4.3 Continuum Parameter Estimation
4.3.1 Discussion of the Continuum Parameter Sets
A common feature of all continuum parameter sets is that the remaining frequency and pressure dependence in the data set is much larger for Cf than for Cs (see for example Figure 4.3 and Figure 4.4 and Appendix E on 177). The structure of the frequency dependence of Cf is not easily inferred from these Figures. In the case of Cf the 153 and 350 GHz data are at the same level while the data taken at 213 and 239 GHz are below (in this order). The data points around 138 GHz are between the 153 and 239 GHz measurements. This structure prevents a simple modification of the quadratic frequency dependence. Furthermore the data shows decreasing slopes (decreasing temperature dependence) with increasing water vapor pressure. This is most pronounced in the 350 GHz data set. These remaining dependences might be an interference of several different factors: (1) the imperfect line absorption model, (2) the insufficient parameterization of the continuum absorption given in Equation (4.1), and (3) measurement inherent effects.
Since this feature is observed for all the different line absorption calculations it seems that imperfections in the line absorption models can not explain these dependences. An-other possibility would be a change of the line shape in the line absorption calculation. More sophisticated line shapes than a Van Vleck–Weisskopf line shape are the Voigt (see Equa-tion (2.54)) and Galatry line shapes. However, the temperature and pressure range of the data sample suggests Voigt parameters y > 5 for all strong water vapor lines in the STHz range. This makes the Voigt line shape already very similar to a Lorentz function. Test calculations with a modified Voigt line shape of Equation (2.55) give in fact indistinguish-able continuum parameter sets compared to the calculations with then Van Vleck–Weisskopf line shape. According toRodgers (1976), the Galatry line shape (Galatry, 1961;Ouyang and Varghese, 1989;Schm¨ucker et al., 1997) is only in a particular area significantly different from the Voigt line shape where the Voigt parameter is between 0.5< y <4 (see Equation (2.56)).
Since the Voigt parameters in this investigation are generally larger than 5, the Galatry line shape would be very close to the Voigt line shape and thus very similar to the Lorentz line shape. This suggests that modifications in the line shape would not significantly reduce the remaining frequency and pressure dependences.
The continuum absorption parameterization is, with respect to the frequency dependence, only a first order approximation. For the mixture of water vapor in nitrogenMa and Tipping (2002b) found a frequency exponent of about 2.0389, a 2 % deviation from the value used above. Test calculations with 2.0389 as frequency exponent yield a similar scatter compared to a quadratic frequency dependence. Ma and Tipping (1990) give an approximation for the STHz range which is essentially proportional to ν2/(ν + 1100) in case of pure water vapor. Furthermore Katkov (1997) derived from atmospheric measurements a similar fre-quency dependence of ν2/(ν+ 330) for the self term. Obviously, the crucial point of such a parameterization is the additive constant in the denominator, which is different in these two models. Moreover the investigations ofMa and Tipping (1990) show that the frequency and temperature dependence of the water vapor continuum is not as separable as assumed in Equation (4.18). In the frequency range of 30-360 GHz the temperature dependence increases with frequency (xs(30 GHz) ≈ 3, xs(360 GHz) ≈ 8)). This increase is not seen in the data, neither in the pure water vapor nor in the water vapor nitrogen mixture subsample. From Figures 4.3 and 4.4 one would suppose the inverse relation, a decreasingxs,f, with increasing frequency.
The pressure dependence of the continuum absorption in Equation (4.18) is in accordance with the quantum mechanical model of Ma and Tipping (1990). For this reason this addi-tional pressure dependence is believed not to be predicted from the theory. Higher order effects which would make the binary collision approximation invalid are not relevant at the
(AAM02/cutoff)
Figure 4.3: Water Vapor self continuum parameter fit result for the full 138-350 GHz data.
The H2O-AAM02line catalog is used in the line absorption calculation in connection with a Van Vleck–Weisskopf line shape plus a cutoff of 750 GHz (VVWC). The solid line is the best fit to the data and the dotted lines indicate the 0.95 confidence interval.
(AAM02/cutoff)
Figure 4.4: Water Vapor foreign continuum parameter fit result for the full 138-350 GHz data.
The H2O-AAM02line catalog is used in the line absorption calculation in connection with a Van Vleck–Weisskopf line shape plus a cutoff of 750 GHz (VVWC). The solid line is the best fit to the data and the dotted lines indicate the 0.95 confidence interval.
measurement conditions of this data sample.
That measurement-inherent effects can cause some scatter is best seen from the NTIA measurements. These two data points are measured with a very similar equipment at nearly the same frequency. But the resulting absorption coefficients differ by approximately 25 %.
