FROM LIMB OBSERVATIONS OF INFRARED RADIANCE UNDER NON-LTE CONDITIONS. A Dissertation LADISLAV REZAC

Transcription

1 SIMULTANEOUS RETRIEVAL OF T(P) AND CO 2 VOLUME MIXING RATIO FROM LIMB OBSERVATIONS OF INFRARED RADIANCE UNDER NON-LTE CONDITIONS A Dissertation By LADISLAV REZAC Submitted to the Graduate College of Hampton University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2011

2 This dissertation submitted by Ladislav Rezac in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Hampton University, Hampton, Virginia is hereby approved by the committee under whom the work has been completed. James M. Russell III, Ph.D. Committee Chair M. Patrick McCormick, Ph.D. William L. Smith, Ph.D. Alexander A. Kutepov, Ph.D. William I. Young, Jr. Ed.D. Interim Dean of the Graduate College Date

3 Copyright by LADISLAV REZAC 2011

4 ABSTRACT Simultaneous Retrieval of T(p) and CO 2 Volume Mixing Ratio From Limb Observations of Infrared Radiance under non-lte Conditions. (May 2011) Ladislav Rezac, B.A., Hampton University; Ph.D., Hampton University Chair of Advisory Committee: James. M. Russell III The kinetic temperature, T k, and carbon dioxide, CO 2, are two very important parameters that characterize and to a degree determine the energetics and dynamics of the Mesosphere and Lower Thermosphere (MLT) region. Hence, there is much interest in obtaining high quality observations of these parameters in order to study the short term variability as well as the long term trends in characteristics of the MLT region. The Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument on board the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite provides simultaneous measurements of Non Local Thermodynamic Equilibrium (non-lte) limb emissions in the 15 and 4.3 µm region from CO 2 molecules, which can be used to infer the T k and CO 2 Volume Mixing Ratio (VMR) in this region. In this study a method is developed to simultaneously and self-consistently invert the SABER measured radiances in the two channels to obtain vertical profiles of T k and CO 2 VMR. The two-channel retrieval architecture is build upon iterative switching between two independent retrieval modules for T k and CO 2. Studies of the inversion algorithm made with synthetic radiances indicate that a stable solution of this non-linear inverse problem can be obtained that is nearly independent of the starting conditions; however, the interdependence of the channels places a limit on the altitude of the lower boundary of the retrieval at 65 km. iv

5 A detailed error analysis of the retrieved parameters shows that the largest uncertainty in the retrieved CO 2 VMR comes from the uncertainty in the O( 1 D) VMR, V-T rate coefficient for CO 2 (ν(3) ν(2)) splitting during collisions with the N 2, and O 2 molecules, and in the V-V coefficient for the CO 2 + N 2 interaction. The retrieved CO 2 VMR is very sensitive to these parameters, especially in regions of high optical thickness, and reaches an uncertainty of 26% at km. The uncertainty is a minimum around km (10%) and grows again to 30% at km altitude. The largest uncertainty in the retrieved T k profile comes from the uncertainty in atomic O VMR and the CO 2 +O V-T rate above the km altitude. In addition, the increased error in CO 2 at km is reflected in the retrieved T k at that altitude. Nevertheless, the T k uncertainties are comparable (above km) to the single channel derived uncertainties reported in the literature. The results presented from the two-channel algorithm applied to the SABER v1.07 data include the seasonal and latitudinal behavior of the multi-year average of the CO 2 VMR at three latitudes (45N, 0, 45S). These results indicate inter-hemispheric differences in the CO 2 VMR profiles that are seasonally symmetric. In addition, the global, long term behavior of the CO 2 VMR in the MLT derived from observations is presented. The eight year time-series ( ) reveals a positive trend in the CO 2 VMR that is altitude dependent, and at km, the CO 2 VMR trend is slightly negative (less than 1 ppmv/year). There is a small (< 1K/year) negative trend in T k throughout most of the MLT. The statistical significance of the T k /CO 2 trends was determined through Student s T-test and indicates that there is a small (< 5%) probability for obtaining these results by chance if the null hypothesis of zero trend was true. However, the determined statistical significance of the trends should be taken as a crude approximation due to the various limitations in the data and the assumptions of the test, which are discussed. v

6 In addition, several case studies of comparison of the zonal means of the retrieved CO 2 VMR between the SABER and the Atmospheric Chemistry Experiment Fourier Transform Spectrometer (ACE-FTS) instrument were performed, which conclude that the SABER v1.07 CO 2 VMR departs from a well mixed value by about 5-7 km higher than the ACE-FTS profile. Furthermore, analysis of the unconstrained two-channel retrieved CO 2 VMR profiles identified several issues and imperfections in the forward model parameters and the SABER 4.3 µm radiance pre-processing. The mis-characterization of the wing of the Field-Of-View (FOV) function for this channel results in un-physical false maxima in CO 2 VMR profiles at km when a strong change in the radiance gradient is present, which usually occurs at high latitudes in polar summer. Another false maxima frequently occurring in the CO 2 VMR at 66 km was also studied. It was found to be correlated with the beta angle of the TIMED satellite, however, it may also result from the imperfect removal of the FOV function or from the uncertainty in the relevant V-V and V-T collisional rate coefficients. Discussion of the many factors contributing to this feature is presented. Finally, case studies of the two-channel retrieval comparing the v1.07 and the new v1.08 SABER data are described. The new v1.08 data corrects several of the issues in both assumed model atmosphere, such as over-estimated O( 1 D) VMR around km, and in the radiance pre-processing, which is shown to yield further improvements in the retrieved CO 2 VMR. vi

7 This dissertation is dedicated to my parents. vii

8 ACKNOWLEDGEMENTS It is my great pleasure to acknowledge the guidance and support I received during my doctoral research. This work was made possible by the National Aeronautics and Space Administration (NASA) that funded the TIMED/SABER project and hence provided the financial support for this work through grants NNX07AB62G, NAG , NNL-04AA32G, NNG-04GE42G, and additional support in the frame of grant NNX06AD73G. My first thanks goes to all the faculty and staff members of the Center for the Atmospheric Sciences at Hampton University, especially to the faculty who kindly served on my dissertation committee: Dr. James M. Russell, Dr. M. Patrick Mc- Cormick, Dr. William L. Smith and Dr. Alex. A. Kutepov who is associated with the Catholic University of America in Maryland. I am thankful to many individuals and collaborators who contributed guidance, assistance and resources during this work. I recognize and thank the members of GATS Inc., especially Tom Marshall and Larry Gordley for the discussions regarding the processing of the SABER signal, retrieval algorithm, and getting me started with the LTE version of BANDPAK. I would like to thank Dr. R. A. Goldberg for inviting and accommodating me at the Goddard Space Flight Center. I would like to acknowledge Dr. K. Walker from the Canadian Space Agency for providing me with ACE-FTS data and Dr. R. R. Garcia for WACCM CO 2 VMR calculations. I would like to extend my thanks to Dr. Ping-Ping Rong, Dr. John McNabb, Dr. Charles Hill, and Dr. Jasper Lewis for their help and the many discussions. Thanks also go to all my fellow graduate students whom it was a pleasure to work with: Kaba Bah, Kevin Leavor, Chris Spells and David Gomez-Ramirez. I would like to express my sincere gratitude to Dr. James M. Russell III for the opportunity to work on this research, his support, patience, encouragement and viii

9 understanding whilst supervising my studies. His commitment and dedication to push this work forward by finding and encouraging outside collaboration with some of the most prominent scientists in the non-lte radiative transfer field is greatly appreciated. In spite of his many commitments he has always been generous with his time, especially putting great care into reading this manuscript and teaching me the proper scientific writing skills, which will be of lasting benefit to me. I would like to also devote a special section of acknowledgement to Dr. Alex A. Kutepov and Dr. Artem. G. Feofilov, which I have had the honor to meet when my research moved into the world of non-lte radiative transfer modeling. This research would be impossible without their willingness to help, and to share their in-depth knowledge and resources, literally going above and beyond what is normally expected to help me when I was just starting out in the field. I may never reach their level of problem solving skills and non-lte knowledge, however, their enthusiasm and approach to science is an inspiration to me, and it has had an enormous influence on my development as a scientist. I am especially grateful to Dr. Kutepov for showing me how to perform scientific research, introducing me to the theoretical analysis and computational methods of non-lte radiative transfer, and for the many discussions on non-lte and atmospheric science in general, which many times led into other topics. There are many unmentioned people without whom this work would not have been accomplished. This work would certainly be impossible without the unwavering support of my family and friends. ix

10 TABLE OF CONTENTS Section Page ABSTRACT iv DEDICATION vii ACKNOWLEDGEMENTS viii TABLE OF CONTENTS x LIST OF TABLES xiv LIST OF FIGURES xv LIST OF ACRONYMS xix 1 INTRODUCTION Non Local Thermodynamic Equilibrium in the MLT Limb infrared sounders of the MLT Non-LTE retrieval schemes Overview of CO 2 and temperature in the MLT Carbon Dioxide Temperature Research statement THE NON-LTE PROBLEM Overview of non-lte Formulation of the problem Complete Thermodynamic Equilibrium x

