Retrieval of carbon dioxide concentration from AIRS thermal emission data

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1 Retrieval of carbon dioxide concentration from AIRS thermal emission data Research Proposition Report Daniel Feldman Advisor: Yuk Yung December 1, 2003 Abstract: The advent of high-resolution infrared sounders is expected to increase the scientific community s knowledge of the spatial distribution of carbon dioxide (CO 2 ) and other trace gases. Preliminary work has been started addressing CO 2 retrieval using data from Atmospheric Infrared Sounder (AIRS). Computer simulations were performed to test the numerical stability of the retrieval algorithm and also to test the algorithm s sensitivity to instrument noise and a priori knowledge of the atmospheric state. With temperature, water vapor, and ozone constrained and cloud-cover and aerosols eliminated from the computer calculations, it was found that CO 2 profile retrieval from a single spectrum is numerically stable and is only moderately sensitive to the instrument noise. In addition, the retrieval was found to be insensitive to the a priori knowledge of the system within the limits prescribed by NOAA s Climate Modeling Diagnostic Laboratory. Finally, it was found that the CO 2 retrieval was accurate to within 2 parts per million by volume in the troposphere and lower stratosphere when the retrievals from several spectra were averaged together. This suggests that it may be possible to use AIRS data to compile a spatial distribution of monthly-averaged CO 2 concentrations. The accuracy of this data will be only slightly inferior to that of the Orbiting Carbon Observatory.

2 Introduction: Understanding the global spatial distribution of carbon dioxide is of critical importance for predicting and perhaps mitigating the extent and severity of global warming. The primary motivation behind the drive to understand the spatial distribution of carbon dioxide (CO 2 ) is the identification of its sources and sinks, which will ultimately determine the course of climate change in the coming decades 1. At present, model estimates of climate change predict between 1.5 and 5.5 C increase at the end of this century and there are serious policy implications contingent upon the severity of the incipient global warming 2. Unfortunately, the current network of CO 2 measurements from ground and aircraft data maintained by the NOAA s Climate Monitoring Diagnostics Laboratory (CMDL) is insufficient to resolve CO 2 sources and sinks with great confidence 3. As such, the tabulation of anthropogenic carbon outputs and natural sinks cannot account for the missing carbon sink of 1.7 Pg/yr, which represents over one-sixth of the anthropogenic output 4. It has been proposed that this discrepancy can be addressed with satellite measurements of sufficient accuracy. Therein lies the drive for the Orbiting Carbon Observatory (OCO) instrument which aims to measure CO 2 column abundance to within 0.3% of the true value 5. Nevertheless, it may be possible to use data from the already operational Atmospheric Infrared Sounder (AIRS) instrument to provide auxiliary CO 2 data to assist in the algorithm improvement for the OCO and to create an independent source of CO 2 data. The AIRS instrument has been operational since November, 2002 and was primarily designed as a satellite to improve weather forecasting 6. The spacecraft travels in a sunsynchronous orbit with an orbital period of around 2 hours at a nominal height of 705 km and uses a highly-advanced grating spectrometer to acquire 2378 spectral channels in the thermal infrared ranging from 650 to 2500 cm -1 (3.7 to 15.4 mm). The instrument also has a 4-channel visible/near-infrared module that senses between 0.4 and 1.0 mm. The infrared channels have a spatial resolution of 13.5 km and the device scans between ±49.5 from nadir every 2.67 seconds. Since launch, AIRS has been performing better than expected with a noise effective dt (NeDT) of less than 0.35 K 7. It has previously been suggested that spectral data from the AIRS instrument could be used to retrieve CO 2 abundances 8, although the accuracy will be less than that of the OCO instrument. One of the most challenging problems facing the OCO project is the development of a sufficiently accurate algorithm for retrieving CO 2 concentrations from radiance measurements. A proper AIRS retrieval faces a similar problem, but the OCO instrument is prone to several problems which AIRS will not face. Bi-directional surface reflection, polarization complicate the retrieval problem 9, and aerosol contamination are major obstacles to the mission s success. Despite the use of data-parsing for cloudy pixels, the scattering by aerosols and thin-cirrus clouds add substantial measurement noise and reduce the accuracy of the retrieval 10. Due to the nature of the bands being measured, OCO is more sensitive to the lower troposphere, while AIRS gathers more information in the middle troposphere and lower stratosphere. The issue of aerosol contamination will still present a challenge to the AIRS data analysis, but it is still possible to analyze cloud-free spectra and it may even be possible to characterize aerosol scattering and correct for it 11. Background: Inverting for the spatial distribution of a gas in the atmosphere from remote sensing measurements is a very challenging task and has been the subject of recent research 12. Atmospheric radiation calculations begin with the equation of radiative transfer: 2

