On the efficient treatment of temperature profiles for the estimation of atmospheric transmittance under scattering conditions

The vertical temperature profile of the atmosphere has an influence on the width and intensity of gaseous absorption lines. In the visible and near infrared part of the spectrum, this poses a problem for the fast forward simulation of the radiative transfer, needed in algorithms for the retrieval of any atmospheric or surface-related parameter from satellite measurements. We show that the main part of the global variability of temperature profiles can be described by their first 2 to 6 eigenvectors, depending on the accuracy requirement, by performing a Principal Component Analysis (PCA) on a global set of temperature profiles from the Global Forecast System (GFS). Furthermore, we demonstrate the possibility to approximate the atmospheric transmittance in the O2 A band for any temperature profile with almost perfect accuracy by a linear combination of the transmittances attributed to each of the significant temperature eigenvectors. For the retrieval of surface pressure from O 2 A band measurements, this reduces the global root mean square error from > 30 hPa to better than 1 hPa by strongly reducing the regional bias of surface pressure, retrieved on the assumption of an average temperature profile. The technique can be applied under scattering conditions to eliminate temperatureinduced errors in, e.g., simulated radiances. In principal, the method can be useful for any problem including gaseous absorption or emission with a significant influence of the temperature profile, such as the retrieval of total water vapour content or sea surface temperature.

Abstract. The vertical temperature profile of the atmosphere has an influence on the width and intensity of gaseous absorption lines. In the visible and near infrared part of the spectrum, this poses a problem for the fast forward simulation of the radiative transfer, needed in algorithms for the retrieval of any atmospheric or surface-related parameter from satellite measurements. We show that the main part of the global variability of temperature profiles can be described by their first 2 to 6 eigenvectors, depending on the accuracy requirement, by performing a Principal Component Analysis (PCA) on a global set of temperature profiles from the Global Forecast System (GFS). Furthermore, we demonstrate the possibility to approximate the atmospheric transmittance in the O 2 A band for any temperature profile with almost perfect accuracy by a linear combination of the transmittances attributed to each of the significant temperature eigenvectors. For the retrieval of surface pressure from O 2 A band measurements, this reduces the global root mean square error from > 30 hPa to better than 1 hPa by strongly reducing the regional bias of surface pressure, retrieved on the assumption of an average temperature profile. The technique can be applied under scattering conditions to eliminate temperatureinduced errors in, e.g., simulated radiances. In principal, the method can be useful for any problem including gaseous absorption or emission with a significant influence of the temperature profile, such as the retrieval of total water vapour content or sea surface temperature.

Introduction
The forward simulation of the radiative transfer in the atmosphere is an essential component of any remote sensing algorithm for the retrieval of atmospheric or surface-related parameters from satellite observations. The requirements regarding the speed of the forward simulation can be hard to meet with state-of-the-art computers and radiative transfer models. This applies even more if both gaseous absorption and scattering from molecules, aerosols and/or clouds in the atmosphere are important, such as in gaseous absorption bands in the visible and near infrared part of the electromagnetic spectrum. Consequently, many forward simulation modules employed in retrieval algorithms in the shortwave region are based on pre-calculated look-up tables of the desired radiative quantity, which is tabulated as a function of the measurement geometry and any influencing geophysical parameter (see e.g. Gao and Kaufman, 2003;Schneising et al., 2009;Lindstrot et al., 2012).
Within absorption bands, the measurement is influenced by the vertical temperature profile through the impact of the pressure and temperature profile on the absorption line widths and intensities (Liou, 2002, see). The Doppler frequency shift caused by the molecular velocities results in a temperature-dependent line width, described by the Doppler line shape: Additionally, absorption lines are broadened by molecular collisions, described by the Lorentz line shape: with the Lorentz line width α L , which is roughly proportional to the number of collisions per unit time: α L ∝ pT −1/2 .
The resulting line can approximately be described by the Voigt line shape, which is obtained from a convolution of both line profiles. In the lower atmosphere, the dominant process is pressure broadening with the resulting inverse influence of the temperature on the line width. An absorption line will therefore be broadened with increasing pressure and decreasing temperature, if the whole vertical column of the atmosphere is traversed.
