- Email: [email protected]
The N2-broadened water vapor absorption line shape and infrared continuum absorption—II. Implementation of the line shape
J. Quant. Spectrosc.Radiat. Transfer Vol.28.No. 2, pp. 103-112,1982
0022-4073/82/08013-10503.00/0 PergamonPressLid.
Printedin GreatBritain.
THE N2-BROADENED WATER VAPOR ABSORPTION LINE SHAPE AND INFRARED CONTINUUM ABSORPTIONtulI. IMPLEMENTATION OF THE LINE SHAPE MICHAELE. THOMAS~t The Johns Hopkins University, Applied Physics Laboratory, Laurel, MD 20707, U.S.A.
and ROBERT J. NORDSTROM Battelle Columbus Laboratories, Columbus, OH 43201, U.S.A. (Received 17 September 1981)
AIMtract--The total line shape model of the previous paper is tested using a set of experimental room temperature H20 continuum measurements of high quality. Parameters of the far wing component of the total line shape are determined from near band experimental data. Grating spectrometer measurements from 300 to 650cm -t are used to determine unknown far wing parameters of the pure rotational band of H20. CO and HF laser measurements taken in the 5 and 3 ~m regions are used to determine the far wing parameters of the v2 and v~, v3 fundamental bands, respectively. The total line shape model is applied to the 10 and 4 ~m transmission windows with encouraging success. A significant increase in the self-broadening ability of H20 over N2 is predicted in the far wing. This procedure allows the proper modeling of the absorption coefficient vs H20 partial pressure dependence in all window regions. A negative temperature is predicted by the model in the continuum. The observed rate of the temperature decrease is not predicted by the model; however, this limitation is related to the approximations made on the interaction potentials and the perturbation expansion of the Hamiltonians. Although the total line shape has limitations, it does demonstrate the importance of considering far wings of absorption lines in continuum absorption. INTRODUCTION
In the preceding paper, we have developed an expression for the pressure-broadened absorption line shape of H20 molecules. Only binary collisions were assumed and both self-broadening and foreign-broadening with N2 were included. The spectral line shape was developed by first considering the time evolution of the correlation function of the energy levels of the colliding molecules. Collisions were categorized by their severity: weak collisions contributed only to the near-line-center of the spectral profile, while strong collisions contributed to the line shape everywhere including the far wings. The study of H20 continuum absorption must include a study of the accumulation of these far wings. Statistical broadening I was used in our development to formulate the far wing component to the spectral line shape. Recently, another line shape development used only the impact approximation, modified to account for variations in collision duration. 2 It is not clear at this time whether there is any significant difference in the line shapes produced by these two separate approaches. In this paper, we demonstrate how the spectral line profile, defined by Eqs. (49), (77), and (78) of the preceding paper, is implemented. Furthermore, comparison between exerimental absorption data and the absorption predicted by this line shape is given. We will show that a line shape formulation is adequate to predict the total behavior of water vapor continuum absorption over the ranges of pressure and temperature of tropospheric interest. Data will be shown for the 8-14 and 3.5-4 ~tm atmospheric transmission windows. HISTORICAL PERSPECTIVE
Figure 1 demonstrates the inability of the Lorentz line shape to model the observed pressure dependence of water vapor continuum absorption near 10/zm for H20 samples pressure broadened to 1 atm with N2. The data were recorded using a frequency-stabilized CO2 laser and a long-path absorption cell described in several references)-5 The curve marked A was tThis work done under ARO Contract DAAG-29-77-C-0010 while both authors were with the Ohio State University, ElectroScience Laboratory, Columbus, OH 43212, U.S.A. *Author to whom correspondence should be sent. 103
104
M . E . THOMASandR. J. No~sTaOM 0.4 Pt24) CO 2
_.-..
940.548 LASER
0.3
q E
0.2
63
B
o1l i o_ ~ o
4
8 Po (Torr)
12
16
Fig. 1. Pressure dependence of water vapor continuum absorption.
generated from data on the AFGL tape 6 using a bound of 20 cm -j. That is, for the calculations at each H20 partial pressure, only the H20 absorption lines within a range of -+20 cm -~ from the measurement frequency were assumed to contribute to the absorption at that frequency. All other H20 absorption lines were assumed to be too far away to make a significant contribution. This bound of 20 cm -1 became a standard in early line-by-line calculation programs because it was believed to be a reasonable compromise between spectral representation and computing time. The curve marked B in Fig. 1 was generated by accumulating the contributions of all H20 absorption lines from 0 to 5000 cm -~. The contributions at the measurement frequency were computed using the Lorentz line shape for all lines listed on the data tape. At low H20 partial pressures, this line shape overestimates the observed absorption, while at high H20 partial pressures it underestimates the observed absorption. Despite this failure, the difference between curve A and curve B aemonstrates that significant contribution to the total absorption can be found in the accumulation of the far wings of strong absorption lines. The Lorentz profile, however, is not an adequate model for the far wings of lines. It is therefore not surprising that curve B fails to provide suitable agreement with the measured absorption. Motivated by the hope that an accurately derived H20 absorption line shape would improve the modeling of continuum absorption, we undertook the work presented in this paper and in the previous paper. THE LINE PROFILE
The absorption coefficient at frequency u (in cm i) is k(~) = ~ Sj(v - v0,;/3,),
(1)
where S~ is the strength of the ith absorption line and j(V-voi;/3i) is the shape of the absorption line centered at voi and which has a set of shape parameters {/3~}.The major features of the total line shape have been derived in the previous paper. It was found that jc(mv) = NldCNLC(Av)p(Av) + jC~w(Av)[ I - p(Av)]},
(2)
where the subscript C shows that the line shape is collision broadened, 1
j,'NLc(av)
O~R
~ (av + af)2+ (aR) 2'
(3)
The N2-broadenedwatervaporabsorptionline shapeand infraredcontinuumabsorption--II • 1 AoE lcrw(AV)=-~{~Ym~(mk, Av)+~
Ym,(mi,Av)} ,
(4)
p(Av)= f 1/2+l/2c°s ( 7 Av)IAvi <5 cm-', 0
105
(5)
la~l _>5 cm-';
N is the normalization constant. Normalization requires that f
jc(Av) dv = 1.