3 m di ( n ) = I(n) - B(n) (1) dt n Measurements made from a satellite platform in a non-scattering atmosphere can be described by integrating the equation (1) from the surface to the satellite s orbiting height as shown in equation (2): I TOA (m,n) = B s (q(0),n)t n (0,z,m) + Ê = B s (q(0),n)exp - t (0, ) ˆ n Á + Ë m Ú 0 Ú0 t n (0, ) B(q(z'),n) T n (z',0,m) z' dz' B(q(t n '),n)k( t n (0, ),t n ',m)dt n ' (2) where I TOA (m,n) is the radiance at wavenumber n measured by the satellite at solar zenith angle m, B s (q(0),n) is the Plank emission at wavenumber n from the surface at temperature q(0), T n (z',z,m) is the transmission between z and z at wavenumber n and solar zenith angle m, t n (0, ) is the optical depth from the surface to the top of the atmosphere at wavenumber n, and B(q(t n '),n) is the Planck emission at wavenumber n from the layer at which the optical depth from the top of the atmosphere to z the equals t n ' and the temperature equals q. K( t n (z, ),t n ',m) is the kernel of the emission integral in optical depth space and is represented by equation (3): ( ) = 1 m exp Ê - ( t '-t (z, )) n n Á K t n (z, ),t n ',m Ë m ˆ (3) It will be shown below that K is central to real-world retrieval calculations because it describes the difference in I TOA (m,n) with different optical depth. If the atmospheric temperature profile were known exactly, it would be possible to invert equation (2) to find the gas concentrations of species that absorb at the wavelength of interest. Unfortunately, equation (2) is ill-conditioned and therefore difficult to solve. The retrieval of absorbing species using a matrix inversion is extraordinarily sensitive to measurement noise 13. In addition, the measurement space which is represented as a vector of radiance values that define the spectrum, is circumscribed by a state space that is not unique, so a correct retrieval is not guaranteed. Nevertheless, it is possible to use ancillary data and a careful analysis of the information content of the measurements to obtain a reasonable profile. Several techniques have been developed to address the difficulties associated with atmospheric remote sensing retrieval 14. A general representation of the inverse problem in the absence of error is shown by equation (4): y = F(x) (4) where y is the measurement and F(x) represents a hypothetical forward model acting on the state x. In general, F(x) is a nonlinear function of x, but can be linearized in the vicinity of a state x o as shown in equation (5): 3

4 y - y o = K( x - x o ) (5) where K = df(x) dx (6) It can be shown that the K in equation (6) is equivalent to kernel as defined in equation (3). For the purposes of numeric calculations, it is convenient to represent the equation (5) discretely with an ensemble of homogeneous layers which transforms x into vector x, y into vector y, and F(x) into vector F(x), and K into K with elements defined by equation (7): K jk = F j (x) x k (7) K represents the linearization of the discretized forward model and is for historical reasons called the weighting function matrix. Problems associated with this linearization will be discussed below. Because measurement error is inevitable, a discretized form of equation (5) is more realistically described by equation (8) y - y o = K( x - x o ) + e (8) where e is the measurement error vector. Various methods have been developed to solve for x in terms of y, K, and e depending on the conditioning of the problem 15. The weighting function matrix rows are of particular importance to profile retrieval methods using wavelengths for which the transmittance between the surface and the satellite is very low. Under these circumstances, the altitude of the peak of the weighting function row for a certain channel corresponds to that layer which contributes most significantly to the channel radiance measured at the top of the atmosphere. Because these weighting functions exhibit relatively narrow peaks, the satellite measurements are almost entirely determined by the contribution from one or two layers in the atmosphere. Figure (1) shows a schematic of how different information is incorporated into a profile retrieval. For a linear radiative transfer problem, it is possible to use a Bayesian approach to retrieve the state vector from the measurement vector. Recall that Bayes theorem states that: P(y x)p(x) P(x y) = P(y) (9) where the term on the left-hand side describes the probability of having a state vector x given the measurement vector y, P(y x) represents the probability of having the measurement vector y given a known state vector x, P(y) describes the statistics of the measurement, and P(x) describes the statistics of the state. P(y) and P(x) are determined in part by covariance matrices which describe both the variability in the ensemble of data in y and x. Assuming that the measurement error values exhibit Gaussian statistics, a straightforward retrieval is possible with the linear case 4