11 Section Page LTE Partial LTE Accelerated Lambda Iteration Method of Solution ALI-ARMS current status SABER EXPERIMENT AND THE FORWARD MODELING OF SABER RADIANCES Chapter overview SABER experiment The forward modeling of SABER radiances Measurement geometry Basic relations for non-lte limb radiance modeling SABER non-lte limb radiances in the 15 and 4.3 µm channels Non-LTE limb radiance in the SABER 15 µm channel Non-LTE limb radiance in the SABER 4.3 µm channel Input parameters for the forward model INVERSE PROBLEM AND RETRIEVAL ALGORITHM Inverse theory and definitions General inversion approaches Relaxation methods review Onion peeling method Chahine method Twomey-Chahine extension Multiplicative reconstruction method Simultaneous T k /CO 2 retrieval algorithm General considerations xi

12 Section Page Architecture of the retrieval scheme Temperature retrieval module CO 2 retrieval module Self-consistent studies The need for self-consistency Self-consistency case studies Sensitivity and error analysis RETRIEVAL RESULTS Daily zonal means for March and July of 2002 at different latitudes Investigating latitude variations: multi-year zonal mean T k /CO 2 profiles Investigating seasonal variations: multi-year zonal mean T k /CO 2 profiles Time series of the T k /CO 2 averaged over seasons and latitudes Comparison of the two-channel retrieved T k /CO 2 with ACE-FTS v3.0 data Comparison of the T k /CO 2 two-channel retrieval using v1.08 and v.107 SABER data SUMMARY AND FUTURE WORK Future research APPENDICES A On the limb radiance sensitivity to the changes in atmospheric parameters xii

13 Section Page A.1 A theoretical description of the radiance sensitivity displacement A.1.1 The limiting assumptions of the argument A.2 FOV effects on the shift of radiance sensitivity A.2.1 Displaced radiance sensitivity impact on T k retrieval A.3 Summary B Error analysis for individual latitudes C Details on the effect of deconvolution of radiances with mis-characterized FOV function and correlation of the retrieved CO 2 VMR with the beta angle at 66km C.1 On the effects of imperfect knowledge of the wings of the FOV function C.2 On the correlation of retrieved CO 2 VMR at 66 km with the TIMED beta angle REFERENCES VITA xiii

14 LIST OF TABLES Table Page 1.1 Summary of observations and modeling of MLT CO SABER pass bands and science motivation for the measurement Mean retrieval errors and their sources Retrieved trends with the 95% confidence interval B , DOY 330, Latitude 73S B , DOY 265, Latitude 45N B , DOY 1, Latitude 20S B , DOY 33, Latitude 0N xiv

15 LIST OF FIGURES Figure Page 1.1 Mean vertical profiles of CO 2 VMR from several different experiments The SEE and RTE coupling Convergence plot for the CO 2 problem in Earth s atmosphere Graphics of TIMED spacecraft and placement of the SABER instrument SABER latitude vs time coverage Limb viewing geometry (scale distorted) CO 2 energy level diagram Vibrational temperatures of the CO 2 (v 2 ) level Fractional LOS contribution of different bands into Ch Ch1 radiance contribution functions Vibrational temperatures for the CO 2 (v 3 ) level Fractional LOS contribution of different bands into Ch Ch7 radiance contribution functions Ch1 and Ch7 sensitivity to global temperature shifts Ch1 and Ch7 sensitivity to global CO 2 VMR shifts Ch1 and Ch7 sensitivity to global temperature shifts with hydrostatic adjustment General scheme of two-channel retrieval method xv

16 Figure Page 4.5 Case 1: self-consistency test of the two-channel retrieval algorithm for a climatological mid-latitude summer atmosphere Case 1: radiance fitting and convergence speed in Ch1 and Ch Case 2: self-consistency test of the two-channel retrieval algorithm for a climatological polar summer atmosphere Case 2: radiance fit and convergence speed in Ch1 and Ch Case 3: self-consistency test of the two-channel retrieval algorithm for a climatological tropical atmosphere with a sine wave perturbation Case 4: self-consistency test of the two-channel retrieval algorithm with fine scale T k structure and complex CO 2 structure Changes in retrieved CO 2 VMR due to applied biases for the most important forward model parameters Daily zonal average March 2002 in the latitude bin 0-3 N Daily zonal average March 2002 in the latitude bin N Daily zonal average March 2002 in the latitude bin -73 S Daily zonal average Jul 2002 in the latitude bin 0-3 N Daily zonal average July 2002 in the latitude bin N Daily zonal average Jul 2002 in the latitude bin N multi-year zonal average of SOPS and two-channel retrieved T k /CO 2 for January multi-year zonal average of SOPS and two-channel retrieved T k /CO 2 for July multi-year zonal average of SOPS and two-channel retrieved T k /CO 2 for March multi-year zonal average of SOPS and two-channel retrieved T k /CO 2 for September multi-year zonal average of SOPS and two-channel retrieved T k /CO 2 at the equator for different months xvi

17 Figure Page multi-year zonal average of SOPS and two-channel retrieved T k /CO 2 at 45N for different months multi-year zonal average of SOPS and two-channel retrieved T k /CO 2 at 45S for different months time-series of the two-channel retrieved and SOPS T k and the CO 2 VMR along with a linear fit at km time-series of the two-channel retrieved and SOPS T k and the CO 2 VMR along with a linear fit at 85 km time-series of the two-channel retrieved and SOPS T k and CO 2 VMR along with a linear fit at km time-series of the two-channel retrieved and SOPS T k and CO 2 VMR along with a linear fit at 95 km time-series of the two-channel retrieved and SOPS T k and CO 2 VMR along with a linear fit at km time-series of the two-channel retrieved and SOPS T k and CO 2 VMR along with a linear fit at km time-series of the two-channel retrieved and SOPS T k and CO 2 VMR along with a linear fit at 120 km Comparisons of zonal mean CO 2 VMR retrieved from the ACE-FTS observations and the SABER the two-channel retrieval for July, 2005 at latitude 45N Comparisons of zonal mean CO 2 VMR retrieved from the ACE-FTS observations and the SABER two-channel retrieval for March, 2005 at the equator Comparisons of zonal mean CO 2 VMR retrieved from the ACE-FTS observations and the SABER two-channel retrieval for November, 2004 at 45S Comparison of v1.07 and v1.08 T k /CO 2 retrievals for Mar 21, 2010 at 15N Comparison of v1.07 and v1.08 T k /CO 2 retrievals for Mar 21, 2010 at 32N xvii

18 Figure Page A.1 Ch1 altitude displacement of radiance sensitivity to the T k change A.2 Illustration of the geometrical argument A.3 FOV effects on Ch1 altitude displacement of radiance sensitivity to the T k change A.4 Two sets of convergence curves for T k retrieval for different FOV functions C.1 Two Ch7 measured limb radiance profiles for high latitude polar summer and high latitude equinox conditions C.2 Ratio of radiance convolved with a Gaussian FOV function of different widths to the original radiance C.3 Correlation between TIMED beta angle and two-channel retrieved CO 2 VMR at 66 km xviii

19 LIST OF ACRONYMS ACE-FTS ALI ALI-ARMS Atmospheric Chemistry Experiment Fourier Transform Spectrometer Accelerated Lambda Iteration Accelerated Lambda Iteration for Atmospheric Radiation and Molecular Spectra CLAES CMAM CTE CRISTA FOV FWHM GCMs GOMOS HITRAN HRDI ISAMS LBL LI Cryogenic Limb Array Etalon Spectrometer Canadian Middle Atmosphere Model Complete Thermodynamic Equilibrium Cryogenic Infrared Spectrometers and Telescopes for the Atmosphere Field-Of-View Full Width at Half Maximum Global Circulation Models Global Ozone Monitoring by Occultation of Stars HIgh-resolution TRANsmission High Resolution Doppler Imager Improved Stratosphere and Mesosphere Sounder Line-By-Line Lambda Iteration xix

20 LIDAR LIMS LOS LTE LRIR MART MIPAS MLT NASA NER non-lte ODF RSS RTE SABER SAMS SCISAT SEE SISI Light Detection and Ranging Limb Infrared Monitor of the Stratosphere Line of Sight Local Thermodynamic Equilibrium Limb Radiance Inversion Radiometer Multiplicative Algebraic Reconstruction Technique Michelson Interferometer for Passive Atmospheric Sounding Mesosphere and Lower Thermosphere National Aeronautics and Space Administration Noise-Equivalent-Radiances Non Local Thermodynamic Equilibrium Opacity Distribution Function Root Sum Squared Radiative Transfer Equation Sounding of the Atmosphere using Broadband Emission Radiometry Stratospere and Mesosphere Sounder SCIentific SATellite Statistical Equilibrium Equation Spectroscopic Infrared Structure Investigation xx