(6)
In order to eliminate the singularities in the far wing line shape functions at Av = 0, we now adopt the formulation of Ref. 7, which includes an additive term'in the denominators of Eq. (4). Furthermore, we now set 6 and mi = 4 as discussed in the previous paper. Therefore,
mk=
1 f 0.3198Aa6
j c ~ ( a v ) = g/lavl'" + x~:6Yt(m,
0.4334Ab4 = 6, Av) + IAvlt.7s +
(Ab4)7/3Y4(mj
Av)}. = 4,
(7)
This modification of the line-shape function does not significantly alter the value of the function when Av is very large. It assures that this function is well behaved at the line center for normalization purposes. Furthermore, in our formulation, the p-function is turning off the far wing line shape near Av = 0 and any constant value of Jcrw(Av) at Av = 0 artificially introduced by this modification is given a weighting of zero. The multiplicative constants which appear in Eq. (7) are due to the normalization introduced by the Varanasi formulation. An evaluation of )%6 and Ab4in far wing data (Av ,> Aat, )tb4) would simply be scaled by these constants. The functions Yt(mk = 6, Av) and Y4(mj = 4, Av) require discussion. As mentioned in the previous section, Fomin and Tvorogovg derived a term of the form
exp { - L
T
J J'
where G is a constant and T is the temperature. An exponential decay of some form is expected in the far wings of absorption lines as discussed by Clough and his coworkers. 2 For self-broadening (m = 6) the above term is proportional to exp (-[Av]°56), and for foreignbroadening this term is proportional to exp (-IAvl°%. We have adopted a simplification of this term. We have set Y6(mk = 6, Av) = exp (- Ga6IAv[0"5)
(8a)
Y4(mj= 4, Av) = exp (-Gb41Av[°5).
(8b)
and
We use the same frequency dependence for both the self- and foreign-broadened terms in order to facilitate the normalization process. The origin of these terms is the level shift contribution which is not fully incorporated in the phase shift approach. Such terms are impossible to derive by perturbation methods since, by definition, in the far wings of i.r. absorption lines the perturbation energy (Av) is no longer small compared to the unperturbed rotational energy levels of the absorbing molecule. The G parameters above are
QSRT VoL 28, No. 2---C
G.6=g.(v)(~) "a'v),
(9a)
Gb4 = g~(v)
(9b)
,
106
M.E. THOMASand R. J. NORDSTROM
where the parameters g,(v), gb(v), ~/,(V), and yb(V) depend on the vibrational quantum numbers. In particular, it can be anticipated that the parameters might be slightly different for the pure rotational band (v = 0) where various rotational levels are more easily accessible through thermal energy than in the case for the vibrational energy levels. Thus we might expect one set of parameters to describe the shape of lines in the rotational band and another set of parameters to describe the shape of absorption lines of the vibrational bands. Equation (7) then becomes
jcvw(Av) =
1 f 0.3198A,6
0.4334Ab4
}
g [IA,,I ' ' + XLb exp (-G~X/IA~I) + lavl,.7: + 17,2 exp ( - Gb4VIAvl) .
(1o)
The temperature and pressure dependences of Gb and Ab4 can be extracted by examining the inputs to Eq. (70a) in the previous paper, viz.
Abj = Ok ~, O~b3'~,
(11)
where the terms have been defined previously. For self-broadening, Eq. (l 1) becomes /~a6 = ~ T ~s
27rl 1/2'61/2F ( - ~ ) C O S ( 4 Psb-~(W6u(Z)-W6l(Z))
)"
(12)
The potential for self-broadening is seen in Eq. (40) of the previous paper to be proportional to liT. Therefore, we write Aa6= AI(v)
p.,
(13)
where Pa is the absorber pressure and Al(v) is a parameter which should have some dependence on the vibrational quantum numbers and 296 K is the reference temperature. The foreign-broadening parameter is X3/4 (14) which can be written simply as /~.b4 -~- A2(v,j)(2-~-)pb,
(15)
where Pb is the broadener pressure and A2(v,J) is a parameter which depends on the vibrational quantum number and on the rotational quantum number J. Because of the statistical averaging over rotations, the H20-H20 potential matrix elements for strong collisions are independent of angle and not a function of the rotation quantum number J. For N2-H20 collision, however, the interaction times are shorter than the rotational times; thus, no averaging over rotations is performed. Thus, A2(v, J) will be a function of the J quantum number. This J dependence can be expressed in terms of the J dependence of the near-linecenter halfwidth ~o9
A2(v,J)=
• 0~0 . 1.125 a 2 ( v ) \!(~.-.-.-.~)
(16)
where ao is the near-line-center halfwidth, and 0.07 represents a mean value of So for pressure broadened water vapor lines in the mid-i.r. The pressure and temperature dependence of the halfwidth a was taken as /296\°83 f
o= otT ) /B
(2~6) °'7
pa+p
} (17)
The N2-broadenedwater vapor absorptionline shape and infrared continuumabsorption--II
107
where p, is the absorber pressure and Pb is the broadener pressure. B is the self-to-foreign broadening ratio which is typically set to 5. The self-broadened component of the halfwidth has a temperature dependence of I/T, while the foreign-broadened component has a temperature dependence of I/T °sa as discussed in the previous paper. The line strength is
S = S o Top - ~ o ( " ~To ) e x1., p[
Ei[T-T°~]\kTTo ]J
{~ :~j-exp(-hcv°lkT)~ {ScoR}.
(11)
The factor written as SCORrepresents a correction to the line strength factor which takes account of the fact that for shifted energy levels the Boltzmann population distribution is not calculated from the energy level separation hvo, but rather from the total energy separation hvo + AE. The correction is important for the rotational band where different rotational levels are populated at room temperature. The correction is far less important for the vibrational bands. The term ScoR can be shown 9 to take the form I v ( 1 - e x p (-hcv/kT)) Vo(1 - exp (-hcvo/kT))
near line center
(12a)
far wing.
(12b)
{ScoR} = (1 + exp (- hcvlkT)) (1 - exp ( - hcvo/kT))
Since these terms are frequency dependent, we can include them in the line shape term. These terms only affect absorption lines below approximately 1000 cm -1. The total line shape can be written as
ToPa @ )
t.5 e El(T- T°/k'V/'°) (1 -
e-h~'~kr) ( 1 - e-~"o'~°) '
s,~ = s,° r-G~o
1 v
( 1 - e -~c/kT)
a"
(13)
JcNLC(V) = 7r V0 (1 -- e-v°hcliT)((Av)2 + (Ct r)2), • 1 ( 1 - e -v"'tkr) [0.3198A,6 e -o*tvla'l 0.4334Abae -Gb'vlAvI'I J~Fw(V)='~(1--e-"Oh~tkT) L IAvll5+x~6 + lavll7'+x ]
where •
Ga6 = ga(v) ~---~]
,
Oto .
