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J. Quant. Spectrosc. Radiat. ¹ransfer Vol. 61, No. 2, pp. 153—184, 1999  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII: S0022-4073(97)00208-2

0022—4073/98 $19.00#0.00

MODEL, SOFTWARE, AND DATABASE FOR COMPUTATION OF LINE-MIXING EFFECTS IN INFRARED Q BRANCHES OF ATMOSPHERIC CO2 —I. SYMMETRIC ISOTOPOMERS R. RODRIGUES-, K. W. JUCKS‡, N. LACOMEm, Gh. BLANQUETn, J. WALRANDn, W. A. TRAUB‡, B. KHALIL#, R. LE DOUCEN#, A. VALENTIN-, C. CAMY-PEYRET-, L. BONAMY** and J.-M. HARTMANN-,-- Laboratoire de Physique Mole´culaire et Applications, UPR 136 du CNRS associe´e aux Universite´s P. et M. Curie et Paris-Sud, Universite´ Paris-Sud (baˆt. 350), 91405 Orsay Cedex, France; ‡ Smithsonian Astrophysical Observatory, Cambridge, MA 02138, USA; m Laboratoire de Spectrochimie Mole´culaire, URA 508 du CNRS, Universite´ P. & M. Curie (baˆt. F-Bte 49), 4 place Jussieu, 75252 Paris Cedex 05, France; n Laboratoire de Spectroscopie Mole´culaire, Faculte´s Universitaires N.D. de la Paix, 61 rue de Bruxelles, B-5000 Namur, Belgique; # Laboratoire de Physique des Atomes, Lasers, Mole´cules, et Surfaces, UMR CNRS, Universite´ de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France and ** Laboratoire de Physique Mole´culaire, URA 772 du CNRS, Faculte´ des Sciences et des Techniques, 25030 Besanion Cedex, France (Received 6 June 1997)

Abstract—A theoretical model based on the energy-corrected sudden approximation is used in order to account for line-mixing effects in infrared Q branches of symmetric isotopomers of CO . Its performance is demonstrated by comparisons with a large number (about 130) of  CO —N and CO —O laboratory spectra recorded by several instrument setup: nine     Q branches of different vibrational symmetries lying between 4 and 17 lm are investigated in wide ranges of pressure (0.05—10 atm) and temperature (200—300 K). The model is used to generate a set of suitable parameters and FORTRAN software (available by ftp) for the calculation of the absorption within CO , CO , and CO infrared Q branches    under atmospheric conditions, which can be easily included in existing radiance/transmission computer codes. Comparisons are made between many (about 280) computed atmospheric spectra and values measured using two different balloon-borne high-resolution Fourier transform instruments: transmission (solar occultation) as well as radiance (limb emission) measurements of seven Q branches between 12 and 17 lm for a large range of atmospheric air masses and pressure/temperature conditions have been used, including the l band of  both CO and CO . The results confirm the need to account for the effects of   line-mixing and demonstrate the capability of the model to represent accurately the absorption in regions which are often used for temperature/pressure sounding of the atmosphere by space instruments. Finally, quantitative criteria assessing the validity of the widely used Rosenkranz first-order approximation are given.  1998 Elsevier Science Ltd. All rights reserved.

I . INT RO DUC TI ON

The influence of line-mixing in CO infrared Q branches on atmospheric transmission or emission has been widely demonstrated (see Refs. 1—4 and those quoted therein) and the need to account for this effect is now established. Since the present temperature sounders, such as AIRS (Atmospheric InfraRed Sounder) and IASI (Infrared Atmospheric Sounding Interferometer), are based on improved resolution instruments, their ultimate performances will heavily depend on the precision of the modeling of CO absorption and accurate predictions of spectra around the pressure and temperature sensitive Q-lines is required. Indeed, the Q-branch shapes can have significant influence on the quality of retrieved vertical profiles of temperature, pressure, as well as volume mixing ratios.

-- To whom all correspondence should be addressed. E-mail: [email protected]. 153

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Line-mixing effects in CO Q branches have been the subject of many laboratory studies in the  4—17 lm infrared region. Measurements have been made for various bands and mixtures of CO  with different collision partners (see Refs. 6—11 and those quoted therein). Nevertheless, only the main isotopomer (CO ) has been studied and, besides Refs. 11—13 which cover the 200—300 K  range, other studies are limited to room temperature. A number of theoretical approaches have been proposed which include simple empirical models, fitting laws,\ and scaling approaches based on the energy-corrected sudden (ECS) approximation.\ To our knowledge, only the ECS model has been widely tested by numerous comparisons\\ with experimental CO absorption spectra. It was shown that it leads to  very satisfactory predictions in wide ranges of total pressure and temperature, for mixtures of CO  with a number of buffer gasses (He, Ar, N , O ), and for bands of different symmetries (&Q&,   &Q%, %Q&, *Q%, %Q*, 2 ). Furthermore, comparisons of observations with calculations of atmospheric transmission spectra in the &Q% Q branch near 721 cm\ have confirmed the interest of the model. The only work which provides self-sufficient data in order to account for line-mixing effects in Q branches of atmospheric CO is, to our knowledge, that of Ref. 24. The parameters proposed are  first-order line-mixing coefficients for a number of bands of CO and the l transition of   CO which have been generated with a fitting law model. Comparisons between measured and  computed atmospheric spectra have demonstrated the quality of this approach. Nevertheless, not all Q branches of atmospheric interest have been treated and use of the first-order approximation is only valid in a limited pressure range. Furthermore, the results of Ref. 4 indicate that the model used in the present paper may be more accurate and those of Ref. 13 show that the fitting law approach used is not well adapted in bands (such as %*) involving both even and odd rotational states. There is thus still a need for data that would enlarge those proposed by Strow et al, fulfill the needs of atmospheric applications and be adapted to all conditions (pressure, temperature, Q branch). The present paper is the final result of an extensive study of line-mixing effects in Q branches of symmetric isotopomers of CO in which several groups have been involved. A comprehensive  model is proposed, which is based on only few parameters, whose quality is assessed by comparison of computed results with many laboratory (nine Q branches, about 130 spectra) and atmospheric (seven Q branches, about 280 spectra) measurements in bands of different symmetries. This model is used to determine quantitative criteria qualifying the accuracy of the widely used first-order approximation. Numerical data and FORTRAN software for the computation of atmospheric absorption in the Q branches of CO , CO , and CO have been    generated and are available to potential users. The remainder of the paper is divided into five parts. Section 2 details the model and data used. The laboratory and atmospheric experiments are presented in Sec. 3. Comparisons between computed absorption and laboratory measurements for CO —N and CO —O are the subject of Sec. 4. The numerical data and software made available for     computation of atmospheric CO Q-branch spectra are described in Sec. 5. The model quality is  tested in Sec. 6 using atmospheric transmissions and radiances measured by high-resolution balloon-borne instruments. Some open questions and limitations of the model are discussed in the conclusion. 2 . THE OR ETI CA L MO DEL

2.1. Absorption coefficients Consider a mixture of CO with a collision partner X (infrared inactive) at temperature ¹ and  total pressure- P, with mole fractions (volume mixing ratios) x and x (with x ;x +1). !- 6 !- 6 Within the binary collision and impact approximations and in the absence of Doppler effects the absorption coefficient a in the infrared, accounting for line-mixing (LM) at wavenumber p is

- Density should be used but, since gases may be considered as ideal in the investigated range, the more common pressure unit is used in the following.

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given by 8n a*+(p, x , P, ¹ )" p[1!exp(!hcp/k ¹ )](x P) !- !- 3hc ; ol (¹ ) dl dl Im +[l'&!L !iPWCO (¹ )]\ " l \ , . (1) Y  \6 l l Y In this expression, the sum extends over all CO lines l and l and Im+ 2 , denotes the imaginary  part. ol and dl are, respectively, the population of the initial level of line and the dipole matrix element of the optical transition. &, L , and W are operators in the Liouville space. The first  !-\6 two are diagonal and associated with the wavenumber p of the calculation and with positions pl of the unperturbed lines, i.e. [l " & " l \"dl l p and [l " L " l \"dl l pl . (2)  Y    All the influence of collisions on the spectral shape is contained in the complex relaxation operator W which depends on the considered band, temperature, and the perturbing species X. Its offdiagonal elements account for interferences between absorption lines, whereas the diagonal terms are related to the pressure broadening (cl , HWMH) and shifting (dl ) coefficients of the lines, i.e. [l " WCO

\6

(¹) " l \"cl!-\6 (¹ )!id!l \6 (¹ ).

