An assessment of recent water vapor continuum measurements upon longwave and shortwave radiative transfer

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jd015505, 2011 An assessment of recent water vapor continuum measurements upon longwave and shortwave radiative transfer D. J. Paynter 1 and V. Ramaswamy 2 Received 16 December 2010; revised 8 July 2011; accepted 19 July 2011; published 18 October [1] Recent measurements of the water vapor continuum have been combined to form an empirical continuum termed the BPS continuum model. This covers the 800 to 7500 cm 1 spectral region for the self continuum and most of the major absorbing spectral regions between 240 and 7300 cm 1 for the foreign continuum. Longwave (i.e., absorption/emission of terrestrial radiation between 1 and 3000 cm 1 ) and shortwave (i.e., using solar radiation as a source and considering atmospheric absorption between 1000 and cm 1 ) line by line (LBL) radiative transfer calculations have been performed for clear sky conditions in three standard test atmospheres using line data from the HITRAN database. This has allowed BPS to be compared to the commonly used CKD and MT CKD continuum models, in addition to conducting a more detailed investigation of the separate roles of the self and foreign continua than previously provided in the literature. Using uncertainties obtained from multiple experimental studies it has been possible to estimate the upper and lower limits of the effects due to the continuum in many spectral regions. The outgoing longwave radiation in a midlatitude summer (MLS) atmosphere calculated by all three continuum models agree to within 0.6 Wm 2 with a ±1.1 Wm 2 estimated uncertainty. The corresponding values for surface downwelling radiation are 1.3 Wm 2 ± 2.5 Wm 2. For shortwave absorption, the different models agree within 1.0%, with an estimated uncertainty of ±1.7%. However, the three models differ in the amount by which the self and foreign continua contribute to shortwave absorption. Citation: Paynter, D. J., and V. Ramaswamy (2011), An assessment of recent water vapor continuum measurements upon longwave and shortwave radiative transfer, J. Geophys. Res., 116,, doi: /2010jd Introduction [2] The water vapor spectrum is characterized by a combination of highly structured and narrow line like features and a much smoother and broader underlying continuum. The line spectra can be modeled using parameters from databases such as the High Resolution Transmission Molecular Absorption Database (HITRAN) [Rothman et al., 2005]. The water vapor continuum consists of two components; the self and foreign. The self continuum results from the interactions between water vapor molecules. When water molecules are mixed with those from other gases there is an additional foreign continuum. The optical depth of the self continuum is proportional to the square of the water vapor pressure, while the foreign continuum is proportional to the product of the water vapor pressure and the sum of the partial pressures of the foreign broadening gases. 1 Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey, USA. 2 Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey, USA. Copyright 2011 by the American Geophysical Union /11/2010JD Motivation [3] The two most commonly used water vapor continuum models are the Clough, Kneizys and Davies (CKD) [Clough et al., 1989] model and its successor, the Mlawer, Tobin, Clough, Kneizys and Davies (MT CKD) model [Clough et al., 2005; Mlawer et al., 1999]. Both models propose that the continuum can be derived as a function of the water vapor line properties. How the continuum is derived from the water vapor line properties differs between MT CKD and CKD, but both models have been fitted to produce agreement with experimental estimates of the continuum. CKD was fitted to agree with the self and foreign continuum coefficients derived from laboratory based water vapor measurements by Burch [1981, 1982, 1985] and Burch and Alt [1984] covering most of the cm 1 region. Although still used in some radiation codes, CKD is now considered to be obsolete and has been superseded by MT CKD. [4] MT CKD was fitted using newer measurement data than CKD, such as the Fourier transform spectrometer (FTS) laboratory water vapor measurements by Tobin et al. [1996] (covering 1200 to 2200 cm 1 ) and the Quality Measurement Experiment (QME) data [Turner et al., 2004] (600 to 1400 cm 1 ). QME compared spectral line calculations with high resolution downwelling FTS longwave atmospheric radiation measurements at Department of Energy 1of13

2 Atmospheric Radiation Measurement (ARM) sites; a similar process was employed for the cm 1 spectral region using downwelling measurements from the Surface Heat Energy Budget of the Arctic Ocean (SHEBA) ice station in Alaska [Tobin et al., 1999]. [5] On numerous occasions the continua of both CKD and MT CKD have been updated to agree with new observations. We will focus on the two most recent updates of MT CKD associated with versions 2.4 and 2.5. Relative to version 1.1, the version 2.4 update increased the self continuum ( 25%) between 1 and 600 cm 1. Additionally, the 2.4 update led to a decrease of the foreign continuum ( 25%) between 1 and 400 cm 1, and a slight increase between 400 and 600 cm 1. These changes were in part due to the Radiative Heating in Underexplored Bands Campaign (RHUBC) [Delamere et al., 2010], which, similar to QME, compared simulated spectra with high resolution downwelling measurements in order to obtain the continuum in 5 micro windows (i.e., the gaps between spectral lines) between 410 and 560 cm 1.At longer wavelengths, microwave radiometer atmospheric measurements of Payne et al. [2011] between 0 and 5.7 cm 1 were also used. The model was improved over the entire 0 to 600 cm 1 region by fitting a Pade function to this new data. [6] MT CKD 2.5 increased the self continuum coefficients at 296 K by approximately an order of magnitude in the cm 1 window. This change was not based upon observations within this window, but instead on interferometric calorimeter laboratory measurements over a narrow spectral range by Fulghum and Tilleman [1991] at 9466 cm 1 and Bicknell et al. [2006] at 4610 cm 1 and 6140 cm 1. All of these measurements suggest that previous versions of MT CKD underestimated the continuum by a factor of about 3 to 10. [7] The small residuals between observation and measurement shown by Payne et al. [2011], Turner et al. [2004] and Delamere et al. [2010] indicate that MT CKD is accurate for spectral regions and atmospheric conditions where it has been validated against, or updated using, measurement data. However, the fact that measurement data was needed to fit and improve MT CKD means that one should be cautious of its accuracy where it has not been verified by, or updated to agree with, measurement data. This is well demonstrated by studies which show the presence of errors in MT CKD in certain spectral regions [e.g., Baranov and Lafferty, 2011; Burch, 