 x 105

A check on the accuracy of the rotational line intensities is to calculate the rotational line intensities for each band, then add the individual line intensities together. The results should agree closely with the original band intensity. This calculation was performed and is discussed in Section 4 where the selection of suitable band intensities is discussed. 19

3. Direct Integration Techniques 3. 1 Introduction Suppose that we have a sample of gas which contains one or more optically absorbing constituents. The transmissivity at any frequency along a given path is a function of both the position and the intensity of the individual absorption lines, as well as the physical characteristics of the sample (temperature, pressure, concentration, etc.). Inhomogenities introduce additional complications to the determination of transmissivity. It is the purpose of this chapter to describe in some detail the methods by which the absorption has been calculated, with particular emphasis on paths through the atmosphere. For theoretical or experimental purposes, we are generally interested in obtaining the transmissivity y averaged over a finite frequency interval AY, weighted by some function ( v ) - Jv~)(W) dv = (), dv(3.1. la) j' ~ (v) dv In the case of a spectrometer with slit function w( v), we calculate -= J 2c W (V V- ) -Y dv (3. 1. lb) X V2 Wc(v': - v) dv i. e., the weighting function (v) is equal to the slit function w( v - v ) It is theoretically possible to use standard quadrature techniques to calculate 'y to any desired accuracy, provided that we can obtain 7V at any frequency in the integration interval. However, typically includes contributions from many spectral lines and because it is a rapidly varying function of frequency, we need many calculations for even 20

a small frequency interval. This precludes hand calculations for all but the simplest situations. Band models were introduced to overcome this difficulty. Using a band model allows one to suitably approximate the true distribution and intensities of the spectral lines in such a manner that analytic expressions for y can be obtained, at least, in the case where the weighting function ( v ) is constant. Several such models of varying complexity and realism have been used (see Goody (1964),Plass (1958) and Drayson (1964), for details). These band models were very important in the development and application of radiative transfer theory, particularly for meteorological applications. In recent years the rapid advancements in the field of digital computing have made the numerical evaluation of Eq. 3. 1. 1 an attractive alternative. Several successful calculations using this approach have been made for infrared active molecules in the atmosphere. [Hitschfeld and Houghton (1961), Shaw and Houghton (1964), Gates et al. (1963), Gates and Calfee (1966), Drayson (1966)3 The results have shown that the same input data may yield substantial differences when the absorption from a band model is compared with the absorption obtained by direct integration [Drayson (1964) and (1966)]. Because of the errors introduced by band models it has been rather difficult to obtain some of the basic band parameters needed for the calculation of absorption. Of these parameters the two most important are the total band intensity and the Lorentz half-width. In a later section it is shown how the direct integration techniques may be used as a tool to deduce more accurate values. 21

3. 2 Homogeneous Paths The transmissivity Ty for monochromatic radiation at frequency v is given by yv = exp (- Jkv du) (3.2. 1) where kv is the absorption coefficient and u is the optical mass, the integral being taken along the absorption path. In general the pressure, temperature and other parameters will change along the path. The absorption coefficient kv may be expressed as the sum of absorption coefficients of the individual spectral lines. k = kv (i) (3.2.2) i The value of kv (i) depends on the shape of the spectral line. For the 15W bands of CO2 there are three line shapes of importance. a) Lorentz line shape, in regions where pressure or collisional broadening dominates. S. aL (i) kv (i) Tr (v- v (- ( L(i)) (3.2.3) i+ CL(i where S. is the intensity of the ith line aL(i) the Lorentz half-width v the frequency of the line center 22

b) Doppler line shape k (i) = k (i) exp (-x ) (3. 2. 4) where (v - v.) x = /1ln 2 a D aD is the Doppler half-width k = V - c) Mixed Doppler-Lorentz line shape +00co 2 k y et k (i) = j 2 -(x (3.2.5) v T 2 2 ' ' o o y + (x -t) where GC (i) y a L V- nT D Strictly speaking, the mixed Doppler-Lorentz line shape should be used at all pressures, but athigher pressures where the Lorentz half width is much greater than the Doppler, the error in using Lorentz broadening is quite small. Similarly, at very low pressures Doppler broadening is adequate. For the calculations described in this report, the Lorentz line shape was employed at pressures higher than 0. 1 atmosphere, and the mixed Doppler-Lorentz at all lower pressures (Drayson (1966)).