(3)

At sufficiently low pressures, a perturbation approximation of the eigenvectors and eigenvalues of the L #iPW operator can be made. Assuming that the imaginary part of W is diagonal  !-\6 !-\6 (see below), Eq. (1) to second-order leads to 8n a*+\(p, x , P, ¹ )" p [1!exp (!hcp/k ¹)](x P) !- !- 3hc



;Im ol (¹)dl l



[1#Pg!l \6 (¹)]#iP½!l \6 (¹) , [p!pl!Pdl!-\6 (¹)#Pkl!-\6 (¹)]!i[Pcl!-\6 (¹)] (4)

which reduces to the approximation proposed by Rosenkranz when limited to first-order !-\6 are directly related to the (i.e., gl!-\6"k l!-\6"0). The parameters ½ l!-\6 , g !l \6 , and k l relaxation operator, and the first-order coefficients are given by d [l " Re[W (¹)] " l\ !-\6 \6 (¹)"2 l Y ½ !. l  d pl!pl l Y Yl l Recall that Eq. (4) is valid when line-mixing is weak, for pressures that satisfy P"[l " WCO

(5)

(¹) " l\ "" pl!pl " (lOl ) (6) Y for any lines l and l that are indeed coupled by collisions. Criteria which enable quantification of the validity of Eq. (4) in CO Q branches are discussed in Appendix A.  \6"g l!-\6"k !\6"0), the usual addition of In the absence of line-mixing (W diagonal, ½!l  l  Lorentzian line contributions is obtained: \

a, *+ (p, x

!-

X



, P, ¹)"p[1!exp(!hcp/k ¹)] Im l

(x P) Sl (¹ ) !- pl [1!exp(!hcpl /k ¹)]



n\ ; , [p!pl!Pdl!-\6 (¹)]!i [Pc l!-\6 (¹)]

(7)

where the integrated intensity Sl of line l is given by 8n Sl (¹)" pl [1!exp(!hcpl /k ¹)]ol (¹)d l . 3hc

(8)

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Finally, recall that Doppler effects are accounted for by convoluting Eqs. (1), (4), and (7) by a Gaussian function. Efficient ways to do this are described in Appendix B. 2.2. Relaxation operator 2.2.1. Real part of ¼. For the computation of the real part of the relaxation operator (see works quoted in the introduction and in Ref. 28) we have retained the model proposed by Bonamy et al. It is based on the ECS approximation and properly accounts for the coupling and relaxation of rotational angular momenta. Tests presented in Refs. 4, 11—13, 22, 23, 29, have shown that it enables very satisfactory predictions of absorption by CO in wide ranges of pressure, temperature,  perturbing species, for bands of various symmetries. In the following, only Q-lines will be considered since they can be considered as isolated from P and R branches; indeed, the pressures investigated here are too low to make absorption in the vicinity of the Q branch sensitive to the elements of W coupling Q to P and R lines. Note that, as discussed in the conclusion, this approximation is inaccurate in the far wing. Coupling between lines of different bands and the influence of vibration on the CO —X interaction are neglected. Finally,  only symmetric CO isotopomers (D group) are treated here; asymmetric CO (group C ) will be     considered in a future work. Within this frame, the element of Re+W, transferring intensity from the Q line to the Q line within the (v v ID v )Q(v vIG v ) band is given by (Y ( D D D G G G ID> [Q " Re+W (¹), " Q \" h IGYID h IGYID IGIGY! !-\6 (¹), (9) ( !-\6 (Y (( (Y(Y ((Y IGYID\ where the dipole reduced elements h IYII are given in terms of 3J symbols, by (( J 1 J h IYI"(!1)(>I (2J#1 . (10) (( k k!k !k






For downward transitions (JQJ'J), the state to state rates IGIGY! !-\6 (¹) are given by ((Y IIY! !-\6 (¹)"(!1)I>IY>(2J#1))!-\6(J, ¹) ((Y J ¸ J J ¸ J ; (2¸#1) k 0 !k k 0 !k *  Q!-\6(¸, ¹) ; (for J(J), (11) )!-\6(¸, ¹)









whereas upward transitions (JQJ(J) are deduced from the detailed balance principle, i.e.

and

IGIGY! !-\6 (¹)"[o (¹)/o (¹)]IGIGY! !-\6 (¹) ((Y ( (Y (Y(

[Q " Re+W (¹), " Q \"[o (¹)/o (¹)][Q " Re[W (¹)] "Q \. (12) ( !-\6 (Y ( (Y (Y !-\6 ( The )!-\6 and Q!-\6 factors in Eq. (11) are energy corrections to the infinite-order sudden (IOS) approximation and the basis rates associated to the collisional transfer from rotational level J"0 to J"¸, respectively. In the present approach these quantities are modeled through analytical laws. )!-\6(J, ¹) is given by







1 u d!-\6 (¹)  \ ((\  )!-\6(J, ¹)" 1# , vN !-\6(¹) 24

(13)

where u , vN !-\6, and d!-\6 are the frequency spacing between level J and J!2, the mean ((\  relative velocity in CO —X collisions, and a scaling length. The basis rates Q!-\6(¸, ¹) are  described by the widely used exponential power (EP) law, i.e.



Q!-\6(¸, ¹)"A!-\6(¹) [¸(¸#1)]\H!-\6 (¹) exp !b!-\6(¹)



hcE * , k ¹

(14)

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where E is the rotational energy of level ¸. The perturber and temperature dependent quantities * d!-\6 , A!-\6, j!-\6, and b!-\6 are the parameters of the final (ECS-EP) model. Knowledge of  these terms enables construction of the off-diagonal part of Re+W, in any band. The approach of Ref. 20 also enables computation of the diagonal elements (pressure-broadened widths), which are given by Eq. (9), with IIY! !-\6 (¹)"[IIYq!-\6 (¹)]\#(!1)I>IY>(2J#1) )!-\6(J, ¹) (( ( J ¸ J J ¸ J Q!-\6(¸, ¹) ; (2¸#1) . (15) k 0 !k k 0 !k )!-\6(¸, ¹) *  The collisional terms IIYq!-\6, which are related to the relaxation of the rotational angular ( momentum and of its associated higher-order tensors, can be directly computed from the parameters d!-\6, A!-\6, j!-\6, and b!-\6 when k"k; on the other hand, for kOk, they  should be determined from other sources, such as absorption spectra, spin relaxation, viscomagnetic effect, etc. (see Refs. 22 and 23). 2.2.2. Imaginary part of W. Computation of the off-diagonal part of Im+W, is a very complex problem which is intractable for systems such as CO —N . Nevertheless, these elements are expected   to be small and have little influence on spectra at moderate pressures. They are generally neglected and Im+W, thus reduces to the collision induced line-shifts [Eq. (3)]. But, due to lack of knowledge of this parameter for Q-lines (see Sec. 2.3), the whole imaginary part of W is neglected below. Available studies on CO Q-branch absorption and the results presented below show that this  approximation is sufficiently accurate.









2.3. Data used The data required for computations using Eq. (1) are, for each Q line l, the line identification (rotational and vibrational quantum numbers), position pl , dipole transition moment dl , energy of the lower level El , and the real part of the relaxation operator for CO —X collisions. The latter  can be computed provided that the ECS-EP parameters d!-\6 , A!-\6, j!-\6, b!-\6, and  \6 are known as a function of temperature. q !( Comparisons with experimental data have shown (see Ref. 11, for instance) that the present approach enables satisfactory predictions of both the Q-branch shape and linewidths. Nevertheless, although discrepancies remain small (a few percent on a and from 0 to about 15% on cl) the error on calculated half-widths [diagonal elements of Re+W, obtained from Eqs. (9) and (15)] may be larger than the uncertainty on direct measurements of cl . Furthermore, the computed broadening parameters differ from those stored in widely used spectroscopic data bases (HITRAN, GEISA, etc.). For these two reasons, we have decided not to use Eq. (15) but rather to use more accurate values (from HITRAN-96 for CO —air for instance) for the diagonal part of Re+W,. The d!-\6 , A!-\6, A  j!-\6, and b!-\6 parameters have then been determined in order to obtain agreement between computed and measured spectra of the (100) Q(010) Q branch near 618 cm\. Calculations '' ' using Eq. (1) thus lead to results similar to those obtained by adding Lorentzian line shapes with parameters from HITRAN-96 at pressures where line-mixing effects are negligible; this insures consistency with line-by-line atmospheric emission/transmission codes that use this data base. The line identification, the spectroscopic parameters, pl , dl , El , and the temperature-dependent width c!l \  (¹) due to CO —air collisions have been taken in the 1996 version of the HITRAN  database. Values of cl for CO —N and CO —O collisions have been deduced from the relation     \6 \  (¹)]&'20,- R6 (l, ¹),   c!(¹)"[c!(16) l l where the temperature and line-dependent ratio R6  of X- to air-broadened values was determined from the measured data of Ref. 35. The values of d due to CO —N and CO —O collisions, which     remain unknown for Q-lines [d!l \  (¹)"0 for all Q-lines in HITRAN-96], were set to zero. The ECS-EP parameters and their temperature dependence are given in Table 1. Their values are different from those of Ref. 11 since, while determining them from absorption in the (100) Q(010) '' ' Q branch, the diagonal terms of Re+W, have been imposed and not calculated from the equations of Sec. 2.2.

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R. Rodrigues et al Table 1. Parameters of the ECS-EP model [Eqs. (13) and (14)] for the reference temperature ¹ "296 K  Parameter

For CO —N  

For CO —O  

A!-\6 (cm\/atm) j!-\6 b!-\6 d!-\6 (As ) 

0.0180(¹ /¹)   0.81(¹ /¹)   0.008 2.2

0.0168(¹ /¹)   0.82(¹ /¹)\   0.007 2.4

These data together with the equations given above enable computation of all quantities needed for the prediction of absorption by Q-lines of symmetric CO within the various approaches  [Eqs. (1), (4), or (7)]. For all other lines (i.e. of asymmetric isotopomers, P and R lines of CO , lines of  other species when atmospheric spectra are considered) line-mixing has been neglected. Their contributions to absorption have been computed using Eq. (7) and the data of Ref. 33. 3. EX PE RI ME NTAL

3.1. Laboratory experiments Many measurements of absorption by CO —(N and O ) mixtures have been used which originate    from the groups and instruments presented in Table 2. The Q branches investigated in the present work are summarized in Table 3. Due to experimental constraints, experiments were made for the most abundant isotopomer (CO ) only. Nevertheless, bands of various symmetries have been  studied and the 200—300 K temperature range has often been covered. Although atmospheric applications are our final aim, pressures up to 10 atm have been investigated in order to enhance line-mixing effects. The variety of conditions (pressure, temperature, collision partner, vibrational symmetries) and number of spectra enable an extensive test of the model for a reliable assessment of its quality. Furthermore, Q branches in which lines are densely [(110) Q(020) '' ' band, " p(Q) )!p(Q ) "+0.610\ cm\], sparsely [(110) Q(000) band, " p(Q )!p(Q ) "+   '' '   1.3 cm\)], and intermediately [(120) Q(010) band, " p(Q )!p(Q ) "+0.25 cm\)] spaced ' '   have been studied. Note that when inter-comparisons of measurements by different groups could be made, there was agreement within a few percents, confirming the quality of experimental data. All experiments were made with small values of x and, in each case, the spectral resolution was !- chosen (as a fraction of the linewidths for the considered pressure/temperature conditions) in order to make the effects of the instrument function small. The latter is thus neglected in all the computations whose results are presented in Sec. 4. Since we are interested in the absorption by Q branches only, the ‘‘perturbing’’ contributions from P and R lines and weak Q branches that lie in the spectral intervals investigated have been numerically removed from experimental spectra. This was done using the model and data of Sec. 2, as explained in Ref. 11 which demonstrates the validity of this procedure. The spectra presented in Sec. 4 are thus only due to the main Q branch within the considered spectral range.

