Retrieval of CO 2 Column Abundances from Near- Infrared Spectroscopic Measurements. A Candidacy Report by Vijay Natraj

Transcription

1 Retrieval of CO 2 Colun Abundances fro Near- Infrared Spectroscopic Measureents A Candidacy Report by Vijay Natraj

2 Abstract A retrieval algorith was developed to obtain the colun averaged dry air ixing ratio of CO 2 (X CO2 ) fro spectroscopic easureents of absorption in three near-infrared (NIR) bands. The radiative transfer coputations were done using the discrete ordinate ethod, and optial estiation theory was used for the inversion. To test the algorith, colun O 2 was retrieved fro easureents of absorption in the O 2 A-band over sea surface. After correcting for instruent calibration effects, excellent fit was obtained with the easured spectra. The colun O 2 was retrieved with an error of around 1%. With spectral averaging using ultiple soundings, it is possible to reduce the error to 0.1%. This suggests the feasibility of retrieving X CO2 with precisions better than 0.3%.

3 Table of Contents 1 Introduction Proble Description Fundaentals of Atospheric Radiative Transfer Spectroscopy and the Underlying Physics of the Measureents 10 2 Retrieval Strategy Discrete Ordinate Method Convolution Inverse Method 20 3 Case Study: Retrieval of Aircraft Measureents 23 4 Future Work 31 5 Conclusion 33 Acknowledgents 34 References 34

4 1. Introduction Atospheric CO 2 is an efficient greenhouse gas. The CO 2 concentration has increased fro 280 to 370 parts per illion (pp) since the beginning of the industrial era [IPCC, 1996; Schnell, 2001]. There is growing apprehension that this will adversely alter the global cliate [Cicerone, 2001; IPCC, 1996]. Measureents fro a global network of surface stations [CDIAC, 2001; Schnell, 2001] indicate that the biosphere and oceans have absorbed alost half of the carbon eitted during the past 20 years. However, the nature and geographic distribution of these CO 2 sinks are not adequately understood, precluding accurate predictions of their response to future cliate or land use changes [Cicerone, 2001; IPCC, 1996]. One ajor concern is that these sinks ay saturate in the future, accelerating the build-up of atospheric CO 2 [Cox, 2000; Friedlingstein, 2001]. Existing odels and easureents also fail to explain why the atospheric CO 2 concentration increases vary fro 1 to 6 gigatons of carbon (GtC) per year in response to steadily rising eission rates (see Fig. 1) [Conway, 1994; Frolking, 1996; Houghton, 2000; Keeling, 1995; Le Quere, 2000; Lee, 1998; Randerson, 1999; Randerson, 1997]. The principal shortcoing of the ground-based network is its sparse spatial sapling. Accurate, tie-dependent, spatially-resolved, global aps of the colun-averaged CO 2 dry-air ole fraction (X CO2 ) will draatically iprove our understanding of its surface sources and sinks. Modelling studies [Rayner, 2001] confir that source-sink inversion algoriths eploying global, space-based easureents of X CO2 will outperfor those using the existing GLOBALVIEW-CO 2 (GV-CO 2 ) data if the space-based easureents 1

5 have accuracies better than 2.5 pp. Figs. 2 and 3 illustrate the resulting iproveents in retrieved carbon flux errors on global and continental (~ k 2 ) scales, respectively, for X CO2 errors of 1 pp (0.3%). Fig. 1: (A) 40-year history of atospheric CO 2 buildup. (B) Observed variations in annual atospheric CO 2 accuulation ( CO 2 ) copared with fossil fuel eissions [CDIAC, 2001; Schnell, 2001]. Significant changes in carbon sequestration occur on annual tie scales. Fig. 2: Error in retrieved global carbon flux (GtC/yr) vs. space-based X CO2 easureent accuracy [Rayner, 2001]. The OCO 1-pp accuracy (circle) is needed to outperfor the GV-CO 2 network (dashed line). 2