As a special case, consider a homogeneous absorption path, i. e. one along which the physical parameters are kept constant. Such conditions are nearly always used in experimental laboratory measurements, and are therefore of great interest and importance, as well as being simplest to calculate. Eq. (3. 2. 1) reduces to 7v = exp (-k v u) (3.2.6) k may be readily evaluated from Eq. (3. 2. 2) using the appropriate line shape. Standard quadrature techniques are used to evaluate (3. 1. 1) 3. 3 Slant Paths In meteorological and other atmospheric applications we are seldom fortunate enough to encounter homogeneous absorption paths. Variations in temperature and pressure, and hence in half-width (Doppler and Lorentz) and in line intensity, must be accounted for in the evaluation of the integral 7 (i) = k (i) du (3. 3. 1) Suppose the atmosphere is divided into horizontal layers, with only a small variation of temperature within the layer. By treating the layer as isothermal and assuming a constant mixing ratio for CO2, it is possible to obtain expressions for Eq. (3. 3. 1) (Drayson 1966).

For Lorentz broadening 2 2 2 cS. o p + ( v - v.) T(i) = - n [~ 2+: a r a p 2 + ( v - v.) o o P l x a (3. 3.2) where p1 and p2 are the pressures of the upper and lower boundaries, a is defined by the relation a L(i)= op, within the layer and c udp du The corresponding expression for the mixed Doppler-Lorentz broadening is more complicated f+00 rP2 -oo P1 k yc Ir 2 -t e y + (x - t) dpdt (3. 3. 3) The result of k c r (i) = o 2Tr y integrating with 2 J -t in - e In -oD respect to pressure is y2 2+(x-t)2 YO P2 2 2+(xPt)2 y0p1 (3. 3. 4) where y is defined by y = yop The value of 'v for a path between two points in the atmosphere is obtained by summing Tv (i) over the absorption lines and over the pressure layers between the points. 25

It is a frequent practice to reduce the absorption over a variable slant path to a homogeneous path by means, of the Curtis-Godson approximation. This method is sometimes quite satisfactory, but can lead to absorption errors as high as 6%. However, these errors can be sharply reduced by approximating the atmosphere by not one but several homogeneous layers, applying the Curtis-Godson approximation over each of these. This method was applied by Gates and Calfee (1966) to atmospheric water vapor absorption. 3.4 Programing Techniques The transmissivity ' defined by Eq. (3. 1. 1) may be readily evaluated by standard quadrature procedures. However, to carry out this task in an efficient manner and produce accurate answers over a wide variety of physical conditions is not a trivial problem. In the course of writing the computer programs a large number of techniques and approximations were investigated before being incorporated into the programs. It is the purpose of this section to describe some of these techniques which may prove useful to others writing similar programs. In evaluating the integral of some function over a finite interval, it is generally more efficient to apply low order quadrature over a number of sub-intervals, rather than higher order quadrature over the whole interval. This is particularly important when the integrand is a rapidly fluctuating function of the integration variable. The evaluation of the integral V = J V dv (3. 4. 1) v 1 26

is an excellent example. In the neighborhood of a line centre, the value of 7Y may be almost zero while remaining near unity away from the line. Moreover, the neighborhood of the line centre may be quite narrow with a very rapid change from complete absorption to negligable absorption. By a process of trial and error it was found that the integration near the line centre could be accomplished by 4-point Gaussian quadrature over intervals formed by points distance 0. 0, 0. 001, 0. 002, 0. 003, 0. 005 and -1 0. 01 cm from the line centre. Higher order quadrature resulted in no differences in '-in the 6th decimal place. Lower order quadrature led to a rapid deterioration, particularly at lower pressures. It should be noted at this point that the consideration of mixed Doppler-Lorentz broadening is an asset rather than a hinderance in choosing the interval subdivision. If pure Lorentz broadening is used at low pressures the line profile becomes very sharp and smaller subdivisions must be employed. Doppler-Lorentz broadening limits the line half-width to about 6 x 10 4 cm, even at very low pressures. In the gaps between the spectral lines the procedure is not so clear-cut. Using 4-point Gaussian quadrature over intervals of length 0. 01 cm 1gave convergent solutions, but is wasteful since a much coarser subdivision is sufficient. A modified form of this, using 0. 011 sub-intervals out of 0. 04 or 0. 05 cm1 from the line centres and 0. 1 cm sub-intervals elsewhere proved adequate, as did one or two similar methods. In the course of testing the programs it was found that round off error in the machine's representation and manipulation of the frequencies 27