It is not feasible to tabulate the simulated quantity as a function of the temperature in each vertical atmospheric layer, since this would result in look-up tables with an excessive, non-manageable number of dimensions. Such effort is indeed not necessary, since the individual layer temperatures are highly correlated and can be described with a much lower number of parameters, such as determined by means of a Principal Component Analysis (PCA, see e.g. Peixoto and Oort, 1992).
A PCA is an orthogonal linear transformation to a new coordinate system, such that the new variables are uncorrelated and the new coordinates carry the total variance in descending order. The technique is frequently used in various fields including atmospheric modeling and remote sensing. In the past, it has for example been applied with great success to reduce the dimensionality of hyperspectral simulations. Natraj et al. (2005Natraj et al. ( , 2010 used PCA to eliminate redundancies in line-by-line simulations of, e.g. the O 2 A band, by optimally grouping wavenumbers with similar optical properties. Liu et al. (2006) and Matricardi (2010) followed a related approach to speed up radiative transfer calculations for hyperspectral measurements from infrared sounders, such as the Infrared Atmospheric Sounding Interferometer (IASI) and the Atmospheric Infrared Sounder (AIRS). We propose using PCA to reduce the dimensionality of temperature profiles and reproduce the top-of-atmosphere (TOA) transmittance, exploiting the largely linear relationship between temperature profile eigenvectors and the transmittance.
The reconstruction of globally occurring temperature profiles is possible on the basis of a few eigenvectors, as demonstrated in Sect. 2.1. Furthermore, the atmospheric transmittance for an arbitrary temperature profile can be approximated in a similar way, as proposed in Sect. 2.2. This is demonstrated by applying the method to the retrieval of surface pressure from coarse spectral resolution O 2 A band measurements and approximating high spectral resolution transmittance for a number of well-mixed and non well-mixed gases (see Sect. 3.1). The applicability for the approximation of radiances in the O 2 A band is demonstrated in Sect. 3.2.

PCA of global temperature profiles
The natural variability of the temperature profile was analysed using data from the Global Forecast System (GFS), providing the temperature on 26 pressure levels at a 1 × 1 degree resolution. A single, global field of temperature profiles from 19 October 2007 was used for the analysis. All following studies and plots are based on this case. Depending on the application, it might be necessary to use a more extended data set, such as one representing both the diurnal as well as the seasonal change of temperature profiles. Furthermore, it is beneficial to use data on an equal-area projection to avoid an overweighting of high latitudes. In the frame of this work, using a single day of GFS data is sufficient, since it merely serves to demonstrate the general validity of the approach. Due to the high correlation of the temperature among neighbouring vertical layers, the number of independent parameters that suffice to reconstruct each vertical profile is significantly smaller than the number of pressure levels. A PCA was performed on the above mentioned global field of temperature profiles, revealing that the three most significant eigenvectors ν j are sufficient to reproduce 96 % of the variability inherent to the data (see Table 1).
In case the statistical basis for the PCA is well-chosen, a random, realistic temperature profile T can be expressed as the sum of the average profile T mean and a linear combination of the significant eigenvectors ν j : where ncomp is the number of eigenvectors used. The weight c j of each eigenvector is obtained from c j = (T − T mean ) · ν j . Figure 1 shows three arbitrary profiles as extracted from GFS data and their reconstruction, using the three most significant eigenvectors. While the dashed red curve shows an almost perfect representation of the vertical temperature profile, the green case reveals that more eigenvectors are needed to reconstruct the vertical fine structure, e.g. in cases of boundary layer temperature inversions. The blue case, representing an arctic atmosphere, shows that an extremely low tropopause height results in some deviations of the reconstructed profile. Here, more eigenvectors are needed to be able to reproduce the vertical temperature profile.