21 SNR SPIRE SSR TIMED TIME-GCM Signal-to-Noise-Ratio Spectral Infrared Rocket Experiment Sum of Squared Residuals Thermosphere Ionosphere Mesosphere Energetics and Dynamics Thermosphere-Ionosphere-Mesosphere Electrodynamics Global Circulation Model UARS VMR WACCM Upper Atmospheric Research Satellite Volume Mixing Ratio Whole Atmosphere Community Climate Model xxi

22 1 CHAPTER 1 INTRODUCTION The observational nature of atmospheric science places high demands on availability of routine, high quality measurements of the atmospheric state and atmospheric composition. These observations serve as evidence against which to test our understanding of the physics, chemistry and the time evolution of planetary atmospheres. This is especially important as research on the physics and chemistry in the middle and upper atmosphere accelerates and modeling complexity increases. Over the past two decades a great deal of new knowledge on the relative importance of various processes driving the thermal structure, energetics and dynamics of the Mesosphere and Lower Thermosphere (MLT) region has been gained; however, there is still a lack of consistent global measurements that allow complete characterization and model validation of the key atmospheric parameters, temperature and CO 2 concentration. The MLT is inherently difficult to study, both theoretically and experimentally. This region can be considered as the lowest boundary of space or upper limit of the atmosphere, as molecular flow starts dominating turbulent mixing. There are numerous other regime changes taking place in the MLT, including atmospheric chemistry changes from neutral to weakly ionized. The ions and electrons become the more dominant species as the energy deposition by particle precipitation and solar radiation lead to the development of the ionosphere. Metallic layers resulting from meteor ablation add to the level of complexity of ion and neutral chemistry of this region. Another regime change profoundly influencing this region is in the wind circulation system. In the middle atmosphere (stratosphere and mesosphere) the zonal wind system is mostly geostrophic, driven by meridional temperature gradients. The ther-

23 2 mosphere, on the other hand, has a non-geostrophic dynamical component driven by the diurnal expansion and contraction of the atmosphere (Brasseur and Solomon 2005). In addition, dynamical features like gravity wave breaking and (solar) tidal wave propagation significantly perturb the temperature structure and chemistry of the MLT and provide a coupling mechanism of this region with processes in the lower atmosphere. 1.1 Non Local Thermodynamic Equilibrium in the MLT Further difficulty in carrying out studies of atmospheric physical processes involving energy transport in the MLT, is due to the departure of molecular level populations from Local Thermodynamic Equilibrium (LTE). This is particularly important for analysis of infrared space based measurements of the upper MLT, but also becomes a relevant issue in today s sophisticated numerical modeling of atmospheric energetics and dynamics that extend well into the thermosphere. Both radiative cooling (i.e. when translational energy is converted into internal energy of radiating species and then subsequently lost from the volume through emission of a photon) and radiative heating (i.e. conversion of solar energy into translational energy of molecules) are important in calculating the neutral temperature profiles in today s Global Circulation Models (GCMs) of the upper atmosphere (Ward and Fomichev 1996, Kutepov et al. 2007). Both, cooling and heating processes are affected by non-lte, and the radiative transfer parametrization schemes of GCMs must include these effects to accurately derive the temperature distribution and pressure gradients in this region. In general, LTE conditions prevail when the time between collisions of molecules is short compared to the radiative de-excitation lifetime so that energy population levels are determined by the local kinetic temperature as described by the Maxwellian statistical distribution of molecular motion. In such cases the chemical, radiative and collisional processes that influence the energy level populations are rendered neg-

24 3 ligible. Under these conditions the energy level populations follow the Boltzmann distribution and the radiative properties of a gas mixture are only a function of the local kinetic temperature. When thermal collisions are not frequent enough to maintain the Boltzmann statistics of the internal energy levels of a particular molecule, non-lte conditions exist. A more detailed formulation of the non-lte problem will be developed in chapter 2 along with a description of a method of solution. A detailed non-lte analysis will be presented in chapter 3 where the behavior of CO 2 vibrational level populations which contribute to the SABER measurements of 15 and 4.3 µm radiance will be discussed. A review of past measurements of middle and upper atmospheric CO 2 and temperature are discussed next. 1.2 Limb infrared sounders of the MLT Direct in-situ measurements in the MLT region can be provided only by rocketborne instruments because these altitudes are above the range of inflatable balloons and/or aircraft and well bellow altitudes where stable satellite orbits are possible. The rocket measurements are by nature limited in space and time, and to obtain long term global information on the neutral temperature and composition of this region, satellite remote sensing technique must be employed. Global satellite data on the temperature distribution can be obtained utilizing various techniques, such as solar absorption (e.g. Russell III et al. 1993), Rayleigh scattering (e.g. Shepherd et al. 1993), spectrally resolved O 2 ( 1 Σ) emissions (e.g. Ortland et al. 1998), and atmospheric thermal emissions from the CO 2 15µm bands. Early satellite limb measurements of thermal emissions from atmospheric trace gases, were performed by several instruments including e.q., the Limb Radiance Inversion Radiometer (LRIR) (Gille et al. 19), Stratospere and Mesosphere Sounder (SAMS) (Drummond et al. 19) and Limb Infrared Monitor of the Stratosphere (LIMS) (Gille and Russell III 1984).

25 4 These experiments served as a proof of concept for the inference of temperature and trace gas densities from infrared measurements. This early instrument technology, however, did not have the sensitivity needed to probe altitudes much higher than km. Hence it was safe to assume LTE conditions in the 15µm CO 2 emission bands used for vertical temperature profile retrievals. Later advances in instrument technology motivated a number of new space infrared experiments that were capable of probing the tenuous atmospheric layers of the MLT with high Signal-to-Noise-Ratio (SNR) well above km for certain molecular bands. As pointed out earlier, the radiation emerging from the upper MLT is strongly influenced by non-lte effects for the majority of the infrared molecular bands. For observations of particular molecular emissions from space, these non-lte effects are important down to the lower stratosphere. Such is the case for 4.3µm emissions used in inversions to retrieve CO 2 concentrations in this study. Interpretation of this remotely sensed data requires adequate accounting of the deviations from LTE and makes the retrieval task much more difficult in several respects, due to the non-local nature of the forward problem. Experiments that have provided information on atmospheric infrared emissions affected by non-lte (for different molecular bands) include e.g., the Cryogenic Limb Array Etalon Spectrometer (CLAES) (Roche et al. 1993), Improved Stratosphere and Mesosphere Sounder (ISAMS) (Taylor et al. 1993), and Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) (Endemann et al. 1993). The more recent instruments, such as, the Cryogenic Infrared Spectrometers and Telescopes for the Atmosphere (CRISTA) (Offermann et al. 1999) have used non-lte modeling extensively to broaden the range of retrievals into the upper MLT. Also, interpretation of SABER (Russell III et al. 1999) measurements relies on a non-lte model as part of the operational inversion algorithm.

26 5 1.3 Non-LTE retrieval schemes Mertens et al. 2001, have described the operational non-lte retrieval algorithm for temperature from SABER measurements of 15µm CO 2 emissions and Gusev et al. 2006, developed a non-lte temperature retrieval method for CRISTA observations. These instruments as well as others have employed non-lte modeling to derive densities of several minor constituents in the MLT including, for example, CO retrieval from ISAMS (López-Valverde et al. 1991), retrieval of O 3 (Mlynczak et al. 2007) and recently H 2 O (Feofilov et al. 2009) from SABER, CO 2 (Kaufmann et al. 2002) and CO (Grossman et al. 2006) from CRISTA measured spectra, and NO x (Funke et al. 2005) from MIPAS data. Even though much effort has been spent to develop and include non-lte radiative transfer models in retrieval schemes for temperature and atmospheric parameters in the MLT, many basic assumptions in the retrieval approach either do not reflect the full nature of the non-lte modeling of population processes or they rely on techniques developed for the LTE inverse problem. These assumptions do not allow the best use of all the available information in the observations in a consistent fashion. This may lead to additional errors and non-negligible biases in the retrieved atmospheric parameters. For example, in LTE, different molecular species only influence each other s emissions when spectral lines overlap. However, under non-lte conditions, even when the molecular spectral lines do not overlap, the population of vibrational states of a particular molecule giving rise to the observed emissions is dependent on energy exchange processes within the molecule as well as on interaction with vibrational excited states of other molecules (this is discussed in detail in chapter 3). Hence, the sequential, single channel retrieval method, where separate retrieval of the trace gas abundances follows once the T k is retrieved, becomes problematic under non-lte conditions. In addition, the unknown and variable CO 2 VMR in the MLT requires that kinetic temperature and CO 2 VMR be retrieved simultaneously

