Fast weighting functions for retrievals from limb scattering measurements

Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 273 – 283

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Fast weighting functions for retrievals from limb scattering measurements Johannes W. Kaiser∗ , John P. Burrows Institute f ur Fernerkundung (ife) und Umweltphysik (iup), Universit at Bremen, 28353 Bremen, Germany Received 9 April 2002; accepted 20 June 2002

Abstract The satellite-borne UV-visible-NIR spectrometers SCIAMACHY and OSIRIS perform operational measurements of the Earth’s limb radiance with global coverage. The vertically resolved atmospheric composition is retrieved from the measurements. We have computed synthetic limb measurements and O3 weighting functions (WFs) with several orders of scattering and surface re=ection. Comparisons reveal the wavelength-dependent contributions of single scattering and the second orders of scattering and surface re=ection. We have also performed test retrievals of the O3 and NO2 pro>les with the program package SCIARAYS. They prove that the single scattering approximation is suAcient for the calculation of the WFs during the retrieval process. We conclude that algorithms for the analysis of limb scattering measurements can be accelerated by neglecting higher orders of scattering in the WF calculations. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: SCIAMACHY; SCIARAYS; Spherical radiative transfer; Atmospheric parameter retrieval; Limb scattering; Weighting functions; Remote sensing; Linearisation

1. Introduction The new satellite-borne spectrometers SCIAMACHY, e.g. Bovensmann et al. [1], and OSIRIS, e.g. Llewellyn et al. [2], make limb measurements of electromagnetic radiation scattered from the atmosphere in the ultraviolet (UV), visible, and near-infrared (NIR) spectral regions. These instruments oFer a novel approach to atmospheric sounding. The geometry of this type of limb measurements is shown schematically in Fig. 1. The instrument observes the atmosphere above the Earth’s horizon. Spectra are recorded at diFerent tangent heights: some 20 – 60 being observed in each scan of the atmosphere. The UV-visible-NIR radiance from the ∗

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274 J.W. Kaiser, J.P. Burrows / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 273 – 283

Fig. 1. Limb observation geometry with single scattering, surface re=ection, and second order of scattering. (The atmosphere’s curvature is exaggerated.)

sun is scattered by the atmosphere into the line of sight of the instrument. In addition to this single scattering (SS), the observed limb radiance may have been re=ected from the Earth’s surface and may have been scattered several times, i.e. experienced multiple scattering (MS). The objective of this type of measurements of the limb radiance is to determine pro>les of trace atmospheric constituents (gases and aerosols) and parameters, such as temperature and pressure. It combines the positive attributes of high intrinsic vertical resolution, similar to that of occultation measurements like SAGE, and rapidly achieving global coverage, comparable to that of nadir-viewing instruments like TOMS and GOME. Two previous missions have demonstrated the potential uses of limb scattering measurements: mesospheric O3 and upper stratospheric NO2 pro>les were obtained from the satellite-borne SME measurements in the early 1980s, cf. Barth et al. [3]. McPeters et al. [4] have retrieved stratospheric O3 pro>les from the SOLSE/LORE measurements obtained aboard NASA’s space shuttle in 1997. The spectrometers OSIRIS and SCIAMACHY have been launched aboard the Swedish satellite ODIN in February 2001 and aboard ESA’s Envisat in March 2002. These instruments have better spectral resolution (0.22–1:48 nm) in larger spectral ranges (280 –800 and 240 –2380 nm) than the previously =own instruments. Therefore, a larger set of retrieval parameters and an extended retrieval height range are anticipated, cf. Kaiser et al. [5]. Radiative transfer models (RTMs) and inversion algorithms for the limb measurements are currently being developed, e.g. [6–8]. Some are intended for the operational, i.e. routine, analysis of the complete data sets while others will be used to study a limited subset in detail. The RTMs must account for the atmosphere’s sphericity. Furthermore, Oikarinen et al. [9] have shown that MS including surface re=ection contributes 5 – 60% of the limb radiance and concluded that a SS model would not be suAcient for the analysis of limb measurements. The large data rates of both SCIAMACHY and OSIRIS also require that the RTMs for the operational analyses execute rapidly. Therefore, they must calculate the weighting functions (WFs), i.e. the derivatives of the radiances with respect to the atmospheric parameters, from analytical formulae. This approach is called quasi-analytical calculation. Since it is a demanding task for MS in a spherical atmosphere none of the published RTMs conforms to all requirements simultaneously. However, Kaiser et al. [10] have demonstrated that the shape of a weighting function may be reproduced qualitatively by an approximating model. The objective of this study is to explore the applicability of WFs calculated with the SS approximation for the analysis of limb measurements. Using the approximation facilitates the development of the retrieval code and accelerates the computations, which is important for operational processing of the large data sets anticipated from SCIAMACHY and OSIRIS.