Table 2. Groups and laboratoy instruments who have recorded the CO —N and CO —O Q-branch spectra used in the     present work GroupP

G1

G2

G3

G4

Group name Address C (front page) Instrument Used resolution Path (cm) P range (atm) ¹ range (K) Spectral region (lm) On instrument, cf. Refs.

LPALMS (5) FT (Bruker IFS120) 0.01—0.03 (cm\) 0.1—7000 0.5—10 200—300 5, 15 [9, 11]

LSM (3) FT (Bruker IFS120) +0.01 10, 80 0.25—1 200—300 5, 15 [36]

LSM (4) TDL +0.001 (cm\) 16, 400 0.05—1 300 15 [29, 37]

LPMA (1) Home Made FT +0.01 3600 1—3 300 +5 [38]

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Table 3. Experimental (laboratory) data on CO —(N or O ) Q-branch absorption used in the present work. (*p is    \ the spectral gap between Q and Q )   Type

Region (cm\)

*p \ (cm\)

P range (atm)

¹ range (K)

X

No. of spectra

Group

Ref.

(110) Q(020) '' '

%Q*

+ 597

0.006

(100) Q(010) '' ' (010) Q(000) ' ' (100) Q(010) ' '

&Q% %Q& &Q%

+ 618 + 667 + 721

0.73 0.95 0.96

(110) Q(020) ' ' (110) Q(000) '' ' (110) Q(000) ' '

%Q* %Q& %Q&

+ 742 +1932 +2077

0.3 1.33 1.04

(120) Q(010) ' '

*Q%

+2093

0.26

(200) Q(010) ' ' 9 Q branches

&Q%

+2130

0.57

1—10 0.05—1 0.5—10 1—10 0.5—10 0.85 0.1—0.4 1—10 1—7 1—10 0.5—1 1—2.5 1—6 1—2.5

200—300 296 200—300 200-300 200—299 222—294 296 200—300 299 200—300 203—298 296 299 296

5 12 26 23 26 4 4 4 3 7 4 7 3 4

G1 G3 G1 G1 G1 G2 G3 G1 G1 G1 G2 G4 G1 G4

[13] [13] [11] [12] [11] This work [29] [13] [12] [12] [12] [39] [13] [39]

0.05—10

200—300

N  N  O ,N   O ,N   O ,N   N  N  N  N  N  N  O ,N   N  O ,N   O ,N  

Band

132 spectra

Table 4. Groups and instruments who have carried the atmospheric measurements used in the present work and CO Q  branches studied LPMA Experiment

SAO Experiment

Main Characteristics of the Instruments Created/Monitored by Address C (front page) FT Spectrometer Resolution Max balloon altitude Technique geometry Refs.

LPMA (1) BOMEM DA2.01 +0.01 cm\ +30 km Solar occultation Looking up/limb [43, 44]

SAO (2) Home Made +0.004 cm\ +40 km Emission Looking up/limb [45, 46]

Main Characteristics of the spectra used in this work Date Place Ballon altitudes Tangent heights No. spectra looking up No. spectra looking down Spectral range CO Q branches  CO Q branches  Interfering features Refs.

22 March 1995 Kiruna, Sweden (67°N, 22°E) 14—30 km 17—30 km 60 24 60—960 cm\ (100) Q(010) (&Q%, +721 cm\) ' ' (110) Q(020) (%Q*, +742 cm\) ' ' (110) Q(100) (%Q& +791 cm\) ' '' (100) Q(010) (&Q%, +722 cm\)* ' ' Many O , few H O and solar lines   [4]

22 May 1994 Ft. Summer, NM, USA (34°N, 254°W) 36.6 km 15.9—36.6 km — 6 80—700 cm\ (110) Q(020) (%Q*, +597 cm\) '' ' (100) Q(010) (&Q%, +618 cm\) '' ' (010) Q(000) (%Q&, +667 cm\) ' ' (010) Q(000) (%Q&, +648 cm\) ' ' H O and N O lines   —

For the sake of completeness, recall that prevous laboratory measurements demonstrating the influence of line-mixing in the 597, 618, 667, 721, 742, 1932, 2077, 2093, and 2130 cm\ Q branches can be found in Refs. 13, 17, 21; 11, 21, 40; 12, 17, 21, 41; 7, 11, 21, 29; 13, 18, 21; 12, 16, 42; 6, 8, 12; 10, 13, 39; and 10, 39, respectively. 3.2. Atmospheric experiments The atmospheric data used here have been recorded by two different balloon-borne highresolution Fourier transform instruments that are briefly described in Table 4 which includes the

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Fig. 1. Temperature and pressure vertical profiles of the atmospheric measurements. The points on the left and right sides of the figures indicate the balloon altitudes of the up-looking recordings (during ascent) and the tangent heights of the down-looking recordings (during float): (a) LPMA experiment, (b) SAO experiment.

main characteristics of the spectra. These experiments again enable reliable assessment of the model quality (including the CO isotopomer) since transmissions and radiances for seven Q branches  of different symmetries and intensities can be used. Very different line-spacings are involved since " p(Q )!p(Q ) " is about 0.6;10\, 0.3, and 1 cm\ in the 597, 742, and 667 cm\ Q branches,   respectively. Note that the atmospheric spectra used show absorption by a number of other minor Q branches (such as the CO *Q' band near 757 cm\); they are not listed in Table 4 since  tests have shown that line-mixing has negligible influence on the computed results for these weak bands. The altitudes and the pressure and temperature profiles involved by the present measurements are given in Fig. 1. Note that spectra have been recorded for line of sight covering a large range of optical thicknesses and temperatures in the tangent layer (195—250 K); this variety of conditions again makes the test of the model quite stringent. The solar occultation spectra have been treated as described in Ref. 4; the procedure used removes all ‘‘continuum-type’’ contributions to absorption (aerosols, far wings of CO (l ) and H O lines,    CCl , etc.). Hence, the results presented in the following are not absolute transmissions but those  only due to lines inside and nearby the considered spectral range. Finally, the calibration used to deduce radiances from atmospheric emission recordings can be found in Ref. 46. For completeness, recall that demonstration of the influence of line-mixing effects using atmospheric measurements has been made previously in the 618, 721,\ 742,\ 791,\ and 1932 cm\ Q branches using spectra recorded by a variety of instruments. Purely computational studies have also been made in Refs. 5 and 47. 4 . TES T S OF TH E MOD EL USI N G L AB OR AT OR Y DATA

Laboratory spectra (Sec. 3.1) are used here in order to assess the quality of the model when applied to CO —N and CO —O . Comparisons are made between measured values and computed results     accounting for as well as neglecting line-mixing between Q-lines. Equations (1) and (7), modified in order to account for Doppler effects, have been used. Spectra of each of the nine Q branches in Table 3, recorded by various groups for different temperature, pressure, and collision partner conditions are presented below. They have been chosen among the numerous measurements (Table 3) in order to illustrate the quality of the model. Global results quantifying the accuracy of our approach considering all available experimental data are detailed at the end of this section. The isotopomer is not included since all results refer to CO and, for convenience, absorption coefficients have 

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Fig. 2. Absorptions in the (100) Q(010) Q branch: (䊉) measured values,( - - - ) computed results neglect'' ' ing line-mixing ( —— ) computed results accounting for line-mixing and corresponding Meas-Calc residuals. (a) CO —N with P"9.88 atm and ¹"300 K (from group G1). (b) CO —O with P"4.93 atm and     ¹"200 K (from group G1).

been normalized by the CO partial pressure [i.e. a(p, x , P, ¹)/(x P), in cm\/atm]. Since  !- !- detailed analyses of the influence of physical parameters (i.e. band structure, inter-branch mixing, temperature, total pressure, collision partner) have been made previously the following is limited to a brief presentation of results. 4.1. &Q% and %Q& Q branches Figures 2—4 illustrate the quality of predictions in three Q branches of &Q% symmetry which are of interest for atmospheric applications. They show that our approach correctly accounts for line-mixing effects regardless of the band, pressure, temperature, and collision partner. Recall that agreement with measurements made in Rennes (group G1) for the (100) Q(010) band (Fig. 2) is '' ' expected since these experimental results have been used to determine the model parameters of Table 1 (see Sec. 2.3). On the other hand, the quality of predictions in the other bands is a successful test of the model. Also worth noting is that use of experimental data from different groups leads to consistent conclusions. Results obtained for three Q branches of %Q& symmetry are displayed in Figs. 5—7. The branch structures and the results are similar to those obtained in &Q% bands and the accuracy of our approach is evident. In particular, the l Q branch, which is of great importance for atmospheric  applications, is correctly reproduced in both the central and wing regions. 4.2. *Q% and %Q* Q branches Figures 8—10 present results in the (120) Q(010) (*Q%), (110) Q(020) (%Q*), and ' ' '' ' (110) Q(020) (%Q*) Q branches, respectively. Again, our approach leads to satisfactory results ' '

Fig. 3.

