2.2 Spectral Line Absorption Theory
3.1.1 Comparison of Oxygen Absorption Models
Figure 3.3: Line strength ratios calculated from theory with the rotational constant of B=43.100543 GHz (Amano and Hirota, 1974). The ratio is calculated with respect to the 3+ and 3−transitions at temperatures of 300 K and 200 K, respectively. The circles and dia-monds denote the ratios (SN∓/S3∓) at a temperature of 300 K and the triangles and squares denote the same ratios at a temperature of 200 K.
S(T) = exp [12B h / kBT] exp [B N(N+ 1)h / kBT]
whereSN± are the transition intensities and B = 43.1 GHz is the rotational constant. Fig-ure 3.3 shows the intensity ratios with respect to the N = 3 triplet9 lines for two different temperatures, 200 K and 300 K. For T = 300 K, the lines above N = 31 are less than 1 % of theN = 3 lines. For T = 200 K, this ratio is already reached for N = 25. Hence for the accuracy level needed in atmospheric radiative transfer models one can neglect lines from triplets with higher rotational number (approximately'35) without introducing noticeable systematic deviations in the absorption calculations.
O2-PWR93 are basically the same as in O2-MPM92 and previous O2-MPM versions but differ by 5 % from the values in O2-MPM93. The first extensive measurements of the line coupling parameters, Yj, were performed by Liebe et al. (1977). Rosenkranz (1988) revised these coupling parameters, incorporating unpublished dispersion data of Liebe and Layton (1987) in his fit. This revised parameter set entered the MPM model with the version O2-MPM87. A second series of laboratory measurements in the frequency range of 54 to 66 GHz followed in 1991 (Liebe et al., 1991). Coupling parameters derived from this latest data set are reported in Liebe et al.(1992), Rosenkranz (1993), and Liebe et al. (1993) and are implemented in O2-PWR98, O2-PWR93, and O2-MPM92. On the other handMPM93 increased these parameters generally by 15 % compared to O2-MPM92. According to Liebe et al. (1993), the simultaneous increase in line broadening and coupling parameters should improve the global fit to theLiebe et al. (1992) laboratory measurements by 7 %.
The main update from O2-PWR93 to O2-PWR98 is the modified temperature depen-dence of the pressure broadening for the 118 GHz line. This modification is based on findings of Schwartz (1997) who analyzed clear sky measurements collected within the CAMEX10 and TOGA-COARE11 campaign with the Microwave Temperature Sounder (MTS). The MTS recorded data at frequencies of 52.5-55.8 GHz and around the 118 GHz line. The find-ings do not reveal any significant change for the 60 GHz band parameters but indicate an additional temperature scaling of the line width of (300 K/T)z with z=0.15±0.03. The up-datedPWR98 model thus includes a global line width scaling factor of 1.0 for the pressure broadening temperature dependence, e.g.:
PWR98 : γ1 = w1·Pd + 1.1·PH
2O
·Θ (3.7)
PWR93 : γ1 = w1·Pd·Θ0.8 + 1.1·PH
2O·Θ (3.8)
where Θ = 300 K/T and Pd, PH2O are the dry air and water vapor partial pressures, re-spectively. Beside this modification the line intensities of theMMW and SMMW lines are updated according to HITRAN96 (Rothman et al., 1998). The parameterization of the 60 GHz band core region is unchanged and hence the same as in O2-PWR93 and O2-MPM92.
10Convection And Moisture EXperiment
11Tropical Ocean Global Atmospheres/Coupled Ocean Atmosphere Response Experiment
Figure3.4:Relativedifferenceintotalsimulatedabsorptionforthefrequencyrangeof40-70GHz.Consideredabsorbersarewatervapor (H2O-PWR98,Rosenkranz(1998)),nitrogen(N2-MPM93,Liebeetal.(1993)),ozone(HITRAN96,Rothmanetal.(1998))andoxygen(O2- PWR98asreferencemodel)asatmosphericconstituents.Usingdifferentoxygenabsorptionmodels,therelativedifferenceintotalabsorption iscalculatedaccordingtoEquation(3.10)withO2-PWR98asreferenceoxygenmodel.Theatmosphericprofileusedisthemeanprofileofthe reducedECMWFatmosphericprofilesample(seeAppendixBandChevallier(2001)).