of the lines and of the quadrature points could introduce considerable noise. This difficulty was overcome in the following way: the line frequencies were read into two decimal places, multiplied by 100 and converted from floating point to integer mode. To calculate the absorption in any 1 cm interval, the frequencies were converted back to floating point, relative to the center of the interval. By calculating all frequencies relative to this point, the difference between neighboring frequencies could be accurately found. The mixed Doppler-Lorentz integral in Eq. (3. 2. 5) was calculated using the method described by Young (1965), with one modification: 4-point Gauss-Hermite quadrature was employed for x > 7. 0, instead of 20-point Gauss-Hermite quadrature for x > 10.0. The same accuracy was maintained with a considerable saving in execution time. In the present calculations the half-width was assumed to be the same for all lines, enabling a further simplification to be made. Since the Doppler half-width is directly proportional to frequency, it varies only slightly over a 1 cm interval. By taking the value of half-width at the centre of the interval, the integral (3. 2. 5) was evaluated for those values of x and y corresponding to the quadrature points in the neighborhood of a line centre (i. e. within 0. 01 of the line centre), for each pressure being calculated. (For slant paths the integral is (3. 3. 3)). These values, multiplied by the appropriate line intensity, can be used for all lines in the 1 cm1 interval. This gives a considerable time saving, especially in those regions (eg.Q-branches) where the line distribution is dense. 28

In addition, the value of x is small for frequencies near the line centre, and it is the small values of x which take the longest time to calculate. The Lorentz line shape is used for Iv - vi I? 0. 2. The above method was further extended to cover those values of v - v I satisfying 0. 2 > v - vi >01. A useful reduction in execution time was obtained in the Q-branches, with smaller savings elsewhere. Let us now consider a 1 cm interval in which transmission calculations are being made. Those lines which are distant from the interval normally contribute only a small amount to the absorption; furthermore their contribution does not vary very much over the interval. Instead of computing their contribution directly at each quadrature point, it is calculated only at the centre and two end points of the 1 cm interval. Lagrange interpolation is used at intermediate points. More specifically, the Lorentz line shape is approximated by aL Si iT (V -. ) 1 and the sum i i (v - Vi) formed, with the index i running over those lines further than some distance, d say, from the interval centre. Test calculations in the 15; CO2 bands showed that do = 3. 5 cm produced no errors out to the 6th significant figure. Smaller values led to rapid deterioration. The sum above must be multiplied by l Tr, using the correct value of qL for the temperature and 29

pressure under consideration. For the slant paths, where the temperature varies, the sum was formed for 6 temperatures, and interpolated for the correct temperature value. The same sort of interpolative procedure may be used over a 0. 1 cm interval. In this case the sum k (i) v is formed at each pressure, for those lines at distance greater than or equal to some minimum, say d., from the centre of the 0. 1 cm interval. Similar tests showed dl =0. 8 cm 1 to be a satisfactory value. Finally, on the smallest scale the interpolation technique may be used in a line neighborhood, ~ 0. 01 from the line centre. The sum Zk (i) i V was again formed, for lines greater than d2 from the line centre. A value of d2 = 0. 1 cm1 proved adequate. These three interpolative procedures are very important for efficient programing, especially the first. They enable the effect of all lines to be accurately included, without calculating directly their contribution at each quadrature frequency. However, the programing is necessarily more complicated using these techniques and requires careful checking. Most of the programing was written in MAD language, which is very flexible and easy to use. The object program produced by MAD is not always very efficient, so that certain critical subroutines were rewritten in UMAP, an assembly program similar to FAP. For homogeneous paths, two completely separate programs were written by different programers. When the 50