Fig. 4.

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Fig. 5. Absorptions in the (010) Q(000) Q branch: same symbols as in Fig. 2. (a) CO —N with ' '   P"2.66 atm and ¹"200 K (from group G1). (b) CO —O with P"0.997 atm and ¹"298 K (from   group G1).

and couplings between even and odd J lines, even—even and odd—odd mixings are correctly modeled. Indeed, the agreement observed in Fig. 9a, where the peak results from even J lines only [" p(Q !p(Q ) "+0.6;10\ cm\] and is insensitive to even—odd mixing, shows that   even—even couplings are correctly predicted. At higher pressures absorption, which becomes sensitive to the overall coupling (even—even, even—odd, and odd—odd) since all lines tend to merge, is still accurately predicted (Figs. 8, 9b and 10). Detailed analysis of these regimes can be found in Refs. 13 and 21. Note that Fig. 9a shows that, even for a pressure of 50 hPa (about 20 km altitude), the central part of the 597 cm\ Q branch cannot be predicted by adding Voigt shapes. 4.3. Overall quality of the model Systematic comparisons between computed and measured absorptions have been made using all the spectra described in Table 3. In order to have an overall view of the quality of the model we have determined the quantities p+  , !* , and !0  from each experimental and theoretical spectrum, / / /

Figures 3 and 4 opposite. Fig. 3. Absorptions in the (100) Q(010) Q branch: same symbols as in Fig. 2. (a) CO —N with P"0.836 atm and ' '   ¹"222 K (from group G2). (b) CO —N with P"0.40 atm and ¹"296 K (from group G3).   Fig. 4. Absorptions in the (200) Q(010) Q branch: same symbols as in Fig. 2. (a) CO —N with P"2.46 atm and ' '   ¹"296 K (from group G4). (b) CO —O with P"1.0 atm and ¹"296 K (from group G4).  

Fig. 6.

Fig. 7.

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Fig. 8. Absorptions in the (120) Q(010) Q branch: same symbols as in Fig. 2. (a) CO —O with ' '   P"1.01 atm and ¹"296 K (from group G4). (b) CO —N with P"6.91 atm and ¹"299 K (from   group G1).

given by [a+  (spectrum)]+   ! "Max[a+   ! (p, spectrum)]"a+   ! (p+  , spectrum), / / a+   ! (p+ !!* , spectrum)" [a+  (spectrum)]+   ! , (17) / /  / a+   ! (p+ #!0  , spectrum)" [a+  (spectrum)]+   ! . / /  / They depend on the conditions (i.e. pressure, temperature, collision partner, Q branch) and quantify the maximum absorption and left and right half-widths at half maximum of the Q branch, respectively; knowledge of these three points of the profile gives a good idea of the absorption shape (i.e. magnitude, width, and asymmetry). The relative difference (i.e. 1-Calc/Meas) between measured and computed values of a+  , !* , / / and !0 are plotted in Fig. 11 where each of the 132 spectra is represented by a point. These results / show that our approach reproduces the characteristics of the Q-branch shape within $5% in most cases of temperature, pressure, Q branch, and collision partner. This test shows that the model correctly couples angular momenta; indeed, although the parameters of Table 1 have been deduced from measurements in a single &Q% band which contains only even J lines, the shape of other

Figures 6 and 7 opposite. Fig. 6. Absorptions in the (110) Q(000) Q branch: same symbols as in Fig. 2. (a) CO —N with P"1.01 atm and ' '   ¹"299 K (from group G2). (b) CO —N with P"9.81 atm and ¹"197 K (from group G1).   Fig. 7. Absorptions in the (110) Q(000) Q branch: same symbols as in Fig. 2. (a) CO —N with P"0.987 atm and '' '   ¹"298 K (from group G1). (b) CO —N with 6.91 atm and ¹"299 K (from group G1).  

Fig. 9.

Fig. 10.

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Fig. 11. Relative errors on the computed values of the a+  , !* , !0  , and !2"!*#!0  / / / / / / parameters [see Eq. (17)] deduced from each of the spectra. 䊉 and 䊊 refer to results for CO —N and   CO —O , respectively. The dashed lines indicate the $5% error levels.  

transitions involving different symmetry and/or J values is correctly computed without introduction of any additional parameters. The values of the mean deviations reported on the figure show that the peak absorption (resp. the width) is slightly overestimated (resp. underestimated). Besides experimental errors, there are three possible explanations to this discrepancy. The first is an overestimation of line-mixing effects by our model. The second would be a slight underestimation of the half-widths of the Q-lines. The last one is the influence of the instrument function on the measured spectra, which was disregarded in our computations but can have slight effects. In most cases, the errors in Fig. 11 are within experimental uncertainty; those on the Q-branch widths (typically 5% of 0.03P(atm) cm\], for instance, are generally lower than the spectral resolution of the measurements. An overall picture of the influence of line-mixing on the spectra used is given by Fig. 12. It shows that the simple addition of individual Lorentzian profiles leads to very inaccurate results. As remarked previously, it overestimates the Q-branch width (and wing) and underestimates peak absorption by a factor of about two at pressures for which the line widths are greater than the

Figures 9 and 10 opposite. Fig. 9. Absorptions in the (110) Q(020) Q branch: same symbols as in Fig. 2. (a) CO —N with P"0.050 atm and '' '   ¹"296 K (from group G3). (b) CO —N with P"9.88 atm and ¹"297 K (from group G1).   Fig. 10. Absorptions in the (110) Q(020) Q branch: same symbols as in Fig. 2. (a) CO —N with P"9.88 atm and ' '   ¹"297 K (from group G1). (b) CO —N with P"3.04 atm and ¹"199 K (from group G1).  

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Fig. 12. Relative errors on the computed values of the a+  , and !2"!*#!0  parameters / / / / [see Eq. (17)] deduced from each of the spectra. 䊉 and 䊊 refer to results obtained accounting for and neglecting line-mixing, respectively. The dashed lines indicate the $5% error levels.

separation between transitions. The error thus depends on the wavenumber interval between lines, as indicated by the particular behaviour of the very narrow (110) Q(020) Q branch. It is '' ' important to recall that, even if the central part of the Q branch is affected only for sufficiently high pressures, it is not the case in the wing; far away from all Q-lines, the »oigt model overestimates absorption by a factor of about two regardless of the total pressure.

5. DATA FO R AT MO SPHE RI C SPECTR A C OM PUT AT ION S

The model and data described in Sec. 2 have been used to compute parameters in order to account for line-mixing effects in Q branches of atmospheric CO . The collisional parameters for  CO —air have then been computed from their values for CO —(O and N ) by     [W, dl , cl , or ½l ]!-\ "0.21[W, dl , cl , or ½l ]!-\-#0.79[W, dl , c l , or ½l ]!-\, .

(18)

All bands of symmetric isotopomers of CO which have Q lines in the HITRAN-96 list (with the  usual selection rule k !k "0,$1) have been treated. This represents about 200 bands for each of  which three files have been generated providing spectroscopic line parameters, first-order line-mixing coefficients, and relaxation operator elements, respectively. Several FORTRAN routines have been developed which may be helpful when updating computer codes to account for line-mixing using the above data and model. These parameters and software are briefly described below and more information is given on the ftp site where they are available (see the end of this section).

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5.1. CO Q branches treated  With a few exceptions,- all Q branches of bands of symmetric isotopomers that have lines appearing in the HITRAN96 data base have been treated; this choice is consistent with today’s spectroscopic knowledge and should, in most cases, be sufficient to compute absorption in the Earth’s atmosphere. The proposed data thus include 147, 53, and 3 bands of CO , CO ,   and CO respectively. The other symmetric isotopomers have natural abundances which are  smaller and are not included in HITRAN. Note that, in most practical cases, accounting for line-mixing effects in many weak Q branches is unnecessary. At elevated pressure when line-mixing effects are important their contribution will often be masked by absorption from other strong lines/bands. At lower pressures, when their lines are discernible among others, line-mixing may be negligible. 5.2. Line spectroscopic data The spectroscopic data of Q lines have been extracted from the HITRAN-96 data base. These include, for each line l, its position pl , dipole transition moments dl , energy of the lower level El , the \  (¹ "296 K), the associated temperroom temperature pressure broadening coefficients c !l   ature dependence coefficient, and the rotational quantum number Jl . In cases where HITRAN does not provide values for all lines needed,‡ new data sets were generated; in the case of a band for which HITRAN-96 provides Q line data up to J&'20, we have generated data up to ( + J "Max(70, J&'20,). This was done by using Eqs. (7), (9), (12) and (14b) of Ref. 49 which express

 + the energies, line positions, and line-intensities in terms of band constants (intensity, vibrational energies, rotational constants, Hermann Wallis constant). The half widths and their temperature dependence have been taken from the HITRAN-96 list, where they depend on J but not on vibrational states. As said previously, the line-shifts have been neglected since their values are zero for all Q-lines in the HITRAN-96 database. The relative abundances of isotopomers were set to the natural values and included in the transition moments dl . The results of this procedure are identical to those given in HITRAN-96 when the considered line is in both data bases. 5.3. Relaxation operators [ Re + W, elements] Since upward transitions can be deduced from downward transitions using Eq. (12), only the [Q " Re[W (¹)] " Q \ for J'J have been computed. This was done using the data of ( !-\  (Y Table 1 and Eqs. (9)—(11), (13), (14), and (18). Their temperature dependence has been modeled by [Q " Re[W (¹)] " Q \"¼ ( !-\  (Y /(\/(Y

¹ \U/(\/(Y  (J(J). ¹




(19)

The parameters ¼ and w have been determined (for ¹ "296 K) from a least-squares fit /(\/(Y /(\/(Y  (¹)]"Q \ computed from the data and model described in Sec. 2 for of values of [Q " Re[W ( !-\  (Y 8 temperatures in the 176—316 K range. Tests have shown that Eq. (19) represents the temperature dependence within better than 1% for all significant elements of W. 5.4. First-order line-mixing coefficients \  have been computed from the relaxation The first-order line-mixing coefficients ½!l  operators for CO —N and CO —O using Eqs. (5) and (18). Their temperature dependence has    

- Only Q branches with Dl "k !k "0,$1 that have more than 10 lines in HITRAN have been treated. Furthermore,   a few bands involving levels (such as 211 and 400 ) in which, due to Coriolis effect, the usual polynomial expression of ''' '' rotational energy is inaccurate, have been omitted. ‡ For weak bands, lines of low and/or high rotational quantum number, which have integrated-intensities at 296 K lower than the cut-off criterion of the database are missing in HITRAN. This is a problem for computations of the W operator since the latter should be constructed including all J values of significant relative population (with respect to the overall population of the considered vibrational state). This last criterion typically leads to J "70 at room temperature.