6 1.2 A) Flask Flux Retrieval Errors (GtC/yr/region) B) Satellite Fig. 3: Global aps of carbon flux errors for 26 continent/ocean-basin-sized zones retrieved fro inversion studies. A) Studies using data fro the 56 GV-CO 2 stations produce flux residuals that exceed 1 GtC/yr in soe zones. B) Inversion tests using global X CO2 pseudo-data with 1 pp accuracy reduce the flux errors to <0.5 GtC/yr/zone [Rayner, 2001]. The Orbiting Carbon Observatory (OCO) ission was thus set up to ake the first spacebased easureents of atospheric CO 2 with the accuracy, precision, resolution, and coverage needed to characterize the geographic distribution of CO 2 sources and sinks and quantify their variability. The OCO satellite will carry three high-resolution, grating spectroeters, designed to easure CO 2 and O 2 near-infrared (NIR) absorptions fro reflected sunlight. Reote sensing retrieval algoriths will process these data to yield estiates of X CO2 with accuracies near 0.3%. Cheical transport odels will use OCO X CO2 data and other easureents to retrieve the spatial distribution of CO 2 sources and sinks on regional scales over two annual cycles. 3

7 1.1 Proble Description The ai of this work is to develop and test algoriths for the retrieval of X CO2 fro spectroetric easureents in three NIR bands. Such a retrieval is a highly nontrivial proble and is the subject of extensive research. Any retrieval proble can be broadly divided into two ain coponents, viz., a forward odel and an inverse ethod. The forward odel is an approxiate schee to describe the radiative transfer in the atosphere. It involves coputation of the absorption coefficients, and using these along with key paraeters such as the scattering and absorption optical depths, the single scattering albedo and the surface reflectance to copute the radiances at specific solar zenith and viewing angles. These ters will be defined in section 1.2. The nuerical schee used for the forward odel will be described in section 2.1. The coputed radiances have to pass through a convolution function to siulate the instruent response. This will be discussed in further detail in section 2.2. The idea behind retrieval is to copare the easured spectru with the coputed spectru, and iteratively iprove the coputed spectru to best atch the observed spectru. The standard ethod used is the Optial Estiation Theory, which was propounded by Rodgers [Rodgers, 2000] and will be considered at length in section

8 1.2 Fundaentals of Atospheric Radiative Transfer [Goody, 1989; Liou, 2002] The specific intensity of radiation, or radiance, denoted by I, is the ain quantitative characteristic of a radiation field. It depends on the frequency of radiation υ, the coordinates r of the point under consideration, and the direction n of the ray. Physically, it refers to the flux of energy at point r in direction n per solid angle per frequency. The dependence of the radiance upon these values is usually denoted as I υ (r,n), where the subscript is usually dropped for econoy of notation. In atospheric transfer probles, it is convenient to use a diensionless quantity called the optical depth, rather than the actual depth. The optical depth τ can be defined as τ = σ n dl (1) where σ is the cross-section of an individual olecule, n is the nuber density of olecules and l is the path length. The product of σ and n is called the extinction coefficient k. The optical depth easures the aount of extinction a bea of light experiences travelling between two points. For historical reasons, the optical depth is defined to be 0 at the top of the atosphere and increases as we go towards the surface. 5

9 The fundaental equation of radiative transfer, which was first postulated by Schwarzschild, is as follows (when the atosphere is approxiated as being vertically stratified and horizontally hoogeneous; coonly called plane-parallel): di( τ, µ ) µ = I( τ, µ ) J ( τ, µ ) (2) dτ where µ is the cosine of the zenith angle, and J is the source function. The first ter on the right hand side represents attenuation due to absorption and scattering of a radiance strea as it propagates through the atosphere, and the source function represents the strengthening of the radiance strea. For solar radiation it arises fro photons scattered in the path fro all other directions. The presence of this scattering source ter ensures that the radiation field is no longer erely a function of local sources and sinks, but of the entire atospheric radiation field and of its transport over large distances. In practice this akes the solution uch ore difficult to obtain. If there is no source function, then the above equation reduces to the failiar Beer s law. Solution of equation (2) has been the subject of intense research over the years, starting fro Chandrasekhar s classical solution [Chandrasekhar, 1950] to various nuerical ethods [van de Hulst, 1980]. The technique adopted in this work will be explained in section 2.1. However, a discussion of the fundaental quantities involved in any radiative transfer proble is in order. 6