Linearity between temperature eigenvectors and transmittance
The strategy pursued in this work is based on the assumption that a linear relation between the temperature profile eigenvectors and the atmospheric transmittance exists. This means that the transmittance associated with any temperature profile can be constructed by a linear combination of the transmittance perturbations associated with the temperature profile eigenvectors, using the same weights c j as in Eq. (1). We define the effect of eigenvector ν j on the atmospheric transmittance, i.e. the first derivative δt δν j , as: where t mean is the transmittance associated to the average temperature profile T mean , t mean+ν j is the transmittance associated to the average profile, modified by eigenvector ν j , and t ν j is the resulting difference in transmittance for eigenvector ν j , all for a given airmass. In accordance with Eqs. (1) and (2), the transmittance t for a random temperature profile T is then constructed by a linear combination of the mean transmittance t mean and the contributions of each eigenvector t ν i , using the same weights c j as determined for the temperature profile eigenvectors: Provided that the requirement of a linear relation between temperature eigenvectors and the transmittance is fulfilled, it is sufficient to calculate, for a given surface pressure, the transmittance t mean and ncomp partial derivatives of the transmittance, with respect to the temperature eigenvectors, to be able to construct the transmittance for any realistic temperature profile. If the transmittance-temperature-relation is non-linear, including the second derivative is suited to minimize the residual errors. The second derivative δ 2 t δν j 2 of t with respect to eigenvector ν j can be approximated from Accordingly, the Taylor series (Eq. 3) is extended by the second derivative, neglecting the mixed partial derivatives: A detailed mathematical description is provided in the Appendix.
The technique has the potential to speed up the calculation of atmospheric transmittance and enable the inclusion of the temperature profile effect without the need to perform the computationally demanding calculation of absorption coefficients for each temperature profile under consideration. The validity of the approach is demonstrated in Sect. 3 for the retrieval of surface pressure from coarse spectral resolution measurements in the O 2 A band. Furthermore, the applicability is tested at high spectral resolution for a variety of wellmixed and non well-mixed gases.

Coarse spectral resolution
In order to test the approach for the approximation of atmospheric transmittance, the global set of GFS temperature profiles was used to calculate the average O 2 A band transmittance t O 2A between 760 nm and 763 nm for a fixed solar zenith angle of 45 • and nadir view, hereby using the HITRAN 2008 database (Rothman et al., 2009). The problem was simplified by assuming a globally constant surface pressure of 1000 hPa, neglecting surface elevation in order to isolate the effect of temperature broadening on the O 2 absorption lines (see Fig. 2 for an example of the transmittance in the considered spectral window). The resulting global transmittance for the considered case is shown in in Fig. 3. The highest transmittance, respectively weakest O 2 absorption, is found in regions with the highest temperatures, namely over tropical land areas. As stated before, this is due to the pressure-and temperature-dependence of the Lorentz line widths, resulting in less broadened absorption lines for higher temperatures. The hypothetical transmittance for the given airmass of 2.41 varies between 0.227 in the Antarctic and 0.265 in the tropics.
For comparison, a transmittance t * O 2A was obtained by a linear combination of the eigenvector contributions, following formula (3). The global root mean square deviation of the derived transmittance is shown in Fig. 4 as a function of the number of eigenvectors used for the approximation of transmittance. The error is reduced from roughly 1 % of transmittance, in case an average temperature profile is assumed, to 0.09 %, by just using the first temperature profile eigenvector to modify the average transmittance. Adding the second eigenvector yields a further reduction of the transmittance error below 0.05 %. The error can be brought down to 0.015 % by using 6 eigenvectors, with this residual error indicating the limit of the approach caused by the simplifying assumption of linearity. Figure 5 shows the global distribution of the errors of t * O 2A as a function of the number of eigenvectors used. In the case of using only the most significant eigenvector, some systematic deviations of the approximated transmittance are apparent, with positive biases found mainly over Antarticta and Africa and negative biases found mainly over the extratropical oceans. These systematic biases are gradually eliminated with increasing numbers of eigenvectors and are almost totally removed in the case four or more eigenvectors are used.