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been modeled by



½!l \  (¹)"½ l 1!y l ln


 ¹  ¹

.

(20)

The parameters ½l and yl (for ¹ "296 K) have been determined, for each line, from a least squares  fit of values of ½!l \  (¹) computed from the data and model described in Sec. 2 for 8 temperatures in the 176—316 K range. Again tests have shown that Eq. (20) is accurate within about 1% for all significant values of ½!l \ .5.5. FORTRAN software proposed FORTRAN subroutines have been developed which help handling the data and may make updating of existing atmospheric absorption/emission computer codes easier. Their main functions are: (1) Reading the necessary data for computations in a given wavenumber range, (2) converting them to given temperature and total pressure conditions and (3) computing absorption coefficients in a given layer for all three models described in Sec. 2.1 (i.e. LM, LM-1st Order, and No-LM). Note that local thermodynamical equilibrium (LTE) is assumed since a single (kinetic) temperature is used to compute the populations of the levels (but adaptation in order to introduce vibrational temperatures are easy to implement). 5.6. How to get them The data and software described above (all together, these ASCII files represent a storage of about 10 MegaBytes) are available by ftp Anonymous in the directory /PºB/CO2 at the address batz.lpma.u-psud.fr (IPC 194.57.26.165). One should first transfer the file ‘‘ReadMe.txt’’ and read it since it gives helpful information on how to get the other files proposed. Further information for the implementation of the data and software on computers as well as indications for the modification of atmospheric transmission/emission codes are given in this file. Problems encountered while transfering or using the package can be reported to J.-M. Hartmann by E-mail (address on front page). 6 . T ES TS OF THE M ODE L U S ING A TMOS PH ER IC DA TA

The quantities deduced from the atmospheric experiments described in Table 4 are transmissions t and radiances I. Within a given model M used to calculate the absorption coefficient, their values where computed from:



>

 



8 ds exp ! a+ [p, x (z), P(z), ¹(z)] (z) dz F (p!p) dp, (21) ? ' dz \ ? 8 > X F (p!p) a+ [p, x (z ), P(z ), ¹(z )] I [p, ¹(z )] I M (p)" ' ? ?      \ X ? X ds ds ;exp ! a+ [p, x (z), P(z), ¹(z)] (z) dz (z ) dz ;dp. (22) ? ? dz dz   ? X For a given spectrum, the integration over the optical path is carried out from outside the atmosphere (altitude z ) to the balloon (altitude z ) following the curvilinear abscissa s(z) along the  line of sight. F and I are, respectively, the normalized instrument function and the blackbody '  radiance. The sum extends over all species ‘‘a’’ that make significant contributions to absorption in the considered spectral range. Two different computer codes have been used to predict values of t and I, respectively. That used to calculate transmissions was developed at LPMA; it is similar to that described in Ref. 4 but now includes computations using Eq. (1). The code used for the computation of radiances was developed previously at SAO and modified recently in order to account for line-mixing effects using the data and software described in Sec. 5. t M (p)"



 










- Note that, due to the approximate nature of Eq. (20), small corrections should be made to ½l!-\  (¹) in order to exactly verify the sum rule in Eq. (B8) of Appendix B. This can be done simply by modifying the value for the lowest J line.

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The contribution of CO (symmetric isotopomers) Q-lines has been computed from the three  models [LM, LM-1st, and No-LM, in Eqs. (1), (4), or (7)] of Sec. 2.1 using the package of Sec. 5. For all other lines (P and R of CO , O lines, etc.) the Voigt line shape was assumed and spectroscopic   parameters from HITRAN-96 were used. The instrument function, the geometry of the optical path, and the variations of temperature, total pressure, and species volume mixing ratios with respect to altitude have been determined previously using a variety of procedures which are summarized in Ref. 4 for the LPMA experiment and in Ref. 46 for the SAO measurements. The CO  mixing-ratio was set to the nominal value 358 ppmv and local thermodynamic equilibrium (LTE) was assumed; indeed, deviations from these assumptions occur above about 70 km or higher and their effects on the spectra considered here are negligible due to the relatively low (balloon and tangent height) altitudes of the present measurements. Comparisons between measured and computed values of t and I (radiances have been normalized by the blackbody value at 277 K) are presented below. A few representative spectra have been chosen for display among the numerous measurements, and global results quantifying the accuracy of our approach considering all available data are presented at the end of this section. Note that tests have shown that use of the first-order line-mixing approximation is always valid under the conditions pertaining to the observed spectra. This is expected for all bands, except for the 597 cm\ Q branch, considering the total pressures involved (Fig. 1), the line-spacing within the Q branches, and the results of Appendix A. This was unexpected in the case of the narrow (110) Q(020) band '' ' where the first-order approximation should only be valid above about 25 km (+20 hPa); the agreement below this altitude results from the fact that the significant errors on the absorption coefficient occur only in spectral regions which are optically thick. 6.1. &Q% and %Q& Q branches of CO  Results obtained for Q branches of &Q% symmetry are plotted in Figs. 13 and 14. Those in Fig. 13 are transmissions in the vicinity of the (100) Q(010) Q branch near 721 cm\ for which ' ' LPMA solar occultation measurements have been used. Figure 14 presents radiances near the (100) Q(010) Q branch (618 cm\) for which SAO emission measurements have been used. The '' ' reduction of residuals resulting from the line-mixing model, which is indicated by arrows on the (measured-calculated) residuals, is clear. It is important beyond the Q-branch heads but more accurate results are also obtained in the troughs between Q-lines (near J"40) on the low-frequency side of the plots. The fact that residuals between measurements and values computed using our model are of the same order inside and outside regions where line-mixing effects are significant, demonstrates the quality of our approach. Neglecting line-mixing can lead to underestimation of transmissions of up to 0.25 (Fig. 13), whereas radiances can be overestimated by a factor of nearly 2 (Fig. 14). These values, which are related to the errors presented in Fig. 12, are discussed at the end of this section. Results obtained for Q branches of %Q& symmetry are plotted in Figs. 15 and 16. They are transmissions (LPMA measurements) in (110) Q(100) Q-branch region near 791 cm\ (Fig. 15) ' '' and radiances (SAO measurements) in the vicinity of the (010) Q(000) Q branch near 667 cm\ ' ' (Fig. 16). Our model and data again lead to satisfactory results. The influence of line-mixing on transmission in the 791 cm\ region is discernible but quite weak, even for the thickest optical path investigated by the LPMA experiment (Fig. 15); this is due to the small intensity (particularly at the 200 K temperature of the tangent layer) and relatively large line spacing of the (110) Q(100) ' '' band. 6.2. %Q* Q branches of CO  Results obtained for Q branches of %Q* symmetry are plotted in Figs. 17 and 18. They are transmissions (LPMA measurements) in the vicinity of the (110) Q(020) Q branch near 742 cm\ ' ' (Fig. 17) and radiances (SAO measurements) in the vicinity of the (110) Q(020) Q branch near '' ' 597 cm\ (Fig. 18). Again, accounting for line-mixing significantly improves the quality of computed radiances. In Fig. 18, the contribution of the very closely spaced even J Q-lines leads to the narrow peak near 597.3 cm\ as in Fig. 9a; odd J lines, which are much sparser and spread towards low wavenumbers, are rather isolated and clearly distinguishable on the plot.

Fig. 13.

Fig. 14.

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Fig. 15. Atmospheric transmissions in the (110) Q(100) Q branch of CO , looking down ' ''  (z "29.9 km, Htg"16.7 km). Same symbols as in Fig. 13.

6.3. Q branches of CO  The main Q branches of CO which appear in the atmospheric spectra considered here belong  to the (100) Q(010) (near 721 cm\ in the solar occultation measurements) and (010) Q(000) ' ' ' ' (near 648 cm\ in the emission measurements) bands. The first clearly appears on the highfrequency side of the CO (100) Q(010) Q branch between 721 and 721.6 cm\ in Fig. 13;  ' ' calculations show that, even for the thickest optical path, line-mixing effects in this Q branch are too small (and masked by absorption due to the strong nearby Q branch of CO ) to affect measured  transmissions. The second Q branch is much more intense and rather isolated. Comparisons between computed and measured radiances, plotted in Fig. 19, show that line-mixing effects are significant and properly predicted with our model and data. This is, to our knowledge, the first demonstration of line-mixing effects in a minor isotopomer of CO . Note that residuals in Fig. 19  mostly result from improper catalogued line positions of OCO in the spectroscopic database.