1985; Paynter et al., 2007, 2009]. In addition, caution is necessary if one is using the continuum model in conditions substantially different from those in which it has been validated. For example, Baranov et al. [2008] demonstrate that the self continuum between 800 and 1200 cm 1 is stronger than that suggested by MT CKD 2.5 at temperatures over 305 K. [8] We have constructed a continuum model from the laboratory measurements of Baranov et al. [2008], Baranov and Lafferty [2011], Paynter et al. [2009] and the atmospheric measurements of Serio et al. [2008], which is termed the BPS continuum model in this study. It may be noted that MT CKD has not been modified with data from these measurements. The BPS model is based on a slightly different conceptual approach to MT CKD, in that the continuum coefficients in wavenumber space come directly from the continuum derived from measurements and not from a fitted analytic form. In addition, using experimental errors an upper and lower limit for the BPS model has been obtained. In contrast, there are presently no uncertainty parameters associated with MT CKD. [9] It should be emphasized that the BPS model is not a complete continuum model like MT CKD, and that there are numerous spectral regions where there is insufficient measurement data to construct the continuum (for example, above 7500 cm 1 ); in these regions this study uses the MT CKD 2.5 continuum. The BPS model should therefore be seen as a first step in constructing an alternative model to MT CKD using new measurement data and not as a finished product. It is envisaged that, as more measurement data become available, it will be possible to increase both the accuracy and spectral coverage of this new model. In section 2.1, there is further discussion about the details of the BPS model. [10] We will assess the impact of the BPS continuum model upon idealized radiative transfer calculations by comparing it to CKD 2.4 and MT CKD 1.1 and 2.5. CKD 2.4 is chosen because it represents the final version of CKD and at present is still used in numerous radiation codes. The two different versions of MT CKD selected reflect how this model has evolved Previous Studies on Continuum Radiative Transfer [11] The importance of the continuum for longwave radiative transfer calculations has been demonstrated in numerous studies [e.g., Clough et al., 1992; Ellingson et al., 1991; Huang and Ramaswamy, 2007; Huang et al., 2007; Schwarzkopf and Ramaswamy, 1999]. Details of the longwave radiative effects of the continuum are summarized in section 3.1 below. Previous studies comparing versions of MT CKD (1.0 and 1.2) and CKD (2.1, 2.2 and 2.4) found good agreement between them at all levels in the atmosphere, with a bias typically less than 1 Wm 2 [Fomin et al., 2004; Kratz, 2008]. [12] Studies have investigated the absorption of solar radiation by the water vapor continuum using line by line (LBL) calculations [e.g., Collins et al., 2006a; Ptashnik and Shine, 2003; Zhong et al., 2001]. Collins et al. [2006a] looked at the effects of the CKD 2.4 continuum in different standard atmospheres and also the individual contribution of both the self and foreign continua. They found that while the self continuum is only important in humid tropical atmospheres, the foreign continuum is important in both dry and humid atmospheres. Zhong et al. [2002] showed, for a water vapor only tropical standard atmosphere (TRO) atmosphere, that when the original version 0 of CKD is used the continuum contributes 24.1 Wm 2 to water vapor shortwave absorption, but when version 2.2 is used this value is reduced to 8.7 Wm 2. Details of the radiative effects of the continuum in the shortwave are summarized in section 3.3. [13] A few papers have also attempted to compare and update the MT CKD or CKD models using measurement data. Zhong and Haigh [1999] used a wideband radiation code to compare global (using a 10 degree latitude longitude grid) clear sky fluxes calculated with CKD 0, CKD 2.2 and an updated version of CKD 2.2 called PKR. This strengthened the temperature dependence of the CKD 2.2 self continuum in the atmospheric window using aircraft measurement data [Rudman et al., 1994]. Although the difference between CKD 2.2 and PKR was minimal when calculating outgoing long- 2of13

3 Table 1. The Spectral Regions in Which Different Measurements Are Used to Make Up the BPS Continuum Model a Region (cm 1 ) Data Set for Continuum Coefficient at 296 K Temperature Dependence Range of Measurement Temperatures (K) Uncertainty Assumed in Coefficient at 296 K (±%) Self MT CKD 2.5 MT CKD 2.5 N/A MT CKD 2.5 MT CKD 2.5 N/A Baranov et al. Baranov et al Exp. Err Paynter et al. MT CKD Exp. Err Paynter et al. Paynter et al Exp. Err Paynter et al. MT CKD Exp. Err BL BL Exp Err Paynter et al. Paynter et al Exp. Err Paynter et al. MT CKD Exp. Err Paynter et al. Paynter et al Exp. Err Paynter et al. MT CKD Exp. Err Paynter et al. Paynter et al Exp. Err Paynter et al. MT CKD Exp. Err MT CKD 1.1 MT CKD 2.5 N/A 50 Foreign MT CKD 2.5 N/A N/A Serio et al. N/A N/A Exp. Err MT CKD 2.5 N/A N/A Paynter et al. N/A Exp. Err MT CKD 2.5 N/A N/A Paynter et al. N/A Exp. Err MT CKD 2.5 N/A N/A Paynter et al. N/A 351 Exp. Err MT CKD 2.5 N/A N/A Paynter et al. N/A 351 Exp. Err MT CKD 2.5 N/A N/A 50 a Paynter et al. refers to Paynter et al. [2009], BL refers to Baranov and Lafferty [2011], Baranov et al. refers to Baranov et al. [2008], Serio et al. refers to Serio et al. [2008] and Exp. Err. denotes experimental error. Despite there being BPS continuum data only up to 7500 cm 1, to show the full contribution of the continuum to solar radiative transfer, shortwave calculations are performed up to cm 1. wave radiation (OLR), there were notable differences in surface downwelling radiation (SDR), with PKR predicting up to 5 Wm 2 (2.5 Wm 2 globally) greater values than CKD 2.2 in mid latitude regions. [14] Firsov and Chesnokova [2010] compared the Baranov et al. [2008] measurements between 800 and 1200 cm 1 (also used in this paper) to MT CKD 1.0 (which is the same as version 2.5 in this region). They showed that despite the weaker temperature dependence of the Baranov et al. [2008] measurements, there was good agreement (<3 Wm 2 ) for all atmospheric profiles tested. They suggested that the difference between the laboratory measurements of Baranov et al. [2008] and the atmospheric measurements of Turner et al. [2004], from which MT CKD is derived, could be the result of the difficulty in separating out the self and foreign continua in atmospheric measurements. 