Transmission measurements made on near-infrared laboratory methane spectra have previously been fit using a Malkmus band model. The laboratory spectra were obtained in three groups at temperatures averaging 112, 188, and 295 K; band model fitting was done separately for each temperature group. These band model parameters cannot be used directly in scattering atmosphere model computations, so an exponential sum model is being developed which includes pressure and temperature fitting parameters. The goal is to obtain model parameters by least square fits at 10/cm intervals from 3800 to 9100/cm. These results will be useful in the interpretation of current planetary spectra and also NIMS spectra of Jupiter anticipated from the Galileo mission

Benner, D. C.

Fink, U.

Giver, Lawrence P.

Kerola, D.

Tomasko, M. G.

English

NASA Technical Reports Server

N90-26758
GAUSSIAN QUADRATURE EXPONENTIAL SUM MODELING OF
NEAR INFRARED METHANE LABORATORY SPECTRA
OBTAINED AT TEMPERATURES FROM 106 TO 297 K.
LAWRENCE P. GIVER*, D. C. BENNER**, M. G. TOMASKO***, U. FINK***,
and D. KEROLA***
*NASA-Ames Research Center, N245-4, Moffett Field, CA 94035-1000.
**College of William and Mary, Physics Dept., Williamsburg, VA 23185.
***University of Arizona, Lunar and Planetary Lab., Tucson, AZ 85721.
ABSTRACT
Transmission measurements made on near-infrared laboratory methane spectra
have previously been fit using a Malkmus band model. The laboratory spectra were
obtained in three groups at temperatures averaging 112, 188, and 295 K; band model
fitting was done separately for each temperature group. These band model parameters
cannot be used directly in scattering atmosphere model computations, so an exponential
sum model is being developed which includes pressure and temperature fitting param-
eters. The goal is to obtain model parameters by least square fits at 10 cm -1 intervals
from 3800 to 9100 cm -1. These results will be useful in the interpretation of current
planetary spectra and also NIMS spectra of Jupiter anticipated from the Galileo mission.
INTRODUCTION
Three sets of about 30 near-infrared spectra of methane have previously been
fit using a Malkmus band model. 1 These spectra were obtained at NASA-Ames using
a refrigerated White cell with path lengths from 3 to 61 meters, pressures of pure
methane from several torr to several atmospheres, and temperatures clustered at 112,
188, and 295 K; at the lowest temperatures the maximum pressure used was limited to
0.5 atmosphere. An Eocom interferometer was used at 1 cm -1 resolution; the quartz
147
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beam-splitter together with the InSb detector produced a useful range for thesespectra
from 3760to 9200cm-1. The laboratory conditions of the spectra used in the Malkmus
band model analyses are listed in Tables 1-3. The transmissions measured at the 10 cm -1
interval centered at 4650 cm -1 are also listed, along with the calculated transmissions
from the Malkmus band model fit and the exponential sum fit, which are described
below.
At pressures sufficiently high that line profiles can be described by the Lorentz
formulation, the Malkmus band model can be written in the form
where T is the mean transmission of the spectral interval centered at wavenumber v, u is
the gas abundance, and P is the pressure. Only two parameters are determined from
least square fits to the laboratory spectra: the absorption coefficient kv, and the pressure