Figure 3.4 shows the relative difference between three O2-PWRversions and O2-MPM93 in the frequency range of 40-70 GHz. The model labeled with O2-PWR88 is basically the O2-PWR93 version but with the line mixing parameters of Rosenkranz (1988). The atmo-spheric profile taken as input for the absorption calculation is the mean profile of the reduced ECMWF sample described in Section B.1.1. A four component atmosphere with water va-por, oxygen, nitrogen, and ozone is assumed. The absorption of H2O is modeled according to H2O-PWR98 (Rosenkranz, 1998), the nitrogen absorption is calculated with N2-MPM93 (Liebe et al., 1993), and for ozone absorption the HITRAN96 (Rothman et al., 1998) catalog is used:
αtot=αO2+αH2O(PWR98) +αN2(MPM93) +αO3(HITRAN96) (3.9) The relative difference in per cent with respect to the reference oxygen model of PWR98 is then calculated as
rel. diff. = 100 %·
αtot−αtot(O2-PWR98) αtot(O2-PWR98)
= 100 %·
"
αXO2−αPWR98
O2
αtot(O2-PWR98)
#
(3.10) where X stands for O2-PWR93, O2-PWR88, and O2-MPM93. As expected, the O2 -PWR98 and O2-PWR93 give identical absorption values throughout this frequency range.
The line mixing parameters of PWR88increase the absorption in the band center and lower the absorption therefore in the near wings of the band. O2-MPM93is generally lower than O2-PWR98. At higher altitudes two spikes are seen at the borders of the band. These spikes are caused by the additional lines included in the MPM model. MPM include lines up to N =±37 while O2-PWR98 encompasses lines up toN =±33 (see discussion in connection with Figure 3.3). How the differences in absorption translate into differences in brightness temperatures,TB, is shown for AMSU-A channels in Figure 3.5. Here the 77 profiles of the re-duced ECMWF profile set are taken as input. For each profile the difference ∆TB is calculated for radiative transfer calculations using different oxygen absorption models but identical wa-ter vapor (H2O-PWR98,Rosenkranz (1998)) and nitrogen (N2-MPM93,Liebe et al.(1993)) absorption models. Ozone is omitted from this calculation since its contribution is negligible.
The brightness temperatures show in the wing of the 60 GHz band (50-53 GHz) the highest variability while at frequencies above 55 GHz this variability decreases to standard deviations of about 5 K (see top left plot of Figure 3.5). In comparison to this atmospheric variabil-ity, the variability due to different oxygen absorption models is quite small with 0.4 K (top right and bottom left and right plots of Figure 3.5). O2-PWR98and O2-PWR93show very similar differences with respect to O2-MPM93. The influence of the updated 118.75 GHz line plays no significant role in the AMSU-A frequency range. Furthermore, O2-MPM93 gives generally higher TB values than O2-PWR98 and O2-PWR93. The mean difference reaches 0.4 K around 53 GHz. In contrast to the newer O2-PWRversions, O2-PWR88 gives on average higher brightness temperatures than O2-MPM93below 56 GHz and lower values above 56 GHz. The mean model differences of up to 0.4 K for the AMSU-A channels are of the same order as the noise equivalent temperature12, NE∆T (National Oceanic and Atmo-spheric Administration, 2002) and hence not negligible. Present state operational radiative transfer models like RTTOV-6 use for the assimilation of AMSU-A radiances the O2-MPM92 model, which in this frequency range is essentially the same as O2-PWR93and O2-PWR98 (Rayer, 2001).
12The noise equivalent temperature is the minimum difference in antenna temperature which can be resolved by the receiver.
Figure 3.5: Simulated brightness temperatures,TB, for the AMSU-A channel region around 60 GHz, using different oxygen absorption models. The upper left plot shows the absolute values of TB, considering water vapor (H2O-PWR98, Rosenkranz (1998)), nitrogen (N2 -MPM93, Liebe et al. (1993)), and oxygen (O2-MPM93) as atmospheric constituents. The dots and vertical bars show the mean and the two standard deviation margin of TB for the 77 atmospheric profiles (see Appendix B) of the reduced ECMWF profile set ofChevallier (2001). The upper right and the lower two plots show the mean and two standard deviation differences between the TB values using identical H2O and N2 absorption models in the radiative transfer calculation but different O2 absorption models. O2-MPM93 is taken as the reference oxygen absorption model. The AMSU-A characteristics are neglected in this calculation. The surface emissivity is generally set to 0.95.
Figure 3.6: Relative difference in total simulated absorption for the frequency range of 70-300 GHz. Considered absorbers are water vapor (H2O-PWR98,Rosenkranz (1998)), nitrogen (N2-MPM93, Liebe et al. (1993)), ozone (HITRAN96 Rothman et al. (1998)) and oxygen (O2-PWR98 is taken as the reference model) as atmospheric constituents. Using different oxygen absorption models, the relative difference in total absorption is calculated according to Equation (3.10) with O2-PWR98 as reference oxygen model. The atmospheric profile used is the mean profile of the reduced ECMWF atmospheric profile sample (see Appendix B andChevallier (2001)).