As an example, satellite measurements in the O 2 A band can be used for the retrieval of surface pressure, as demonstrated by e.g. Barton and Scott (1986), Mitchell and O'Brien (1987), Breon and Bouffies (1996)   profile, resulting in regional surface pressure biases of up to 50 hPa, as shown in Fig. 6, left panel.
If the temperature profile variability is accounted for using the technique described above, these large systematic errors can be eliminated, as shown in Fig. 6, right panel. The corresponding global root mean square error of surface pressure due to temperature profile effects is reduced from 35 hPa to less than 0.5 hPa, if six or more eigenvectors are used (see Fig. 7). Other error sources, such as the uncertainty due to aerosol loading (∼ 10 hPa if a mean optical depth is assumed) or the MERIS spectral calibration (0-50 hPa for a calibration uncertainty of 0.1 nm, depending on channel position), result in considerable errors in derived surface pressure (Lindstrot et al., 2009(Lindstrot et al., , 2010. Therefore, using two eigenvectors to describe the temperature profile variability is sufficient to account for the temperature-dependence of the O 2 absorption for the case of surface pressure retrievals from coarse spectral resolution measurements in the O 2 A band, since the accepted error of ∼ 1.5 hPa is well below the errors introduced by other sources of uncertainty.

High spectral resolution
In the previous section, the technique was tested at coarse spectral resolution for the well-mixed gas O 2 . In order to evaluate the validity of the approach for different at-mospheric species and at high spectral resolution, a close look was taken at the hyperspectral transmittance in a variety of spectral intervals. Figure 8 shows the transmittance and the associated error in the approximated transmittance for a random tropical profile for absorption lines of O 2 (762 nm-764 nm), CO 2 (2008 nm-2010 nm), CH 4 (3201 nm-3208 nm) and water vapour (900 nm-902 nm, 940 nm-942 nm, 1780 nm-1782 nm). Both the linear correction (see Eq. 3) as well as the correction including the second derivative (see Eq. 5) are shown. The main conclusions are as follows: -For O 2 absorption (panel A), the initial error in transmittance, caused by assuming an average temperature profile, reaches maxima at the line flanks in the range of 0.03-0.04 of absolute transmittance. Applying the linear correction (using 4 eigenvectors) results in small residual errors below 0.005. Accounting for the second derivative does not improve the accuracy of the approximated transmittance.
-Similar numbers are found for CO 2 (panel B). However, here the second-derivative correction improves the accuracy, with the residual error in transmittance being reduced by half. This hints at a weak non-linearity between ν j and t.
-The situation for CH 4 (panel C) is similar to the CO 2 case.
-For water vapour, a non well-mixed gas, generally the errors in transmittance are larger in comparison to the other gases. This is in part due to the fact that the maximum temperature differences between T mean and the temperature profile under consideration are found in lower atmosphere, as is the bulk of water vapour. In the ρσ τ -absorption band, the intial errors are up to 0.1 with peaks around the centers of unsaturated absorption lines. Using the principal components eliminates the errors, except for some spectral ranges, such as found at 901 nm (panel D) or 940.6 nm (panel E). Here, including the second derivative further improves the accuracy of the approximated transmittance.
-The largest errors are found for water vapour absorption around 1780 nm (panel F): in the line wings, the transmittance is overestimated by up to 0.2, when the average profile is assumed, with the peaks of the errors located in the centers of unsaturated and the edges of saturated absorption lines. Correcting for the linear effect of four eigenvectors is sufficient to eliminate large parts of the deviation, however, those regions with the highest initial errors are still not corrected completely (see 1781.6 nm-1781.9 nm range). Including the second derivative for correction further improves the approximation.
The residual errors of the approximated transmittance, mainly found in regions of large optical depth, raise the question whether better results can be obtained by applying the PCA in optical depth-space rather than reproducing the transmittance directly. This means that instead of the scalar transmittance t, the profile of the optical depth τ is approximated and the transmittance is calculated afterwards. Consequently, Eqs.