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By simulating realistic limb spectra, we also obtain the wavelength-dependent contributions of SS, surface re=ection and higher orders of scattering. 2. Methodology In order to achieve our objective, we have simulated an ensemble of 50 synthetic limb radiance measurements with several orders of scattering and surface re=ection for an assumed atmospheric state and measurement setup. Parameter pro>les have subsequently been retrieved from the measurements using consistent radiance, but approximate WF calculations. Finally, the retrieved pro>les are compared to the assumed ones. The whole process is repeated for three diFerent spectral >t windows with 81 equidistant spectral points each. The pro>les of NO2 and O3 are retrieved simultaneously from each window. The choice of the >t windows is motivated by the strong diFerential absorption of O3 in the UV and NIR and of NO2 in the visible. Focusing on dedicated >t windows is necessary as a large number of retrievals from the whole spectral range would be too time consuming even for modern computers. On the other hand, retrievals from wavelength pairs or triplets, which have successfully been applied to SOLSE/LORE measurements by Flittner et al. [8], would neglect the information obtained by SCIAMACHY’s and OSIRIS’ high spectral resolution. All calculations have been performed with the computer program package SCIARAYS, cf. Kaiser [11]. It contains an RTM, an instrument model, and inversion routines for limb measurements. 2.1. Forward calculation The RTM in SCIARAYS models the >rst two orders of scattering and surface re=ection in a spherical, horizontally strati>ed, cloud-free atmosphere. In this manner all ray paths shown in Fig. 1 are included. Refractive bending is adequately accounted for. The RTM also calculates the WFs for all atmospheric parameters quasi-analytically. The full eFect of MS is simulated in this study by doubling the contribution of the second order of scattering arti>cially and assuming the maximum albedo of unity. The detailed setup is summarised in Table 1. The solar coordinates re=ect the worst-case scenario with the maximum MS to be encountered by SCIAMACHY, cf. Figs. 3 and 8 of Oikarinen et al. [9]. The instrument model in SCIARAYS computes signal-to-noise values of SCIAMACHY for a given radiance. It has been developed using measured instrument parameters. We have created 50 synthetic limb measurements by simulating one limb measurement with the RTM and adding 50 diFerent noise vectors with SCIAMACHY’s signal-to-noise characteristics to it. 2.2. Inversion The applied inversion algorithm within SCIARAYS is based on the optimal estimation method by Rodgers [13]. During the inversion, the radiances are modelled consistently with the measurement simulation, i.e. simulating the eFect of MS, but the weighting functions are approximated by SS.

276 J.W. Kaiser, J.P. Burrows / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 273 – 283 Table 1 Measurement scenario and retrieval setup

Fit window Tangent heights Solar zenith angle Azimuth angle Albedo Trace gases Assumed pro>les Assumed aerosols Retrieved pro>les Pro>le spacing A priori pro>les A priori std. dev.

UV 320 –340 nm

Visible 420 –460 nm

NIR 820 –900 nm

0; 3; 6; : : : ; 60 km 25◦ 100◦ 1 O3 , NO2 , BrO, OCIO, CIO, SO2 , NO3 , HCHO, H2 O, O2 , CH4 , CO2 , CO, N2 O 45◦ N [12] Stratospheric background, 23 km tropospheric visibility O3 ; NO2 3 km 80% of assumed pro>le 30%