1.125
Gb4 = gb(V)
As mentioned earlier, the near-line-center profile and the far wing profile were blended through the intermediate wing region by using the p-function
k(v) = ~ S~N{jcr~LC(V- vo, ; fli)p(Av) + ]crw(v - vo,; fli)(1 - p(Av))}.
(14)
The normalization constant for each line is determined by imposing Eq. (6) on that line. COMPARISON WITH EXPERIMENT A computer program was written which incorporated this line shape in a line-by-line calculation of absorption in the i.r. Before calculations of the contribution of far wings to H20 continuum absorption could be
108
M. E. THOMASand R. J. NORDSTROM Table 1. The far wingparameters. AI (Rotational)
= 2,35
A2 (Rotational)
= 0.18
ga (Rotational)
= 0.065
Ya (Rotational)
= 0.67
gb (Rotational)
= 0.115
y b (Rotational)
= 0.67
AI (V2) = 1.73
ga (v2) = 0.07
'fa (v2) = 0,67
i
A2 (~2) = 0.153
gb (v2) = 0.084
~b ( " 2 )
i
A1 (~1' ~3 ) = 1.95
ga (Vl" v3) = 0.045
"~a (Vl' \~3) = 0.67
A2 (~1' ~3 ) ~ 0 . [ 6 5
gb (Vl' v3) = 0.088
*b (~i' v3 ) = 0.67
= 0.67
I
made, the various parameters had to be evaluated. This was done by fitting predicted absorption coefficients to measured absorption coefficients in spectral regions where line absorption clearly dominates. It must be mentioned here that absorption data in the window regions where continuum can be measured was not used in this evaluation of the parameters. The regions chosen for the parameter evaluation were located in the near absorption band regions (i.e., 3, 5, and 25/~m) where far wings are recognized to exist. Single frequency measurements located between lines were used since the absorption is dominated by the wings of local lines and strong absorption bands. Data provided by Burch m were used to evaluate the parameters which define the rotational band. Data recorded with a CO laser at our laboratory9 were used to evaluate parameters of the v2 vibrational band near 6.6/~m, and HF laser measurements made by White et al. n were used to estimate the parameters in the v], v3 spectral region near 3/~m. In all cases it was clear that insufficient temperature data existed to make an accurate estimate of certain temperature controlling parameters such as y,(v) and yh(v). For these parameters, it was decided to vary them only slightly if necessary from the value 0.67 given by Fomin and Tvorogov. 8 No attempt at a least square analysis was made since the data do not yet possess the accuracy and completeness to make such an analysis meaningful. Table 1 lists the resulting parameters for each H20 absorption band. As can be seen, the largest deviation between parameter values occurs between the rotational band and the rest of the absorption bands. Again, it is important to emphasize that absorption data in the window regions were not used in the evaluation of these parameters. Once the parameters wer.e estimated, we used the total line shape to calculate the absorption at CO2 laser frequencies in the 10 t~m atmospheric transmission window. Figure 2 shows the results of the calculation at room temperature for the pressure dependence of the absorption coefficient at one of the laser lines. A bound of 1500 cm-' was used, although we found that increasing this made no significant difference to the calculated absorption coefficient. The 0.4~ P~24) CO z
940.548 LASER
0.3-I
i
,/
D
E -~
0.2-m
0.1 X
o
I O
I
4
8 Po
J
I
i
12
16
(Torr)
Fig. 2. Roomtemperaturemodellingof watervaporpressuredependence.
The N2-broadenedwater vapor absorption line shape and infrared continuum absorption--lI 1 0 .20
I
10-21
-
I
~
l
i
I
109
I
VAN VLECK-WEISSKOPF
E
EXPERIMENTAL + -I+
+
T E
SL
10-22
OCO (D
~
4 4-
+
H
I
.,.+ ÷+~...~,...,~" ~ J
S
WORK
FULL LORENTZ 10-23
~SIMPLE
LORENTZ
I
I
1200 C"
1
I
1000 WAVENUMBER (cm-1)
i
1
800
;
1'0 '1'1 1'2 1'3 1'4 WAVELENGTH (microns) Fig. 3. Experimental results in the 8 to i4 pm region.
predicted absorption is in very good agreement with the measured absorption at all pressures. It should be pointed out that the curvature in the data (seen as a p2 dependence) is well modelled by this line shape theory. This is a manifestation of the increased importance of H20-H20 interactions over H20--N2 interactions for strong collisions (see previous paper). In contrast, when the Lorentz profile is used (as in Fig. 1) the curvature in the data is not well modelled. Our results in modelling the overall continuum absorption throughout the 8-14/~m atmospheric window can be seen in Fig. 3. Measurements of the self-broadened coefficient Cs made by Burch '2 were compared to predictions calculated with various classic line shape formalisms. Using our line shape formalism, a very good fit to the data was achieved. We have also applied the line shape theory to the 3.5-4 t~m transmission window. DF laser measurements of Ref. 13 did not confirm earlier spectroscopic measurements made of Ref. 14. The measurements of Ref. 13 were consistently higher than the absorption values reported in Ref. 14 and the curvature as a function of frequency in Ref. 13 throughout this region was much greater than in Ref. 14. Figure 4 shows the results of our calculations in the 3.5-4 #m region. The squares are the total absorption calculated using water vapor absorption lines and a partial pressure of 15 torr in an environment of 745 torr N2. This spectral region contains many absorption lines of the HDO molecule. Burch was careful to avoid regions close to the centers of these absorption lines when he made his measurements. We can accomplish this in our calculations by subtracting the local line contribution within a few wavenumbers. The triangles show the residual continuum absorption predicted by our line shape calculation. Agreement with the data of Ref. 14 is quite good. Modelling the temperature dependence is much more difficult because a precise understanding of intermolecular interactions does not exist (particularly for the rotational band) and accurate temperature data do not exist for all of the absorption bands of H20 in the i.r. Further, the exponential term of the level shift part of the statistical line shape as derived by Fomin and Tvorogov8 comes from a perturbation solution. Thus, for the very far wing the formalism breaks down. The temperature dependence of this term in particular becomes highly suspect. In fact, it counters the negative dependence of the phase shift part in the far wing which was solved without using perturbation techniques; it represents the essential physics of the leading term in the interaction potentials. The consequences of this inadequacy has limited the ability of the line shape model to predict the observed temperature dependence of wing absorption as successfully as the frequency and pressure dependence. For example, the near rotational band temperature data of Burch 1°is not well modelled. These measurements of absorption are known to be wing dominated and, therefore, verify the suspicions of inadequate temperature modelling. To see the effect of a correctly derived temperature dependence in the model, the
--
--
0.11
0.10
~
-
-1
--
-
--
--
0 2470
0.01
0.02
0.03 ~-
0.04
0.05
0.06
IE 0.07 _=
0.08 --
0.09
--
0.12
0.13
t
2550
o~
[]
o
I
2600
[]
o
CALCULATED DATA TOTAL ABSORPTION
O
0
CALCULATED DATA FAR W I N G S
EXPERIMENTAL
A
BURCH DATA
pNz- 7 4 5 T o r t
0
f
n
2650 V ( cm-I)
0
I
O
2700
0
O
L
2750
[]
El
I
I
2800
o
Fig. 4. Comparison of experimental and theoretical frequency dependence of the absorption coefficient from 2470 cm -I to 2870 cm -t.