2. Methodology 2.1. Constructing the BPS Continuum Model [15] The BPS model consists of the continuum formulation derived from measurements (or extrapolations from measurements at higher temperatures) of the self and foreign continuum coefficients at 296 K in most of the 240 to 7500 cm 1 region. The continua used in BPS are derived from FTS measurements, making them suitable for the construction of an empirical continuum model, as they are derived at high resolution over a broad spectral range. To derive the continuum from spectral measurements, the removal of the spectra of other absorbers is required. This is performed using databases such as HITRAN [Rothman et al., 2005]. Due to the effects of saturation and uncertainty of the line parameters it is only possible to derive the continuum in some microwindows. For all measurement sets, the gaps between the suitable micro windows are normally less than 20 cm 1.In the BPS model we have assumed that the continuum varies linearly between the micro windows. [16] A continuum model necessarily requires the ability to predict the continuum as a function of vapor pressure, foreign molecular pressure and temperature. The first two properties are well known (see section 1), but there still is no consensus about the exact form of the temperature dependence. For the self continuum derived from the Baranov et al. [2008] and Paynter et al. [2009] measurements, it was possible to fit a temperature dependence coefficient using an exponential inverse temperature law of the same form as CKD and MT CKD (see Appendix A, equation (A4)). This successfully predicts the continuum at different measurement temperatures to a reasonable degree of accuracy (within 5%). It should be stressed that, given the current lack of theoretical understanding about temperature dependence, the agreement for measured temperatures (which are generally warmer than those found in the atmosphere) does not imply that estimates at lower temperatures will also be accurate. Additionally, in the window regions above 2000 cm 1 there is now evidence to suggest that the temperature dependence may be stronger than allowed by the MT CKD form [Baranov and Lafferty, 2011]. [17] However, in all regions where suitable measurement data exist (see Table 1), the BPS model temperature depen- 3of13

4 Figure 1. The CKD 2.4 (red), MT CKD 2.5 (green), BPS (blue), upper BPS and lower BPS (gray shaded) (a) foreign and (b) self continuum models used in this work in the longwave. (c) The temperature dependence coefficient (see Appendix A) of the self continuum of MT CKD/CKD (red) and BPS (blue). (d, e) The cm 1 and (f, g) cm 1 regions clarifying the contrast between the models. Note: as BPS is based upon MT CKD, the blue and green lines overlap in certain regions. dence coefficients (see Figures 1c and 2c) are obtained by fitting the MT CKD form to the continuum derived from measurement data. In regions where there is no suitable measurement data, the coefficients of MT CKD are used. We have assumed that the foreign continuum has no temperature dependence other than that associated with the radiation field (see Appendix A). Paynter et al. [2009] and Serio et al. [2008] have shown this to be a reasonable assumption, but this does not preclude the existence of a weak temperature dependence. [18] Table 1 lists the details of the spectral regions where each measurement set contributes, while Figures 1 ( cm 1 ) and 2 ( cm 1 ) compare both the continuum and temperature coefficients for BPS, MT CKD 2.5 and CKD 2.4. What follows is a brief summary of the measurement data used in the BPS model and how it differs from MT CKD and CKD. Unless versions are specified in the text, this analysis refers to MT CKD 2.5 and CKD 2.4. Appendix A provides additional information about how the continuum is included in the radiative transfer calculations. [19] Serio et al. [2008] performed atmospheric measurements of the foreign continuum between 240 and 590 cm 1. These were the first observations made in the cm 1 region and showed that MT CKD 1.1 overestimates the continuum in most regions, sometimes by up to 40%. However, as noted in section 1.1, MT CKD 2.4 is based upon measurements by Delamere et al. [2010], bringing it closer in value to the Serio et al. [2008] findings (Figure 1d). Delamere et al. [2010] suggest that the remaining disparity is likely due to the different line parameters used to derive the continuum rather than differences in the actual measurements. [20] Baranov et al. [2008] derived the self continuum from measurements in the cm 1 window region at six different temperatures between 311 and 363 K. The 4of13

5 Figure 2. Same as Figures 1a 1c but for the cm 1 region. results show that both MT CKD and CKD underestimate the continuum coefficients at these measured temperatures, but overestimate the strength of the temperature dependence (Figure 1c). At 296 K, these two effects cancel each other out to some extent, leading to very similar continuum coefficient predictions by all models (Figure 1b). [21] The self continuum derived from the measurements by Baranov and Lafferty [2011] in the cm 1 window show that CKD and MT CKD (up to version 2.4) underestimate the continuum by an order of magnitude between 311 and 363 K. These latter measurements demonstrate that the temperature dependence coefficients are underestimated by around 30% (Figure 1c). As mentioned earlier, they also suggest that the form of the MT CKD temperature dependence is too weak in this region. This could mean that the continuum coefficients given at 296 K in the BPS model may be too small, by up to 20% in this region. Uncertainties associated with measurement data, in addition to the absence of measurements below 310 K, make it difficult conclude anything more precise at present. In MT CKD 2.5 the continuum coefficients between 2000 and 3000 cm 1 at 296 K have been increased (but not the temperature dependence coefficients) and are now closer to (but still smaller than) the Baranov and Lafferty [2011] measurements extrapolated to 296 K. [22] Measurements [Paynter et al., 2009] also indicate that MT CKD underestimates the continuum in the and cm 1 window regions, but there are large uncertainties associated with this data (see Figure 2b). There is also no reliable temperature dependence data for these window regions at present, although given the Baranov and Lafferty [2011] result it is likely that the MT CKD coefficients used in BPS are too small. If this is true it would also imply that the BPS continuum coefficients in these regions, which are extrapolated from the continuum derived from measurements by Paynter et al. [2009] at 351 K, are too small. [23] The Burch [1985] measurements in the cm 1 band showed noticeable differences from both the CKD and MT CKD estimates, with peaks in the continuum detected around 3600 and 3800 cm 1 which are not included in either formulation. In the same region, the Burch [1985] measurements also suggest that the foreign continua of MT CKD and CKD are each too large by up to 50%. The Burch [1982] and Tobin et al. [1996] measurements in the cm 1 band region are in good agreement with the MT CKD and CKD foreign continua, but show that both MT CKD and CKD underestimate the self continuum. The measurements used to construct the BPS model of Paynter et al. [2007, 2009] generally agree with the Burch [1982, 1985] and Tobin et al. [1996] data. However, these newer high resolution measurements also reveal finer structure in the continuum within the bands than captured by MT CKD. At present it is clear that some of this structure is a property of the continuum, but it is unclear how much is the result of incorrect spectral line data in HITRAN. Figures 1f and 1g compare the foreign and self continua respectively in the cm 1 region. [24] In the cm 1 region, measurements by both Ptashnik et al. [2004] and Paynter et al. [2009] also show both MT CKD and CKD to underestimate the self continuum, but with better agreement in the cm 1 region. In the and cm 1 regions the lower estimate of the foreign continuum by MT CKD agrees better with the measured data than the estimate given by CKD. Above 8000 cm 1 there are currently no published measurements of the self or foreign continua over a large wavenumber region viz., suitable for an empirical continuum. Measurements at 10,611 cm 1 [Reichert et al., 2007] and 9466 cm 1 [Fulghum and Tilleman, 1991] provide evidence that there is continuum at higher wavenumbers, with 5of13