27 6 and self-consistently. Yet, the single channel, single parameter retrieval has been the standard for inversion of non-lte radiances even for instruments with spectral channels selected for self-consistent measurement of the relevant information on kinetic temperature (T k ) and gas abundances. Such is the case, for example, for CRISTA non-lte retrieval of T k and CO 2, and SABER T k retrieval for the latest version, 1.07 data. There are numerous factors playing a role in the lack of development of formulations that include a physical basis for a multi-parameter self-consistent retrieval method from non-lte atmospheres. There have been only a few instruments with high enough SNR to observe multiple molecular emissions in the MLT; and there is only one, i.e. SABER that is providing long-term observations on a number of parameters whose emissions are influenced by non-lte. Also, solving the non-lte task for multilevel ro-vibrational populations of several molecules is a computationally daunting task, especially for broad band radiances, such as those provided by SABER. In addition, lack of global a-priori information on atmospheric parameters (and their statistics) along with the non-linearity of the forward and inverse problem precludes the use of statistical inversion techniques based on linearization of the radiative transfer equation (as described in Rodgers 2000). The first analysis of a simultaneous T k /CO 2 non-lte retrieval approach for a broad band instrument can be found in the paper by Mertens et al. (2003). The described algorithm was applied for daytime profile retrievals while at night the T k was retrieved with CO 2 VMR supplied by the Thermosphere-Ionosphere-Mesosphere Electrodynamics Global Circulation Model (TIME-GCM). This effort yielded version 1.06 SABER data. However, because of severe non-linearities, the retrieval algorithm had trouble converging and to avoid total instability, several constraints were imposed. This resulted in a retrieved CO 2 VMR profile that was highly-smoothed and constrained (Marshall 2009). In addition the day/night profiles did not reproduce the

28 7 expected tidal features in the T k profile (Remsberg et al. 2008). For these reasons, for the latest SABER T k data, v1.07, the CO 2 VMR was supplied from averaged day/night profiles of the Whole Atmosphere Community Climate Model (WACCM). Another non-lte scheme, used for NO and NO 2 VMR retrieval from high resolution spectrum measured by MIPAS was described by Funke et al. (2005). In this scheme, some of the non-lte parameters, such as the nascent vibrational NO 2 distribution, are treated as unknown in the retrieval and derived along with the VMR. Kostov and Timofeyev (2001) developed and analyzed a different paradigm for simultaneous determination of atmospheric parameters under non-lte conditions. Their approach allows simultaneous retrieval of temperature, pressure, vibrational populations, and trace gas densities from high spectral resolution data. It is a radically different approach, that does not require non-lte population modeling as part of the forward radiance model. However, it requires general a-priori information on the unknown atmospheric parameters and their statistics, including those derived from results of the non-lte modeling of populations of molecular states. This technique is suitable only for data measured by high spectral resolution instruments and has not yet been used in operational retrievals. 1.4 Overview of CO 2 and temperature in the MLT Carbon Dioxide Carbon dioxide, becomes a dominant greenhouse gas above the tropopause, due to diminishing water vapor opacity (Lindzen 2007). Its dominant role in the energy budget of the Earth s stratosphere, mesosphere and lower thermosphere, (altitude range about km), is due to infrared radiative cooling from emissions in the 15µm band (Roble and Dickinson 1989). Investigations of ice core data have revealed a steady CO 2 VMR increase over the past 200 years (Oeschger and Siegenthaler 1987)

29 8 in the troposphere, going from about 2-2 (ppmv) in 10 to about 3 (ppmv) in 2009 (Tans 2010). This increase has been attributed mainly to anthropogenic activities related to fossil fuel burning and deforestation. The rising CO 2 levels will affect middle and upper atmospheric layers through enhanced cooling and subsequent contraction of the atmosphere (Akmaev 2002), with a time lag of about 5-6 years due to the time it takes for CO 2 near the surface to propagate to levels in the upper stratosphere (Bischof 1985). The significance of the knowledge of CO 2 concentrations in the MLT region goes beyond the radiative cooling. It is also a long lived tracer molecule influenced by dynamical and mixing processes, and hence it is suitable for monitoring the global height distribution of the turbopause. Below the turbopause, eddy (turbulent) mixing is many times stronger than the process of molecular diffusion and tends to vertically separate heavier and lighter constituents. Early observations led to a belief that the atmosphere is well mixed up to about km (Wallace and Hobbs 1977). However, this belief has been put into question by several recent CO 2 measurements (Girard et al. 1988; López-Puertas and Taylor 1989; Zaragoza et al. 2000; Kaufmann et al. 2002; Beagly et al. 2010), that seem to be more consistent with much weaker eddy mixing, suggesting that the turbopause can be as low as 75- km (see Figure 1.4.1). The general shape of the CO 2 VMR profiles in Fig is determined by only a few processes. In the lower troposphere CO 2 has a weak seasonal cycle that arises mainly due to biological processes (Brasseur and Solomon 2005). The CO 2 VMR is almost constant with altitude in the lower and middle atmosphere due to eddy mixing and the absence of significant sources or sinks. In the upper mesosphere ( km) molecular diffusion and shortwave ultraviolet radiation cause the CO 2 VMR to decrease with height (López-Puertas et al. 2000). Model calculation of the CO 2 distribution also manifest a strong stratosphere/mesosphere exchange driven by advection, with downward motion in the winter hemisphere and upward motion in the

30 9 Altitude (Km) 120 ACE-FTS (Global mean) CRISTA 1 (Global mean) CRISTA 2 (Averge NH) ISAMS (Global mean) SAMS (Global mean) ROCKET (38 N) GRILLE (30 N) CO 2 (ppmv) Figure 1.1: Mean vertical profiles of CO 2 VMR from several different experiments. ACE- FTS, CRISTA-1, ISAMS, SAMS are global mean profiles. SAMS and ISAMS profiles constitute an average of different latitudes, seasons and years. CRISTA- 2 is a mean CO 2 profile over the Northern Hemisphere (0- N), and the GRILLE represents a mean of two profiles at 30 N. The rocket profile is based on the Aladdin 74 profile (June 1974, 38 o ). Source: Figure 13. in Kaufmann et al. (2002) and Figure 2. in Beagly et al. (2010) for the ACE-FTS. The GRILLE profile was taken from Figure 11. in López-Puertas and Taylor (1989) summer hemisphere (López-Puertas et al. 2000). The exact altitude of the departure from the well-mixed value is still in question as is its spatial and temporal variation. Rocket (mass-spectrometer) measurements (Offermann and Grossmann 1973; Trinks et al. 1978) suggested a uniformly mixed CO 2 VMR up to about - km. The derived theoretical profiles (Wintersteiner et al. 1992) as well as early model simulations (Rodrigo et al. 1986) further supported this conclusion. However, satellite data on CO 2 derived from solar occultation experiments (Grille spectrometer) (Girard et al. 1988) and ATMOS (Rinsland et al. 1992) suggest a rapid decrease in CO 2 VMR starting at about 75 km. Additional data on the MLT CO 2 content came from experiments observing daytime limb emissions from the 4.3µm bands. A few rocketborne emission experiments provided information on non-lte processes affecting the 4.3 µm bands [see for example Spectral Infrared Rocket Experiment (SPIRE) (Nebel

31 10 et al. 1994) and Spectroscopic Infrared Structure Investigation (SISI) (Vollmann and Grossmann 1997)]. With this information non-lte kinetic models were applied to limb radiances measured by SAMS on NIMBUS 7 (López-Puertas and Taylor 1989) which showed that the CO 2 VMR starts to decrease around km from the well mixed value. The later successor of this instrument, ISAMS on board the Upper Atmospheric Research Satellite (UARS), confirmed that finding, i.e., CO 2 is well mixed up to km followed by sharp decrease above this altitude (Zaragoza et al. 2000). In addition, there appeared to be no latitudinal or seasonal variations in the derived data, that could be discerned above the errors in the measurement. The CRISTA instrument flew twice on the space shuttle (Novermber 1994 and August 1997) (Offermann et al. 1999) and provided new information on the global distribution of CO 2 in the MLT. CRISTA observed emissions in the 4.3µm bands were used to determine the CO 2 VMR and the authors (Kaufmann et al. 2002) showed that it deviates from a well mixed value between -75 km. The data analysis demonstrated significant spatial and temporal variation of the CO 2 throughout the MLT. The latest MLT CO 2 VMR data have been inferred from solar occultation observations by the ACE-FTS instrument on the SCIentific SATellite (SCISAT) 1 satellite. As part of the retrieval validation, the data were also compared with the Canadian Middle Atmosphere Model (CMAM) simulations (Beagly et al. 2010). The CO 2 VMR profiles from ACE-FTS show a decrease starting at altitudes between 75- km and the CO 2 profile lies between the CRISTA and the ROCKET profile. The CMAM simulations were not able to reproduce the observed values, especially the altitude where deviations from the well mixed value begins. Although numerous investigations that varied several of the model parameters governing the CO 2 VMR have been performed, the model could not be reconciled with the observed data. This has led