I
2500
1~
K
Po " 15 T o r t
T " 296*
A
[]
I 2850
[]
zx
E!
o
=o
z
.=
o
E p~
The Nz-broadened water vapor absorption line shape and infrared continuum absorption--II
P (20) t
944.1945 CO z
?_,
cm " t
LASER o OSU
o
111
0
•
DATA
SOVIET
DATA
eo
i °----__<______ \l CALCULATED
2
0
LORENTZ NO BOUND
(1)
•
\
0
285
I
I
305
325
I 345
5,5
T - Kelvin
Fig. 5. Comparison of experimental and theoretical temperature dependence of C, at 944.1945 cm-~.
temperature dependence of the X,b term was changed to allow agreement with Burch's near band data. With the values of the line shape parameters listed in Table 1, we calculated the absorption coefficient of the P(20) CO2 laser line at 944.1945 cm-t. Data which exist3"15on the absorption in this region indicate a strong, negative temperature dependence. The values of ~,a and ~,b were fixed to 0.67. Figure 5 shows the results of our modelling of the temperature dependence. A negative temperature dependence is shown by the line shape model. Without modifications to the model, a negative temperature dependence is observed in the 10 and 4 t~m window regions; however, it is weaker than the measured data particularly for the 10 #m window. The point to be made is that a proper far wing model should predict the observed temperature data, as does the model of hydrogen bonding. An adequate line shape model for atmospheric H20 is important, not only for modelling the continuum absorption but also for calculating the effects of overlap between H20 absorption line features and weak absorptions from a possible trace gas in the atmosphere. Recently, Menyuk et al) 6 used the differential absorption lidar technique to detect trace amounts of NO in the atmosphere. Interference from strong absorption lines of H20 was not adequately compensated for by two separate programs provided by AFGL. We were asked to calculate the absorption coefficient of pressure-broadened water vapor at their measurement frequencies. Using the results which we provided to them, they recomputed the effects of H20 interference in their experiments and found much better agreement with their experimental results. CONCLUSIONS
In these papers, we have developed and discussed the shape of pressure-broadened H20 absorption lines in the i.r. spectral region. The water vapor continuum in both the 8-14 ~m and in the 3.5-4 tzm atmospheric windows is quite well modelled by this absorption line shape. The strength of this model is in predicting the observed frequency and pressure dependence of water vapor absorption throughout the entire i.r. This work represents initial efforts in describing the continuum absorption water vapor in terms of far wings. This is a very complex problem, however, and the theory is embryonic in nature with many limitations. Nonetheless, the line shape model as presented in our two papers is quite successful in demonstrating the importance of far wings in continuum absorption. REFERENCES 1. R. G. Breene, The Shift and Shape of Spectral Lines. Pergamon Press, Oxford (1961). 2. S. A. Clough F. X. Kneizys, R. Davies, R. Gamache, and R. Tipping, "Theoretical Line Shape for H20 Vapor: Application to the Continuum", in Atmospheric Water Vapor (Edited by Adarsh Deepak, T. D. Wilkerson, and L. H. Ruhnke). Academic Press, New York (1980). 3. J. C. Peterson, "A Study of Water Vapor Absorption at CO2 Laser Frequencies Using a Differential Spectrophone and White Cell", Ph.D, Dissertation, The Ohio State University, June 1978.
112
M. E. THOMASand R. J. NORDSTROM
4. R. J. Nordstrom, M. E. Thomas, J. C. Peterson, E. K. Damon, and R. K. Long, Appl. Opt. 17, 2724 (1978). 5. J. C. Peterson, M. E. Thomas, R. J. Nordstrom, E. K. Damon, and R. K. Long, Appl. Opt. 18, 834 (1979). 6. R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, and J. S. Garing, U.S. Air Force Research Laboratories, AFCRL-TR-73-0096, Bedford, Mass. (1973). 7. P. Varansia, S. Chou, and S. S. Penner, JQSRT 8, 1537 (1968). 8. V. V. Fomin and S. D. Tvorogov, Appl. Opt. 12, 584 (1973). 9. M. E. Thomas, "Tropospheric Water Vapor Absorption in the Infrared Window Regions", gep. 784701-5, Aug. 1979,The Ohio State University ElectroScience Laboratory, Department of Electrical Engineering; prepared under Contract DAAG-29-77-C-0010 for U.S. Army Research Office. 10. D. A. Gryvnak and D. E. Burch, "Infrared Absorption by CO2 and H20", Aeronutronic Publication U-6417, Air Force Geophysics Laboratories, Contract F19628-76-C-0302 (May 1978). 11. W. R. Watkins, R. L. Spellicy, K. O. White, B. Z. Sojka, and L. R. Bower, Appl. Opt. 18, 1582 (1979). 12. D. E. Burch, Aeronutronic Publication No. U-4784, Semi-Annual Technical Report, AFCRL Contract No. F19628-69-C0263, U.S. Air Force (1970). 13. K. O. White, W. R. Watkins, C. W. Bruce, R. E. Meridth, and F. F. Smith, Appl. Opt. 17, 2711 (1978). 14~ D. E. Burch, D. A. Gruyvnak, and J. D. Pembrook, Aeronutronic Publication No. U-4897, "Investigation of the Absorption of Infrared Radiation by Atmospheric Gases: Water, Nitrogen, Nitrous Oxide". Air Force Cambridge Research Laboratories, Contract No. F19628-69-C-0263, Jan. 1971. 15. V. N. Aref'ev and V. I. Dianov-Klokov, Opt. Spectrosc. 42, 488 (1977). 16. N. Menyuk, D. K. Killinger, and W. E. DeFeo, Appl. Opt. 19, 3282 (1980).