An important detail of the MPMmodel – which is not explicitly discussed in its conse-quences in theMPM documentation – is the applied cutoff for the oxygen line absorption.
Negative line absorption values caused by the line mixing in the 60 GHz band are rigorously set to zero:
αMPMl =
( αMPMl : αMPMl >0
0 : αMPMl <0 (3.11)
This cutoff is most important in the far wing regions of the 60 GHz band. Especially in the window regions this cutoff is noticeable. Up to about 165 GHz the resonant line absorption remains positive, from 165 GHz to about 355 GHz it has negative values. Figure 3.6 shows the consequences of this cutoff and the increased line mixing parameters in O2-MPM93compared to O2-PWR98. The calculated total absorption with O2-MPM93 is, at all four pressure levels, lower than the total absorption with O2-PWR98for frequencies up to 170 GHz. Above this frequency the cutoff is noticeable in the way that O2-MPM93 absorbs more than O2 -PWR98. The oxygen model exchange causes a difference of up to−20/+10 %, respectively.
Westwater et al. (1990) and Danese and Partridge (1989) mention that the frequency range between the 60 GHz band and the 118 GHz line is especially susceptible to model pa-rameters. Westwater et al.(1990) report discrepancies of about 5 K in brightness temperature between their ground based radiometer data at 90.0 GHz and the O2-MPM87version (Liebe and Layton, 1987). They got better agreement between model and measurement by
replac-Figure 3.7: Simulated brightness temperatures, TB, for a ground based radiometer, using different oxygen absorption models. The radiative transfer calculation considers water vapor (H2O-PWR98,Rosenkranz (1998)), nitrogen (N2-MPM93,Liebe et al. (1993)) and oxygen (O2-PWR98, reference model) as atmospheric constituents. The dots and vertical bars show the mean and the two standard deviation margin of the brightness temperature differences,
∆TB, for the 77 atmospheric profiles (see Appendix B) of the reduced ECMWF profile set of Chevallier (2001). The additionally considered oxygen absorption models are O2-MPM93, O2-MPM92 (without the cutoff of Equation (3.11)), O2-MPM87, and O2-PWR88. The radiometer characteristics are neglected in this calculation.
ing all the line mixing parameters with those of Rosenkranz (1988). Danese and Partridge (1989) have inferred some O2-MPM85model adjustments from their ground based radiome-ter data set. From the brightness temperature difference measured at 33 and 90 GHz, they proposed a general increase in 60 GHz line intensity of the order of 15 %13. Both observations are conform with Figure 3.7, where again the reduced ECMWF sample is used to calculate brightness temperatures for a ground based radiometer. The mean difference in brightness temperature of these 77 atmospheric profiles can reach±8 K. Using the O2-MPM92model without the cutoff of Equation (3.11) yields comparable brightness temperatures to using O2-PWR98. This underlines the increased affinity of this frequency range to O2 absorption models.
This point will further be investigated in Chapter 5 where ground based radiometer measure-ments at 89 and 150 GHz are analyzed. The window between the 118 GHz oxygen line and the 183.3 GHz water vapor line shows also high differences in absorption. Hence a combined analysis at 89 and 150 GHz could be one possibility to discriminate between the different oxygen absorption models.
The update of the 118 GHz line parameter in O2-PWR98 is of interest particularly for the Odin channel around this line (Eriksson, 1999). This channel is used to derive tem-perature and pointing information. Figure 3.8 shows the ratio in absorption due to this update compared to O2-PWR93. In the line wing, the difference follows the atmospheric
13This recommendation was not followed in subsequent O2-MPMversions.
Figure 3.8: Ratio of simulated total absorption around the 118.75 GHz oxygen line for the two O2-PWR98 and O2-PWR93 oxygen absorption models (ratio = αtot(O2-PWR93) / αtot(O2-PWR98)). The water vapor absorption is calculated with H2O-PWR98 while the nitrogen absorption is calculated with N2-MPM93. For the absorption of ozone the HI-TRAN96 (Rothman et al., 1998) line catalog is used. The 77 atmospheric profiles are taken from the reduced ECMWF sample (Chevallier, 2001) as described in Appendix B. The solid line is the mean ratio while the two grey shaded regions indicate the 1σ and 2σ range.