Over all, it can not be quantified to which extent each of the above-mentioned explanations contribute to the scatter seen in Figure 4.4. To better discriminate between measurement and model-inherent sources of this scatter it would be necessary to a have an estimate of the measurement uncertainty.
Figures 4.3 and 4.4 and the corresponding plots in Appendix E are moreover useful for a discussion of the validity of the calculated confidence intervals. At first sight the confidence intervals ofCsseem to overestimate the uncertainty while forCfthe opposite seems to be true.
One simple reason for this difference is the approximately inverse square root dependence of
∆Cs,f on the sample size. For the regression fit of Cs only 26 data points are available while for the Cf fit 64 data points are used. Equal sample sizes would reduce ∆C
s,f to a level of 10-15 % for Cs which is close to the level of the Cf confidence interval. Since the regression fit performed is a one dimensional regression fit with a modified temperature as the single variable, it is not possible to cover the additional dependence on frequency and pressure simultaneously in the fit. But before a multidimensional regression fit is applied it is advantageous to get more precise theoretical predictions for the frequency and pressure dependence in theSTHz range, which are not available at the moment. Together with the unknown measurement uncertainties a multidimensional regression fit of the considered data sample is questionable and therefore omitted.
Comparison of the Parameter Sets
TheCbso parameters of Table 4.6 are within a range of 8.53 to 9.00 dB/km/hPa2GHz2 which is a variability of about 6 %. The corresponding temperature coefficients ˆxs vary by 7 %.
With respect to the given confidence intervals the variability of these parameters due to the different line catalogs and line shapes is relatively small. This small variability has its origin partly in the low pressure level of the pure water vapor measurements where the pressure broadening of the water lines is comparatively small. The measurement frequencies are in the window region so that the narrow lines contribute only a small fraction to the measured total absorption.
As expected, the parameters Cbso and Cbfo compensate the reduced line absorption in the cases where a cutoff is applied in the line shape. The self termCbso increases by 0.5-3 % while the foreign term Cbfo increases by 4-20 %. The 20 % increase is seen for the three largest line catalogs with 645 lines up to 2.5 THz. The comparatively small increase of 4 % is for the H2O-AAM02 line catalog with 34 lines, all in the STHz range. From this result one can infer that the water vapor continuum in theSTHz range is strongly influenced but not dominated by the far wings of far infrared lines. It is furthermore interesting to note that the temperature coefficients ˆxs and ˆxf depend on the applied line shape too. In case of ˆxs this dependence is moderate (0.4-4 %) but for ˆxf the change is of the same order as for Cbfo (4-20 %). But the main difference is that the temperature coefficients decrease in contrast to the continuum coefficients if a cutoff is used in the line shape.
The continuum parameter sets for the H2O-MPM93and H2O-AAM02(VVW line shape) line catalogs are very similar although some differences exist in the line broadening and line intensity parameters. The main difference between these models is seen in ˆxfwith 9 % while theCbfo are virtually identical.
Comparison of Models and Measurements
The relative differenceRDbetween the measured (αdtot) and modeled (αmtot) total water vapor absorption is calculated according to
RD = 100%· αmtot−αdtot αdtot
!
(4.32) where αmtot is the sum of the modeled line and associated continuum absorption. The data set of Table E.1 is for this comparison extended with the additional USTL data around the 183.3 GHz line (measurement range 183.71-190.31 GHz, Bauer et al. (1986); Bauer and Godon (1991)). This extended data sample is then divided into three subsamples depending on frequency: 150-160 GHz, 170-200 GHz, and 200-350 GHz. The first and last interval are window regions while the second interval covers the 183.3 GHz line. Since the measurements around the 183.3 GHz line are not considered in the fit of the continuum parameters the model calculations are independent of these measurements.
The detailed list of values for the mean and standard deviations of the relative differences is given in Section E.3 of Appendix E. In Figure 4.5 the mean and standard deviations are shown for pure water vapor and in Figure 4.6 for the water vapor mixed with nitrogen.
In the 150-160 GHz window region the model calculations are 4-5 % (H2O-H2O) and 7-13 % (H2O-N2) below the measurements, respectively. Looking to Figures 4.3 and 4.4, one can see that the measured absorption at 153 GHz is especially at low temperatures higher than the modeled absorption including the best fit of the continuum parameters. In the second window region from 200-350 GHz, the differences are smaller since the model absorption is between the 350 GHz and 213-240 GHz absorption measurements. Hence the relative difference between model calculation and measurement is much lower than in the 150 GHz window, i.e. <1 % (H2O-H2O) and<4 % (H2O-N2), respectively.