5 which maximizes P(x y). It can be shown that the retrieved state vector ˆ x can be calculated using equation (10): ˆ x = (K T S ē 1 K + S ā 1 ) -1 (K T S ē 1 y + S ā 1 x a ) (10) where S e is the error covariance matrix of the measurement vector, S a is the a priori covariance matrix of the state vector, x a is the a priori estimate of the state vector, and the (-1) superscript denotes a matrix inverse and the (T) superscript denotes a matrix transpose. However, there are more complicated and accurate approaches to the linear retrieval where the problem is more ill-conditioned, but many rely on calculating the most-likely state vector given the measurements and other ancillary data 16. Unfortunately, the hypothetical forward model of equation (4), which exactly describes the radiative transfer equation, is not possible, and thus the error in the forward model must be acknowledged along with the recognition that the forward model parameters are not always tuned properly. In addition, the discretization of the radiative transfer is imperfect and carries with it some associated error. Therefore, a complete retrieval must account for the smoothing error associated with discretization, the forward model parameter error, the forward model error, and the measurement error 17. For the purposes of this study, only measurement error was analyzed, but the other 3 sources of retrieval error will need to be incorporated in future studies. Another source of difficulty in the retrieval problem arises from the assumption of linearity. In thermal emission radiative transfer problems, this assumption unfortunately is unwarranted. The Planck function is highly-nonlinear in the infrared and the transmittance is low so the forward model linearization does not hold for mapping the a priori estimate of the state vector to the retrieved state vector. To address the nonlinear problem, several ad hoc methods have been developed, though all of them exhibit deficiencies either in their requirement for massive computation or in their inability to retrieve a state vector that is satisfactory 18. These techniques start with an a priori estimate of the state vector and then iterate a recursive zerofinding method by adding an update value D to find a state that closely simulates the measurement vector. This D is a function of the current state x i, and x a, S a, S e, y, and F(x i ). The Gauss-Newton method is a multi-dimensional approach using a least-squares zero-finding and is the standard routine. Unfortunately, it can take many iterations to converge satisfactorily if at all. There is also the steepest-descent method which chooses D such that the difference between the measurement vector and the measurement vector calculated from the forward model is minimized. The general method is shown in equation (11) with the Gauss-Newton method shown in equation (12) and steepest descent method shown in equation (13): ( ) (11) [ ] -1 F(x i ) (12) [ ] (13) x i+1 = x i + D x i,x a,s a,s e,y,f(x i ) x i+1 = x i - x F(x i ) x i+1 = x i + ak T y - F(x i ) where a is a scaling parameter that varies from problem to problem. Several other methods utilize Gauss-Newton and steepest-descent including the Levenberg-Marquardt revised method (L-Mr). This technique combines the Gauss-Newton and steepest-descent methods using a hybridization parameter g i. As g i approaches 0, the L-Mr approaches the Gauss-Newton method and as g i gets very large, the L-Mr approaches the method of steepest-descent. By judiciously 5