6 Table 2. The Decrease (Wm 2 ) in OLR and Increase (Wm 2 )in SDR Due to the Inclusion of Various Continuum Models and the Upper (U) and Lower (L) Limits of the BPS Continuum Model a H 2 O, CO 2,O 3,O 2, H 2 O Only CH 4,N 2 O TRO MLS SAW TRO MLS SAW No continuum OLR No continuum SDR CKD OLR CKD SDR MT CKD OLR SDR MT CKD OLR SDR BPS OLR BPS SDR BPS U OLR BPS U SDR BPS L OLR BPS L SDR a HITRAN 2004 is used for all calculations. The initial values without any continuum present are shown for reference. These calculations are performed for both a water vapor only atmosphere and one including different absorbers as specified in the table. the latter study suggesting that MT CKD needs to be increased in the windows above 7500 cm 1. [25] Paynter et al. [2009] found that, within the bands from 1200 to 7300 cm 1 (Figures 1c and 2c), MT CKD and CKD generally did well at modeling the temperature dependence of the self continuum, but failed to capture some of the spectral structure Uncertainty in the BPS Continuum Model [26] In order to investigate how uncertainty in the continuum impacts radiative transfer, lower and upper estimates of the BPS model have been made (shown in Figures 1 and 2 by the gray area). Where measurements exist, the associated uncertainty in the measurement data was used. Where there is no data for the self continuum, between 1 and 600 cm 1 a ±25% error at 296 K is assumed, while beyond 7500 cm 1 a ±50% error at 296 K is assumed. In the cm 1 region an error of ±10% is assumed to reflect the constraints placed on the continuum by the measurements of Turner et al. [2004]. These other values were chosen to be consistent with both the uncertainties and the deviation from MT CKD, as observed by Baranov et al. [2008], Baranov and Lafferty [2011] and Paynter et al. [2009] in nearby regions. In the windows above 4000 cm 1 the measurements by Paynter et al. [2009] used in the BPS model show up to an order of magnitude uncertainty in the self continuum (see Figure 2). This implies that in the windows beyond 7500 cm 1, the uncertainty in the continuum is underestimated. However, as the absorption in this area is rather weak, the uncertainty should not have a significant impact upon the radiative transfer results. [27] For the foreign continuum, other than where measurement data exist, a value of ±50% at 296 K is assumed for the uncertainty. This is larger than for the self continuum in some spectral regions because the foreign continuum is weaker than the self continuum and thus there are larger experimental uncertainties. The ±50% error is also consistent with the deviation seen between the MT CKD model and measurements of both Paynter et al. [2009] and Serio et al. [2008]. In the microwave region between 0 and 5 cm 1, the recent work of Payne et al. [2011] suggests only a 4% uncertainty in the self and foreign continua. However, due to a lack of additional measurements it is not clear how well constrained the MT CKD model is up to 240 cm 1. [28] It should be added that neither uncertainty in the temperature dependence coefficient or the form of temperature dependence have not been considered here and no assumption is made. The inverse temperature dependence of the self continuum means that at temperatures lower than 296 K the difference between BPS and its upper and lower limits will increase, reflecting to some degree our increased lack of confidence in the continuum at lower temperatures Radiative Transfer Model [29] This work uses an LBL code with an output resolution of 0.1 cm 1 and assumes a clear sky plane parallel atmosphere. Calculations are performed for 3 standard atmospheres, the Midlatitude Summer (MLS), Tropical (TRO) and Subarctic Winter (SAW). The spectral line data is obtained from the HITRAN 2004 [Rothman et al., 2005] database. A 73 level atmosphere is used, divided into 20 mb layers between the surface and 20 mb and then into finer layers above that. The Reference Forward Model (RFM) (A. Dudhia, Reference Forward Model version 4: Software user manual, 2005, available at RFM) LBL code is used to calculate the optical depth for each of these layers. [30] Two sets of calculations have been conducted in what will be referred to as the longwave and shortwave spectral regions. The longwave calculations consider absorption/ emission between 1 and 3000 cm 1 for both the Earth s surface and atmosphere. The shortwave calculations consider the absorption of the atmosphere between 1000 and cm 1, for both the direct solar radiation and reflected solar radiation. [31] For calculations in the shortwave, scattering is ignored and it is assumed that the incident solar radiation can be modeled as a homogeneous beam with a fixed solar zenith angle. By comparing results using DISORT with Rayleigh scattering, it can be demonstrated that this approximation agrees within 0.5% for solar zenith angles less than 30 degrees. This approximation creates a correspondingly similar uncertainty for the effect of the continuum. Unless stated otherwise, a solar zenith angle of 30 degrees and a zero surface albedo are assumed. Good agreement is observed between the results of the radiation code used in this work and those of other researchers presented by Collins et al. [2006b]. [32] For all cases, the contribution of the continuum is calculated as the difference in flux/heating rate when that continuum is removed. We have not considered the impact of the continua of other absorbers (e.g., CO 2, O 2 ) nor included them in the calculations. 3. Results and Discussion 3.1. Summary of Continuum Longwave Radiative Transfer [33] Table 2 provides a detailed analysis of the integrated fluxes for the MLS, TRO and SAW atmospheres, giving the total changes in outgoing longwave radiation (OLR) 6of13