In the 170-200 GHz intervals the model absorption is in most cases lower than the mea-surements, about 6-12 % in case of pure water vapor and about 10-17 % in case of water vapor mixed with nitrogen. Best agreement can be seen for the HITRAN00 catalog where the differ-ence is about−5 % (H2O-H2O). The main difference between the line catalogs with respect to the 183.3 GHz line broadening parameters are the temperature coefficients NH2Os,k and NN2f,k. HITRAN00 sets both parameters to 0.64 whileMMHIT-A,MMHIT-B, and H2O-AAM02 use values of 0.85 and 0.74, respectively. The influence of the temperature coefficients is only seen in pure water vapor (RD=5 %). For H2O-N2 the difference between the different line catalogs is not noticeable.
In H2O-MPM93, Liebe increased the line intensity by 5 % compared to HITRAN00 (see Figure 3.16). However, this increase does not improve the match with the data significantly.
Summarizing this comparison, the degree of agreement between the USTL measurements and the different absorption models does not significantly differ between the absorption mod-els. For the H2O-N2 mixture the agreement between measurement and models is better than 15 % in the window regions and better than 17 % around the 183 GHz water vapor line. Con-cerning pure water vapor, the agreement is better than 5 % in the window regions and better than 12 % around the 183.3 GHz line. With the exception of the 200-350 GHz window region the absorption models generally underestimate the measured absorption.
Comparison with other Models
A Comparison of the self continuum coefficients,Cso, of Table 4.6 with other corresponding model values of Table 4.7 shows that the differences are of the order of 20 %, which is the level of the confidence interval given for the Cso. For the foreign continuum coefficients, Cfo, the differences are up to 40 %. In general the continuum coefficients derived in this
Figure 4.5: Mean and standard deviation of the relative difference between model calculations and measurements. The line catalog used in the line absorption calculation is indicated at the left side. Additionally the line shape is stated in brackets: Van Vleck–Weisskopf (VVW), Van Vleck–Weisskopf with a cutoff of 750 GHz (VVWC). The associated continuum parameter sets for the calculation of the continuum absorption are given in Table 4.6. The symbols indicate the mean value and the horizontal bars the standard deviation of the three subsamples for pure water vapor measurements. The subsamples are defined by the frequency intervals of the laboratory measurements: 130-160 GHz (squares), 170-200 GHz (triangles), and 200-350 GHz (circles).
Figure 4.6: Mean and standard deviation of the relative difference between model calculations and measurements. The line catalog used in the line absorption calculation is indicated at the left side. Additionally the line shape is stated in brackets: Van Vleck–Weisskopf (VVW), Van Vleck–Weisskopf with a cutoff of 750 GHz (VVWC). The associated continuum parameter sets for the calculation of the continuum absorption are given in Table 4.6. The symbols indicate the mean value and the horizontal bars the standard deviation of the three subsamples for H2O-N2measurements. The subsamples are defined by the frequency intervals of the laboratory measurements: 130-160 GHz (squares), 170-200 GHz (triangles), and 200-350 GHz (circles).
investigation are larger than the values of other existing models. This is partly due to the different laboratory data used for the determination of the individual continuum parameter sets. Another effect is the different line absorption term implemented in the models. As can be seen from Table 4.6 the line catalog and the applied cutoff in the line shape results in large differences ofCso and Cfo. Furthermore in the case of H2O-PWR98 a constraint is applied on the temperature dependency ofCfo (xf=0.0) which influences the result of the parameter fit.
The continuum coefficients of H2O-MPM93∗ 7 are about 20 % larger compared to H2 O-MPM87. Large discrepancies are also found for the temperature coefficients xs and xda
between these two H2O-MPM versions. This difference can be explained by the different sources used for the determination of the continuum parameters. H2O-MPM87 is based on laboratory measurements at 138 GHz (Liebe and Layton, 1987) while for H2O-MPM93 additional laboratory measurements ofBecker and Autler (1946) and Godon et al. (1992) is used.
Comparing the continuum parameters of H2O-MPMwith those of H2O-AAM02 in Ta-ble 4.6 for a V V W line shape function reveals differences of about 3-15 % (compared to H2O-MPM93∗) and 26-37 % (compared to H2O-MPM87) for the continuum coefficients.
The temperature coefficients of H2O-AAM02 agree with those of H2O-MPM93∗ but not with H2O-MPM87.
The differences between the continuum coefficients of H2O-MPM93∗ and those derived with the H2O-MPM93 line absorption model (see last row in Table 4.6) are about 3-14 %.