same input parameters were used, both programs gave the same answers to six significant figures, providing a good check on the accuracy. A further check was made by calculating the absorption due to a single isolated line with Lorentz line shape, using the programs developed and also using the Ladenburg-Reiche formula; the same accuracy was attained. In making practical absorption calculations such high numerical accuracy is rarely, if ever, needed. Some of the convergence criterion were relaxed to produce a faster but slightly less accurate program. To give some indication of the speed of the program the following example is given. Input 2137 lines, transmissivity calculated at a resolution of 0. 1 wave numbers for a total of 20 homogeneous paths at 4 pressures, three of which require mixed Doppler-Lorentz broadening, for the 390 cm1 interval from 489.5 to 879.5 cm. Time taken: about 58 minutes on an IBM 7090. The 5 cm interval about the main Q-branch took just over 4 minutes. A typical 1 cm interval in the main R-branch containing 5 absorption lines took about 6. 5 seconds. These times compare quite favorably with execution times required for band models, where they can be applied. The programs described were developed specifically for the 15u region, but are quite general in scope and have also been used to calculate the absorption in other regions (egthe oxygen A-band). Because the input parameters differ from band to band, the approximations should be retested for validity. Where a lower accuracy is sufficient, some of the conditions may be relaxed to give a faster calculation. The numerical errors in calculation are certainly negligable compared to the uncertainty in band intensities, half-widths and line shapes.

4. Homogeneous Path Transmissivities 4. 1 Calculation of Rotational Line Positions and Intensities We have discussed in Section 2 the methods for calculating the rotational line positions and strengths. In this section we will discuss the practical details of the computation. The calculation of the rotational line positions poses no difficulty since the relevant parameters are fairly well known, the values listed in Section 2 were used. The calculation of the rotational line intensities is, in principle, not difficult. However, we need to know accurate values for the band intensities, quantities which are not very well known. Determining accurate band intensities from experimental measurements is difficult, we will discuss this point later. The band intensities that we used are listed in Table 4. 1. The integrated intensity for all the bands is 241. 24 cm (atm. cm) 1 at 300~K. It is interesting to compare this value with some of the estimates which have appeared in the literature. Table 4. 2, taken in the most part from Varanasi and Lauer (1966), lists most of the recent estimates for the integrated band intensity. It is interesting to note the wide range of values quoted in the light of the various error estimates. The value that we used is a little higher than most of the estimates, we will discuss this point later. In the present calculations we have neglected the influence of the Coriolis vibration-rotation interaction on the intensity distribution within a particular band. Due to the uncertainties in the band intensities we feel that attempting to introduce this additional correction is not justified, particularly since the necessary parameters are not too well known. 52

TABLE 4. 1. Band Intensities Used in Calculating Rotational Line Intensities LEVEL BAND INTENSITY Lower Upper Center (cm (atm cm) at 300 K) (atmc) at K) 000:0 010:1 667. 379 194(1) 010:1 020:0 618. 033 4.27(1) 010:1 100:0 720. 808 6. 2(2) 010:1 020:2 667. 750 30(1) 020:0 030:1 647. 054 1. 0( 020:0 110:1 791.447 0. 022(2) 020:2 030:1 597. 337 0. 14(1) 020:2 110:1 741.730 0. 14(2) 020:2 030:3 668.151 0.85(2) 100:0 110:1 688.672 0. 3(3) 100:0 030:1 544.279 0.004(1) 030:3 040:2 581.62 0. 0042(2) 030:3 120:2 757.47 0. 0059(2) 030:1 120:2 828. 284 0. 00049(2) 030:1 120:0 738. 364 0. 014(2) (1) Madden (1961) (2) Yamamoto and Sasamori (1958) (3) Yamamoto and Sasamori (1964)

TABLE 4. 2. Integrated Intensity of the 15 - Micron CO2 Bands Integrated Intensity (cm' (atm. cm) 1 at 3000K) Burch, Gryvnak and Williams (1962) Weber, Holm and Penner (1952) Kaplan and Eggers (1956) Thorndike (1947) Overend, Youngquist, Curtis and Crawford (1959) Schurin (1960) Eggers and Crawford (1951) Varanasi and Lauer (1966) National Environmental Satellite Lab., E.S.S.A., (1966) 330 ~ 90 170 ~ 34 217 ~ 5 170 ~ 18 218 ~ 5 217 ~ 5 146 ~ 18 200 ~ 10 225 ~ 7 34