32 11 the authors to speculate on new loss process for CO 2 in the mesosphere, including a possible role played by meteoric dust (Beagly et al. 2010). Table 1.1 summarizes the available data and main conclusions on the MLT CO 2 VMR distribution from the few GCMs simulations and measurements. The entries are sorted according to the type of measurement technique used (solar occultation, mass spectrometer measurements, limb emission observations) and model calculations Temperature Our knowledge of the thermal structure and its distribution in the MLT region is more comprehensive than that of the CO 2 VMR. In part, this is a result of the numerous methods that can be used to obtain information on the temperature distribution in the MLT. Broadly speaking, there are 8 experimental methods that have been used to measure MLT temperature: 1) hydroxyl airglow, OH, rotational temperature, 2) O 2 airglow rotational temperature, 3) Doppler temperature derived from the atomic oxygen line, 4) rocketsonde temperature sensors, 5) Rayleigh Light Detection and Ranging (LIDAR), 6) sodium LIDAR, 7) rocket grenade and falling spheres, 8) satellite probing (after Beig et al. 2003). A number of important climatological features in temperature structure in the MLT have been discovered in the past two decades. The so called two-level mesopause (She and Zahn 1998) was revealed from the LIDAR data. It turns out that the mesopause altitude occurs either at around 86 km or at around km, but rarely in between. The lower altitudes appear in the summer hemisphere at latitudes above 23. Model studies of the two-level mesopause further support these observations (Berger and von Zahn 1999) but, not all GCMs could reproduce this feature, including for example, MSISE- and CIRA-86 (Ratnam et al. 2010). However, later satellite observations provided more evidence to support the two-level mesopause (see for example, the High Resolution Doppler Imager (HRDI) [Ortland et al. 1998]; CRISTA

33 Table 1.1: Summary of observations and modeling of MLT CO 2. Ref. / Year Source 1 Lat. 2 Type Main conclusion Muller et al. Occultation Meas N (1985) (Spacelab 1) CO 2 volume mixing ratio is homospheric up to km. Girard et al. Occultation Meas N (1988) (Spacelab 1) CO 2 volume mixing ratio decreases in mesosphere starting at 75km. McHugh and Occultation Meas. Global CO Rinsland (2003) (ATMOS) 2 mixing ratios decline due to diffusive effects above about km. Distinctive difference between winter and summer latitudes. Rinsland et al. (1992) Beagly et al. (2010) Meas N Meas. Global Occultation (AMOS 3) Occultation (ACE-FTS) CO 2 VMR nearly constant between -km followed by rapid decline between - km. No evidence of non-lte effects in ν 2 mode below km. CO 2 profile departs from well mixed about 75-km but is higher than CRISTA. The obtained profile cannot be reproduced by the GCM CMAM, especially the height where the fall off in CO 2 occurs. Offermann (1973) Meas. 40N Rocket CO 2 volume mixing ratio is in diffusive equilibrium up to an altitude km. Trinks (1974,78) Meas. 38N Rocket CO 2 densities are consistent with complete mixing up to km. Limbemission Zaragoza et al. Meas. Global (2000) (ISAMS) Kaufmann et al. (2002) Lopez-Puertas et al. (1989) Rodrigo et al. (1986, 1991) Wintersteiner et al. (1992) Kaufmann et al. (2002) Meas. 0-N Meas. Global Limbemission (CRISTA2) Limbemission (SAMS) The CO 2 VMR profile is found to be constant below km and decreases strongly above this altitude. Any change in CO 2 between equatorial and polar latitudes is small and does not exceed the measurement error. CO 2 begins to deviate from lower atmospheric concentrations between -75km. Strong longitudinal and latitudinal variation is found. CO 2 mixing ratio begins to deviate from well mixed as low as km in altitude based on best fit to observed radiances. Mod. Global GCM CO 2 is significantly depleted only above km. Mod. n/a Empirical profile Constant CO 2 up to km Mod. Global GCM CO 2 is significantly depleted only above km. Difference between model and CRISTA observation unlikely to be due to spatial or temporal coverage. 1 Meas. stands for measurement and Mod. stands for model simulations. 2 Global indicates latitude coverage including high, middle and low latitudes for both hemispheres. Measured by mass spectrometer. 12

34 13 [Gusev et al. 2006] and SABER [Ratnam et al. 2010]). In addition, the large number of observations have led to steady theoretical progress in understanding various dynamical processes influencing the temperature structure and its seasonal variation (gravity waves, planetary waves and tidal oscillations). For instance, comprehensive analyses of tidal signatures in satellite temperature data have been conducted and published (see e.q. Forbes et al and reference therein. However, just as for CO 2 in the MLT, an important question that has sparked much interest is whether there are any systematic temperature changes and trends in the MLT region. A comprehensive review and near complete synthesis of all the available observations are presented in the paper by Beig et al. (2003). One of the main conclusions derived from available observations is a zero temperature trend in the altitude range - km. However, there are a few sites that report a negative trend -7 to -10 K/decade (Beig et al. 2003) and none that report any positive trend in the long-term observations. In addition, model calculations done by WACCM predict a temperature trend in the range -7 to -10 K/decade (Garcia 2010b). However, the limited size and number of observations do not permit extrapolating the observational results globally and neither can all decadal variability factors modulating these trends be discriminated. It is only when longer data sets become available that these limitations can be overcome. It is clear from the previous review of the MLT CO 2 and T k that some of these gaps in our knowledge can be filled only through reliable global, long-term observations. The SABER instrument is on track to be the first infrared limb emission satellite sensor to complete a full solar cycle of observations and provide an unprecedented opportunity to study this region. It is imperative that the retrieved quantities be derived from the information in the radiances in a self-consistent and reliable manner to take advantage of this opportunity. SABER provides broad band radiances that are strongly affected by non-lte, and therefore developing a retrieval algorithm is

35 14 very challenging as there is no multi-parameter retrieval precedent. The goal of this study is to investigate various methods for accomplishing this task, develop a selfconsistent simultaneous retrieval of T k /CO 2 from non-lte emissions, and conduct initial scientific investigations of long-term changes in there parameters. 1.5 Research statement In this study a detailed analysis of a self-consistent simultaneous retrieval algorithm for temperature and CO 2 VMR will be presented using spectrally un-resolved limb infrared radiance measurements of the non-lte atmosphere. A focus of this study is on investigating sensitivity, stability and convergence of the algorithm and the potential inherent limitations. This technique will be applied to infer the vertical structure of temperature and CO 2 VMR in the MLT region from observed limb infrared radiances obtained by the SABER instrument on board the TIMED satellite. The SABER instrument provides global day and night observations of limb infrared radiance profiles in 10 spectral channels ranging from 1.6 to 17 µm and it is the longest-term data set available providing information on the temperature structure and CO 2 concentrations in the MLT region. The narrow ( cm 1 ) 15 µm channel will be used for retrieval of kinetic temperatures up to 105 km altitude and the 4.3 µm channel ( cm 1 ) for CO 2 VMR retrievals up to 115 km. The focus will be only on daytime retrievals, at solar zenith angles less than 75 o, as the nighttime 4.3 µm measured signal is substantially weaker than for daytime, and the nighttime 4.3 µm non-lte modeling of ro-vibrational states is still not well characterized (see for example López-Puertas et al. 2004). Nighttime retrievals also require additional a-priori information which further limits their accuracy. In chapter 2 a formulation of the multilevel vibrational-rotational non-lte problem for a mixture of gasses is provided along with a method of solution that is used in the Accelerated Lambda Iteration for Atmospheric Radiation and Molecu-

36 15 lar Spectra (ALI-ARMS) radiative transfer code. Chapter 3 describes the SABER experiment, its measurement technique and the forward problem under non-lte conditions. Chapter 4 provides details of the development of a self-consistent retrieval method for simultaneous temperature and CO 2 VMR retrievals from the measured limb radiances. Extensive sensitivity and error analyses are included. Chapter 5 includes inversion results using measured signals for daytime conditions, followed by a preliminary analysis. Chapter 6 contains a summary of the important conclusions and then a discussion of future direction in this research.

37 16 CHAPTER 2 THE NON-LTE PROBLEM 2.1 Overview of non-lte The importance of accounting for deviations from Local Thermodynamic Equilibrium (LTE) for interpretation of limb infrared radiation originating in the MLT region has been touched upon in the first chapter. This chapter presents a general formulation of the non-lte problem, i.e., determining the populations of ro-vibrational levels for a mixture of molecules relevant to the Earth s atmosphere. A method of solution for the coupled equations that arise from this problem is an important component in non-lte modeling and will also be discussed in this chapter. The following mathematical formulation of the non-lte problem and the solution technique serve as the theoretical basis for the calculations of non-lte populations as implemented in the Accelerated Lambda Iteration for Atmospheric Radiation and Molecular Spectra (ALI-ARMS) code (Gusev 2002), that is used in this study for the forward modeling of the SABER limb radiance. The discussion in this chapter will begin with a general consideration of when LTE applies and when it does not. The distribution of internal energy among the various ro-vibrational modes of a particular molecule is influenced by numerous processes at any instant. The most important in the Earth s atmosphere are: 1) thermal collisions, 2) spontaneous emission, 3) absorption of external radiation (either solar radiation or upwelling earth radiation), 4) chemical or photochemical reactions, and 5) vibrationto-vibration (V-V) energy transfer as well as electronic-to-vibrational (E-V), (for more discussion see (López-Puertas and Taylor 2001). The process of stimulated emission is included in the calculations for completeness, but it is unimportant for determining the non-lte level populations for conditions in Earth s atmosphere. When collisions