Figures 13 and 14 opposite. Fig. 13. Atmospheric transmissions in the (100) Q(010) Q branch of CO for a balloon altitude z "15.1 km looking ' '  up (elevation angle "8.5°). The upper part gives the measured values and the deviations with respect to calculations accounting for and neglecting line-mixing. The lower part presents details with the symbols: 䊉 measured values (LPMA experiments). (——) computation accounting for line-mixing; (- - - -) computation neglecting for line-mixing. Fig. 14. Atmospheric radiances (normalized to the blackbody value at 277 K) in the (100) Q(010) Q branch of CO '' '  for a balloon altitude z "36.6 km looking down (tangent height Htg"24.2 km). The upper part gives measured values and the deviations with respect to calculations accounting for and neglecting line-mixing. The lower part presents details with the symbols: 䊉 measured values (SAO experiments); (——) computation accounting for line-mixing; (- - - -) computation neglecting for line-mixing.

Fig. 16.

Fig. 17.

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Fig. 18. Atmospheric radiances in the (110) Q(020) Q branch of CO , looking down (z "36.6 km, '' '  Htg"15.9 km). Same symbols as in Fig. 14.

6.4. Overall quality of the model Systematic comparisons between computed and measured atmospheric spectra have been made using all the data described in Table 4. In order to have an overall picture of the quality of the models we have determined the quantity *t+  (for solar occultation data) or *I+  (for emission / / data) for each measured and computed spectra, given by Meas (p+ , spectrum)!t Calc (p+ , spectrum) ", *t Max Q (spectrum)"" t

(23)

Meas *IMax (p+ , spectrum)!I Calc (p+ , spectrum)]/I Meas (p+ , spectrum) , Q (spectrum)"[I

(24)

where p+  is the wavenumber where line-mixing has the maximum effects on the computed quantity [i.e. such that (t Calc\LM!t Calc\No LM) or ( " I Calc\LM!I Calc\No LM "/I Calc\LM ) is maximum]. The values of *t+  are plotted in Fig. 20 vs &.  , which quantifies the thickness of the optical / / path in a Q-branch wing and is given by



&. " n\ /



X  ds x P(z)Sl [¹(z)]c l [¹(z)] (z) dz . ! dz /  l X



(25)

Figures 16 and 17 opposite. Fig. 16. Atmospheric radiances in the (010) Q(000) Q branch of CO , looking horizontally (z "Htg"36.6 km). ' '  Same symbols as in Fig. 14. Fig. 17. Atmospheric transmissions in the (110) Q(020) Q branch of CO , looking down (z "29.9 km, ' '  Htg"18.6 km). Same symbols as in Fig. 13.

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Fig. 19. Atmospheric radiances in the (010) Q(000) Q branch of CO , looking down (z "36.6 km, ' '  Htg"31.6 km). Same symbols as in Fig. 14.

The values of *I+  are displayed in Fig. 21 as a function of tangent height; all available data have / been used so that each of the measured spectra is represented by a point (84 and 6 for each Q branch in Figs. 20 and 21, respectively) on the plots. These results call for the following remarks: (i) the quality of our model is confirmed in all cases, whatever the Q branch and line of sight. Indeed, in most cases, discrepancies remain of the same order inside and outside regions where line-mixing effects are significant. Nevertheless, one should note that uncertainties on the temperature and pressure profiles contribute to residuals. Indeed, an error of only $2 K would modify computed absorptions due to the 721 cm\ Q branch by about 5%. This introduces an additional uncertainty of $0.025 for t"0.5. Furthermore, a 1% error on total pressure adds $0.01 to the uncertainty on computed results. (ii) Neglecting line-mixing leads to large errors, which depend on the line of sight and reach #0.25 on transmissions and !100% on radiances. These limit values are analyzed thereafter. The effects of line-mixing on transmission can be understood recalling that use of purely Voigt line-shapes leads to overestimation of the absorption coefficient in the Q-branch wing by a factor of about two (Fig. 12 and Refs. 11, 24, 47 and 48). Since the instrument function has little influence in the slowly varying wing, Eq. (21) shows that t,\*+ is roughly the square of t*+ and the absolute error on transmissions, assuming that only Q-lines contribute, is *t(p)"t LM (p)!t No\LM (p)+t LM (p)!t LM (p)t LM (p).

(26)

The maximum residual, obtained for t*+(p)"0.5, is of the order of #0.25 as confirmed by Fig. 20. Note that the maximum error is generally lower, particularly for thin optical paths; this is due to

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Fig. 20. Maximum absolute error on computed atmospheric transmissions [see text and Eq. (23)] deduced from each of the LPMA solar occultation spectra in the three Q branches of CO considered. 䊉 and  䊊 refer to errors when line- mixing is accounted for and neglected, respectively. The dashed lines indicate the typical value of residuals in nearby spectral regions unaffected by line- mixing.

contributions of local lines for which the Voigt model is accurate or, for weak Q branches, to the fact that absorption in the Q-branch wing is negligible. Analysis of the effects of line-mixing on radiances is slightly more complex and the results of Ref. 5 show that neglecting them may lead to both underestimation and overestimation of emission depending on the conditions. Let us again neglect the instrument function and assume that only Q-lines contribute. With simplified but self-explaining notations, Eqs. (22) and (24) lead to  I0 (p, s)[a*+(p, s)t*+(p, 0Qs)!a, *+(p, s)t, *+(p, 0Qs)]ds *I(p)" \  I 0 (p, s)[a*+(p, s)t*+(p, 0Qs)] ds \

(27)

where t(p, 0Qs) is the transmission from the current point to the balloon following the curvilinear abscissa s along the path. In the wing, a, *++2a*+ and t, *++t*+ t*+, and Eq. (27) becomes  I0 (p, s)[a*+(p, s)t*+(p, 0Qs)][1!t*+(p, 0Qs)] ds *I(p)"!1#2 \ .  I 0 (p, s)[a*+(p, s)t*+(p, 0Qs)] ds \

(28)

This equation shows that neglecting line-mixing can underestimate emission by at most a factor two (*I"!1), in cases of optically thin paths [t(p, 0Qs)+1]; this value is obtained for weak bands and high tangent heights as illustrated by the results of Figs. 14 and 21. When atmospheric absorption is significant, the error is smaller due to the positive value of the second term in Eq. (28); this is the consequence of compensation of the errors resulting from the overestimation of local

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Fig. 21. Maximum relative error on computed atmospheric radiances [see text and Eq. (24)] deduced from each of the SAO emission spectra in the four Q-branches considered. Same symbols as in Fig. 20.

emission and underestimation of the transmission to the balloon. The error can then be zero (‘‘by chance’’) but may also be positive in the case of rather thick paths [t(p, 0Qs)+0] for which layers far away make significant contribution to measured radiance. The preceding analysis is illustrated by the results of Fig. 21 and Ref. 5 which confirm that the underestimation of radiances by the Voigt model decreases with increasing optical thickness (i.e. tangent height or/and Q-branch intensity). Other examples of negative values of *I can be found in Ref. 47. The present spectra do no lead to positive values of *I, which are typical of nadir viewing or very low tangent heights, but illustrations can be found in Refs. 3, 5 and 24. 7 . C ON CL US I O N

The results presented in this paper which, for the first time, include many Q branches and a minor isotopomer of CO , have demonstrated the quality of the line-mixing model and associated data  proposed. Comparisons with numerous spectra measured in the laboratory and the atmosphere have shown that absorption, transmission, and emission are predicted with a very satisfactory accuracy; that is residuals are of the same order in all regions of the spectra, showing no particular signature in spectral ranges where the influence of line-mixing is significant. This is quite a success considering the few parameters of the model and the different vibrational transitions considered. Numerical data and associated FORTRAN software are available which enable a complete treatment of infrared Q branches of symmetric isotopomers within the full (relaxation matrix) and first-order line-mixing approaches. The quality of the widely used first-order approximation has been studied and criteria quantifying its validity have been given. Nevertheless, a number of questions and limitations of the model remain, which are discussed thereafter.

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The accuracy of the model in the case of bands for which comparisons with experiments have not yet been made due to lack of measurements should be checked. This is the case for %—% bands and for transitions involving levels with large values of the vibrational angular momentum. More tests of the model in the case of symmetrical isotopomers other than CO would also be of interest.  Nevertheless, we expect our approach to remain sufficiently accurate in these cases. Furthermore, errors in the modeling of line-mixing effects in such (weak) bands would have little influence on atmospheric spectra since, at pressure where they are important, absorption by these transitions will likely be masked by that of other (stronger) bands. Finally, an interesting and still unstudied problem, is absorption in Q branches of asymmetric CO isotopomers and particularly of *%  transitions in which two Q lines are associated with each J value. This study, which is of importance ( for atmospheric applications (the l Q branches OCO and OCO appear in the SAO  spectra near 663 cm\), will be the subject of a forth coming paper (Part II). The main limitation of the model and data proposed is that they should not be used to compute absorption too far away from the considered Q-lines. In practice, calculations remain accurate only within about $5 cm\ around the Q branches, as already remarked in Ref. 2. This restriction, which also applies to the Voigt model, results from two assumptions that have been made. The first is the impact approximation, which has the convenience of a relaxation operator independent of wavenumber. It is valid for detunings that satisfy " p!pl "(cq )\, 

(29)

where q is the (typical) duration of efficient collisions. For CO —air (dominated by the CO —N     quadripole—quadripole interaction) at atmospheric temperatures (cq )\ is of the order of  30 cm\. In the far wings of lines, Eq. (29) is not fulfilled and large errors on computed absorptions are obtained when the impact approximation is assumed (see, for instance, Ref. 56). The second reason for the breakdown of our approach in the far wing of Q-lines is that the effects of Q(P#R) and PR couplings have been neglected. Voigt shapes are assumed for P and R lines, thus strongly overestimating absorption in the wing (e.g., Refs. 56—59). First principles models accounting for the effect of the finite duration of collisions (i.e., q O0) as well as that of inter- and intra-branch  couplings on far wing absorption have been proposed. Nevertheless, they are practically limited to CO -atom systems and their accuracy remains significantly lower than that of measure ments. For this reason, empirical approaches based on simple line-shape correction functions determined from measured spectra are generally used (see, for instance, Refs. 2, 56, 58 and 59). As far as atmospheric Q branches are concerned, only that of the l band of CO is sufficiently intense   to require correct treatment of its far line wings. This need is demonstrated in Ref. 2 where a model is proposed but few laboratory measurements are yet available. Study of this problem and of its atmospheric implications will be the subject of a future work. Acknowledgements—The authors from LPMA are grateful to the Centre National d’Etudes Spatiales (CNES) for supporting part of this work under contract No. 96/CNES/0248 in the frame of spectroscopic studies in support of the IASI mission.