The measurement vector comprises the diFerential spectra of all tangent heights and the retrieval parameter vector comprises the retrieval trace gas pro>les. Thus the spectral and spatial inversions are simultaneously performed in an iterative process, which is terminated by a 2 -criterion. The state vector adjustment in each iteration step is reduced to 33% in order to accelerate and stabilise the approach to convergence. The diFerential spectra are obtained by subtracting a polynomial, in this case of fourth order, from each tangent height’s spectrum. This feature resembles the DOAS technique, cf. Richter et al. [14]. It compensates for known and unknown broadband eFects of the atmosphere and the instrument calibration. The a posteriori variances of the retrieval are also calculated from the WFs, cf. Eq. (18) of [13]. The square roots of the variances are de>ned in this study as the theoretical retrieval precisions. 2.3. Ensemble inspection The ensembles of retrieved trace gas pro>les may be characterised by their mean values and standard deviations. These are compared to the a priori standard deviations and the theoretical retrieval precisions to judge the convergence of the inversion algorithm. 3. Results 3.1. Radiance Fig. 2, top plot, shows the simulated ratio of the radiance/irradiance, which is de>ned as the sun-normalised radiance. It is given in SCIAMACHY’s spectral range for several tangent heights. The ratio is observed to decreases with increasing tangent height and wavelength. This is explained by the atmospheric density at the tangent height and the wavelength dependence of the scattering by

J.W. Kaiser, J.P. Burrows / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 273 – 283 277

Fig. 2. Simulated sun-normalised limb radiances at selected tangent heights (top) and relative contributions of single scattering and the second orders of surface re=ection and scattering to the total radiances (bottom). All restricted to SCIAMACHY’s spectral range.

molecules and the assumed aerosols. In addition, strong gaseous absorption features are readily seen, and are attributed to O3 Hartley-Huggins and Chappuis bands below 330 nm and around 600 nm, O2 A-, B-, and -bands (760, 690, 630 nm), various H2 O bands, and strongly varying line absorption in the channels above 1900 nm. Fig. 2, bottom plot, shows the relative contributions of the modelled ray paths to the total simulated radiances. It can be seen that SS dominates in the regions of the strong absorption features, where light is scattered relatively high in the atmosphere. In the visible range, it contributes about 45 – 65%. Since this is in agreement with other recent investigations, e.g. Oikarinen et al. [9], it justi>es our approach of doubling the contribution of the second order of scattering for simulating the eFect of higher orders.

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Fig. 3. Full (left) and approximate (right) O3 weighting functions at 330 nm and SCIAMACHY’s tangent heights for absolute (top) and relative (bottom) changes of the sun-normalised radiance, i.e. radiance/irradiance.

The contribution of the re=ected radiance is small in the UV and becomes dominant (70%) in the NIR. This particular value is surprisingly larger than 50% because of the chosen maximum albedo and the dominating scattering angle close to 90◦ for SS. The contribution drops below 50% for other solar coordinates. The contribution of the second order of scattering peaks near 335 nm, then decreases with wavelength, and becomes almost negligible in the NIR. This suggests that modelling the radiances with SS and surface re=ection alone could be suAciently accurate for retrievals from SCIAMACHY’s NIR channels. 1 3.2. Weighting functions The O3 pro>le WFs at 330 nm, calculated for tangent heights between 6 and 52 km at a spacing of 3 km, are plotted in Fig. 3. The results shown in the top left plot simulate the eFect of MS. The sharpness of the step-like feature at each tangent height provides the information which determines in large part the vertical resolution of the retrieved O3 pro>le near the corresponding height level. The feature gradually disappears for tangent heights below 15 km. This indicates that little information on O3 at lower altitudes is obtained from this wavelength in the UV. The top right plot of Fig. 3 shows the same O3 pro>le WFs calculated with the SS approximation. In contrast to the WFs calculated with the full scattering scheme, these WFs vanish completely below their tangent heights. Above, they are reduced to about 50%. This is consistent with the SS contribution to the total radiance, cf. Fig. 2. However, the general shapes and relative strengths of both the WFs are remarkably similar for the two sets of calculations. Thus the approximate WFs point well in the correct direction in the 1

This approximation was already employed for the retrieval of stratospheric wind >elds from HRDI aboard NASA’s UARS [15], while SCIAMACHY and OSIRIS aim at the atmospheric composition.