6 updating the g i parameter, it is possible to approach a solution without excessive iteration and with reasonable accuracy 19. The original Levenberg-Marquardt method was prone to numerical instability, so a revised method was developed 20 and is described by equation (14): x i+1 = x i + (( 1+ g i )S 1 ā + K T i S 1 ē K i ) -1 * ( K T i S 1 ē K i *( y - F( x i )) - S 1 ā *( x i - x a )) (14) where the L-Mr allows for multiple updates to the weighting function matrix. In practice, however, the weighting function matrix does not need to be updated at every iteration. In order to test the improvement that each iteration of the L-Mr adds to the previous estimate, a cost-function test is commonly used. The cost-function, or c 2 test, is a measure of the probability that the measurement vector from the current state of the L-Mr is not significantly different from the actual measurement. The retrieval algorithm seeks to minimize equation (15): c 2 y = [ y - F(x i )] T S 1 ē [ y - F(x i )] (15) which is a more generalized form of the signal-to-noise ratio. Once the retrieval has minimized the c 2 y value, the quality of the retrieval must be analyzed. This is accomplished with two tests. (1.) The c 2 y value of the final retrieved state must be less than that of the a priori estimate. This question asks whether the retrieved state is more indicative of the measurements than the a priori state. (2.) The number of degrees of freedom of the retrieved state vector must be less than the number of degrees of freedom of the a priori estimate. The second question asks whether the information from the retrieval algorithm has been effectively incorporated into the updated state 21. Consequently, the quantity in equation (16) must be less than the quantity in equation (17) in order to have any confidence in the retrieval: c 2 x = [x a - x ˆ ] T S -1 a [x a - x ˆ ] (16) df x = E [x a - x ˆ ] T S -1 a [x a - x ˆ ] ( ) (17) Data and Methods For the purposes of this simulation, the L-Mr was used in the context of a customized Matlab retrieval code. The forward model used in this exercise was developed by Scott Hannon of the University of Maryland-Baltimore County in collaboration with AIRS team member Larrabee Strow 22. The Stand-Alone Radiative Transfer Algorithm (SARTA) v1.03 is an optimized Fortran-77 code which uses a 98-layer regression profile using inputs of temperature and 5 variable gases: water, ozone, carbon monoxide, methane, and CO 2. With this information on the atmospheric state, the program calculates and convolves monochromatic transmittances and combines them for the effective layer transmittances. Finally, it performs another regression on the effective layer transmittances to obtain transmittance coefficients and predictors 23. This data can be used for fast forward-model calculations and the model performs well when compared with line-by-line codes such as the Discrete Ordinate Radiative Transfer (DISORT) and the kcompressed Atmospheric Radiative Transfer Algorithm (kcarta) 24. Currently, the model does not compute scattering so only cloud-free AIRS data can be analyzed. Carbon dioxide weighting function matrix columns were calculated according to equation (7) by perturbing each element of the state vector by 5% and computing a finite difference 6

7 derivative using the perturbed and original spectra. The noise covariance matrix was diagonal with each diagonal entry corresponding to the entire band s NeDT. In order to calculate the a priori covariance matrix, the code used a simple scheme suggested by Rodgers [Ch. 2, 2000] which assumed that the CO 2 volume mixing ratio at each level was known to within 30 ppmv. With the diagonal elements of the matrix known, the off-diagonal row elements were filled using a Markov description of the atmospheric profile as shown by equation (17): Ê S ai, j = S ai,i *exp - Dz ˆ i, j Á (18) Ë H where S ai, j is the individual matrix element, S ai,i is the variance of the state vector entry at level i, H is the scale height for the atmosphere, and Dz i, j is the difference in altitude between the i th and j th levels. Originally all 2378 channels were utilized in the retrieval, but this approach left the retrieval problem quite ill-conditioned because most AIRS channels are not sensitive to atmospheric CO 2 concentration. Consequently, only 43 channels were utilized in this study, and the selection of these channels was based on the Optimal Sensitivity Profile which chooses channels based on their sensitivity to CO 2 concentration, the altitude location of their weighting function peaks, and the noise associated with each channel 25. Only CO 2 was retrieved in this exercise, though it is not difficult to modify the algorithm to retrieve other state vector elements including temperature, water vapor, and ozone. This will significantly add to the complexity of the retrieval, but these controlled exercise were designed to test the retrieval algorithm s performance for a simple case. In order to test the ability of the model to retrieve CO 2, the code started with a initial concentration profile of 370 ppmv uniformally distributed. This value is approximately the atmospheric mixing ratio in the present atmosphere. The spectrum was calculated using the forward model and then a vector of normally-distributed random numbers with zero mean and a variance equal to the NeDT channel values was added to the true spectrum. Finally, the retrieval algorithm was run using a uniform a priori estimate of the state vector and results were compared to the true state vector using the c 2 criteria outlined in equations (15), (16), and (17). Results: The first tests that were run with the retrieval algorithm focused on examining the sensitivity of the retrieval to the a priori estimate of the state vector. A true spectrum was created from the true state vector and then perturbed with NeDT noise. As seen in Figure (2), there was very little variation in the retrieval output where the a priori estimate was varied between 360 and 380 ppmv. Although those a priori estimates are excessively erroneous as compared to the CMDL data set 26, the retrieval algorithm was still able to obtain an appropriate state vector. The next test of the retrieval algorithm involved arithmetically averaging the retrievals of 20 different spectra that were created from the true state vector and modified with NeDT noise. The normally-distributed noise values will not skew an ensemble of measurement vectors, but the nonlinear nature of the retrieval has the potential to create biases that would skew the average of the retrieved state vectors. Fortunately, as seen in Figure (3), the retrieval algorithm performed well and the average of retrievals deviated from the true state vector by less than 1 7