Moreover, the differences between the temperature coefficients is about 3-13 % and with respect to the uncertainty level in good agreement.
In the case of the theoretical model ofMa and Tipping (1990) and the operational model of Katkov (1997) the frequency dependence of the self continuum term is not quadratic as assumed here and in all the other operational models. This non-quadratic dependence is reflected in the significantly lower self continuum coefficients. TheCairo value stated byKatkov (1997) is significantly below the values of the other models although this model assumes a quadratic frequency dependence for the foreign continuum term like the other operational models. Katkov (1997) derived the value of Cairo from a best fit to several measurements in the MMW and SMMW range while the other operational models rely on laboratory measurements in the MW and MMW range for Cairo . Good agreement for Cairo is seen betweenKatkov (1997) and the theoretical model of Ma and Tipping (2002b). But one has to keep in mind thatMa and Tipping (2002b) state a frequency dependence ofν2.0389 for the foreign continuum term which compensates the lowCairo value.
A comparison can be made between the continuum parameter set of H2O-PWR98 and the corresponding set in Table 4.6. The line absorption is the same in both models. Hence the different continuum parameters result in the partially different measurements considered for the estimation. The 350 GHz data was not available at that time but instead theBecker and Autler (1946) and Liebe and Layton (1987) data was used. Additionally Rosenkranz (1998) forced xair to be zero without estimating it. The Cso and Cfo values inferred from the regression fits are 15 % and 21 % higher than the values of H2O-PWR98 and xs is 11 % higher. These differences can serve as an estimate of the systematic uncertainty of these three continuum parameters. With respect to the uncertainty ofxair a conservative guess is 0.0< xair<1.5 for the true value ofxair.
7approximation of H2O-MPM93to fit Equation (4.1), see Appendix D.2.3
Cso Cairo Cfo xs xair reference [10−8(hP a GHz)(dB/km)2] [1] [1]
- 0.181 0.195 3-8 1.6 Ma and Tipping (1990) Ma and Tipping (2002b)
7.3-8.7 - - - Becker and Autler (1946)
9.85 0.25 0.27 0.5 -0.5 Liebe (1984) 9.85 0.25 0.27 2.5 -0.5 Liebe (1985)
6.50 0.206 0.22 7.5 0.0 Liebe and Layton (1987) 7.73 0.253 0.27 4.6 1.6 Liebe et al.(1993) 7.80 0.236 0.25 4.5 0.0 Rosenkranz (1998)
3.6 10−5ν2
ν+330 0.17(17) 0.18 9 0.0(5) Katkov (1997)
Table 4.7: List of other continuum parameter sets. The theoretical model of Ma and Tip-ping (1990) predicts a temperature coefficient for the self term for the frequency range of 30-360 GHz. The temperature dependence increases with increasing frequency. The mea-surements ofBecker and Autler (1946) are around the 22 GHz line of water vapor. For the estimation of Cso the data at ν =34.8 and 40.2 GHz are taken. Since the dry air contribu-tion to the total absorpcontribu-tion is only a few percent, the listed absorpcontribu-tion coefficients are not corrected for this effect. The different versions of Liebe’s H2O-MPM model are based on several different absorption measurements and water vapor line catalogs. The H2O-MPM93 is approximated to the continuum absorption parameterization of Equation (4.1). The ap-proximation is described in Appendix D. The parameter set of Katkov (1997) incorporates some measurements of Russian groups in the free atmosphere in the MMW and SMMW range. In the models of Katkov (1997), Ma and Tipping (1990), Ma and Tipping (2002b), and Liebe et al. (1993) the frequency dependence is not quadratic. For a comparison with the continuum parameter sets in Table 4.6 theCairo values are multiplied by 1.08 according to the same ratio for the pressure broadening parameters discussed above.
Independence of the Self Term
The estimation procedure described before assumes implicitly that the self continuum param-eterCso is the same for pure water vapor and for water vapor in nitrogen. This implies that the self and foreign absorption terms are additive and do not have any cross-correlation. One check of this assumption is to compare the measured self term of the total absorption (term proportional to PH2
2O) for pure water vapor and hetero-molecular mixtures, H2O-X. Espe-cially useful for this check are measurements in the window regions, like those at 239 GHz, where the Van Vleck–Weisskopf line shapes of all theSTHz water vapor lines is essentially proportional to the pressure broadened line width. In this case the total absorption is well described by a second order polynomial with respect to the water vapor partial pressure PH2O:
H2O−H2O : αd
tot = aν(T)·PH2
2O (4.33)
H2O−X: αd
tot = ˆaν(T)·PH2
2O+bν(T)·PH2O (4.34)
whereaν(T), ˆaν(T), bν(T) are fitted polynomial coefficients to the data which depend only on temperature for a fixed measurement frequency and fixed buffer gas partial pressure. All three coefficients have contributions from the resonant line and continuum absorption but are independent of any absorption model. If the above assumption is correct, i.e. if the quadratic self absorption term is the same irrespective of the gas mixture, the identityaν(T) = ˆaν(T) should be reproduced in the data within the measurement uncertainty. Table 4.8 summarizes the published values for the ratio
r = ˆaν(T = 296 K)
aν(T = 296 K) (4.35)
of different gas mixtures, all performed at a measurement frequency of 239 GHz. However, the differences betweenaν and ˆaν is with the exception of H2O-CO2 (+37 %) around −25 %.