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linear space of the retrieval parameters, i.e. trace gas pro>les, even though their magnitudes are too small. The relative changes of the radiance induced by a change in the O3 concentration is obtained by normalising the WFs with their corresponding sun-normalised radiances. The results for full and approximate WFs are shown in the bottom row of Fig. 3. Evidently, shape and magnitude are both well described by the SS approximation. This is explained by the fact that the additional ray path segments of surface re=ection and higher order scattering are small compared to the long segment along the line of sight, cf. Fig. 1. It may be concluded that the approximate WFs can be used for parameter retrievals from the logarithms of the observed radiances. We will demonstrate in the remaining sections that even retrievals directly from the observed radiances converge correctly with the approximate WFs (Fig. 3, top right). 3.3. NO2 test retrievals 3.3.1. Individual retrieval Fig. 4, top plot, illustrates an example of an individual NO2 pro>le retrieval from the visible >t window. The retrieved pro>le with its theoretical precision is shown along with the assumed pro>le and the a priori pro>le with its precision. The retrieved NO2 pro>le is highly accurate in the stratosphere. Between 15 and 36 km height, the retrieved pro>le matches the assumed pro>le within the theoretical precision bounds, while the a priori values lie outside these bounds. Below 10 km, the retrieved values and their precisions approach the a priori values and precisions. Thus, in this height region, little information on a clean background troposphere is gained from a single measurement in the visible >t window. 3.3.2. Ensemble of retrievals Fig. 4, bottom plot, depicts the properties of the ensemble of 50 NO2 retrievals from the visible >t window. The ensemble is characterised by its mean values (retrieval mean error) and standard deviations (retrieval std. dev.). Additionally, the a priori standard deviation (a priori prec.) and the theoretical retrieval precision (retrieval prec.) are shown. The height range on which information is contained in one measurement can be identi>ed by comparing the a priori and the retrieval precisions: between 12 and 39 km, the retrieval precision is signi>cantly smaller than the a priori precision, indicating substantial information gain from the measurement. Outside this region, the retrieved pro>les are dominated by the a priori. This is consistent with the mean retrieval error of −20% as the a priori pro>le is 20% smaller than the assumed. When there is no information in the measurement, the retrieval always converges to the a priori pro>le and the standard deviation approaches zero. Between 12 and 39 km, the retrieval mean error lies within the bounds given by the retrieval standard deviation. This proves that the retrievals converge to an adequately correct result without loss of precision even though the WFs are approximate. Furthermore, the theoretical retrieval precision is systematically larger than the retrieval standard deviation. Thus the algorithm actually overestimates its own error variance and the theoretical retrieval precisions are conservative estimates. This occurs because the approximation decreases the

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WFs, cf. Fig. 3. Smaller WFs imply larger a posteriori variances, thus also larger theoretical retrieval precision values. 3.4. O3 test retrievals The results for the ensembles of O3 pro>les, retrieved from the UV and NIR >t windows are illustrated in Fig. 5. The height regions with pro>le information from one measurement exhibit the same features as in Fig. 4, bottom plot: the retrieval precision is much smaller than the a priori precision and the retrieval standard deviation is non-vanishing. Information from the UV clearly yields precise pro>le information throughout the stratosphere, while that in the NIR makes retrievals from the upper troposphere feasible. Combined retrievals from both >t windows will yield the

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hyperspectral advantage of a large retrieval range extending from the upper troposphere to the upper stratosphere. In both cases, the retrievals converge correctly since the retrieval mean errors are bounded by the retrieval standard deviations in the regions with pro>le information in the measurements. The theoretical retrieval precisions are conservative estimates as the actual retrieval standard deviations are smaller everywhere. 4. Conclusions and outlook We have simulated limb scattering measurements similar to those expected from SCIAMACHY and examined the contributions of diFerent ray paths to the total radiance and the WFs. The