7 Table 3. The Change (Wm 2 ) in the OLR and SDR Due to the Inclusion of the Self (S) and Foreign Continua (F) of BPS and MT CKD 2.5 H 2 O, CO 2,O 3,O 2, H 2 O Only CH 4,N 2 O TRO MLS SAW TRO MLS SAW MT CKD (S) OLR MT CKD (S) SDR MT CKD (F) OLR MT CKD (F) SDR BPS (S) OLR BPS (S) SDR BPS (F) OLR BPS (F) SDR and surface downwelling radiation (SDR) that arise from including the continuum in both a water vapor only atmosphere and an atmosphere that contains other important greenhouse gas absorbers (CO 2 (380 ppmv), CH 4 (1700 ppbv), N 2 O (315 ppbv), O 3,O 2 and N 2 ). In Table 3, the continuum is broken down into the self and foreign components. The contribution of the continuum as a function of wavenumber is presented in Figures 3 and 4. In Figure 3, the reduction in OLR (due to the atmosphere, i.e., the greenhouse effect) and increase in SDR due to water vapor are shown with, and without, the continuum included. This is performed for both SAW and TRO atmospheres. For these atmospheres, Figure 4 shows the contributions of the total (self + foreign) continuum and self continuum to the reduction in OLR and increase in SDR, but with the greenhouse gases specified in Table 2 also included. These results are averaged over 100 cm 1 to make the broadband features of the continuum clearer. From Figure 4 it can be inferred to a good approximation that the contribution of the foreign continuum is the difference between the total and self continua, although this is not strictly valid due to the effect Figure 3. The clear sky decrease in OLR due to a water vapor only atmosphere (i.e., surface upwelling OLR) (a) with and (b) without the MT CKD 2.5 continuum for the TRO/SAW atmospheres. (c, d) Same as Figures 3a and 3b, showing the increase in SDR due to the atmosphere is shown. Figure 4. (a, b) The clear sky decrease in OLR due to the atmosphere as a function of wavenumber averaged over 100 cm 1 (in order to make the general broadband features clearer) caused by the BPS (solid blue line) and MT CKD 2.5 (solid green line) models. The shaded gray area is the uncertainty estimated using the upper and lower limits of the BPS model and the separate contribution of the self continuum (dotted lines) is also presented. The contribution of the BPS model in a water vapor only atmosphere (solid purple line) is shown. In all other cases the greenhouse gases detailed in Table 2 are included. (c, d) As in Figures 4a and 4b, but the increase in SDR is presented. of overlap. The cooling rate of the total continuum and the self and foreign components are shown in Figure 5. [34] Similar to the results from Clough et al. [1992], it can be seen that in the TRO atmosphere between 400 and 1200 cm 1, the continuum accounts for up to 100% of the atmospheric reduction of the OLR (Figure 3). The foreign continuum dominates below 500 cm 1 (i.e., within the rotational band), but this changes such that the self continuum causes most of the reduction between 800 and 1200 cm 1 (Figure 4). In the drier SAW atmosphere, the reduction in OLR due to the continuum is smaller and occurs principally in the cm 1 spectral region; this is almost solely due to the foreign continuum (Figure 4). [35] It is well known [e.g., Clough et al. 1992] that the self continuum has a greater impact on SDR than OLR (see Table 3). This is because most of the absorption/emission is in the lower troposphere, which has a similar temperature to the surface. For the same atmosphere, the continuum contribution to SDR and OLR occurs in slightly different spectral regions (Figures 3 and 4). This is most apparent when comparing the SDR to the OLR for the SAW atmosphere (Figures 3b, 3d, 4b and 4d). Here there is a far higher percentage contribution from the continuum to OLR from the spectral region below 500 cm 1 than to SDR. The self continuum contributes far more (as a percentage total) to SDR than OLR, even in the dry SAW atmosphere (Figures 3b, 3d, 4b and 4d). [36] The different radiative transfer of the foreign and self continua can be explained by the fact that the optical depth of the former scales as the product of water vapor pressure and 7of13

8 Figure 5. (a, b) The clear sky cooling rate due to the MT CKD 2.5 (green) and the BPS models (blue). For both models the separate contributions of the self and foreign continua are presented. The error bars give the uncertainty estimated using the upper and lower limits of the BPS model. dry atmospheric pressure, while the latter scales with the square of vapor pressure. Thus, for the relatively low vapor pressures present at higher altitudes in the TRO atmosphere, and throughout the SAW atmosphere, only the optical depth of the foreign continuum is significant. Near the surface in the TRO atmosphere, the water vapor content is high enough for the self continuum optical depth to have a significant influence. It is also important to take into account the fact that per molecule the self continuum is far stronger than the foreign (see Figure 1), especially in the atmospheric window between 800 and 1200 cm 1, which thus explains the dominance of the self continuum in this region. [37] The gaps between water vapor lines within the bands (micro windows) become narrow and saturate near the surface, as both atmospheric and vapor pressures increase. Thus, at levels where there is sufficient water vapor for the self continuum optical depth to become significant, the micro windows are saturated. However, the optical depth of the micro windows and the foreign continuum vary similarly as a function of both vapor and dry atmospheric pressure. Therefore the foreign continuum can play an important role within the bands. This is most apparent in the upper troposphere, where the foreign continuum optical depth starts to contribute because the micro windows are not saturated. In the lower atmosphere, the micro windows are saturated and accordingly there is far less contribution by the foreign continuum. This explains why the foreign continuum contributes to SDR and OLR in slightly different spectral regions and also why the inclusion of the foreign continuum causes additional cooling above an area of reduced cooling (Figure 5), because it creates the same effect as adding more water vapor to the atmosphere (i.e., increasing the optical depth linearly) and thus increases the emission height. [38] The impact of the continuum is reduced if absorbers other than water vapor are included. For instance, the reduction in OLR caused by the continuum is decreased by 35% and 34% in the MLS and TRO atmospheres respectively and by 7% in the SAW atmosphere (Table 2). Likewise, the increase in SDR for the TRO atmosphere is reduced by about 30% and 37% for the MLS and SAW atmospheres respectively. Table 4 shows the reduction in the contribution of the continuum due to the inclusion of different absorbers in an MLS atmosphere. Here, CO 2 can be seen to have the greatest effect, although O 3 and CH 4 do cause small reductions. The wavenumber analysis of OLR and SDR in Figures 3 and 4 reveals that the greatest reduction due to CO 2 is around 700 cm 1 (compare the purple and blue lines in Figure 4). In these regions the addition of CO 2 causes the contribution of the continuum to drop to virtually zero. In the SAW atmosphere the continuum does not overlap as strongly with CO 2 and thus the reduction is far less Comparing BPS, MT CKD and CKD in the Longwave [39] Water vapor content is important in determining how good the agreement is between models. For example, in the TRO and MLS atmospheres BPS estimates a 6% to 8% smaller reduction in OLR due to the continuum compared to Table 4. The Decrease (Wm 2 ) in the Contribution of the MT CKD 2.5 Continuum, to Both OLR and SDR, for the MLS Atmosphere Due to the Inclusion of Different Absorbers (CO 2 (380 ppmv) CH 4 (1700 ppbv) N 2 O (315 ppbv) and O 3 ) a All CO 2 CH 4 O 3 N 2 O OLR SDR SWA MT CKD SWA CKD a The reduction in the contribution of the continuum to SWA (Wm 2 ) due to the inclusion of different absorbers is also given. This has been calculated for the MLS atmosphere using both the MT CKD and CKD continua. HITRAN 2004 is used for all calculations. 8of13