(2)-(5) become: and, if the second derivative is included 2 τ ν j = δ 2 τ δν j 2 = τ mean+ν j + τ mean−ν j − 2τ mean (8) Then, the transmittance is calculated as usual, with where m denotes the airmass. Comparing Figs. 8 and 9 allows for the following conclusions: -For O 2 , CO 2 and CH 4 , no considerable improvement is found when the gaseous optical depth profile is reproduced instead of the transmittance.
-The accuracy of the approximated transmittance is improved in regions of water vapour absorption, at least in the 1780 nm region. Here, the error of the approximated transmittance (including the second derivative) is improved from −0.0034 ± 0.0056 to −0.0011 ± 0.001. The residuals in regions of high optical depth are erased.
At least for water vapour absorption, it therefore seems advantageous to act in τ -space. Generally, it is important to note that the influence of the temperature profile on the transmittance differs among the individual atmospheric constituents and the considered spectral ranges. It is therefore crucial to assess the needed number of eigenvectors and the optimal correction approach for each individual application. A further improvement of the results can be expected from including the mixed partial derivatives in Eqs. (5) and (9), respectively, rather than from increasing the number of eigenvectors used. Furthermore, the required accuracy of reproduced transmittance spectra is constrained by the uncertainty of the assumed temperature profile and the associated error in transmittance. For the water vapour case, temperature profiles extracted from NWP models are likely to exhibit some uncertainty in cases of boundary layer temperature inversions, causing an uncertainty in the derived absorption coefficients and resulting transmittance. Reliable estimates of this uncertainty can help to identify the required accuracy of the transmittance approximation and thus the adequate correction strategy.

Radiance approximation
We have demonstrated the validity of the approach for the approximation of transmittance in the O 2 A band. We hypothesize that the technique can be as equally well used under scattering conditions to approximate the TOA radiances for any given temperature profile from the corresponding TOA radiances associated with each temperature profile eigenvector. By replacing the transmittance t with radiance L, Eq. (3) becomes: where of course L ν j denotes the sensitivity of the radiance L to eigenvector ν j : This means that the radiance for any atmospheric profile, observing geometry and geophysical conditions, can be represented by a linear combination of the tabulated L mean and ncomp radiances.
In order to test this hypothesis, we have performed full radiative transfer calculations of a MERIS-like mediumspectral-resolution channel within the O 2 A band, using the Matrix Operator Model (MOMO, Fell and Fischer, 2001;Hollstein and Fischer, 2012). The spectral channel was assumed to have a Gaussian shape with a width of 3 nm and shifted through the absorption band to visualize the spectral stability of the approach. In order to isolate the effect of scattering on the accuracy of the method, two general cases were studied: A. A case with a strong influence of atmospheric scattering, i.e. a high aerosol loading (aerosol optical depth AOD 550 nm = 0.5) over dark Lambertian surface (albedo α = 0.1).
The approximation of the TOA radiance was performed following Eq. (11) for a tropical and a polar temperature profile with a surface pressure of 1000 hPa (see red and blue profiles in Fig. 1). Figure 10 shows the angular dependency of the differences found between the TOA radiances simulated using the average temperature profile and the tropical profile. If the average temperature profile is used, the deviation of the TOA radiance from the truth, L mean , is between −4 % (Case A) and −3 % (Case B) for low zenith angles (sensor and sun close to zenith) and rises to −8 % (Case A) and −7 % are found for the polar case. When the first two eigenvectors are used for correcting the radiance, the error, L 2 , is well below 1 % for both cases at all viewing geometries. There is no significant influence of the observation geometry on the accuracy of the method, which is able to eliminate the overall bias and its angular dependence. Figure 11 shows the influence of the number of eigenvectors on the approximation error. For both cases and both profiles, the initial error of several percent is reduced to well below 1 % by using the first temperature eigenvector and slightly improved by accounting for the second eigenvector. The addition of further eigenvectors does not significantly change the result. The elimination of the temperatureinduced radiance bias works for both cases with the same accuracy, indicating that for this purpose atmospheric scattering is not a critical issue.