As we noted in Section 2, a check on the accuracy of the rotational line intensity calculation is to add together the calculated rotational line intensities in a given band; the result should equal the band intensity. This calculation was carried out for all the bands, excellent agreement being obtained. In the calculation of the rotational line intensities we neglected lines with intensities less than 10 cm (atm. cm) at 300~K; nevertheless, we ended up with about 7000 rotational lines. Since the calculation of transmissivities using this number of lines would be an exceedingly lengthy task, we decided to reduce the number of lines using a suitable criterion. Examination of the line intensities showed that most of the lines had intensities in the range 10 to 10 cm (atm. cm). Also near strong lines very weak lines would have little influence except at very low pressures. We divided the spectral range into 5 cml intervals and retained only those lines with intensities lying in the range formed by the strongest line and 10-3 times the intensity of the strongest line in the interval. By imposing this criterion we reduced the number of rotational lines to around 2000, a more manageable number. The criterion we have used has the advantage that in spectral regions where there are only weak lines they are retained while in spectral regions with strong and weak lines we retain the strong lines and some of the weak lines, the very weak lines being neglected. We performed the line intensity calculations for six temperatures 300, 275, 250, 225, 200 and 175~K. Intensities for other temperatures may be readily obtained by interpolation. This information has been stored on punched cards and is available, on request, in that form, or on magnetic tape, but is not reproduced in this report. 35

4. 2 Half-Widths and Line Shapes In order to calculate transmissivities, as noted in Section 3, we need to know the Lorentz or collisional half-width and the line shape. The most generally used value for the nitrogen-broadened Lorentz half-width is 0. 064 cm lat 1 atm. and 298~K (Kaplan and Eggers, (1956)). It is very difficult to obtain reliable values for the half-width from the technique used by Kaplan and Eggers (1956). In the case of self-broadened carbon dioxide the half-width varies with the rotational quantum number. Madden (1961) obtained a half-width of 0. 126 cm1 (1 atm., 300~K) for the J = 4 line of the P-branch of the v2 fundamental and 0. 06 cm for the J = 56 line of the same band and branch. Burch et al. (1965) also present some data on the variation of half-width with rotational quantum number for the CO2 bands in the 1 to 1. 25 micron region. They obtained a mean value of 0. 10 cm1 (1 atm., 3000K) for the self-broadened half-width which agrees quite well with the values given by Madden. If we use the self-broadening coefficient 1. 3 given by Burch et al. (1962) then we obtain 0. 077 cm as the value for air-broadened CO2. It is interesting to note that Gray and McClatchey (1965) used a value of 0.07 cm in their calculations involving the 4. 3 micron bands, this value being provided by Kaplan. In the light of the above considerations we have used 0. 08 cm (1 atm., 300~K) for the Lorentz half-width, but in later calculations we intend to study the effect of changing this value. Because the variation of half-width with rotational quantum number for air-broadened CO2 is not known at the moment, we used a constant value. As noted in Section 3 we used the Lorentz line shape for pressures greater than 0. 1 atm. and the Doppler-Lorentz (Voigt) profile for pressures

less than 0. 1 atm. Winters et al. (1964) has discussed the departure of the line shape from Lorentzian in the wings of lines for self-broadened CO2, they studied the 4. 3 micron band. They concluded that the absorption in the wings of lines drops off more rapidly than calculated from the Lorentz formula. However since no results are available for the 15u region or for air (or nitrogen) broadened CO2 we have used the Lorentz formula for the wings. 4. 3 Comparison of Calculated and Experimental Transmissivities The transmissivities for the homogeneous paths were calculated using the procedures outlined in Section 3. The pressures and optical masses for which the calculations were made were chosen to agree with those used by Burch et al. (1962) in their measurements. Table 4. 3 lists the pressures and optical masses considered as well as the calculated and experimental equivalent widths; the calculations extend from 490. 0 to 880. 0 cm 1 and the measurements from 495 to 875 cm. The optical masses given in the report Burch et al. (1962) are for S. T. P., we have converted them so that they apply for a temperature of 300~K. The calculations were performed for four pressures, the pressures around 760 mm Hg were run at a pressure of 760 mm Hg, those around 65 mm Hg were run at 65 mm Hg, those around 15 mm Hg at 15. 6 mm Hg and the last group at 0. 39 mm Hg. The errors introduced by these approximations are small. We have prepared plots of absorptivity versus frequency and a selected number of these are reproduced in Figure 4. 1. We have been very fortunate to have had access to the original spectrophotometer traces of Burch et al. (1962); after suitable normalization some of these have been plotted for comparison in Figure 1. In the case of D 101, D 95, and D83