0022-4073/82/08013-10503.00/0 PergamonPressLid.
Printedin GreatBritain.
THE N2-BROADENED WATER VAPOR ABSORPTION LINE SHAPE AND INFRARED CONTINUUM ABSORPTIONtulI. IMPLEMENTATION OF THE LINE SHAPE MICHAELE. THOMAS~t The Johns Hopkins University, Applied Physics Laboratory, Laurel, MD 20707, U.S.A.
and ROBERT J. NORDSTROM Battelle Columbus Laboratories, Columbus, OH 43201, U.S.A. (Received 17 September 1981)
AIMtract--The total line shape model of the previous paper is tested using a set of experimental room temperature H20 continuum measurements of high quality. Parameters of the far wing component of the total line shape are determined from near band experimental data. Grating spectrometer measurements from 300 to 650cm -t are used to determine unknown far wing parameters of the pure rotational band of H20. CO and HF laser measurements taken in the 5 and 3 ~m regions are used to determine the far wing parameters of the v2 and v~, v3 fundamental bands, respectively. The total line shape model is applied to the 10 and 4 ~m transmission windows with encouraging success. A significant increase in the self-broadening ability of H20 over N2 is predicted in the far wing. This procedure allows the proper modeling of the absorption coefficient vs H20 partial pressure dependence in all window regions. A negative temperature is predicted by the model in the continuum. The observed rate of the temperature decrease is not predicted by the model; however, this limitation is related to the approximations made on the interaction potentials and the perturbation expansion of the Hamiltonians. Although the total line shape has limitations, it does demonstrate the importance of considering far wings of absorption lines in continuum absorption. INTRODUCTION
In the preceding paper, we have developed an expression for the pressure-broadened absorption line shape of H20 molecules. Only binary collisions were assumed and both self-broadening and foreign-broadening with N2 were included. The spectral line shape was developed by first considering the time evolution of the correlation function of the energy levels of the colliding molecules. Collisions were categorized by their severity: weak collisions contributed only to the near-line-center of the spectral profile, while strong collisions contributed to the line shape everywhere including the far wings. The study of H20 continuum absorption must include a study of the accumulation of these far wings. Statistical broadening I was used in our development to formulate the far wing component to the spectral line shape. Recently, another line shape development used only the impact approximation, modified to account for variations in collision duration. 2 It is not clear at this time whether there is any significant difference in the line shapes produced by these two separate approaches. In this paper, we demonstrate how the spectral line profile, defined by Eqs. (49), (77), and (78) of the preceding paper, is implemented. Furthermore, comparison between exerimental absorption data and the absorption predicted by this line shape is given. We will show that a line shape formulation is adequate to predict the total behavior of water vapor continuum absorption over the ranges of pressure and temperature of tropospheric interest. Data will be shown for the 8-14 and 3.5-4 ~tm atmospheric transmission windows. HISTORICAL PERSPECTIVE
Figure 1 demonstrates the inability of the Lorentz line shape to model the observed pressure dependence of water vapor continuum absorption near 10/zm for H20 samples pressure broadened to 1 atm with N2. The data were recorded using a frequency-stabilized CO2 laser and a long-path absorption cell described in several references)-5 The curve marked A was tThis work done under ARO Contract DAAG-29-77-C-0010 while both authors were with the Ohio State University, ElectroScience Laboratory, Columbus, OH 43212, U.S.A. *Author to whom correspondence should be sent. 103
104
M . E . THOMASandR. J. No~sTaOM 0.4 Pt24) CO 2
_.-..
940.548 LASER
0.3
q E
0.2
63
B
o1l i o_ ~ o
4
8 Po (Torr)
12
16
Fig. 1. Pressure dependence of water vapor continuum absorption.
generated from data on the AFGL tape 6 using a bound of 20 cm -j. That is, for the calculations at each H20 partial pressure, only the H20 absorption lines within a range of -+20 cm -~ from the measurement frequency were assumed to contribute to the absorption at that frequency. All other H20 absorption lines were assumed to be too far away to make a significant contribution. This bound of 20 cm -1 became a standard in early line-by-line calculation programs because it was believed to be a reasonable compromise between spectral representation and computing time. The curve marked B in Fig. 1 was generated by accumulating the contributions of all H20 absorption lines from 0 to 5000 cm -~. The contributions at the measurement frequency were computed using the Lorentz line shape for all lines listed on the data tape. At low H20 partial pressures, this line shape overestimates the observed absorption, while at high H20 partial pressures it underestimates the observed absorption. Despite this failure, the difference between curve A and curve B aemonstrates that significant contribution to the total absorption can be found in the accumulation of the far wings of strong absorption lines. The Lorentz profile, however, is not an adequate model for the far wings of lines. It is therefore not surprising that curve B fails to provide suitable agreement with the measured absorption. Motivated by the hope that an accurately derived H20 absorption line shape would improve the modeling of continuum absorption, we undertook the work presented in this paper and in the previous paper. THE LINE PROFILE
The absorption coefficient at frequency u (in cm i) is k(~) = ~ Sj(v - v0,;/3,),
(1)
where S~ is the strength of the ith absorption line and j(V-voi;/3i) is the shape of the absorption line centered at voi and which has a set of shape parameters {/3~}.The major features of the total line shape have been derived in the previous paper. It was found that jc(mv) = NldCNLC(Av)p(Av) + jC~w(Av)[ I - p(Av)]},
(2)
where the subscript C shows that the line shape is collision broadened, 1
j,'NLc(av)
O~R
~ (av + af)2+ (aR) 2'
(3)
The N2-broadenedwatervaporabsorptionline shapeand infraredcontinuumabsorption--II • 1 AoE lcrw(AV)=-~{~Ym~(mk, Av)+~
Ym,(mi,Av)} ,
(4)
p(Av)= f 1/2+l/2c°s ( 7 Av)IAvi <5 cm-', 0
105
(5)
la~l _>5 cm-';
N is the normalization constant. Normalization requires that f
jc(Av) dv = 1.