R EFE RE NC ES

1. Rinsland, C. P. and Strow, L. L., Appl. Opt., 1989, 28, 457. 2. Edwards, D. P. and Strow, L. L., J. Geophys. Res., 1991, 96, 20 859. 3. Clough, S. A., Worsham, R. D., Smith, W. L., Revercomb, H. E., Knuteson, R. O., Anderson, G. P., Hoke, M. L. and Kneizys, F. X., in IRS ’88: Current problems in Atmospheric Radiation, eds. Lenoble and Geleyn. A. Deepak Publishing, Hampton, VA, 1989, pp. 376—379. 4. Kochel, J. M., Hartmann, J. M., Camy-Peyret, C., Rodrigues, R. and Payan, S., J. Geophys. Res., 1997, D102, 12891. 5. Funke, B., Stiller, G. P., von Clarmann, T., Eschle, G. and Fisher, H., JQSR¹, 1998, 59, 215. 6. Gentry, B. and Strow, L. L., J. Chem. Phys., 1987, 86, 5722. 7. Huet, T., Lacome, N. and Levy, A., J. Mol. Spectrosc., 1989, 138, 141. 8. Bonamy, L., Bonamy, J., Temkin, S., Robert, D. and Hartmann, J. M., J. Chem. Phys., 1993, 98, 3747. 9. Tonkov, M. V., Boissoles, J., Le Doucen, R., Khalil, B. and Thibault, F., JQSR¹, 1996, 55, 321. 10. Margottin-Maclou, M., Rachet, F., Boulet, C., Henry, A. and Valentin, A., J. Mol. Spectrosc., 1995, 172, 1.

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11. Rodrigues, R. Khalil, B., Le Doucen, R., Hartmann, J. M. and Bonamy, L., J. Chem. Phys., 1997, 107, 4118. 12. Rodrigues, R., Khalil, B., Le Doucen, R., Lacome, N., Hartmann, J. M. and Bonamy, L., Temperature, pressure, and perturber dependences of line-mixing effects in CO infrared spectra. III. Coupling and angular momentum relaxation in %Q& Q branches, paper under preparation. 13. Rodrigues, R., Le Doucen, R., Blanquet, Gh., Walrand, J. and Hartmann, J. M., J. Mol. Spectrosc., 1997, 186, 256. 14. Hartmann, J. M., Boulet, C., Margottin-Maclou, M., Rachet, F., Khalil, B., Thibault, F. and Boissoles, J., JQSR¹, 1995, 54, 705. 15. Tonkov, M. V., Filippov, N. N., Timofeyev, Yu. M. and Polyakov, A. V., JQSR¹, 1996, 56, 783. 16. Strow, L. L. and Gentry, B. M., J. Chem. Phys., 1986, 84, 1149. 17. Hoke, M. L., Clough, S. A., Lafferty, W. J. and Olson, B. W., in IRS’88: Current problems in Atmospheric Radiation, eds. Lenoble and Geleyn. A. Deepak Publishing, Hampton, VA, 1989, pp. 368—371. 18. Margottin-Maclou, M., Henry, A. and Valentin, A., J. Chem. Phys., 1992, 96, 1715. 19. Green, S., J. Chem. Phys., 1989, 90, 3603. 20. Bonamy, L. and Emond, F., Phys. Rev. A, 1995, 51, 1235. 21. Boissoles, J., Thibault, F. and Boulet, C., JQSR¹, 1996, 56, 835. 22. Lavorel, B., Fanjoux, G., Millot, G., Bonamy, L. and Emond, F., J. Chem. Phys., 1995, 103, 9903. 23. Rodrigues, R., Hartmann, J. M., Boulet, C. and Bonamy, L., Temperature, pressure, and perturber dependences of line-mixing effects in CO infrared spectra. II. Coupling and angular momentum relaxation  in &Q& bands, J. Chem. Phys., (in press). 24. Strow, L. L., Tobin, D. C. and Hannon, S. E., JQSR¹, 1994, 52, 281. 25. Ben-Reuven, A., Phys. Rev., 1966, 145, 7. 26. Smith, E. W., J. Chem. Phys., 1981, 74, 6658. 27. Rosenkranz, P. W., IEEE ¹rans. Antennas Propag., 1975, AP-23, 498. 28. Levy, A., Lacome, N. and Chackerian Jr., C., In Spectroscopy of the Earth Atmosphere and Interstellar Medium. 1992, Academic Press, New York, pp. 261—337. 29. Ozanne, L., Bouanich, J. P., Rodrigues, R., Hartmann, J. M., Blanquet, G. and Walrand, J., JQSR¹, 1998, 59, 337. 30. DePristo, A. E., J. Chem. Phys., 1980, 73, 2145. 31. DePristo, A. E., Augustin, S. T., Ramaswamy, R. and Rabitz, H., J. Chem. Phys., 1979, 71, 850. 32. Monchick, L., J. Chem. Phys., 1991, 95, 5047. 33. Rothman, L. S., (HAWKS 1996) The HITRAN atmospheric workstation, ASA meeting, Reims, France, 4—6 September 1996. 34. Jacquinet-Husson, N., The GEISA system in 1996, ASA meeting, Reims, France, 4—6 September 1996. 35. Margottin-Maclou, M., Dahoo, P., Henry, A., Valentin, A. and Henry, L., J. Mol. Spectrosc., 1988, 131, 21. 36. Frichot, Fr., Lacome, N. and Hartmann, J.-M., J. Mol. Spectrosc., 1996, 178, 52. 37. Blanquet, Gh., Walrand, J., Hartmann, J. M. and Bouanich, J. P., JQSR¹, 1996, 55, 289. 38. Valentin, A., Spectrochim. Acta, 1995, 51A, 1127. 39. Rachet, F., Margottin-Maclou, M., Henry, A. and Valentin, A., J. Mol. Spectrosc., 1996, 175, 315. 40. Eng, R. S and Mantz, A. W., J. Mol. Spectrosc., 1979, 74, 331. 41. Armstrong, R. L., Appl. Opt., 1982, 21, 2141. 42. Dokuchaev, A. B., Pavlov, Yu. A., Stroganova, E. N. and Tonkov, M. V., Opt. Spectrosc., 1986, 60, 585. 43. Camy-Peyret, C., Spectrochim. Acta, 1995, 51A, 1143. 44. Camy-Peyret, C., Jesek, P., Hawat, T., Durry, G., Payan, S., Berube´, G., Rochette, L. and Huguenin, D., The LPMA balloon-borne FTIR spectrometer for remote sensing of atmospheric constituents, Proc. 12th ESA Symp. on Rocket and Balloon Programs & Related Research, Lillehammer, Norway, 1995. 45. Traub, W. A., Chance, K. V., Johnson, D. G. and Jucks, K. W., J. Soc. Photo Opt. Instrum. Eng., 1991, 1491, 298. 46. Johnson, D. G., Jucks, K. W., Traub, W. A. and Chance, K. V., J. Geophys. Res., 1995, 100D, 3091. 47. Strow, L. L. and Reuter, D., Appl. Opt., 1988, 27, 872. 48. Rodrigues, R. and Hartmann, J. M., JQSR¹, 1997, 57, 63. 49. Rothman, L. S., Hawkins, R. L., Wattson, R. B. and Gamache, R. R., JQSR¹, 1992, 48, 537. 50. Traub, W. A. and Stier, M. T., Appl. Opt., 1976, 15, 364. 51. Houghton, J. T., Meira Filho, L. G., Bruce, J., Hoesung Lee, Callander, B. A., Haites, E., Harris, N. and Maskell, K. (eds.), Climate Change 1994: Radiative Forcing of Climate Change and An Evaluation of the IPCC IS92 Emission Scenarios. Cambridge Universty Press, Cambridge, 1994 p. 43. 52. Rinsland, C. P., Gunson, M. R., Zander, R. and Lopez-Puertas, M., J. Geophys. Res., 1992, D97, 20 479. 53. Stiller, G. P., Gunson, M. R., Lowes, L. L., Abrams, M. C., Raper, O. F., Zander, R. and Rinland, C. P., J. Geophys. Res., 1996, D100, 3107. 54. Lopez-Puertas, M., Wintersteiner, P. P., Picard, R. H., Winick, J. R. and Sharma, R. D., JQSR¹, 1994, 52, 409. 55. Kostsov, V. S., Fisher, H., Timofeyev, Yu. and Oelhaf, H., Numerical modeling of the influence of Non-LTE effects on the MIPAS balloon and satellite Limb radiance measurements, ASA meeting, Reims, France, 4—6 September 1996. 56. Hartmann, J. M. and Boulet, C., J. Chem. Phys., 1991, 94, 6406.