Figure 3.9: Brightness temperatures for the Odin 118 GHz channel, using different oxygen absorption models. The individual plots show the ratio of the calculated brightness tempera-ture with one O2 model to theTB using the reference model of O2-PWR98. The absorption of water vapor is calculated with H2O-PWR98 and that of nitrogen with N2-MPM93(see below for details). The 77 atmospheric profiles are taken from the reduced ECMWF sample (Chevallier, 2001) as described in Appendix B. The dots are the mean ratios while the bars indicate the 2σ range for the 77 profiles.
temperature profile, e.g. the difference is most pronounced around the tropopause. In the line center region, the same altitude behavior can be seen but with the opposite sign. In the wing region, O2-PWR98 yields more absorption whereas in the center region, O2-PWR93 absorbs more. This can be understood by the temperature dependence of the line broadening parameterization and its relation to the absorption ratio in the vicinity of the line center:
r = αtot(O2-PWR93) αtot(O2-PWR98) ∝ γ
PWR98 γPWR93
∝ Θ
Θ0.8 = Θ0.2 (3.12)
where Θ = 300 K/T. Temperatures of 190 K around 100 hPa can lead to values of r=1.10 at 118.75 GHz. With decreasing temperature the oxygen line at the same pressure level becomes broader in the case of O2-PWR98 compared to O2-PWR93. This increases the absorption in the wing and decreases simultaneously the absorption in the line center in O2-PWR98. The brightness temperature seen by Odin for four different tangent altitudes (pencil beam calculation, neglecting radiometer characteristics) is shown in Figure 3.8. As indicated by Figure 3.8, the differences are most pronounced in the cold tropopause and the lower stratosphere. Hence the difference inTB is very low in the troposphere where also the opacity increases and in the upper stratosphere where the temperature is relatively high. The differences are most pronounced in the lower stratosphere where in the line wings differences in brightness temperatures of several Kelvin can be reached. Such high ∆TB values could be large enough to analyze Odin data especially with respect to this model difference. In connection with such a study it has to be noted thatTretyakov et al.(2001) measured a line shift of -0.14±0.06 MHz/hPa for this oxygen line. This line shift is neglected in the above model calculations but has to be considered in a comparison between Odin data and model calculations.
The last point of this model discussion comes back to the phenomenon of positive and negative line absorption as frequently mentioned previously. One has to emphasize that only the total oxygen absorption is a measurable quantity which is always positive at atmospheric conditions14. However, as already mentioned in connection with Equation (3.4), the effective line absorption term, αl,tot, can yield positive as well as negative absorption due to positive and negative line coupling parameters (see Figure 3.10). Negative values of αl,tot are most likely in the far wing of the 60 GHz band at MMW and SMMW frequencies. By dividing αl,tot into one term which covers all the positive line absorption (α+l >0) of the individual lines of the 60 GHz band and one term which covers all the negative line absorption (α−l <0) one can write
αl,tot = α+l +α−l (3.13)
αl,tot = α+l +α−l (3.14)
1 = α+l αl,tot
+ α−l αl,tot
=SR1 +SR2 (3.15)
yielding a positive ratio SR1 and a negative ratio SR2 with the sum (SR1 +SR2) = 1.
Table 3.1.1 gives some values for the ratios SR1 and SR2 as well as for the effective line absorption, the continuum absorption and the resulting total absorption. From these cal-culations one can see that αl,tot becomes negative already at ν = 250 GHz, approximately 200 GHz away from the 60 GHz band. Near the band, e.g. at 22 GHz and 90 GHz,αl,tot has positive values. Furthermore, far in the 60 GHz band wing the effective line absorption is a small difference of the two large termsα±l , underlining the importance of the individual line coupling parameters in the calculation ofαl,tot. Figure 3.11 illustrates this situation for the
14Negative absorption coefficients have to be considered in connection with masers and lasers which consti-tute strong non-LTE situations which are far from atmospheric conditionsRead (1980).
Figure 3.10: Line coupling parameter at a temperature of 294 K for the individual lines of the 60 GHz band for three different absorption models (O2-MPM93, O2-PWR93, and O2-PWR88). The dry air and water vapor partial pressures are pdry=1013 hPa, and e=24.92 hPa, respectively. These values correspond to a zonal mean mid-latitude summer surface condition. The numbers at the top of the plot are the rotational quantum numbers N±. The abscissa indicates the individual line center frequency.
effective line absorption at 625 GHz. Additionally Figure 3.11 shows that lines from both branches (P-branch: N−, R-branch: N+) contribute to α+l as well as α−l .