(6)
In order to eliminate the singularities in the far wing line shape functions at Av = 0, we now adopt the formulation of Ref. 7, which includes an additive term'in the denominators of Eq. (4). Furthermore, we now set 6 and mi = 4 as discussed in the previous paper. Therefore,
mk=
1 f 0.3198Aa6
j c ~ ( a v ) = g/lavl'" + x~:6Yt(m,
0.4334Ab4 = 6, Av) + IAvlt.7s +
(Ab4)7/3Y4(mj
Av)}. = 4,
(7)
This modification of the line-shape function does not significantly alter the value of the function when Av is very large. It assures that this function is well behaved at the line center for normalization purposes. Furthermore, in our formulation, the p-function is turning off the far wing line shape near Av = 0 and any constant value of Jcrw(Av) at Av = 0 artificially introduced by this modification is given a weighting of zero. The multiplicative constants which appear in Eq. (7) are due to the normalization introduced by the Varanasi formulation. An evaluation of )%6 and Ab4in far wing data (Av ,> Aat, )tb4) would simply be scaled by these constants. The functions Yt(mk = 6, Av) and Y4(mj = 4, Av) require discussion. As mentioned in the previous section, Fomin and Tvorogovg derived a term of the form
exp { - L
T
J J'
where G is a constant and T is the temperature. An exponential decay of some form is expected in the far wings of absorption lines as discussed by Clough and his coworkers. 2 For self-broadening (m = 6) the above term is proportional to exp (-[Av]°56), and for foreignbroadening this term is proportional to exp (-IAvl°%. We have adopted a simplification of this term. We have set Y6(mk = 6, Av) = exp (- Ga6IAv[0"5)
(8a)
Y4(mj= 4, Av) = exp (-Gb41Av[°5).
(8b)
and
We use the same frequency dependence for both the self- and foreign-broadened terms in order to facilitate the normalization process. The origin of these terms is the level shift contribution which is not fully incorporated in the phase shift approach. Such terms are impossible to derive by perturbation methods since, by definition, in the far wings of i.r. absorption lines the perturbation energy (Av) is no longer small compared to the unperturbed rotational energy levels of the absorbing molecule. The G parameters above are
QSRT VoL 28, No. 2---C
G.6=g.(v)(~) "a'v),
(9a)
Gb4 = g~(v)
(9b)
,
106
M.E. THOMASand R. J. NORDSTROM
where the parameters g,(v), gb(v), ~/,(V), and yb(V) depend on the vibrational quantum numbers. In particular, it can be anticipated that the parameters might be slightly different for the pure rotational band (v = 0) where various rotational levels are more easily accessible through thermal energy than in the case for the vibrational energy levels. Thus we might expect one set of parameters to describe the shape of lines in the rotational band and another set of parameters to describe the shape of absorption lines of the vibrational bands. Equation (7) then becomes
jcvw(Av) =
1 f 0.3198A,6
0.4334Ab4
}
g [IA,,I ' ' + XLb exp (-G~X/IA~I) + lavl,.7: + 17,2 exp ( - Gb4VIAvl) .
(1o)
The temperature and pressure dependences of Gb and Ab4 can be extracted by examining the inputs to Eq. (70a) in the previous paper, viz.
Abj = Ok ~, O~b3'~,
(11)
where the terms have been defined previously. For self-broadening, Eq. (l 1) becomes /~a6 = ~ T ~s
27rl 1/2'61/2F ( - ~ ) C O S ( 4 Psb-~(W6u(Z)-W6l(Z))
)"
(12)
The potential for self-broadening is seen in Eq. (40) of the previous paper to be proportional to liT. Therefore, we write Aa6= AI(v)
p.,
(13)
where Pa is the absorber pressure and Al(v) is a parameter which should have some dependence on the vibrational quantum numbers and 296 K is the reference temperature. The foreign-broadening parameter is X3/4 (14) which can be written simply as /~.b4 -~- A2(v,j)(2-~-)pb,
(15)
where Pb is the broadener pressure and A2(v,J) is a parameter which depends on the vibrational quantum number and on the rotational quantum number J. Because of the statistical averaging over rotations, the H20-H20 potential matrix elements for strong collisions are independent of angle and not a function of the rotation quantum number J. For N2-H20 collision, however, the interaction times are shorter than the rotational times; thus, no averaging over rotations is performed. Thus, A2(v, J) will be a function of the J quantum number. This J dependence can be expressed in terms of the J dependence of the near-linecenter halfwidth ~o9
A2(v,J)=
• 0~0 . 1.125 a 2 ( v ) \!(~.-.-.-.~)
(16)
where ao is the near-line-center halfwidth, and 0.07 represents a mean value of So for pressure broadened water vapor lines in the mid-i.r. The pressure and temperature dependence of the halfwidth a was taken as /296\°83 f
o= otT ) /B
(2~6) °'7
pa+p
} (17)
The N2-broadenedwater vapor absorptionline shape and infrared continuumabsorption--II
107
where p, is the absorber pressure and Pb is the broadener pressure. B is the self-to-foreign broadening ratio which is typically set to 5. The self-broadened component of the halfwidth has a temperature dependence of I/T, while the foreign-broadened component has a temperature dependence of I/T °sa as discussed in the previous paper. The line strength is
S = S o Top - ~ o ( " ~To ) e x1., p[
Ei[T-T°~]\kTTo ]J
{~ :~j-exp(-hcv°lkT)~ {ScoR}.
(11)
The factor written as SCORrepresents a correction to the line strength factor which takes account of the fact that for shifted energy levels the Boltzmann population distribution is not calculated from the energy level separation hvo, but rather from the total energy separation hvo + AE. The correction is important for the rotational band where different rotational levels are populated at room temperature. The correction is far less important for the vibrational bands. The term ScoR can be shown 9 to take the form I v ( 1 - e x p (-hcv/kT)) Vo(1 - exp (-hcvo/kT))
near line center
(12a)
far wing.
(12b)
{ScoR} = (1 + exp (- hcvlkT)) (1 - exp ( - hcvo/kT))
Since these terms are frequency dependent, we can include them in the line shape term. These terms only affect absorption lines below approximately 1000 cm -1. The total line shape can be written as
ToPa @ )
t.5 e El(T- T°/k'V/'°) (1 -
e-h~'~kr) ( 1 - e-~"o'~°) '
s,~ = s,° r-G~o
1 v
( 1 - e -~c/kT)
a"
(13)
JcNLC(V) = 7r V0 (1 -- e-v°hcliT)((Av)2 + (Ct r)2), • 1 ( 1 - e -v"'tkr) [0.3198A,6 e -o*tvla'l 0.4334Abae -Gb'vlAvI'I J~Fw(V)='~(1--e-"Oh~tkT) L IAvll5+x~6 + lavll7'+x ]
where •
Ga6 = ga(v) ~---~]
,
Oto .