8 percent. Therefore, it may be possible to retrieve robust CO 2 profiles by taking multiple AIRS spectra and averaging them over the same pixel. As shown in Figures (2) and (3), the retrieved profile exhibits a great deal of variability from one altitude level to the next, but this is largely an artifact of the retrieval algorithm. In order to address this issue, one can average layers together; while this approach reduces vertical resolution, it makes the retrieval problem better-conditioned by properly constraining the retrieval to fit the measurements and a priori knowledge. Therefore, in the final test, the 98 original layers upon which the forward model operates were combined so that various layers were simply averaged together. The retrieval algorithm was tested to find the number of layers for which the c 2 value of the measurement vector decreased most substantially relative to its initial value. The algorithm worked best where it used 44 layers and this wasaccomplished by averaging every 2 levels and then averaging in the top and bottom remainder levels. The results of the retrieval can be seen in Figures (4) and (5) and although the presence of noise in the measurement signal still dramatically affects the retrieval, there is a marked improvement from the a priori estimate to the retrieved state vector. Discussion: The retrieval tests done over the course of this exercise were limited in scope and meant to test the feasibility of obtaining CO 2 profiles from AIRS data in the absence of other complicating factors. It was found that the retrieval algorithm worked adequately to recover a CO 2 profile from a single spectrum and that the retrieval was insensitive to the a priori estimate. In addition, the averaging of several spectra allowed for a retrieval that was accurate to within 2 ppmv accuracy in the troposphere and lower stratosphere. However, there is more work to be done on the retrieval code and many more complicating factors to consider. First, the code performed adequately, but can be improved by averaging more of the upper layers together. Currently, there are 35 layers in the stratosphere and 9 layers in the mesosphere. Over most of these layers, the weighting function coverage is very poor and as such, most of these layers will have to be averaged together to improve the retrieval code s performance. Also, the a priori covariance matrix was not realistically constructed. A 30 ppmv CO 2 variance may be valid at lower altitudes, but the CO 2 variance is much less for the upper layers, and the off-diagonal entries of the a priori covariance matrix will require modification. The forward model predictor data set, which is used to perform quick calculations of layer transmittances, also needs to be modified to test various atmospheric conditions. The predictor data set that was used was calculated for winter-time conditions at sub-tropical latitudes which is insufficient for a global retrieval undertaking. Moreover, the forward model is currently designed for clear-sky calculations and does not account for aerosol scattering. A method which is not overtly computationally-intensive will need to be developed to address this scattering, or the cloudy data will have to excluded from the global retrieval protocol. Finally, the constraints on temperature, water vapor, and ozone that were imposed during this exercise will need to be partially- or fully-lifted in order to perform a robust CO 2 retrieval. That is, the present exercise assumed that temperature, water vapor, and ozone were known with exceptional accuracy, but these assumptions are unwarranted. While the AIRS instrument is well-designed to retrieve water vapor concentrations, all measurements are exquisitely sensitive to temperature, and it remains to be seen whether the L-Mr retrieval becomes mathematically unstable when it attempts to retrieve multiple state components simultaneously. 8