Negative values of αl,tot are in contradiction to the local thermodynamic equilibrium condition which is implicitly assumed in the calculation ofαl,tot. The main reason forαl,tot <0 in the far wing is the insufficient line shape. The line shape Equation (3.3) is derived within the impact approximation and thus most accurate in the line center region. One semi-empirical approach to remove the inaccuracy in the far wing would be to include an extra term in the line shape which accounts for the finite duration of the molecular collisions (Birnbaum and Cohen, 1976). This extra-term causes a steeper decrease of the line shape in the far wing which removes much of the far wing absorption of the 60 GHz band15. The drawback of this approach is that the line shape calculation gets more complicate, involving the modified Bessel function of the second kind and some additional line parameters have to be introduced. Nonetheless this approach is more physical than a strict cutoff as applied in the O2-MPMmodel since this cutoff is not only arbitrary but violates additionally the sum rule of the line mixing parameters as stated in Equation (2.41).
The uncertainty of the oxygen absorption in the far wing of the 60 GHz band is of special interest for SMMW limb sounder with a high spectral resolution. For example JEM/SMILES16 has two bands around 625 GHz (bands A and B) and one band around 650 GHz (band C). Besides other trace gases, JEM/SMILES is aimed to retrieve volume mixing ratios of O3, HCl, and ClO. Comparing the absorption of O3, O2, and HCl in band A and B reveals that the absorption of HCl is 15-450 times larger than the absorption of oxygen (mid-latitude summer, altitude range: 20-30 km). In band C the absorption of ClO is
15For exampleTobin (1996) applied this approach for the line mixing of CO2 in the 4.3µm band.
16JEM/SMILES is a Superconducting subMIllimeter-wave Limb-Emission Sounder on the Japanese Experiment Module JEM of the international space station ISS.
freq. alt. αtot αc αl,tot SR1 SR2
[GHz] [km] [1/m] [1/m] [1/m] [1] [1]
625 0 0.362·10−6 0.152·10−5 -0.116·10−5 -46 47 625 10 0.888·10−7 0.224·10−6 -0.135·10−6 -63 64 625 20 0.542·10−8 0.122·10−7 -0.682·10−8 -69 70 625 30 0.202·10−9 0.503·10−9 -0.301·10−9 -64 65 250 0 0.268·10−6 0.152·10−5 -0.125·10−5 -45 46 250 10 0.690·10−7 0.224·10−6 -0.155·10−6 -57 58 250 20 0.421·10−8 0.122·10−7 -0.803·10−8 -62 63 250 30 0.157·10−9 0.503·10−9 -0.346·10−9 -58 59 90 0 0.721·10−5 0.152·10−5 0.569·10−5 17 -16 90 10 0.117·10−5 0.224·10−6 0.945·10−6 17 -16 90 20 0.656·10−7 0.122·10−7 0.533·10−7 16 -15 90 30 0.264·10−8 0.503·10−9 0.213·10−9 17 -16 22 0 0.273·10−5 0.152·10−5 0.213·10−8 7 -6 22 10 0.405·10−6 0.224·10−6 0.181·10−6 8 -7 22 20 0.222·10−7 0.122·10−7 0.995·10−8 8 -7 22 30 0.911·10−9 0.503·10−9 0.407·10−9 8 -7
Table 3.1: total, continuum, and Line absorption of oxygen at three different frequencies and four different altitudes. The resulting line absorption αl,tot is according to Equation (3.13) divided into two terms which gives the positive and negative fractional line absorption. The subtraction of SR2 and SR1 gives always 1. The total oxygen absorption is denoted by αtot and the continuum absorption term by αc. For each frequency the calculation (using O2-PWR93) is repeated for four altitudes between the ground and 30 km. The atmospheric input is taken from a mid-latitude summer scenario.
0.15-80 times the absorption of oxygen (mid-latitude summer, altitude range: 20-30 km). The absorption of ozone is in all JEM/SMILES bands much larger than the oxygen absorption.
Estimating from Table 3.1.1 the uncertainty of the oxygen absorption to about 1-2.5 times the actual total oxygen absorption17shows that the uncertainty in the total oxygen absorption is comparatively small with respect to HCl but is not negligible in case of ClO in the lower half of the stratosphere. However, the uncertainty in the nearly constant oxygen absorption can be included as an additional offset in retrieval schemes which is thus not crucial for trace gas VMR retrievals. Nevertheless, for instance in laboratory measurements for line broadening parameters with oxygen as buffer gas this uncertainty could be of importance.