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Cousin, C., Le Doucen, R., Boulet, C. and Henri, A., Appl. Opt., 1985, 24, 3899. Burch, D. E., Gryvnak, D. A., Patty, R. R. and Bartky, C. E., J. Opt. Soc. Am, 1969, 59, 267. Menoux, V., Le Doucen, R., Boissoles, J. and Boulet, C., Appl. Opt., 1991, 30, 281. Davies, R. W., Tipping, R. H. and Clough, S. A., Phys. Rev. A, 1982, 26, 3378. Boulet, C., Boissoles, J. and Robert, D., J. Chem. Phys., 1988, 89, 625. Boissoles, J., Menoux, V., Le Doucen, R., Boulet, C. and Robert, D., J. Chem. Phys., 1989, 91, 2163. Ma, Q. and Tipping, R. H., J. Chem. Phys., 1994, 100, 8720. Gryvnak, D. A., Burch, D. E., Alt, R. L. and Zgonc, D. K., Infrared absorption by CH , H O, and CO ,    Report AFGL-TR-76-0246, US Air Force Geophysics Laboratory, Hanscom AFB 1976. 65. Armstrong, B. H., JQSR¹, 1967, 7, 61. 66. Humlicek, J., JQSR¹, 1979, 21, 309.

57. 58. 59. 60. 61. 62. 63. 64.

A PPE NDI X A: ON TH E P ERTUR BA TI ON DEV EL OPM EN TS The perturbation developments in Eq. (4), have the advantages, when compared with Eq. (1), that the absorption coefficient has a ‘‘line by line’’ expression in which influence of total pressure is simple and that storage of only few parameters for each line is required. Nevertheless, these developments are valid in a limited pressure range which depends on the considered Q branch as is illustrated in Figs. 22 and 23. The following presents criteria used to quantify the validly range of the developments for CO Q branches. 

A.1. First-order development \6"k !\6 (¹)"0] has the advantage that, since the ½l coefficients The first-order development [Eq. (4) with g !l  l  verify the sum rules: ol (¹)dl ½ !l \6 (¹)"0, l

\6 (¹)" ol (¹) dl dl [l " W (¹) " \, ol (¹)dl pl Y!l  Y !-\6 l l l Y

(A1)

exact results are obtained in the far wing of the Q branch, for wavenumbers that verify P" [l "W (¹) " l\ ""p!pl " for all Q lines l. On the other hand, large errors appear in the central part of the branch at !-\6 elevated pressure (see Figs. 22 and 23) and computed absorption can even be negative.

Fig. 22. Absorptions and Meas-Calc residuals in the (100) Q(010) Q branch for CO —N at ¹"250 K '' '   and P"1 atm: 䊉 experimental values measured by group G1. Computed values using the models (——) W operator and Eq. (1), ( ) ) ) ) ) ) second-order approximation and Eq. (4), (- - - -) first-order approximation and Eq. (4).

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Fig. 23. Absorptions and Meas-Calc residuals in the (110) Q(020) Q branch for CO —N at ¹"300 K '' '   and P"0.1 atm. Measurements are from group G3 and the symbols of Fig. 22 are used. Due to their large errors, results obtained with the second-order approximation are plotted separately.

In order to determine quantitative criteria of the validity of the first-order development, let us consider the case of a band involving & and % states and introduce the parameters p>"(p!p )/"*B" and P>"P/"*B", o> l "ol /( o ), and  I I d> l "dl /( (o d) where p is the band origin and *B"B !B is the difference between rotational constants in the I I  D G I upper and lower states. The relative error resulting from the first-order approximation is then given by



 



 

a*+-(p>, P>, ¹) 1#iP>½> l (¹) 1! " 1!Im o> Im o> l (¹)(d> l ) l (¹)d> l d> l Y a*+(p>, P>, ¹) [p>]!i[P>c l!-\6 (¹)] l l l Y ;[l "[&>!L>!iP>W (¹)]\"l\  !-\6



,

(A2)

where ½> l "½l *B and L>"(L !& )/*B. Since Hermann Wallis effects are small and vibration has little influence on the    rotational energies, assuming that o> l , d> l , ½> l , L> , and W do not depend on the considered band are realistic  approximations.- The relative error in Eq. (A2) is thus practically independent on the considered & and % states. Its maximum values (over all p> values), plotted in Fig. 24 versus ¹ and P>, show that the first-order approximation is valid up to P>+100. For P"0.1 atm, for instance, first order can be used for the 721 and 618 cm\ Q branch ("*B"+10\ cm\) but not for the 597 cm\ Q branch ("*B"+5;10\ cm\). Equation (4) should thus be used with care and the full W approach in Eq. (1), which is always valid and involves only little more computer time (see Appendix B), may be preferable. Note that the values in Fig. 24 are ‘‘pessimistic’’ since they are maximum errors; in many practical cases they lie in regions where absorption is saturated and use of the first-order line-mixing may still be valid in the spectral range of interest (in the wing, for instance).

A.2. Second-order development The improvement brought by the second-order development is important in a limited pressure range only, when the \6 and Pk l!-\6) are small but significant when compared with the zero and first-order pressure squared terms (i.e., Pg!l  contributions. In this case, which is illustrated in Fig. 22, more accurate results are obtained than when a first-order approximation is used. At higher pressures very large errors can result from the introduction of the rapidly increasing \6 and Pk l!- terms, as shown in Fig. 23. Furthermore, due to truncature to second-order, a P contribution remains Pg !l  in the far wing absorption. Again, the interest of the second-order approach is not very clear and use of the full W approach in Eq. (1) seems preferable.

- Tests have shown that errors in %Q* Q-branches are, at moderate pressure, about the third of those in &Q% bands for the same P> due to coupling between even and odd J lines.

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Fig. 24. Maximum errors (over all wavenumbers) of the first-order approximation versus P>"P/" DB " (cm atm). (——, - - - -) Computed results for ¹"190 K, ¹"310 K (the lower curve for 310 K is for %Q* bands). The two arrows indicate the values at 0.1 atm for the (100) Q(010) and (110) Q(020) bands ' ' '' ' centered near 721 and 597 cm\. 䊉, 䊊 from experimental room temperature CO —N data in the (618 and   721 cm\) &Q% and (597 cm\) %Q* Q branches.

A PPE NDI X B: CO MP UTA TI ONA L TOO LS Accounting for the influence of the Doppler effect on the collisional absorption shape can be done by introducing the complex probability function W(z"x#iy), defined as i for y'0 W(z)" n

> e\R  2i 8  dt"e\X 1# eR dt , z!t (n  \









(B1)

for which efficient FORTRAN routines are available (see for instance Ref. 66).

B.1. Within the first-order and Lorentzian models Introduction of Doppler effects in Eq. (4), when limited to first-order is straight forward since it is simply the convolution of a complex Lorentzian shape by a Gaussian profile. Noting that W(!zN )"W M (z) and a little mathematics lead to









hcp a*+- -\" (p, x , P, ¹)"8n/3hc x Pp 1!exp ! !- !- k ¹



; ol (¹)dl l

1 c (p, ¹) "



;Re (1!iP½l (¹)) W

ln(2) n

p!pl!Pdl (¹) Pcl (¹) #i c (p, ¹)/(ln (2) c (p, ¹)/(ln (2) " "



,

(B2)

where c (p, ¹) is the CO Doppler width (HWMH). In the absence of line-mixing (½l"0) the usual Voigt shape is obtained. " 

B.2. Within the relaxation operator model Let us note e (P, ¹) the eigenvalues of the [L #iPW(¹)] operator and E(P, ¹) the matrix of the eigenvectors (the mth K  column of E is then the eigenvector associated with eigenvalue e ). Simple mathematics shows that one can write

[&!L !iPW(¹)]\"E(P, ¹)D(p, P, ¹)E(P, ¹)\, 

(B3)

where D is a diagonal operator whose elements are given by [m " D(p, P, ¹) "m \"d [p!e (P, ¹)]\. KKY K

(B4)

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Introducing these equations into Eq. (1) leads to








!-

8n hcp , P, ¹)" p 1!exp ! 3hc k ¹





S (P, ¹) K P)Im , [p!e (P, ¹)] K K where the sum extends over the ‘‘equivalent Q-lines’’ whose (pressure-dependent) ‘‘line strengths’’ S are given by K a*+(p, x

(x

!-

(B5)

S (P, ¹)" ol (¹) dl dl [lY " E(P, ¹) " m\ [m"E(P, ¹)\"l\. (B6) K Y l l Y This technique is much more efficient than inversion of the &!L !iPW(¹) operator for each wavenumber since, once the  e (P, ¹) and S (P, ¹) have been computed, the absorption simply results from addition of complex Lorentz shapes. K K Introduction of Doppler effects in Eq. (B5) then leads to









8n hcp , P, ¹)" x Pp 1!exp ! 3hc !- k ¹

 

1 ln(2) c (p, ¹) n " p!eN (P, ¹) K ;Re SM  (P, ¹)W , (B7) K c (p, ¹)/(ln (2) K " where SM  and eN are the complex conjugates of S and e . K K K K Use of the preceding decomposition makes the W operator approach only slightly more time consuming than the first-order model. Indeed, the difference lies in the diagonalization and inversion of operators and computation of the ‘‘equivalent’’ line intensities for each new pressure; in many applications the CPU time required will be negligible when compared to that involved by the computation of the absorption coefficient from all line contributions due to the large number of lines and spectral points. a*+\" (p, x

!-



B.3. Important constraint for computations As remarked in Refs. 5 and 48 it is essential for correct computations that all (or none) of the Q-line (real or equivalent lines) contributions be accounted for in the sums in Eqs. (B2) and (B7). Indeed, partial troncature of the sum (disregarding the lines that are centered far away from the spectral point considered, for instance) results in the breakdown of the fundamental rules:





\6 (¹)"0 or Im S (P, ¹) "0. (B8) ol (¹)dl ½ !l  K l K Large errors (predicted absorptions may even be negative) may then appear in the Q-branch wing. This constraint has dramatic consequences in terms of CPU consummation since accounting for one Q-line implies that all should be accounted for at the considered wavenumber.