TABLE 4. 3. Comparison of Calculated and Experimental Equivalent Widths Identification No. (Burch et al. ) Equivalent Pressure (mm Hg) D139 D133 D127 D120 D80 D74 755 768 764 762 768 767 Dl 35 D89 D77 D71 D46 D40 DP0 65 65 67. 2 63. 6 64. 5 67 63. 5 Optical Mass (atm cm) 212.10 106.20 51. 00 25. 60 12. 60 6. 30 212.10 26. 37 12. 64 6. 30 0.82 0.42 0. 10 51.00 6. 30 0.20 1.26 0. 64 0. 32 0. 16 199. 3 178.6 157.8 139. 0 120.4 102.1 148. 9 95. 9 76. 92 60. 14 24.46 17. 36 7.59 Equivalent Width Calc. Exp. 183 164 141 125 113 95. 3 143 91. 6 70. 7 54. 7 22. 6 16. 4 7.11.-:. 22 D69 D31 15. 6 15. 6 15. 9 80. 71 37. 10 7. 33 71. 5 34. 6 7.46 D156 D101 D95 D83 0. 39 0. 39 0. 39 0. 39 4.84 3. 64 2. 72 2. 04 5.82 3. 37 2. 66 1.88

Frequency Figure 4.1. Comparison of calculated absorption (solid line) with experimental values of Burch et al. (1962), (dashed line). 't. 0 < L Frequency Figure 4.1. Continued. 39

.4 - o. <. 6 680 Frequency Figure 4.1. Continued. >._._ oQ. 1.0 - 580 600 620 640 660 680 700 Frequency Figure 4.1. Continued. 720 740 760 780

660 680 700 Frequency Figure 4.1. Continued. 0_ -.6 DC Frequency Figure 4.1. Continued.

780 Frequency (cm ) Figure 4.1. Continued. 0. 0. 0. >c 0 _0. an 4< Vex -0 —. / I 1 2 -3,4 5 - (h) D 95 I I I I I I I 0. 0a 580 600 620 640 660 680 700 Frequency (cm1) Figure 4.1. Continued. 720 740 760 780 42

0. 0. A 0.,-0. 0 1- ik 2 -3 - 4 -5- (i) D 83 0. 0. 580 600 620 640 660 680 700 Frequency (cm1 ) 720 740 760 780 Figure 4.1. Concluded. 6, 0 5.0 -I. U:3 CD 40 ' I I I I I I I ' I X Calculated o Experimental (Burch et al., 1962) 0 X X 0 - I I I I I I I I A I I I I 3.0 2.0 I 1.0 0.1 0.2 0.3 0.4 0.6 0.8 1.0 Optical Mass (Atm. cm. ) 1.5 2.0 Figure 4.2. Comparison of calculated and experimental absorption at 0.39 mm Hg. 43

the experimental curves have been taken from the report of Burch et al. (1962). In the plots obtained from the original spectrophotometer traces frequency calibration is not too satisfactory and at the moment we have just centered the calculated and.experimental curves with respect to the Q-branch of the v 2 fundamental. We calculated our absorptivity profile using triangular slit functions with half-widths of 5 cm1 and 4 cm. The experimental profiles were taken using programmed slits so that the slit function half-width varied across the spectral region. The calculated curves displayed in Figure 4. 1 were obtained using the 4 cm1 half-width; we note that it does not provide quite enough smoothing at the higher frequencies but appears adequate at the lower frequencies. Examinations of Table 4. 3 and Figure 4. 1 shows that in our calculations we have overestimated the absorptivity. There is one serious discrepancy in Table 4. 3, D 156, where the calculated equivalent width is considerably less than the experimental value. We believe the error is in the experimental equivalent width since if we plot a curve of growth for the four observations D 156, D 101, D 95 and D 83, Figure 4. 2, we find that D 156 lies considerably to one side of where we would expect it to lie. Figure 4. 1 shows that we have overestimated the absorptivity for certain bands particularly the v 2 fundamental. We thus propose in further studies to reduce the intensities of certain bands and recalculate the absorptivity profiles to compare with the experimental profiles. One factor which influences the absorptivity calculations is the Lorentz half-width, but so far we have not studied how changes in this will influence our absorptivity profiles; this is, again, something which we intend to do in the near future.