coefficient Yr. Our notation is similar to that used by Fink, eta/. _ for the Mayer-Goody
band model. Some of our spectra were obtained at low pressures, making it necessary to
include an approximation to the Voigt line profile in our computations of the Malkmus
band model parameters. But since we adopted mean pressure broadening coefficients
for methane lines at our three temperatures, there were still only two parameters, kv
and yv, determined from least square fits to the spectra in each temperature group.
Methane bands are major features in the spectra of all the outer planets and
the satellites Titan and Triton. Having obtained these laboratory spectra, determining
parameters describing the methane spectrum that can be used in modeling the moderate
resolution near-infrared spectra of these planets and satellites becomes a major goal.
The Malkmus bandmodel parameters have been extrapolated to conditions applicable
for modeling Triton's spectrum, assuming a clear atmosphere, s But scattering processes
are important in the other atmospheres, and although spectral band models have been
an accepted approach for nearly 50 years, 4 the radiative transfer equation in multi-layer
scattering model atmospheres cannot be solved with band model formulations of gaseous
148
absorption. Additionally, the Malkmus band model parameters were determined only
at three specific temperatures; a representation of the temperature dependenceat each
interval is also neededfor atmosphericmodeling computations.
For applications in scattering atmospheres,it is preferred that the gaseousabsorp-
tion be expressed as a weighted sum of exponentials. For example, McKay, eta/. 5 used
the following expression to transform the room temperature Mayer-Goody band model
parameters of Benner and Fink e into exponential sum parameters at intervals of about
500 cm -1 for use in Titan greenhouse model computations:
L/
T --Z .
Here w_ are weights, and k_ are the absorption coefficients, scaled by a pressure factor
pn, where the exponent n is a fitting parameter; this exponential sum model thus has
8 parameters to be determined from least square fits.
EXPONENTIAL SUM MODELS
In a similar fashion, Tomasko, eta/. 7 made some preliminary transformations of the
Malkmus band model parameterization of the Ames spectra for use in modeling certain
intervals of the Titan spectrum of Fink and Larson. s For computational efficiency in
the scattering model atmosphere, Tomasko preferred an 8 term exponential sum using
fixed Gaussian quadrature weights. Despite the higher number of terms, this model has
about the same number of parameters to determine as the 4 terms used by McKay, et
al., since the weights are fixed: wz = ws = 0.0506, w2 = w7 = 0.1112, ws = we = 0.1569,
and w4 = ws = 0.1813. In this model the absorption coefficients are required to increase
in order with the term number, i.
Instead of determining exponential sum parameters from transmission values com-
puted from band model fits, a more direct approach should be to determine the exponen-
tial sum parameters directly from the measured transmission values of the laboratory
149
spectra. Pressureand temperature parametersshould be included; with the large num-
ber of parameters, tbetter fits should be obtained than from 2 parameter band models.
The exponential sum model should have the following form:
where u is the methane abundance, fl(P) is a function of pressure, and fl(T) is a function
of temperature.
This expression requires that a number of decisions be made before final results