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shape of the latter is hardly changed by the single scattering approximation. The magnitude of the relative sensitivity of the radiance w.r.t. the trace gas concentration is also well reproduced by the approximation. Thus the approximation may be applied to WF calculations in retrieval algorithms. We have performed test retrievals of NO2 and O3 pro>les from selected wave-length windows using approximate quasi-analytical WFs. The mean errors and standard deviations of the retrieved pro>les demonstrate that an inversion algorithm which approximates the WFs by single scattering converges correctly and without loss of precision in the UV, visible, and NIR spectral ranges. By using approximate WFs, existing radiative transfer models, e.g. CDI by Rozanov et al. [6], may be used with relatively little extra eFort for the analysis of real limb measurements. The approximation also reduces the execution time of the radiative transfer model, which is crucial for analysing the whole limb scattering data sets provided by SCIAMACHY and OSIRIS. Therefore, such “fast” WFs are highly desirable. Calculating the retrieval precision from the approximate WFs overestimates the retrieval error. Such precisions can therefore be regarded as conservative estimates, e.g. for sensitivity studies. (Note that real SCIAMACHY measurements may allow for more precise retrievals than the ones presented since only 1% of its spectral points have been used for each of our calculations.) In the NIR spectral range, the radiance due to the second order of scattering is much smaller than the one due to single scattering and surface re=ection. We will examine the possibility of approximating the radiance as well as the weighting functions in a forthcoming publication. Acknowledgements We thank Vladimir and Alexei Rozanov and Kai-Uwe Eichmann for fruitful discussions. This work has been funded by the BMBF via the GSF/PT-UKF and the DLR-Bonn and by the University of Bremen. References [1] Bovensmann H, Burrows JP, Buchwitz M, Frerick J, NoTel S, Rozanov VV, Chance KV, Goede APH. SCIAMACHY: mission objectives and measurement modes. J Atmos Sci 1999;56(2):127–50. [2] Llewellyn EJ. OSIRIS: an application of tomography for absorbed emissions in remote sensing. In: Lampropoulos G, Lessard R, editors. Applications of photonic technology, vol. 2. New York: Plenum, 1997. p. 627–32. [3] Barth CA, Rusch DW, Thomas RJ, Mount GH, Rottman GJ, Thomas GE, Sanders RW, Lawrence GM. Solar mesosphere explorer: scienti>c objectives and results. Geophys Res Lett 1983;10(4):237–40. [4] McPeters RD, Janz SJ, Hilsenrath E, Brown TL, Flittner DE, Heath DF. The retrieval of O3 pro>les from limb scatter measurements: results from the Shuttle Ozone Limb Sounding Experiment. Geophys Res Lett 2000;27(17): 2597–600. [5] Kaiser JW, Rozanov VV, Burrows JP. Theoretical precisions for SCIAMACHY limb retrieval. Adv Space Res 2002; 29(11):1837–42. [6] Rozanov AV, Rozanov VV, Burrows JP. A numerical radiative transfer model for a spherical planetary atmosphere: combined diFerential-integral approach involving the picard iterative approximation. JQSRT 2001;69:491–512. [7] GriAoen E, Oikarinen L. LIMBTRAN: a pseude three-dimensional radiative transfer model for the limb-viewing imager OSIRIS on the ODIN satellite. J Geophys Res 2000;105(D24):29,717–30. [8] Flittner DE, Barthia PK, Herman BM. O3 pro>les retrieved from limb scatter measurements: theory. Geophys Res Lett 2000;27(17):2601–4.

J.W. Kaiser, J.P. Burrows / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 273 – 283 283 [9] Oikarinen L, Sivhola E, KyrTolTa E. Multiple-scattering radiance in limb-viewing geometry. J Geophys Res 1999;104(D24):31,261–74. [10] Kaiser JW, Rozanov AV, Rozanov VV, Burrows JP. Evaluation of approximative radiative transfer models intended for retrievals from limb measurements. In: Smith WL, Timofeyev YM, editors. IRS 2000: current problems in atmospheric radiation. Hampton, Virginia: A. Deepak Publishing, 2001. p. 417–20. [11] Kaiser JW, Atmospheric parameter retrieval from UV-vis-NIR limb scattering measurements. Ph.D. thesis, University of Bremen, Institute of Remote Sensing, 2001. URL http://uni-bremen.de/∼johannes/diss. [12] ESA. De>nition of observational requirements for support to a future earth explorer atmospheric chemistry mission. Draft edition, Final Report on ESA contract 1-3379/98/NL/GD, June 2000. [13] Rodgers CD. Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation. Rev Geophys Space Phys 1976;14(4):609–24. [14] Richter A, Wittrock F, Eisinger M, Burrows JP. GOME observations of tropospheric BrO in northern hemispheric spring and summer 1997. Geophys Res Lett 1998;25(14):2683–6. [15] Hays PB, Abreu VJ. Absorption line pro>les in a moving atmosphere: a single scattering linear perturbation theory. J Geophys Res 1989;94(D15):18,351–65.