Even with the claimed measurement uncertainty of 10 % foraν and ˆaν (M. Godon, 2000) the tabulated ratios are more than 2 standard deviations away from the expected ratio of unity.
This deviation might thus raise the question if other mechanisms occur additionally in the H2O−X sample than resonant and continuum absorption of water monomers. One possible additional mechanism is the absorption due to weakly bound H2O•X complexes of a water and a buffer gas molecule (WBC hypothesis). The difficulties to investigate the possibility of WBC are twofold. On one hand one needs the thermochemical information to estimate the concentration of H2O• X complexes and on the other hand one needs spectroscopic information about the absorption features of such complexes. The WBC partial pressure is proportional to the water vapor and buffer gas partial pressures. This enhances thebν-term at the expense of the ˆaν-term in Equation (4.34). The formation of WBC has additionally a more direct influence on the ˆaν-term due to the reduction of the nominal water vapor partial pressure: PH0
2O =PH2O−Pwbc, where Pwbc is the partial pressure of the WBC. Calo and Narcisi(1980) have given a simplified model for the estimation of the concentration of weakly bound complexes under atmospheric conditions. This model is based on the reduced second virial coefficient of bound (Bb∗) and metastable (Bm∗) complexes. According to this model the partial pressure of weakly bound complexes can be estimated to
Pwbc = −[Bb∗(T∗) +Bm∗(T∗)]·
"
2π NAσ3 3
#
·
2·Ptot·VMR[H2O]·VMR[X] (4.36)
whereNAis Avogadro’s constant andPtot the total pressure in the absorption cell. The vol-ume mixing ratios of the buffer gas and water vapor are denoted by VMR[X] and VMR[H2O].
The reduced temperature is defined asT∗=kBT /and the reduced second virial coefficients for a (6-12) Lennard-Jones potential are taken from Stogryn and Hirschfelder (1959). For some molecules of interest the Lennard-Jones parametersand σ are tabulated in Table A.4 of Appendix A.2.2. Additionally the water dimer concentration is calculated, using the model ofEvans and Vaida (2000):
Pdimer PH
2O
=PH2O·3·10−5·e1647/T −1
T (4.37)
where the pressure is given in units of hPa.
For typical measurement conditions, i.e. T=296 K, PH2O=10 hPa, and PX=1000 hPa, the concentrations of weakly bound complexes are of the order of one per cent or less and for water dimers less than one per thousand of the nominal water vapor partial pressure (see Table 4.8). One reason for these low concentrations is the relatively high measurement temperature since the complex concentration decreases with increasing temperature. The low concentration must be compensated by strong absorption features around 239 GHz if such complexes should have a major contribution to the discrepancy in the ˆaν/aν ratio. Reported transitions of water dimers inCoudert and Hougen (1990) andBraly et al.(2000) are clustered at frequencies of 10-40 GHz, 600-800 GHz, 2.6 THz, and 3.2 THz. Similar data for H2O•X complexes are sparse. For the case of H2O•N2 complexes, measurements were performed around 20 GHz (Leung et al., 1989) and vibrational absorption bands are predicted in the infrared region (Svishchev and Boyd, 1998). Observed absorption bands for the complex of water molecules and argon lie at 635 and 740 GHz (Hutson, 1990). It seems therefore that at least for water dimer and H2O−N2 and H2O−Ar mixtures the WBC hypothesis is not an obvious explanation of the discrepancy seen in the ratio of ˆaν/aν.
If the WBC hypothesis is not able to explain the differences of ˆaν and aν satisfactorily, another possibility to elucidate this difference is to reconsider the stated measurement uncer-tainty of 10 % for ˆaν and aν. An increase of the measurement uncertainty from 10 to 15 % would bring the ratios r = ˆaν/aν of Table 4.8 into a two sigma uncertainty range with the predicted value ofr = 1. This seems so far the most natural solution for this discrepancy.