9 MT CKD (Table 3); this difference increases to 14% in the SAW atmosphere. Note that this result, along with others quoted in this section, includes the other gaseous absorbers listed in Table 2. Likewise, for the increase in SDR due to the continuum the percentage difference between BPS and MT CKD becomes greater as the atmosphere becomes drier, from 1.4% for the TRO atmosphere to 11% for the SAW atmosphere. The larger deviations in the SAW atmosphere arise due to the difference between the MT CKD 2.5 foreign continuum and the measurements of Serio et al. [2008] in the cm 1 region. This is evident from Table 3, where the differences between MT CKD and BPS are broken down into the self and foreign continuum components. For OLR and SDR, the differences for the self continuum are less than 5%, but are up to 20% for the foreign continuum. Figure 4 supports this point and shows the greatest deviation between all models occurs below 500 cm 1. In the SAW atmosphere, the reduction in OLR between 240 and 590 cm 1 due to the foreign continuum is 0.67 Wm 2 for BPS versus 0.90 Wm 2 for MT CKD 2.5. These values increase slightly in the TRO atmosphere to 1.02 Wm 2 and 1.32 Wm 2 respectively. For the same spectral region in a TRO atmosphere the continuum increases SDR by less than 0.1 Wm 2, but in the SAW atmosphere the increase is 2.75 Wm 2 for BPS versus 3.27 Wm 2 for MT CKD. [40] The difference in using the Baranov et al. [2008] self continuum data rather than MT CKD in the cm 1 region is smaller in percentage terms, but still notable in humid atmospheres. For instance, in the TRO atmosphere the self continuum of BPS estimates a 5.52 Wm 2 reduction in OLR versus 5.84 Wm 2 for MT CKD, while the increase in SDR is 41.2 Wm 2 versus 42.0 Wm 2 respectively. At the measurement temperatures (311 K to 363 K) Baranov et al. [2008] recorded a greater self continuum than that predicted by MT CKD, but they also measured a weaker temperature dependence. These two effects somewhat cancel each other out so that, for the atmospheric temperatures at which the self continuum is effective ( K), MT CKD and BPS are in quite good agreement. This also explains why the percentage difference between BPS and MT CKD is greater for OLR than SDR, as the typical emission temperature will be colder in the case of OLR. [41] In the cm 1 region the MT CKD 2.5 and BPS foreign continuum models are largely similar, although BPS predicts a larger self continuum (see Figure 1). For the foreign continuum this fact is reflected in the changes to SDR and OLR. For example, in a TRO atmosphere the BPS foreign continuum increases SDR by 0.15 versus 0.15 Wm 2 estimated by MT CKD 2.5; for OLR the respective values are 0.54 versus 0.58 Wm 2. However, for SDR the BPS self continuum in a TRO atmosphere predicts 1.41 versus 1.47 Wm 2 due to MT CKD 2.5; for OLR the respective values are 0.30 versus 0.31 Wm 2. The reason for the similar reduction due to the self continuum of BPS and MT CKD is that in the cm 1 spectral region, the self continuum contribution is dominated by the cm 1 region and here both models are very similar. The differences in the self continuum in the cm 1 region only have an impact in humid atmospheres and even then the differences are fairly small. For example, when calculating the extra SDR due to the self continuum in a TRO atmosphere BPS gives 0.38 Wm 2 versus 0.21 Wm 2 for MT CKD 2.5. Likewise, for OLR these values become 0.11 Wm 2 versus 0.06 Wm 2 respectively. [42] There is also good agreement in the cooling rates (Figure 5) predicted by all models. However, BPS predicts slightly less cooling in the lower troposphere due to the smaller self continuum. There is also less cooling in the upper troposphere around 200 mb due to the smaller foreign continuum (most clearly seen in Figure 5b for the SAW atmosphere). [43] The range between the upper and lower limits of BPS is larger than the differences between MT CKD 2.5 and BPS. In the TRO atmosphere with water as the only absorber the ranges are 4.2 Wm 2 and 6.6 Wm 2 for OLR and SDR respectively (Table 2). When other absorbers are included these differences decrease to 2.3 and 4.5 Wm 2 respectively. This change occurs because the greatest differences between the upper and lower limits of BPS for the TRO and MLS atmospheres exist in the strong CO 2 bands between 500 and 800 cm 1, where default ±10%, ±25% or ±50% errors are assumed (see Figure 4). The cooling rates are most influenced by the large uncertainty assumed in the foreign continuum at low wavenumbers, which has the greatest effect at altitudes between 100 and 500 mb (Figure 5) Summary of Continuum Shortwave Radiative Transfer [44] The shortwave calculations were performed for clear sky conditions with a solar zenith angle of 30 degrees and with a zero surface albedo. These conditions hold for all results described in the next two subsections unless otherwise stated. Figure 6 provides an overview of the shortwave absorption (SWA) due to water vapor both with and without the continuum in the TRO (Figure 6a) and SAW (Figure 6b) atmospheres. The contributions of the self and foreign continua are also given for the same atmospheres (Figures 6c and 6d). Figure 7 compares the SWA of different continuum models as a function of wavenumber for the TRO (Figure 7a) and SAW (Figure 7b) atmospheres. The associated heating rates are presented in Figure 8. [45] In the TRO atmosphere the foreign continuum results in very little additional absorption in the bands between 1000 and 7000 cm 1 (Figure 6c), even though the foreign continuum coefficients are greater in this region than at higher wavenumbers (Figure 2). Here, saturation of the water vapor spectral lines makes the contribution of the continuum irrelevant when calculating SWA. However, where the micro windows are not saturated (toward the edge of the band centers below 7500 cm 1 and within the bands over 7500 cm 1 ) the foreign continuum contribution becomes important. This is somewhat reversed in the SAW atmosphere (Figure 6d) where the micro windows within the bands at lower wavenumbers are not completely saturated. This allows the foreign continuum to have a notable impact; whereas at higher wavenumber (above 11,000 cm 1 ) and between the bands, the optical depth of the foreign continuum is too small to contribute significantly. [46] Similar to the situation in the longwave, the self continuum only contributes in the lower troposphere because its optical depth scales with the square of humidity. However, in the lower troposphere the water vapor band centers are already saturated. Thus, the total contribution of the self continuum to the SWA in the TRO atmosphere 9of13