5. Slant Path Transmissivity The comparison of the calculations for homogeneous paths with laboratory spectra has shown that the band parameters have to be adjusted to give better agreement between the experimental and theoretical absorption. Despite this limitation, a calculation of slant path absorption was made to compare the theoretically determined radiance with that recorded by the Satellite Infra Red Spectrometer in a balloon flight test, at Palestine, Texas in September, 1964. First, the transmissivity was computed and averaged over 0. 1 cm -1 intervals from 660 to 720 cm. The program used the Curtis-Godson approximation over the thin layers (see Chapter 3), and computed the transmissivity between the float pressure (10. 0 mb) and 23 other pressure levels, using the temperature structure deduced from radiosonde data, Fig. 5.1. A weighting function was applied to the transmissivities to simulate the approximate triangular response function of the SIRS. The results are shown in Tables 5. 1 and -1 5. 2. The former is for a fixed resolution of 5 cm and the latter for the variable resolution indicated at each frequency. The resolution is unimportant except in the Q-branch at 669. 0 cm, where the higher resolution gives appreciably more absorption. Fig. 5. 2 is a comparison of the transmissivity calculated above (Table 5. 1) with that used by the U. S. Weather Bureau. The greatest difference is in the Q-branch. The two curves are quite similar down to about 25 mb, but below this level the theoretical calculation gives much more absorption. The opposite is true of the channel in the strongest part of the R-branch of the fundamental, at 677. 5 cm. The Weather Bureau data 45

cm 80 90 -100 20 200 _ \ 300 400 \ 500 600 700 800 1888 180 200 220 240 260 280 Temperature ("K) Figure 5.1. Temperature structure for balloon flight. 46

201 669.0 30 40 50 60 70 - f 697.0 ~I a-)0 400 00 o k _ 6.0SWeathe 500 Oryx^^^ ^^^ ___ High Altitu 600 V l Laboratory Figure 5.2. Comparison of slant path transmission for U. S. Weather Bureau balloon flight, Palestine, Texas, September, 1964. 700. Figure 5.2. Comparison of slant path transmission for U.S. Weather Bureau balloon flight. Palestine, Texas, September) 1964.,r Bureau Data de Engineering Data

TABLE 5.1. Vertical Path Transmissivity from 10.0 mb for SIRS FREQUENCY 669.0 677.5 691.0 697.0 703.0 709.0 PRESS(MB) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10.00 0.8396.9638.9783.9864.9920.9954 10.34 1.7568.9342.9603.9739.9841.9903 11.03 2.6955.9035.9416.9610.9760.9852 12.07 3.6547.8788.9266.9508.9694.9811 13.10 4.5693.8181.8895.9255.9531.9708 16.21 5.4959.7561.8515.8991.9362.9601 20.00 6.4228.6839.8071.8679.9163.9474 25.00 7.3656.6191.7667.8394.8982.9360 30.00 8.2788.5065.6926.7866.8647.9148 40.00 9.2151.4135.6249.7375.8329.8947 50.00 10.1459.3030.5347.6710.7893.8673 65.00 11.0975.2190.4567.6124.7504.8428 80.00 12.0547.1367.3679.5431.7032.8128 100.00 13.0208.0597.2560.4440.6292.7641 130.56 14.0066.0210.1623.3413.5392.7010 163.64 15.0017.0056.0911.2417.4356.6219 200.00 16.0002.0007.0364.1378.3042.5077 250.00 17.0000.0000.0120.0689.1942.3922 302.50 18.0000.0000.0011.0145.0710.2165 400.00 19.0000.0000.0001.0019.0198.0994 500.00 20.0000.0000.0000.0000.0013.0178 660.00 21.0000.0000.0000.0000.0001.0033 800.00 22.0000.0000.0000.0000.0000.0003 1000.00 23 RESOLUTION 5.0 5.0 5.0 5.0 5.0 5.0 TABLE 5.2. Vertical Path Transmissivity from 10.0 mb for SIRS FREQUENCY 669.0 677.5 691.0 697.0 703.0 709.0 PRESS(MB) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10.00 0.8310.9639.9783.9864.9920.9954 10.34 1.7441.9344.9603.9738.9841.9903 11.03 2.6801.9037.9416.9609.9759.9852 12.07 3.6378.8791.9266.9507.9694.9811 13.10 4.5500.8184.8895.9254.9531.9708 16.21 5.4752.7565.8515.8989.9361.9600 20.00 6.4016.6843.8071.8677.9162.9473 25.00 7.3446.6195.7667.8391.8982.9359 30.00 8.2596.5068.6926.7864.8646.9146 40.00 9.1984.4137.6249.7372.8328.8945 50.00 10.1330.3031.5347.6707.7892.8670 65.00 11.0878.2191.4567.6122.7503.8425 80.00 12.0487.1367.3679.5429.7030.8124 100.00 13.0181.0597.2560.4438.6290.7636 130.56 14.0056.0210.1623.3412.5391.7004 163.64 15.0014.0056.0911.2416.4358.6213 200.00 16.0002.0007.0364.1378.3047.5072 250.00 17.0000.0000.0120.0689.1949.3919 302.50 18.0000.0000.0011.0145.0718.2167 400.00 19.0000.0000.0001.0019.0203.1000 500.00 20.0000.0000.0000.0000.0014.0184 660.00 21.0000.0000.0000.0000.0001.0035 800.00 22.0000.0000.0000.0000.0000.0003 1000.00 23 RESOLUTION 4.6 4.8 5.0 5.1 5.2 5.3