can be determined by least square fits. Since a pn dependence is not realistic at very
low pressures, we introduced a second pressure parameter, P':
Since the spectra were obtained in three temperature groups, we initially selected a
2 parameter temperature dependence:
where To = 273 K, and a and b are fitting parameters. An additional term-dependent
temperature parameterl e, was introduced when it was realized this would permit
better fits in the lower temperature groups:
PRELIMINARY RESULTS
The exponential sum model in equation (6) above has 13 fitting parameters; the
Ames transmission measurements of the 10 cm -1 spectral interval centered at 4650 cm -1
150
was fit quite well by this model. Comparisons of the observed transmissions, the ex-
ponential sum model computations, and the Malkmus band model computations are
presented in Tables 1-3. This spectral interval has both a substantial pressure depen-
dence and temperature dependence; being high in the R branch of the v2 + vs band, the
absorption decreases dramatically at low temperatures.
The fitting parameters for equation (6) determined at 4650 cm -1 are: k1=0.0227,
k_-0.0319, ks=0.0415, k4:0.0501, ks:0.1173, k6:0.4775, k7--I.159,ks:17.0, a:-0.845,
b:0.472, c:0.085, P':0.0383 atm., and n:0.615.
DISCUSSION
Several problems must be overcome before determining exponential sum model
parameters at all the spectral intervals where Malkmus band modeling was done. The
forms of the pressure and temperature functions must be chosen which can adequately
represent the transmission data at all spectral intervals. Spectra were obtained at
pressures from several Torr to several atmospheres; the 2 pressure parameters P' and
n of equation (4) may be insufficient to fit the data well at all intervals. Similarly, a
temperature dependence may be found to improve equation (5). We have made a few
tests to see if fits could be improved by allowing the pressure exponent, n, to vary with
the term index, i, in equation (6). Fits were significantly improved, but more fitting
parameters increase the risk of coupling between parameters, raising concerns about the
uniqueness of the fits, and significantly increasing computation time.
Even to get unique values for all 8 absorption coefficients, transmission values
must range from over 0.95 to less than 0.05. This requirement was readily met by the
Ames spectra at 4650 cm -1, as shown in Tables 1-3, but certainly not at all spectral
intervals. In regions of strong absorption, it will be useful to combine the Ames room
temperature spectra with some of the room temperature spectra obtained by Benner
and Fink 6 at the Lunar and Planetary Laboratory. The conditions of some of these
spectra are listed in Table 4. Since the abundance and pressure conditions of some of
151
Spectrum
Number
A002
A003
A004
A005
A006
A007
A012
A013
A014
A015
A016
A017
A026
A027
A028
A029
A030
A034
A036
A041
A042
A043
A044
A054
A055
A056
A057
A058
A059
Table i.
Ames Room Temperature Spectra
Methane Transmission at 4650 cm -l
Abund. Pressure Temp. Malkmus Exp. Sum
m-amgts, atm. Kelvln measured model model
0.2825 0.0971 296 0.946 0.952 0.939
0.7274 0.2500 296 0.875 0.888 0.871
1.66 0.5697 296 0.747 0.767 0.738
4.048 1.3878 296 0.520 0.527 0.513
8.806 3.0070 296 0.260 0.249 0.262
19.53 6.6260 295 0.043 0.046 0.043
1.145 0.1003 295 0.867 0.879 0.871
2.854 0.2500 295 0.725 0.741 0.727
6.498 0.5684 295 0.497 0.513 0.522
16.04 1.4010 295 0.228 0.195 0.232
34.55 3.0070 295 0.039 0.030 0.035