38 17 are frequent enough so that the molecular energy levels are strongly connected to the kinetic energy pool, the distribution of excited states of a molecule is a function of a single parameter, the kinetic temperature. This defines the LTE condition where the distribution of excited states is given by the Boltzmann distribution. When the thermal collisions are not frequent enough to dominate the external excitation mechanisms influencing the level populations, the condition is referred to as non- LTE. Because the collisional frequency falls off with altitude as pressure decreases, non- LTE effects are expected to be stronger higher up in the atmosphere. However, the altitude at which non-lte effects become significant depends on a particular internal energy level of a particular molecule. For instance, a transition with a large energy jump, generally requires a larger average number of collisions to keep the level populations in equilibrium. Then there are the above mentioned processes that may cause the energy populations to depart from LTE. However, for conditions found in planetary atmospheres, complete non-lte for all energy states of a molecule is never found. It is rather a general rule that some of the internal energy states of a particular molecule depart from LTE while others remain in LTE (Kutepov et al. 1998)). In atmospheric science, the first studies of molecular infrared bands affected by non-lte were carried out in the in 1950s (Curtis and Goody (1956)). Since then, many investigators (Kutepov and Shved 1978, Shved et al. 1978, Kumer and James 1982, López-Puertas et al. 1986, Kaye and Kumer 1987, Winick et al. 19, Mlynczak 1991, Wintersteiner et al. (1992), Edwards et al. 1993) turned their attention to solving the non-lte problem for various bands of the radiatively active minor molecular constituents of planetary atmospheres. However, the theoretical basis and methods for solution of non-lte radiation transfer for atomic lines were first developed in the context of stellar atmospheres, starting in 1930 by E. Milne (López-Puertas and Taylor 2001). Since then, stellar astrophysicists developed powerful techniques to deal

39 18 with non-lte radiative transfer, accounting for many complex processes with computational efficiency (Kutepov et al. 1998). A description of one such technique, adopted from stellar astrophysics and applied to the general multilevel non-lte problem for infrared ro-vibrational molecular bands, is presented below following the formalism of (Kutepov et al. (1998), Gusev and Kutepov (2003)). 2.2 Formulation of the problem The multi-level non-lte radiative transfer problem for molecular lines involves two primary components: 1) the Statistical Equilibrium Equation (SEE), which prescribes the rate of change of population and depopulation of molecular levels, accounting for all the microscopic processes involved and 2) the Radiative Transfer Equation (RTE), that describes how radiation interacts with matter and gives a solution to the radiation field at spatial grid points along the Line of Sight (LOS). If the information for all the excited populations is available the radiative transfer equation can be solved. Let us consider a gas mixture obeying the Maxwellian distribution of molecular velocities for a local kinetic temperature at each vertical grid point. When the rate of excitation and de-excitation of the levels does not change with time, it is the so called steady state and the SEE can be conveniently written as ) n i ( j R ij + C i = j n j R ji + C i + Y i. (2.2.1) Where n i is the population of a molecular level, i (in units of molec cm 3 ), R ij are the radiative coefficients for transitions from energy level i j, where the energy relation is E i > E j. The Ci and Ci are total rates of de-population and population of level, i, respectively, due to a variety of collisionally induced energy transfer processes. Y i in Eq. (2.2.1) represents a source term for the rate of excitation of level, i, by processes

40 19 other than collisions or absorption of radiation arising from within the atmosphere, such as absorption of solar and/or ground radiation, chemical or photochemical reactions, etc. C i is the total loss rate for level, i, through collisional processes and is written C i = j C ij + jkl n k C ij,kl. (2.2.2) The first term defines the rate for vibration-rotation energy transfer to translational energy, (V-T), in it C ij = β n β k β ij, (2.2.3) where k β ij are the rate constants for the V-T transition i j by collisions with atoms and molecules of constituent-β of the gas mixture with density n β. The second term in Eq. (2.2.2) describes the loss rate of level i through energy exchange processes of type vibration-vibration (V-V). There, C ij,kl are rate coefficients for the transition i j in one and the transition k l in another of the two colliding molecules and/or atoms. Accounting for Eq. (2.2.2), one can write the term C i on the right hand side of the (2.2.1) as C i = j n j C ji + jkl n j n l C ji,lk. (2.2.4) The collisional rate coefficients are related by the detailed balance relations n ic ij = n jc ji and n in kc ij,kl = n jn l C ji,lk, (2.2.5) where, n are the LTE populations.

41 20 The radiative rate coefficients in the SEE are given by R ij = A ij + B ij Jij B ij Jij if i > j if i < j (2.2.6) where J ij = J ji is the integrated mean intensity, defined as J ij = 1 4π dω I νµ ϕ ij (ν)dν. (2.2.7) Its obvious from Eq. (2.2.7) and (2.2.6) that in order to solve the SEE equation (2.2.1), the radiation field must be known at each altitude grid point. The differential equation of radiative transfer describing the emission, absorption and movement of photons along some line of sight and in a plane parallel geometry can be written as µ di νµ dz = χ µ(ν)i νµ + η µ (ν), (2.2.8) where I νµ is specific intensity of atmospheric radiation, and χ µ (ν) and η µ (ν) are total opacity and emissivity, respectively, at a frequency ν with a directional cosine µ. For each pair of levels connected by a line transition from i j, where i > j, the emissivity η ij and opacity χ ij completely characterize the radiative properties of the gas being considered. If the macroscopic velocity field can be neglected, which is obviously a good approximation for any molecular radiative transfer problem in planetary atmospheres, these quantities do not depend on µ and are given by η ij (ν) = hν ij 4π n ia ij ϑ ij (ν), (2.2.9) χ ij (ν) = hν ij 4π (n jb ji n i B ij )ϕ ij, (ν)

42 21 where ν ij is the line center frequency, A ij, B ij and B ji are Einstein coefficients, ϕ ij (ν) and ϑ ij (ν) are the line profile functions normalized such that ϕ ij (ν)dν = 1. Under the assumption of complete frequency redistribution within a line, the profile functions ϕ(ν) and ϑ(ν) are equivalent, and therefore, the source function for the transition is frequency and angle independent and given by, S ij = η ij(ν) χ ij (ν) = n i A ij (n j B ji n i B ij ). (2.2.10) Even though it is possible to account for angle and frequency variations, this is not required in the solution and it will not be considered here. If the conditions for overlapping of different lines within a band or even between lines of different bands have to be considered, the total emissivity and opacity coefficients are summed over all transitions of all molecules making contributions at frequency, ν η(ν) = ij η ij (ν) + η cont. (ν), (2.2.11) χ(ν) = ij χ ij (ν) + χ cont. (ν), (2.2.12) where for generality, the background continuum contributions η cont. (ν) and χ cont. (ν) are included. The continuum coefficients are assumed to be known a-priori and that they do not change during iterations. In the formulation of equations (2.2.9) the effects of Rayleigh and Mie scattering are not included, as they are not relevant in our study of infrared limb emissions. Accounting for equations (2.2.11) and (2.2.12), the total source function is just S(ν) = η(ν) χ(ν). (2.2.13)

43 22 Note that equations (2.2.1) and (2.2.8) form a coupled set of equations. The way these quantities depend on each other is shown in Figure 2.1 RTE S ν Source function calculation n i I ν Mean intensity calculation J SEE Figure 2.1: The SEE and RTE coupling. Flow diagram for the coupled equations of non- LTE radiative transfer in molecular lines. See text for symbols definition and detailed explanation. The mean intensity is found by integrating the source function (2.2.10). The source function, extinction and emission coefficients depend on the level populations (2.2.9). These in turn depend on the intensities (2.2.1) and (2.2.7). To solve this set of equations, the radiative field and level populations must be determined simultaneously. Since the RTE, Eq. (2.2.8), couples the level populations at different altitudes to each other, the SEE system, Eq. (2.2.1), for the level populations is non-local (and non-linear). While the densities, n β, are considered to be known, the (V-T) coefficients in Eq. (2.2.2) do not introduce the non-linearity into SEE. However, additional (local) non-linearity enters SEE through the second terms in Ci and Ci, representing the (V-V) processes. In the following discussion, more detail on several special steady states of the molecular gas will be considered, including Complete Thermodynamic Equilibrium (CTE), (i.e. when the conditions for all three types of equilibria [mechanical, chemical, thermal] are satisfied and there is no tendency for any change of the state), LTE and partial LTE, when the rotational sub-levels for each vibrational transition remain in LTE. In general, the discussions presented in (Mihalas 1978) and (Ivanov 1973)