While L mean shows a strong dependency on the central wavelength with a clear maximum around 762.5 nm, L 2 is close to zero and the systematic spectral signature is replaced by an apparently random residual (see Fig. 11). Since, depending on the instrument construction type, the central wavelengths of spectral channels can vary with time or the view angle (as an example see Delwart et al., 2007), it is important to account for these spectral effects to avoid temporal or view angle dependent biases in derived products.

Conclusions
We have presented a method to efficiently approximate the effect of the temperature profile on the atmospheric transmittance. The first two to six eigenvectors are sufficient to describe the global variability of temperature profiles, depending on what accuracy is required. The temperature eigenvectors can serve to calculate the corresponding variations in optical depth or transmittance directly and thus to approximate the atmospheric transmittance for any realistic temperature profile.
As shown in an exemplary application for the case of O 2 absorption around 0.76 µm, the such approximated transmittance is in almost perfect agreement with the truth if a sufficient number of eigenvectors is used. In the shown case, an average O 2 transmittance of 0.25 between 760 nm and 763 nm for an airmass of 2.41 and a surface pressure of 1000 hPa, the residual error of the approximated transmittance is in the range of 0.0015 in absolute transmittance, which is 0.06 %. At high spectral resolution, it could be demonstrated that the technique works to strongly reduce the bias occurring in transmittance calculated for the average temperature profile. For some gases, such as water vapour, including the second derivative of transmittance with respect to the eigenvectors can improve the approximation accuracy. Furthermore, for water vapour the optimal correction strategy was found to be a reproduction of the optical depth profile rather than the transmittance directly.
Depending on the nature of the application, the number of eigenvectors needed may vary. As demonstrated for the retrieval of surface pressure from MERIS, using two eigenvectors is sufficient since the resulting residual error of ∼ 1.5 hPa is much smaller than that of other error sources. The technique can readily be used for the elimination of any temperature-induced bias in simulated radiances. Scattering at molecules and particles does not affect the applicability or accuracy of the method.
The benefit of the presented method can either be seen as -a major speed up of the calculation of transmittance or radiance, for which the method works with sufficient accuracy, in any forward simulation module used to model radiative transfer in atmospheric absorption bands, or -the avoidance of temperature-induced errors in calculated radiances at the cost of a linear increase of the amount of simulations to be performed.
It is important to state that the number of eigenvectors needed and the linearity between temperature profile eigenvectors and the transmittance need to be assessed for each potential application, since the transmittance exhibits differing sensitivities to the temperature profile for different atmospheric species and spectral ranges. An upper boundary for the required accuracy of the reproduced transmittances is given by the uncertainty resulting from the limited knowledge of the temperature profile itself, that might be provided by NWP models or temperature soundings.

Appendix A
Let F : R n → R be a function which is partially differentiable at x. The best linear approximation of F near x is: if F is totally differentiable at x. Best linear approximation means that: being the total derivative of F . Necessary and sufficient conditions for total differentiability are, that all partial derivatives exist and that they are continuous. In our case, where F is the atmospheric transmission t as a function of the air temperature at different levels T , the conditions are fulfilled, since there are no discontinuities in t due to small changes of T . The total derivative of t is: Consequently, the linear aproximation of t at T mean + T is: In particular, when T mean is changed by an eigenvector ν j (see Eq. 2), the approximation is: A principal component analysis is an orthogonal linear transformation to a new coordinate system, such that the new variables are uncorrelated, and that the new coordinates carry the total variance in descending order. A PCA can be done by an eigenvector decomposition of the variance-covariance matrix of the mean-centered data, whereat the eigenvectors define the new coordinate system. In this work, the atmospheric temperature profiles T are decomposed: with the eigenvectors ν j and the projections in the eigenspace If the original coordinate system (here the temperature at different levels) has correlated variables, the PCA can be used for a dimensionality reduction (n → ncomp) without losing significant information.
In order to account for non-linear effects, we can extend the approximation of t by its second derivative with respect to ν j : with δ 2 t δν j 2 = t mean+ν j + t mean−ν j − 2t mean = 2 t ν j (A19) and dν 2 j = c 2 j (A20)