76.87 6.6460 295 0.000 0.000 0.000
0.295 0.0052 295 0.960 0.962 0.961
0.817 0.0144 295 0.930 0.929 0.920
2.243 0.0395 295 0.849 0.862 0.847
5.672 0.i000 295 0.704 0.724 0.693
14.19 0.2500 295 0.462 0.466 0.464
32.35 0.5697 295 0.202 0.179 0.196
170.9 2.9930 295 -0.003 0.000 0.000
2.558 0.i000 295 0.795 0.812 0.799
6.399 0.2501 296 0.609 0.614 0.602
14.59 0.5699 296 0.369 0.336 0.366
35.76 1.3950 295 0.092 0.070 0.084
0.5775 0.1004 295 0.911 0.921 0.909
1.438 0.2499 295 0.801 0.825 0.812
3.276 0.5696 295 0.637 0.652 0.637
8.081 1.4010 295 0.378 0.351 0.379
17:33 2.9930 295 0.124 0.107 0.126
44.7 7.6600 295 0.001 0.003 0.002
152
Spectrum
Number
A063
A064
A065
A066
A067
A068
A069
A072
A073
A074
A078
A081
A082
A083
A084
A087
A090
A091
A092
A093
A096
A099
AI00
AI01
AI02
A105
A108
AI09
All0
All1
All2
All5
All6
AI20
Table 2.
,,
Ames Methane Spectra near T=188 K.
Methane Transmission at 4650 cm -l
Abund. Pressure Temp. Malkmus Exp. Sum
m-amgts, atm. Kelvin measured model model
0.293 0.0637 187 0.965 0.968 0.966
0.767 0.1725 194 0.920 0.923 0.910
1.705 0.3771 191 0.822 0.839 0.827
4.257 0.9296 189 0.634 0.647 0.630
9.518 2.0610 189 0.387 0.379 0.389
61.47 12.2450 187 0.000 0.002 0.001
24.44 5.1770 189 0.073 0.084 0.081
0.59 0.0646 186 0.938 0.950 0.943
1.479 0.1620 186 0.869 0.884 0.879
3.467 0.3791 186 0.732 0.750 0.747
19.22 2.0880 187 0.222 0.206 0.235
1.147 0.0647 190 0.908 0.925 0.913
2.87 0.1609 189 0.808 0.828 0.822
6.803 0.3788 188 0.623 0.642 0.640
16.81 0.9276 187 0.368 0.337 0.373
38.93 2.1429 188 0.084 0.080 0.093
2.666 0.0650 184 0.863 0.879 0.868
6.605 0.1636 187 0.708 0.732 0.717
15.24 0.3828 190 0.489 0.486 0.492
37.07 0.9378 192 0.200 0.172 0.189
84.92 2.0880 188 0.019 0.019 0.015
0.302 0.0642 183 0.962 0.968 0.966
0.765 0.1633 184 0.916 0.924 0.916
1.782 0.3818 185 0.822 0.834 0.828
4.346 0.9330 186 0.640 0.643 0.63,1
9.619 2.0820 189 0.396 0.376 0.385
0.311 0.0036 191 0.965 0.979 0.976
0.842 0.0096 191 0.952 0.961 0.945
2.306 0.0263 191 0.912 0.917 0.892
5.674 0.0647 191 0.806 0.822 0.790
14.38 0.1622 189 0.591 0.616 0.587
26.47 0.3726 189 0.405 0.376 0.385
66.58 0.9342 189 0.081 0.086 0.086
192.8 2.6780 190 0.000 0.001 0.000
153
Spectrum
Number
A123
A124
A125
A126
A127
A128
A129
A132
A133
A134
A135
A136
A137
A138
AI41
A142
A143
A144
A145
A146
A147
AI50
AI51
A152
A153
A154
A155
A156
A159
AI60
AI61
A162
A163
A164
A165
Table 3.
Ames Methane Spectra near T=II2 K.
Methane Transmission at 4650 cm "l
Abund.
m-amgts.
0.1295 0.0020
0.3652 0.0054
1.009 0.0146
2.575 0.0382
6.27
14.21
35.08
0.0608
Pressure Temp. Malkmus Exp. Sum
atm. Kelvln measured model model
115 0.993 0.996 0.997
112 0.989 0.990 0.993
109 0.978 0.977 0.982
112 0.945 0.946 0.947
0.0931 112 0.875 0.878 0.881
0.2109 112 0.728 0.745 0.754
0.5208 112 0.477 0.486 0.496
0.0020 113 0.999 0.998 0.999
0.0053 I09 0.997 0.994 0.997
0.0147 107 0.989 0.986 0.992
0.0371 106 0.979 0.968 0.977
0.1147 107 0.924 0.909 0.919
0.2338 107 0.846 0.825 0.843
0.5237 115 0.663 0.666 0.646
0.0020 112 0.984 0.992 0.995
0.0054 113 0.980 0.983 0.985
0.0150 115 0.954 0.962 0.958
0.0369 114 0.906 0.916 0.908
0.0929 113 0.793 0.806 0.807
0.2112 113 0.618 0.615 0.615
0.5250 113 0.317 0.300 0.313
0.0020 114 0.993 0.999 0.999
0.0055 114 0.992 0.997 0.998
0.0147 114 0.989 0.992 0.995
0.0371 114 0.978 0.982 0.985
0.1158 115 0.944 0.948 0.942
0.2333 114 0.895 0.897 0.887
0.5204 113 0.784 0.785 0.765
0.1626
0.464
1.179
3.612
7.36
15.34
0.295
0.799
2.183
5.41
13.73
31.27
77.68
0.0294
0.0823
0.2194
O.552
1.709
3.472
7.815