44 23 for similar situations of an atomic gas in the astrophysical context is followed. For our discussion it is assumed that the index i corresponds to the pair of indices vj, which specify the ro-vibrational state of a molecule; here v and j are the indices of a vibrational level and its rotational sub-level, respectively. For a triatomic molecule, v represents the combination of three vibrational quantum numbers along with an associated angular momentum quantum number. Therefore the sums over this index in our treatment imply summation over all available combinations of these four indices Complete Thermodynamic Equilibrium In CTE, the kinetic temperature is the only parameter that completely describes this state. The populations follow the Boltzmann law n v j n vj = g ( v j exp E v j E ) vj, (2.2.14) g vj kt where k is the Boltzmann constant and g v j and g vj represent the degeneracy of the ro-vibrational state. The primed quantities designate the upper level for the transition. The radiative intensities are given by the Planck function, here expressed for the line center frequency as ν v j = 1 h (E v j E vj) (2.2.15) J v j,vj = 2hν3 v j,vj c 2 [ exp ( ) 1 hνv j,vj 1] (2.2.16) kt Moreover, in CTE all processes obey the principle of detailed balance; each process is exactly compensated by its inverse. In particular, the number of radiative transitions from v j vj is exactly equal to the number of radiative excitations vj v j. n v j (A v j,vj + B v j,vj J v j,vj ) = n vjb vj,v j Jv j,vj. (2.2.17)

45 24 And for collisional transitions the detailed balance has the form n v j C v j,vj = n vjc vj,v j (2.2.18) or with accounting for Eq. (2.2.14) C vj,v j = C v j,vj ( g v j exp E v j E ) vj. (2.2.19) g vj kt LTE As long as the velocity distribution of all molecules remains Maxwellian, the Boltzmann distribution of excited states Eq. (2.2.14) is valid. Such conditions occur when the collisions among the molecules are frequent enough that collisional terms in the SEE dominate over radiational and source terms, in which case one can neglect them all together and then the equilibrium populations are in fact the only solution. The population probability W vj of ro-vibrational level vj of a specific molecular species α can be written as W vj = n vj n α, (2.2.20) where n α is the total number density of the species in question. The star symbol ( ) below denotes LTE values. Accounting to the Boltzmann relation, the expression can be written as W vj = g ( vj Q (T) exp E ) vj, (2.2.21) kt where Q (T) = vj ( g vj exp E ) vj kt (2.2.22) is the total ro-vibrational internal sum, such that vj W vj = 1. (2.2.23)

46 25 The vibrational probability, Wv, that a molecule is in the vibrational state, v, is given by: n v = j n α W v, (2.2.24) where n v is the population of vibrational level v. When deriving the expression for Wv, two facts should be kept in mind; a) the energy of a given ro-vibrational level can be expressed as the sum of the vibrational energy E v of state v and the rotational energy E j,v relative to that in j = 0 of sub-level j of state v E vj = E v + E j,v ; (2.2.25) b) the statistical weight of a given ro-vibrational level is a product g vj = g v g v (j) (2.2.26) of statistical weight g v of level v and degeneracy g v (j) of its rotational sub-level, j. It can be shown from the definition of Eq. (2.2.21) and accounting for (2.2.25) and (2.2.26) that Wv = g Q v (T) ( v Q (T) exp E ) v kt (2.2.27) and introducing the rotation partition function for the level v Q v (T) = j ( g v (j) exp E ) j,v. (2.2.28) kt Finally, an expression for the population probability of a rotational sub-level, j, of vibrational level v can be written as W v (j) = g v(j) Q v ( exp E ) j,v. (2.2.29) kt

47 26 With these definitions equation equation (2.2.21) can be rewritten as: Wvj = W v W v (j). (2.2.30) It follows from (2.2.24) that the LTE ratio of two vibrational populations are related by the expression: n v n v = W v W v = g ( v exp E v E ) v Q v (T). (2.2.31) g v kt (T) Q v Partial LTE In planetary atmospheres, strict LTE or non-lte conditions are never present. At low altitudes and relatively high density, collisional processes dominate the population of molecular levels which leads to LTE. At higher altitudes, the density decreases and the highest vibrational levels start to depart from the Boltzmann distribution. This results from the V-T and V-V processes being less frequent and not able to maintain the coupling of these levels to the other thermalized levels against the influence of radiative transitions. As the density decreases further, even lower vibrational levels start to depart from the LTE distribution. However, the rotational substructure of all vibrational levels can still remain thermalized, ie, their distribution follows the Boltzmann law, until they encounter the most tenuous part of the atmosphere. Such a case defines vibrational non-lte assuming all rotational level are in LTE. This greatly reduces the number of unknown variables in the SEE since the rotational probability Wv (j) is known for each vibrational level, v, and only the vibrational populations, n v need to be found. This state can be described in terms of probability as W vj = W v Wv (j), (2.2.32)

48 27 where W v (j) is still given by (2.2.29) but W v is now W v = g v Q v (T) Q ( exp E ) v kt v (2.2.33) with excitational temperatures T v = T for levels that remain in LTE and T v T for those that do not. The total partition function Q that replaces Q (T) in equation (2.2.33) is no longer a function of a single variable, T, i.e. Q = v ( g v exp E ) v Q v (T). (2.2.34) kt v The SEE still has the form of (2.2.1), but summed over j. Since transitions between rotational sub-levels of vibrational level v cannot change the total populations, n v the corresponding terms in (2.2.1) cancel, which results in ( ) R v v + C = v v n v v n v R vv + C v + Y v (2.2.35) where the radiative coefficients will have the form R v v = j j W v (j)r v j,vj = A v v (T) + B v v (T) J v v, (2.2.36) with A v v (T) = j j W v (j)a v j,vj (2.2.37) which is the spontaneous de-excitation rate per molecule for the entire ro-vibrational band v v. For v < v, A v v = 0. B v v J v v is the stimulated emission (de-excitation) or absorption (excitation)rate for a ro-vibrational band per molecule in state v. The Einstein coefficients (B v v (T),B vv (T) and A v v (T)) for the vibrational transition are

49 28 related by the expressions A v v = 2hν3 v v c 2 B v v, g v B v v = g vb vv (2.2.38) where the vibrational transition frequency ν v v is defined as ν v v = 1 h (E v E v). (2.2.39) It should be noted that the temperature dependence of the band averaged Einstein coefficients is rather weak for the temperature range found in planetary atmospheres and it is usually ignored. This temperature dependence results from summation of the temperature independent coefficients for ro-vibrational transitions with the temperature dependent weighting function Wv (j) (Eq ). The J v v in equation (2.2.36) has the form J v v = 1 B v v Wv (j)b J v j vj v j vj. (2.2.40) j j It replaces J ij in a similar expression [Eq. (2.2.6)] for the radiative rate coefficients for a single line and therefore represents the integrated mean intensity in the rovibrational band. The C and v C v still have the same structure as similar expressions in Eq. (2.2.1) but involve only vibrational populations. The collisional (V-T) coefficients between the vibrational levels C v v = j j W v (j)c v j vj, (2.2.41) are related through the detailed balance relation ( W v C v v = W v C vv or C vv = C g v v v exp E v E ) v Q v g v kt Q v (2.2.42)

50 29 which is obtained by summing the detailed balance equation (2.2.5) over j for rovibrational transitions while accounting for Eq. (2.2.21). Rotational LTE for vibrational levels of a ro-vibrational band is rather similar to the case of a complete frequency redistribution in a single line. In the latter case the frequency of the emitted photon is not correlated with that of the absorbed photon. As a result, the line absorption and emission profiles coincide. Similarly, in the case of rotational LTE, there is no correlation between the rotational lines in which the photon was absorbed and subsequently emitted. This independence is caused by collisions which the molecule undergoes during its radiative lifetime. In rotational LTE, these collisions are frequent enough to keep the rotational structure of the vibrational levels thermalized. In other words, rotational LTE provides an efficient additional large scale (compared to redistribution in a single line) frequency redistribution mechanism, in which photons absorbed in optically thick lines in the cores of band branches may be emitted in weak lines in the band wings. An additional available approach to reduce the dimension of the SEE system (2.2.35) is to treat rotationally thermalized vibrational levels as super-levels when these levels are closely spaced in energy, and although they are out of equilibrium with the ensemble, they are in equilibrium with each other. This happens since the intra-molecular V-V energy transfer between these levels is a few orders of magnitude larger than that for V-T exchange. Therefore, even when the density is low, the collisional processes are able to keep these levels in LTE. A typical example of such a group of levels is for a three-atom molecule such as CO 2 whose vibrational level can be described with four quantum numbers (v 1 v2 lv 3) which may have the same 2v 1 + v 2 and v 3 values, with the only difference being in the l numbers.