indicates rather more absorption below the 50 mb level. It should be noted that the present calculations do not give a cross-over of the absorption curves for these two channels. The four channels from 691. 0 to 709. 0 cm form a family with similar characteristics. The curves from the two sources are very similar in shape, only slightly displaced from each other. The present calculations -l -1 give less absorption at 691. 0 cm, almost the same at 697. 0 cm,and rather more absorption at 703. 0 and 709. 0 cm. The transmissivity values have been used to calculate the vertical component of radiance at the balloon level, and are shown together with the observed value in Table 5. 3. The Q-branch channel at 669. 0 cm did not perform properly during the instrument flight, so the measured value should be ignored. The present calculations shuw good agreement at 677. 5, 691. 0 and 697. 0 cm, but considerable divergence at 703. 0 and 709. 0 cm1 This result was anticipated, since the higher frequencies are quite sensitive to errors in estimation of the absorption, whereas the lower ones are comparitively insensitive. The present calcuiations are known to overestimate the absorption and hence underestimate the radiance for the four channels from 691. 0 to 709. 0 cm. When more realistic values of the band parameters become available, it is expected that the calculated radiance will come quite close to the observed values, at least to within the experimental error. It has been shoown that the present calculations overestimate the absorption for all of the six channels. For some of the channels this makes 49

little difference to the computed radiances, for the temperature profile used. If other profiles are used, however, the differences may become much greater, so that it becomes a matter of importance to obtain more accurate values for all the channels. 50

Table 5.3. Vertical components of radiance at 10 mb for U. S. Weather Bureau Texas balloon flight Frequency Radiance (ergs cm 2 sec. strdn. /cm1) (cmMeasured esent Calculated fro Measured Present Calculated from Calculation U. S. Weather Bureau Data 669.0 (49.6) 48.0 47.3 667.5 42.4 41.6 41. 5 691.0 40.1 40.2 39.3 697.0 43.6 43.5 42.9 703.0 52.2 50.2 52.3 709.0 64.9 60.6 64. 0 51

6. Conclusions The main conclusion that we can draw from the previous discussion is that we must re-estimate our band intensities and it is evident that most of them are too strong. The value of Lorentz half-width (0. 08 cm at 1 atm. and 300 K) that we used, while appearing to give satisfactory results, should be varied in subsequent calculations so that its influence on the transmissivity calculations can be comprehensively studied. One important consideration that became evident during our work was that it is very important to obtain good agreement between calculated and experimental absorptivity profiles for many pressures and optical masses and not just a selected few. One factor which has caused us some trouble in our calculations has been the sparsity of experimental measurements to check with our calculations. We obviously need more low and high resolution measurements, the low and high resolution measurements complementing each other. Finally, we have not studied the influence of the 14-micron ozone band and the rotational water vapor lines on atmospheric transmissivities in the 15-micron region. No reliable estimates of their influence are available and we suggest that this area might be profitably studied. 52

Acknowledgments The authors are grateful for the constant help and encouragement given by F. L. Bartman, Project Supervisor and L. M. Jones, Project Director. D. L. Childs and W. Lee provided invaluable assistance during the preparation of the programs. We are very grateful for the information we received during our conversations with the staff of the Meteorological Satellite Laboratory, in particular Dr. D. Q. Wark. Dr. John Shaw of Ohio State University gave freely of his time and let us use some of the original spectrophotometer tracings for carbon dioxide transmissivities which were taken in his laboratory. We also received valuable advice fron Dr. W. S. Benedict of The Johns Hopkins University. This work is a continuation of research originally sponsored by NASA under Contract No. NASr-54(03) and by NSF under Grant No. G-19131. 55

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