1.125
Gb4 = gb(V)
As mentioned earlier, the near-line-center profile and the far wing profile were blended through the intermediate wing region by using the p-function
k(v) = ~ S~N{jcr~LC(V- vo, ; fli)p(Av) + ]crw(v - vo,; fli)(1 - p(Av))}.
(14)
The normalization constant for each line is determined by imposing Eq. (6) on that line. COMPARISON WITH EXPERIMENT A computer program was written which incorporated this line shape in a line-by-line calculation of absorption in the i.r. Before calculations of the contribution of far wings to H20 continuum absorption could be
108
M. E. THOMASand R. J. NORDSTROM Table 1. The far wingparameters. AI (Rotational)
= 2,35
A2 (Rotational)
= 0.18
ga (Rotational)
= 0.065
Ya (Rotational)
= 0.67
gb (Rotational)
= 0.115
y b (Rotational)
= 0.67
AI (V2) = 1.73
ga (v2) = 0.07
'fa (v2) = 0,67
i
A2 (~2) = 0.153
gb (v2) = 0.084
~b ( " 2 )
i
A1 (~1' ~3 ) = 1.95
ga (Vl" v3) = 0.045
"~a (Vl' \~3) = 0.67
A2 (~1' ~3 ) ~ 0 . [ 6 5
gb (Vl' v3) = 0.088
*b (~i' v3 ) = 0.67
= 0.67
I
made, the various parameters had to be evaluated. This was done by fitting predicted absorption coefficients to measured absorption coefficients in spectral regions where line absorption clearly dominates. It must be mentioned here that absorption data in the window regions where continuum can be measured was not used in this evaluation of the parameters. The regions chosen for the parameter evaluation were located in the near absorption band regions (i.e., 3, 5, and 25/~m) where far wings are recognized to exist. Single frequency measurements located between lines were used since the absorption is dominated by the wings of local lines and strong absorption bands. Data provided by Burch m were used to evaluate the parameters which define the rotational band. Data recorded with a CO laser at our laboratory9 were used to evaluate parameters of the v2 vibrational band near 6.6/~m, and HF laser measurements made by White et al. n were used to estimate the parameters in the v], v3 spectral region near 3/~m. In all cases it was clear that insufficient temperature data existed to make an accurate estimate of certain temperature controlling parameters such as y,(v) and yh(v). For these parameters, it was decided to vary them only slightly if necessary from the value 0.67 given by Fomin and Tvorogov. 8 No attempt at a least square analysis was made since the data do not yet possess the accuracy and completeness to make such an analysis meaningful. Table 1 lists the resulting parameters for each H20 absorption band. As can be seen, the largest deviation between parameter values occurs between the rotational band and the rest of the absorption bands. Again, it is important to emphasize that absorption data in the window regions were not used in the evaluation of these parameters. Once the parameters wer.e estimated, we used the total line shape to calculate the absorption at CO2 laser frequencies in the 10 t~m atmospheric transmission window. Figure 2 shows the results of the calculation at room temperature for the pressure dependence of the absorption coefficient at one of the laser lines. A bound of 1500 cm-' was used, although we found that increasing this made no significant difference to the calculated absorption coefficient. The 0.4~ P~24) CO z
940.548 LASER
0.3-I
i
,/
D
E -~
0.2-m
0.1 X
o
I O
I
4
8 Po
J
I
i
12
16
(Torr)
Fig. 2. Roomtemperaturemodellingof watervaporpressuredependence.
The N2-broadenedwater vapor absorption line shape and infrared continuum absorption--lI 1 0 .20
I
10-21
-
I
~
l
i
I
109
I
VAN VLECK-WEISSKOPF
E
EXPERIMENTAL + -I+
+
T E
SL
10-22
OCO (D
~
4 4-
+
H
I
.,.+ ÷+~...~,...,~" ~ J
S
WORK
FULL LORENTZ 10-23
~SIMPLE
LORENTZ
I
I
1200 C"
1
I
1000 WAVENUMBER (cm-1)
i
1
800
;
1'0 '1'1 1'2 1'3 1'4 WAVELENGTH (microns) Fig. 3. Experimental results in the 8 to i4 pm region.
predicted absorption is in very good agreement with the measured absorption at all pressures. It should be pointed out that the curvature in the data (seen as a p2 dependence) is well modelled by this line shape theory. This is a manifestation of the increased importance of H20-H20 interactions over H20--N2 interactions for strong collisions (see previous paper). In contrast, when the Lorentz profile is used (as in Fig. 1) the curvature in the data is not well modelled. Our results in modelling the overall continuum absorption throughout the 8-14/~m atmospheric window can be seen in Fig. 3. Measurements of the self-broadened coefficient Cs made by Burch '2 were compared to predictions calculated with various classic line shape formalisms. Using our line shape formalism, a very good fit to the data was achieved. We have also applied the line shape theory to the 3.5-4 t~m transmission window. DF laser measurements of Ref. 13 did not confirm earlier spectroscopic measurements made of Ref. 14. The measurements of Ref. 13 were consistently higher than the absorption values reported in Ref. 14 and the curvature as a function of frequency in Ref. 13 throughout this region was much greater than in Ref. 14. Figure 4 shows the results of our calculations in the 3.5-4 #m region. The squares are the total absorption calculated using water vapor absorption lines and a partial pressure of 15 torr in an environment of 745 torr N2. This spectral region contains many absorption lines of the HDO molecule. Burch was careful to avoid regions close to the centers of these absorption lines when he made his measurements. We can accomplish this in our calculations by subtracting the local line contribution within a few wavenumbers. The triangles show the residual continuum absorption predicted by our line shape calculation. Agreement with the data of Ref. 14 is quite good. Modelling the temperature dependence is much more difficult because a precise understanding of intermolecular interactions does not exist (particularly for the rotational band) and accurate temperature data do not exist for all of the absorption bands of H20 in the i.r. Further, the exponential term of the level shift part of the statistical line shape as derived by Fomin and Tvorogov8 comes from a perturbation solution. Thus, for the very far wing the formalism breaks down. The temperature dependence of this term in particular becomes highly suspect. In fact, it counters the negative dependence of the phase shift part in the far wing which was solved without using perturbation techniques; it represents the essential physics of the leading term in the interaction potentials. The consequences of this inadequacy has limited the ability of the line shape model to predict the observed temperature dependence of wing absorption as successfully as the frequency and pressure dependence. For example, the near rotational band temperature data of Burch 1°is not well modelled. These measurements of absorption are known to be wing dominated and, therefore, verify the suspicions of inadequate temperature modelling. To see the effect of a correctly derived temperature dependence in the model, the
--
--
0.11
0.10
~
-
-1
--
-
--
--
0 2470
0.01
0.02
0.03 ~-
0.04
0.05
0.06
IE 0.07 _=
0.08 --
0.09
--
0.12
0.13
t
2550
o~
[]
o
I
2600
[]
o
CALCULATED DATA TOTAL ABSORPTION
O
0
CALCULATED DATA FAR W I N G S
EXPERIMENTAL
A
BURCH DATA
pNz- 7 4 5 T o r t
0
f
n
2650 V ( cm-I)
0
I
O
2700
0
O
L
2750
[]
El
I
I
2800
o
Fig. 4. Comparison of experimental and theoretical frequency dependence of the absorption coefficient from 2470 cm -I to 2870 cm -t.