51 Accelerated Lambda Iteration Method of Solution Consideration in this section will be given to only one approach to the solution of the non-lte problem as implemented in the ALI-ARMS code (Gusev 2002), namely the Accelerated Lambda Iteration (ALI) method. López-Puertas and Taylor (2001) provide a detailed description of another available approach, the so called Curtis- Matrix method. Gusev (2002), and Kutepov et al. (1998) give an in depth analysis of both techniques as well as a comparison in terms of the computational burden. As was discussed earlier, the radiative rate coefficients in SEE Eq. (2.2.1), (2.2.35) not only make the problem non-linear, but also non-local, as the J v v depends on the level populations at all altitudes in the atmosphere. In addition, local non-linearity is introduced by molecular collisions because their rates usually depend on the level populations of the molecule. A simple and obvious solution to this system is to iterativelly evaluate all equations as illustrated in Figure 2.1 following the arrows. This method is called the Lambda Iteration (LI). This approach has been used in atmospheric science applications by Wintersteiner et al. (1992) in solving the non- LTE problem for CO 2 15 µm levels. LI involves an iterative evaluation of RTE and SEE until the system converges. The fact that this process of iteration can be mathematically expressed in the form a of Lambda operator, gives it its name. This approach has been investigated in detail in the astronomical context since the 1920s (Unsöld 1968). Even though this technique is not used in our approach it is briefly discussed to give an introduction to the ALI method. The lambda operator represents the entire procedure of computing mean intensities J ij from the source function. It involves a formal solution to the radiative transfer integral, it includes both angle and frequency integration, and it is formally defined as: J ij = Λ µν [S ν ], (2.3.1)

52 31 or J ij = 1 4π dω Λ µν [S ν ]dνϕ ij (ν) (2.3.2) In lambda iteration, the new populations are found from the solution of SEE n i (R ij + C ij) = j j n j (R ji + C ji) + Y i, (2.3.3) in which the radiative coefficients are evaluated from the populations given by a previous iteration R ij = A ij + B J ij (2.3.4) where J ij = 1 4π Ω Λ µν [S ν ]dν. (2.3.5) In these expressions the dagger sign indicates quantities evaluated with the old variables from the previous iteration. For the time being it is assumed that the collisional rate coefficients and source terms do not vary in the course of iteration. In this way the non-linearity from the collisional rate coefficients is avoided. LI is attractive as it involves matrices no larger than L L, where L is the total number of molecular levels, and is actually a numerical simulation of the process of multiple scattering of photons in lines and bands. Unfortunately, if for some transitions, the atmosphere is optically thick, the photons can be effectively trapped in cores of optically thick lines. Thus a large number of iterations is required to propagate these photons over some distance before the final state is reached. The convergence rate can be so slow that false solutions appear to be stable, or the accumulation of numerical errors will cause the process to diverge. The ALI scheme is similar to the LI scheme, with the difference being that the equations are pre-conditioned to speed up the convergence. The family of accelerated lambda iteration (ALI) methods utilize iteration with approximate (or accelerated)

53 32 lambda operators (see the detail treatment in Mihalas 1978, Rybicki and Hummer 1991, 1992, and Pauldrach et al. 1994). Efficient ALI methods are based on operator splitting into the local self-coupling contribution and the remainder: Λ ij = Λ ij + (Λ ij Λ ij), (2.3.6) with Λ ij being the lambda operator between levels i and j, and Λ ij is the diagonal (local) or tri-diagonal (local+nearest neighbor) part of the full lambda operator. This approach leads to the iterative scheme I µν = Λ µν [S ν] + (Λ µν Λ µν )[S ν ] = Λ µν [S ν] + I eff µν, (2.3.7) where I eff µν = (Λ µν Λ µν)[s ν]. (2.3.8) Although this equation is only approximate at each stage of the iteration, it becomes exact for the converged solution S ν = S ν, Λ µν = Λ µν and Λ µν = Λ µν (Hubeny et al. 2002). The principal advantage of the diagonal approximation is that SEE remains completely local, whereas more sophisticated approximations, such as the tridiagonal, make the equations non-local, they become harder to solve, and may be more unstable. Nevertheless, tridiagonal operators may offer much faster convergence, and have been used successfully in a number of astrophysical applications. Eq.(2.3.7) gives I µν in terms of populations. It is important to note that S ν as well as the operators Λ µν and Λ µν are constructed from the old populations. However the new populations are still present here through the source function S µν. The direct implementation of this expression for the intensity in the radiative rate coefficients of the SEE will cause these equations to become nonlinear. The distinguishing feature of

54 33 the ALI approach is that the linearity of the SEE is easily restored by preconditioning. Rybicki and Hummer (1991, 1992) described several strategies for this procedure. For instance, preconditioning within the same transition only, is designed to handle the case when lines do not overlap or the overlap is weak i.e. only the cores of a few lines overlap, while for the remainder, at most only wings of lines overlap. This leaves the form of the SEE (Eq ) unchanged, while the radiative rate coefficients R ij are replaced by R eff ij = A ij (1 Λ ij ) + B eff ij J ij. (2.3.9) Λ ij eff Here and J ij are obtained from Λ µν and Ieff µν by applying the operator Ω dνϕij (ν). The conditioning of the SEE is improved because much of the transfer in the core of the line (described by the local part of the lambda operator) is canceled analytically. These preconditioned equations are still linear in the molecular level populations, notwithstanding the presence of stimulated emission terms. These modified equations automatically guarantee non-negative solutions for the new populations (Rybicki and Hummer 1991, 1992) and thus they ensure a stable iteration process. This formalism has been presented as a method of solution for the general non- LTE problem, in which the system of equations Eq.(2.2.1), is solved for all molecular level populations. If only ro-vibrational excitation is involved, populations of all rovibrational levels are found. Additionally, if rotational LTE is valid, the radiative rate coefficients R v v in the SEE (Eq ) will be replaced by Reff. These are v v obtained by summing, R eff Eq. (2.3.9) over j and, as was done in deriving R v j vj j v v. 2.4 ALI-ARMS current status The ALI-ARMS code in its current version can treat an arbitrary number of molecules of arbitrary structures in a given planetary atmosphere provided by the pre-

55 34 scribed format inputs of (a) planetary atmosphere properties (pressure, temperature, VMRs of molecules), (b) solar spectra, (c) vibrational and ro-vibrational energies, (d) spectroscopic information (Einstein coefficients for ro-vibrational or rotational transitions, line half-widths), (e) collisional rate coefficients for specified sets of V-T and V-V transitions. The radiative transfer equation for a plane-parallel atmosphere may be solved by a number of different methods (Gusev 2002, Gusev and Kutepov 2003), among them being the long and short characteristics of zeroth and first order, the modified Feautrier algorithm, and the discontinuous finite element algorithm. In addition to the operator splitting technique, the ALI-ARMS code implements another option for iteration improvement using a scheme called Ng-acceleration (Ng 1974). This is one of the methods that improves convergence of any linearly converging iteration scheme. The Ng-method is a purely numerical scheme and is completely separate from the ALI technique. The Ng algorithm uses previous results for molecular level populations to estimate the new accelerated set of populations according to some minimization criteria. It is found that after every four iteration steps, the Ng-acceleration can be applied to yield reliable and significant acceleration. This is illustrated in Figure 2.2 as well as in the Figures 1 and 2 of Kutepov et al. (1998) where the number of iterations is plotted versus the convergence.

56 Figure 2.2: Convergence plot for the CO 2 problem in Earth s atmosphere. The labels on the curves stand for various iteration schemes applied in combination with the name of the method of solution of the RTE: (DFE) Discontinuous Finite Element, (MF) is Modified Feutrier method, Short-0 stands for Short characteristic of zeroth order and Short-1 for the first order. NG indicates that Ng-acceleration was applied and LI stands for Lambda iteration. Figure adapted from Figure 4.5 in (Gusev 2002) 35

57 36 CHAPTER 3 SABER EXPERIMENT AND THE FORWARD MODELING OF SABER RADIANCES 3.1 Chapter overview In this chapter, an overview of the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument is presented, including its main objectives along with a description of the measurement technique. Then, a description of the forward model (part of which has been described in chapter 2) used for simulating the observed radiance follows. The behavior of populations of bending mode, (v 2 ), and asymmetric stretch mode, (v 3 ), of the CO 2 molecule from which the 15µm and 4.3µm (respectively) observed emissions originate is examined. Finally, a brief discussion of the sources for the input parameters needed for modeling SABER radiances is given. 3.2 SABER experiment The main scientific objective of the SABER experiment is to explore the MLT region on a global scale to improve our knowledge of the fundamental processes governing the thermal structure, energetics, chemistry and dynamics of this region, including the transport of trace gases (Russell III et al. 1999). The SABER instrument (see Figure 3.1) was launched on board the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite from Vandenberg Air Force Base on December 7, It was placed into a nominal circular orbit at 625 km, inclined at 74 with an orbital period of 97 minutes. SABER makes simultaneous measurements of infrared radiances in 10 broadband channels by scanning the limb of the Earth s atmosphere. The scanning sequence starts at the surface going to

58 37 about 350 km and then back down to the surface in about seconds with vertical sampling every 0.35 km and 2 km FOV. Figure 3.1: Graphics of TIMED spacecraft and placement of the SABER instrument. The SABER instrument is oriented to observe the Earth s limb perpendicular to the TIMED satellite velocity vector. Measurements are made both, day and night, with latitudinal coverage alternating between 83 N-52 S and 83 S-52 N approximately every 60 days, as shown in Figure 3.2. This is a result of a yaw maneuver which has to be performed as the orbit precesses in order to keep the SABER side of the TIMED spacecraft away from direct sun light. SABER measures about 1400 radiance profiles per channel per day. For a complete description of the SABER instrument, tests, and calibration see (Russell III et al. 1999). Table 3.1 summarizes the science motivation as well as the channel pass bands for each of the SABER channels.