0.0149
0.0426
0.1164
0.2911
0.0019 ii0 0.995 0.999 1.000
0.0054 109 0.998 0.998 0.999
0.0147 109 0.995 0.996 0.998
0.0368 109 0.987 0.990 0.993
0.747
1.673
4.015
0.0937 108 0.972 0.974 0.978
0.2117 109 0.948 0.944 0.939
0.5314 114 0.870 0.869 0.839
154
Table 4.
L.P.L. Methane Spectra
Methane
Spectrum Abund. Pressure Temp.
Number m-amgts, atm. Kelvln
D
170A 0.856 0.905 294
170C 0.851 0.901 294
140A 0.477 0.505 294
120A 0.260 0.275 294
120C 0.260 0.275 294
IIOA 0.129 0.137 294
105A 0.0653 0.0691 294
102A 0.0255 0.0270 294
H70A 0.439 0.911 294
H40A 0.254 0.528 294
H40C 0.254 0.528 294
H20A 0.125 0.259 294
HIOA 0.0609 0.126 294
H05A 0.0322 0.0668 294
Q70A 0.224 0.922 294
Q40A 0.129 0.528 294
Q20A 0.0641 0.263 294
QIOA 0.0317 0.130 294
Q05A 0.0163 0.0671 294
MIIA 0.590 0.0083 297
MI2A 1.06 0.0150 297
MI3A 1.99 0.0280 297
MI5A 9.57 0.1347 297
M21A 1.18 0.0083 297
M22A 2.13 0.0150 297
M23A 3.98 0.0280 297
M31A 2.35 0.0083 297
M32A 4.26 0.0150 297
M33A 7.96 0.0280 297
M41A 4.71 0.0083 297
M42A 8.52 0.0150 297
M51A 9.42 0.0083 297
155
these spectra are quite different from the spectra listed in Table 1, combining these
2 sets in exponential sum modeling will increase the number of regions where good fits
can be determined directly from the transmission data.
In the weak absorption regions between the main bands, the number of parameters
that can be determined directly from the transmission measurements on the spectra will
be greatly reduced. Nevertheless, some of these weak absorption regions are of primary
interest in the spectra of the outer planets and Titan, and exponential sum model pa-
rameters are needed. Our original expectation was that exponential sum parameters
could best be determined from the measured transmissions, as compared to computed
transmissions based on band model parameters. 7 In regions of strong absorption with
sufficient lab data, this approach should give the best results. But in the weak absorp-
tion regions it will be necessary to augment the measured transmissions with computed
transmissions based on the Malkmus model parameters in order to determine complete
sets of exponential sum parameters. Such a procedure could become quite arbitrary un-
less it is closely coupled to intended applications. At this time methane exponential sum
parameters are needed for modeling Titan's spectrum, and thus improving knowledge of
Titan's haze, cloud elevations, and methane abundance. In six years Galileo will be in
orbit around Jupiter, and the NIMS instrument should be producing many interesting
spectra, requiring intensive modeling of various regions of the Jovian atmosphere.
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1. L.P. Giver, D. C. Benner, and R. W. Boese,
Bull. Amer. Astron. Soc. 16, 711 (1984).
2. U. I;'ink, D. C. Benner, and K. A. Dick,
J.Q.S.R.T. 18, 447 (1977).
3. D.P. Cruikshank, R. H. Brown, L. P. Giver, and A. T. Tckunaga,
Science 245, 283 (1989).
4. R. Goody, R. West, L. Chen, and D. Crisp,
J. Q.S.R.T. in press (1989).
5. C. P. McKay, J. B. Pollack, and R. Courtin,
Icarus 80, 23 (1989).
6. D. C. Benner and U. Fink,
Bull. Amer. Astron. Soe. 12, 697 (1980).
7. M. G. Tomasko, S. K. Pope, D. Kerola, P. H. Smith, and L. P. Giver,
Bull. Amcr. Astron. Soe. 21,961 (1989).
8. U. Fink and H. P. Larson,
Astrophys. J. 223, 1021 (1979).
156


Gaussian quadrature exponential sum modeling of near infrared methane laboratory spectra obtained at temperatures from 106 to 297 K

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