10 Figure 6. The clear sky SWA due to water vapor with and without the MT CKD 2.5 continuum for the (a) TRO and (b) SAW atmospheres with a 30 degree solar zenith angle. The increase in SWA due to the MT CKD 2.5 model in the (c) TRO and (d) SAW atmospheres. The individual self and foreign components are also shown. (Figure 6c) is even less in the band centers than is the case for the foreign continuum. Instead, the self continuum mainly absorbs between the bands where there is little or no overlap with water vapor line absorption. In these regions, the self continuum dominates over the foreign because it has a larger continuum coefficient than the latter (i.e., for the same reason it dominates in the atmospheric window regions in the longwave). In the TRO atmosphere, most of the contribution is in the windows between the stronger bands in the cm 1 region (Figure 6). In the SAW atmosphere, even in these windows, the self continuum optical depth is too small to contribute, meaning that the self continuum can essentially be neglected. This is also reflected in the SWA values presented in Table 5. [47] These inferences also explain the different heating rate patterns of the self and foreign continua (see Figure 8). In the upper troposphere of the TRO atmosphere nearly all of the heating attributed to the continuum results from the foreign continuum. This contribution decreases in the lower troposphere as the band centers become saturated, but some contribution from weaker bands at higher wavenumbers Figure 7. The clear sky SWA of different water vapor continuum models assuming a 30 degree solar zenith angle. The contributions of MT CKD 2.5, BPS and CKD 2.4 are shown for (a) TRO and (b) SAW atmospheres. The shaded gray area is the uncertainty estimated using the upper and lower limits of the BPS model. Note, as BPS is based upon MT CKD, the blue and green lines overlap in certain regions. Figure 8. The clear sky daily heating rates of different water vapor continuum models assuming a 30 degree solar zenith angle. The total contribution of MT CKD 2.5, BPS and CKD 2.4, along with the self components are shown for (a) TRO and (b) SAW atmospheres. The error bars represent the uncertainty in the BPS model. 10 of 13

11 Table 5. The Increase in Water Vapor SWA (Wm 2 ) Due to the Inclusion of Various Continuum Models a TRO MLS SAW H2O CKD CKD (S) CKD (F) MT CKD MT CKD (S) MT CKD (F) MT CKD MT CKD (S) MT CKD (F) BPS BPS (S) BPS (F) BPS U BPS U (S) BPS U (F) BPS L BPS L (S) BPS L (F) a HITRAN 2004 is used for all calculations. The initial values without the continuum present are shown for reference. The increase for only the self (S) or foreign (F) continuum of each model is also shown. Calculations are for a 30 degree solar zenith angle for clear sky conditions. remains. In the SAW atmosphere, the impact of the foreign continuum on the heating rate peaks at a lower altitude, due to the drier atmosphere. However, the strong bands also become saturated nearer the surface, resulting in a decrease in the contribution. The self continuum mainly contributes to heating the lower troposphere, because only in this region of the atmosphere is the optical depth between the bands (in both TRO and MLS atmospheres) significant. [48] The effect of other absorbing gases upon the continuum absorption is less significant in the shortwave than in the longwave. The inclusion of other absorbing gases in the MLS atmosphere (Table 5) reduces the continuum by only 0.22 Wm 2 for both CKD and MT CKD. Approximately 80% of this reduction is due to the overlap with CO 2 and CH 4. BPS, with its larger self continuum between the bands, is somewhat more affected by overlap. This is most apparent in the TRO atmosphere where the overlap causes a 0.53 Wm 2 reduction in SWA versus 0.32 Wm 2 for MT CKD 1.1 and 0.38 Wm 2 for MT CKD 2.5. [49] As the overlap of other absorbers has a much smaller influence than in the longwave, detailed calculations of their effects have not been performed for the shortwave Comparing BPS, MT CKD and CKD in the Shortwave [50] An important difference between the continuum models is that in numerous shortwave bands CKD 2.4 has larger foreign continuum coefficients than either MT CKD or BPS (Figure 2). In addition, the BPS self continuum coefficients are greater than those of MT CKD or CKD 2.4 between the bands below 7500 cm 1 (Figure 2). Despite this, BPS and CKD 2.4 predict similar amounts of SWA in the TRO and MLS atmospheres (Table 5). This is due to the differences between the models in the self and foreign continua somewhat cancelling each other out. Accordingly, the SWA of the BPS foreign continuum is smaller than CKD 2.4 and similar to that of MT CKD 2.5, but the BPS self continuum yields greater absorption than either MT CKD 2.5 or CKD 2.4 (Figure 7 and Table 5). Likewise, in the drier SAW atmosphere, where the foreign continuum dominates, it is CKD 2.4 which has the most continuum absorption (Figure 7 and Table 5). [51] The larger self continuum coefficients of BPS also cause extra heating in the lower troposphere of the TRO atmosphere (Figure 8) compared to the other models. In the SAW atmosphere, where the self continuum is insignificant, the SWA and heating predicted by MT CKD 2.5 and BPS are in close agreement, but are overestimated by CKD 2.4. In fact, as the foreign continuum is important in both the SAW and TRO atmospheres, we observe that CKD 2.4 overestimates the heating in the upper troposphere in both cases (Figure 8). [52] Apart from the increased absorption by the self continuum between 2000 and 3000 cm 1 MT CKD 2.5 and MT CKD 1.1 are identical. The extra absorption causes a 0.3 Wm 2 increase in the integrated flux in the TRO atmosphere and a 0.2 Wm 2 increase in the MLS (Table 5). In this spectral region, MT CKD 2.5 is still smaller than BPS and has weaker temperature dependence coefficients. The smaller water vapor column in the SAW atmosphere makes it insensitive to the MT CKD update. [53] The large uncertainty in the contribution of the continuum is demonstrated by the differences between the SWA when the upper and lower values of BPS are considered (Table 5). Furthermore, a plot of the SWA of the self continuum as a function of wavenumber (Figure 7) highlights how the self continuum between the bands contributes to this uncertainty. However, because the optical depth in these regions is only significant in humid atmospheres, this uncertainty is only important in the MLS and TRO atmospheres. In addition, as Table 5 illustrates, there is a smaller, but still considerable, uncertainty resulting from the foreign continuum which affects results in all three atmospheres. This uncertainty arises from the lack of accurate measurements within the water vapor bands (not shown). 