I
2500
1~
K
Po " 15 T o r t
T " 296*
A
[]
I 2850
[]
zx
E!
o
=o
z
.=
o
E p~
The Nz-broadened water vapor absorption line shape and infrared continuum absorption--II
P (20) t
944.1945 CO z
?_,
cm " t
LASER o OSU
o
111
0
•
DATA
SOVIET
DATA
eo
i °----__<______ \l CALCULATED
2
0
LORENTZ NO BOUND
(1)
•
\
0
285
I
I
305
325
I 345
5,5
T - Kelvin
Fig. 5. Comparison of experimental and theoretical temperature dependence of C, at 944.1945 cm-~.
temperature dependence of the X,b term was changed to allow agreement with Burch's near band data. With the values of the line shape parameters listed in Table 1, we calculated the absorption coefficient of the P(20) CO2 laser line at 944.1945 cm-t. Data which exist3"15on the absorption in this region indicate a strong, negative temperature dependence. The values of ~,a and ~,b were fixed to 0.67. Figure 5 shows the results of our modelling of the temperature dependence. A negative temperature dependence is shown by the line shape model. Without modifications to the model, a negative temperature dependence is observed in the 10 and 4 t~m window regions; however, it is weaker than the measured data particularly for the 10 #m window. The point to be made is that a proper far wing model should predict the observed temperature data, as does the model of hydrogen bonding. An adequate line shape model for atmospheric H20 is important, not only for modelling the continuum absorption but also for calculating the effects of overlap between H20 absorption line features and weak absorptions from a possible trace gas in the atmosphere. Recently, Menyuk et al) 6 used the differential absorption lidar technique to detect trace amounts of NO in the atmosphere. Interference from strong absorption lines of H20 was not adequately compensated for by two separate programs provided by AFGL. We were asked to calculate the absorption coefficient of pressure-broadened water vapor at their measurement frequencies. Using the results which we provided to them, they recomputed the effects of H20 interference in their experiments and found much better agreement with their experimental results. CONCLUSIONS
In these papers, we have developed and discussed the shape of pressure-broadened H20 absorption lines in the i.r. spectral region. The water vapor continuum in both the 8-14 ~m and in the 3.5-4 tzm atmospheric windows is quite well modelled by this absorption line shape. The strength of this model is in predicting the observed frequency and pressure dependence of water vapor absorption throughout the entire i.r. This work represents initial efforts in describing the continuum absorption water vapor in terms of far wings. This is a very complex problem, however, and the theory is embryonic in nature with many limitations. Nonetheless, the line shape model as presented in our two papers is quite successful in demonstrating the importance of far wings in continuum absorption. REFERENCES 1. R. G. Breene, The Shift and Shape of Spectral Lines. Pergamon Press, Oxford (1961). 2. S. A. Clough F. X. Kneizys, R. Davies, R. Gamache, and R. Tipping, "Theoretical Line Shape for H20 Vapor: Application to the Continuum", in Atmospheric Water Vapor (Edited by Adarsh Deepak, T. D. Wilkerson, and L. H. Ruhnke). Academic Press, New York (1980). 3. J. C. Peterson, "A Study of Water Vapor Absorption at CO2 Laser Frequencies Using a Differential Spectrophone and White Cell", Ph.D, Dissertation, The Ohio State University, June 1978.
112
M. E. THOMASand R. J. NORDSTROM
4. R. J. Nordstrom, M. E. Thomas, J. C. Peterson, E. K. Damon, and R. K. Long, Appl. Opt. 17, 2724 (1978). 5. J. C. Peterson, M. E. Thomas, R. J. Nordstrom, E. K. Damon, and R. K. Long, Appl. Opt. 18, 834 (1979). 6. R. A. McClatchey, W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman, and J. S. Garing, U.S. Air Force Research Laboratories, AFCRL-TR-73-0096, Bedford, Mass. (1973). 7. P. Varansia, S. Chou, and S. S. Penner, JQSRT 8, 1537 (1968). 8. V. V. Fomin and S. D. Tvorogov, Appl. Opt. 12, 584 (1973). 9. M. E. Thomas, "Tropospheric Water Vapor Absorption in the Infrared Window Regions", gep. 784701-5, Aug. 1979,The Ohio State University ElectroScience Laboratory, Department of Electrical Engineering; prepared under Contract DAAG-29-77-C-0010 for U.S. Army Research Office. 10. D. A. Gryvnak and D. E. Burch, "Infrared Absorption by CO2 and H20", Aeronutronic Publication U-6417, Air Force Geophysics Laboratories, Contract F19628-76-C-0302 (May 1978). 11. W. R. Watkins, R. L. Spellicy, K. O. White, B. Z. Sojka, and L. R. Bower, Appl. Opt. 18, 1582 (1979). 12. D. E. Burch, Aeronutronic Publication No. U-4784, Semi-Annual Technical Report, AFCRL Contract No. F19628-69-C0263, U.S. Air Force (1970). 13. K. O. White, W. R. Watkins, C. W. Bruce, R. E. Meridth, and F. F. Smith, Appl. Opt. 17, 2711 (1978). 14~ D. E. Burch, D. A. Gruyvnak, and J. D. Pembrook, Aeronutronic Publication No. U-4897, "Investigation of the Absorption of Infrared Radiation by Atmospheric Gases: Water, Nitrogen, Nitrous Oxide". Air Force Cambridge Research Laboratories, Contract No. F19628-69-C-0263, Jan. 1971. 15. V. N. Aref'ev and V. I. Dianov-Klokov, Opt. Spectrosc. 42, 488 (1977). 16. N. Menyuk, D. K. Killinger, and W. E. DeFeo, Appl. Opt. 19, 3282 (1980).