9 Conclusion: With the successful deployment of the AIRS instrument in 2002, there is great potential for improving upon the current understanding of the spatial distribution of CO 2 in the terrestrial atmosphere. However, because AIRS was not specifically designed to measure CO 2 and because CO 2 retrieval from a satellite platform is a non-trivial task, the task for converting satellite measurements to gas concentration profile remains a challenge. This exercise demonstrated that CO 2 retrieval using AIRS data is feasible given the proper constraints, that the retrieval is insensitive to the a priori estimate, and that proper averaging of satellite measurement can be used obtain a CO 2 profile to within 2 ppmv. Nevertheless, before AIRS spectra are analyzed for CO 2, this algorithm must be greatly expanded to account for many of the complicating factors listed discussed above. Figures: Figure (1): Schematic of simple gas profile layer retrieval shown at far right from a single channel satellite measurement (I TOA (m,n)), temperature profile (q(z)), and channel weighting function (K n (z)). Only where the weighting function peaks is there information on CO 2 concentration. Figure (2): 98-layer retrieval comparison where different a priori estimates were used (color denotes a priori value) 9

10 Figure (3): Result of averaging layer retrievals from noisy spectra compared to the true value (a) (b) Figure (4) (a): Single 44-layer retrieval in the absence of noise. Figure (4) (b): Single 44-layer retrieval in the presence of noise. Acknowledgements: The author would like to thank the following individuals for their help: Professor Yuk Yung of Caltech; Luke Huang of Caltech; Run-Lie Shia of Caltech; Vijay Natraj of Caltech; Luke Chen of JPL; and Dr. Clive Rodgers of the University of Oxford. 10

11 References: 1 IPCC, Third Assessment Report Climate Change 2001, IPCC/WMO/UNEP, Cambridge University Press, Ibid. 3 Rayner, P. J. et al. The utility of remotely sensed CO 2 concentration data in surface source inversions. Geophysical Research Letters. vol : Schlesinger, W.H. Biogeochemistry: An analysis of global change. San Diego: Academic Press, Ch. 11: Orbiting Carbon Observatory Products: Aumann, H. H., et al. AIRS/AMSU/HSB on the Aqua Mission: Design, Science Objectives, Data Products, and Processing Systems. IEEE Transactions on Geoscience and Remote Sensing. vol. 41, no : February, AIRS Instrument Specifications Aumann, H. H., et al. AIRS/AMSU/HSB on the Aqua Mission: Design, Science Objectives, Data Products, and Processing Systems. IEEE Transactions on Geoscience and Remote Sensing. vol. 41, no : February, Rayner, P. J. et al. Global observations of the carbon budget: Initial assessment of the impact of satellite orbit, scan geometry, and cloud on measuring CO2 from space. Journal of Geophysical Research, vol. 107, no. D21. ACH 2(1-7): Stephens, G.L. et al. Molecular Line Absorption in a Scattering Atmosphere. Part I: Theory. Journal of Atmospheric Science, vol. 57, no : Huang, X. L. et al. High-resolution thermal IR detection and characterization of cirrus from the retrieval point of view. In press: Applied Optics, Rodgers, C.D. Inverse Methods for Atmospheric Sounding: Theory and Practice. Singapore: World Scientific, Inc., Ch. 1: Milman, A.S. Mathematical Principles of Remote Sensing: Making Inferences from Noisy Data. Chelea, MI: Sleeping Bear Press, Ch : Rodgers, C.D. Inverse Methods for Atmospheric Sounding: Theory and Practice. Singapore: World Scientific, Inc., Ch. 1: Ibid., Ch Ibid., Ch Ibid., Ch Ibid., Ch Ibid.. Ch Marquardt, D. "An Algorithm for Least-Squares Estimation of Nonlinear Parameters." SIAM Journal of Applied Mathematics. vol : Fletcher, R. Practical Methods of Optimization. Chichester, New York: John Wiley & Sons, Rodgers, C.D. Inverse Methods for Atmospheric Sounding: Theory and Practice. Singapore: World Scientific, Inc., Ch. 12: Strow, L. L. et al. An Overview of the AIRS Radiative Transfer Model. IEEE Transactions on Geoscience and Remote Sensing. vol. 41, no : February, Ibid. 24 Ibid. 25 Crevoisier, C., et al. AIRS channel selection for CO 2 and other trace-gas retrievals. Quarterly Journal of the Royal Meteorological Society. vol : GLOBALVIEW-CO2: Cooperative Atmospheric Data Integration Project - Carbon Dioxide. CD-ROM, NOAA CMDL, Boulder, Colorado [Also available on Internet via anonymous FTP to ftp.cmdl.noaa.gov, Path: ccg/co2/globalview],