4. Conclusions [54] This paper has compared two widely used continuum models (MT CKD and CKD) to an empirical continuum model (the BPS continuum) that was constructed from recent measurements of the continuum by Baranov et al. [2008], Baranov and Lafferty [2011], Paynter et al. [2009] and Serio et al. [2008]. These models have been compared for three standard atmospheres (MLS, TRO and SAW) in clear sky conditions for both longwave and shortwave spectra. [55] In the longwave, although the contribution of the continuum is large, especially to the SDR, the agreement between the models is very good. In fact, the choice of continuum model leads to less than 0.4% variation in the SDR and OLR. [56] It should be emphasized that this good agreement does not imply that knowledge of the continuum is well constrained in the longwave, as the accuracy of any model cannot be asserted. A better guide comes from using the upper and lower estimates of BPS. This gives an estimated uncertainty of ±0.7% for SDR and ±0.4% for OLR in a TRO/ MLS atmosphere. A certain amount of caution should be 11 of 13

12 attached to these values, since in those spectral regions where there is no BPS measurement data (i.e., cm 1 for the self, 1 220/ cm 1 for the foreign) a somewhat arbitrary ±10%, ±25% or ±50% uncertainty is assumed. In these regions, the impact of the continuum is predicted to be fairly small and thus, unless an error far greater than the upper and lower estimates employed here is discovered (say, an order of magnitude), the continuum should not cause substantial uncertainties in longwave radiative transfer. However, all data used in the BPS model was obtained from measurements above 293 K (sometimes as high as 351 K), and hence uncertainty about the form of the self continuum temperature dependence (not investigated in this paper) could also affect the accuracy of results at lower temperatures. Therefore, for calculations requiring high absolute accuracy (i.e., <±1% error) the radiative transfer community should be aware of the potential errors associated with the continuum. The uncertainties in particular regions may also have an impact upon remote sensing applications. [57] In the shortwave compared to the longwave, the three continuum models show greater disagreement among each other. However, because the continuum contributes less to the total water vapor absorption this translates to a variation of less than 1.0% of the total water vapor SWA for a TRO atmosphere and 1.7% for a SAW atmosphere; the difference between the foreign continua in the MT CKD and CKD models accounts for the larger variation in a SAW atmosphere. It is also apparent that, due to the lack of reliable measurements, there is still great uncertainty about the contribution of the continuum. In particular, this concerns the self continuum between the bands and the foreign continuum within the bands above 5000 cm 1. For example, in a TRO atmosphere the continuum could contribute as much as 6%, or as little as 2%, of the total shortwave water vapor absorption. Similar to the longwave, the fact that the self continuum was derived from measurements as high as 351 K means that uncertainties in temperature dependence, not investigated here, could also become important at lower temperatures. [58] Thus, the continuum contributes notably to our lack of complete understanding of shortwave absorption. There is a need for improved measurements of the continuum in the shortwave to help constrain these values better. Until this is done, there is a caution in performing shortwave radiative transfer calculations, since the continuum could contribute up to a ±2% uncertainty in water vapor absorption. This also highlights the importance of incorporating a continuum in the shortwave radiative transfer, something that is not included in some radiation codes [Collins et al., 2006b]. Appendix A: Including the Continuum in Radiative Transfer Calculations [59] To include the CKD/MT CKD/BPS continuum in radiative transfer calculations the optical depth of each water vapor line is calculated up to 25 cm 1 from the line center and any absorption further from the center is considered to be part of the continuum. The reason for this cutoff is first to reduce computational expense and second because there is evidence of sub Lorentzian behavior far from the line center. What is known as the base term is also removed from the line absorption. The base term is defined for a spectral line as the absorption calculated at 25 cm 1 from the line center using a Lorentzian line shape which is subtracted from the line absorption between ±25 cm 1 from the center. [60] Therefore, in the CKD/MT CKD/BPS models the continuum is defined as the absorption beyond 25 cm 1 from the line center, any absorption in addition to that predicted by the Voigt line shape inside of 25 cm 1 and the base term. [61] The optical depth of the self continuum (t s ) can be calculated as s ð; TÞ ¼ P wl P w kt P 0 T0 T C s ð; TÞ: ða1þ Likewise the optical depth of the foreign continuum (t f )is given by f ð; TÞ ¼ P wl P f T0 C f ð; TÞ: ða2þ kt T P 0 where C s and C f are the self and foreign continuum coefficients respectively, P w is the partial pressure of water vapor, P f is the total pressure of all gases present other than water vapor, T is temperature, v is wavenumber, k is Boltzmann s constant, L is the atmospheric layer path length, T 0 = 296 K and P 0 = 1 atm. [62] Hence the total optical depth of water vapor is given by ¼ P wl kt þ P f P 0 X i¼1 T0 S i F i þ P w T0 C s ð; TÞ P 0 T! C f ð; TÞ ; ða3þ T where there is a total of X water vapor lines, S is the line strength and F is the line shape function cut off at 25 cm 1 and excluding the line base term. [63] When modeling the self continuum it is necessary to take its inverse temperature dependence into account. Due to controversy over the origins of the continuum there is little consensus about the exact form of the temperature dependence. CKD and MT CKD use the form tanh C2 C s ðtþ ¼ C s ðt 0 Þ 2T 0 expð v ðt T 0 ÞÞ; ða4þ tanh C2 2T where C 2 = cm K. The tanh term represents the effect of the radiation field and only has a significant contribution at low wavenumber values ( <500 cm 1 ). s is the temperature dependence coefficient and varies as a function of wavenumber. The same form is assumed in the BPS model. [64] For the foreign continuum, both MT CKD and CKD assume the following temperature dependence, tanh C2 C f ðtþ ¼ C f ðt 0 Þ 2T 0 tanh C2 2T : ða5þ Hence, other than the radiation field at low wavenumber ( <500 cm 1 ), the foreign continuum is assumed to be temperature independent. 12 of 13

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