10 The four ain quantities influencing the radiation field are the optical depth, the surface reflectance, the single scattering albedo and the phase function. The surface reflectance, denoted by a, is the ratio of the intensity reflected fro the surface to that incident on it. It is a function of wavelength. A further coplication arises because it can also depend on the zenith and aziuthal angles. This dependence is called the bidirectional reflectance distribution function (BRDF), and will be discussed further in section 3. The single-scattering albedo ω 0 represents the fraction of energy scattered to that reoved fro the radiance strea under consideration. For conservative scattering, ω 0 = 1, and for pure absorption, ω 0 = 0. In an inhoogeneous atosphere, the single scattering albedo is a function of optical depth. When light strikes a particle with an index of refraction different fro its environent, the light is refracted. The angle at which the light is bent is a function of the size and shape of the particle as well as the wavelength of the incident light and the incidence angle of the light. In general, each particle will have a different scattering profile. This scattering profile is called the phase function. For siplicity an average phase function which adequately describes the ost iportant features of the scattering process is used. This average phase function is further constrained by assuing that the probability of scattering fro one direction into another is a function only of the angle between the two directions. The phase function P then describes the aount of light scattered fro the incident direction into the scattered direction. 7

11 Using spherical trigonoetry cosine laws, the scattering angle Θ can be obtained in ters of the directions of the incident and scattered radiation: 2 2 cosθ = µ µ ' + 1 µ 1 µ ' cos( φ φ') (3) where µ and µ are the cosines of the incident and scattered zenith angles, and φ and φ ' are the incident and scattered aziuthal angles respectively. The convention is adopted that µ > 0 refers to upward radiance streas and µ < 0 refers to downward radiance streas. There are a nuber of ways in which the phase function ay be noralised, but the ost natural is that used by the astrophysicists. They treat the phase function as a probability distribution; consequently, their noralisation condition requires the integral of the phase function over all angles to equal unity. π 2π P (cosθ)sin ΘdΘ = 1 (4) 0 The siplest phase function is the isotropic phase function 1 P (cos Θ) = (5) 4π 8

12 The factor of 1/4π results fro the noralisation condition and the fact that there are 4π steradians in a sphere. Particles that are very sall in coparison to the wavelength of the radiation (such as oxygen and nitrogen olecules) exhibit a type of scattering called Rayleigh scattering. The Rayleigh phase function is given by the following relation: 3 P (cosθ) = (1 + cos 2 Θ) (6) 4 If the phase function is not isotropic, then a paraeter called the asyetry paraeter is used to describe the degree of anisotropy of the phase function. This paraeter is often denoted by g and is defined as the integral over all angles of the phase function ultiplied by the cosine of the angle. In other words, it is the first oent of the phase function. A single-paraeter analytic for for an anisotropic, scattering phase function is the Henyey-Greenstein phase function: 2 1 g P (cosθ) = (7) 2 3 / 2 (1 + g 2g cosθ) 9

13 1.3 Spectroscopy and the Underlying Physics of the Measureents High-resolution (R = λ λ ~ 20,000) NIR spectroetry of coplete rotation-vibration bands was chosen to obtain X CO2 [Kuang, 2002]. This technique has the advantage that, at NIR wavelengths, theral eission fro the surface, the atosphere, and the instruent are all negligible copared to reflected sunlight, siplifying radiative transfer and instruent calibration. Also, absorption of sunlight by the NIR CO 2 bands is ost sensitive to the CO 2 concentration in the boundary layer, where the effects of sources and sinks are ost readily detected. Theral infrared soundings can yield observations of CO 2 fro space, but these easureents have liited sensitivity to CO 2 concentrations near the surface (see Fig. 4) and their accuracy (~1%) is insufficient to eet the 0.3% requireent described earlier. Solar occultations also lack the ability to sense the boundary layer. LIDAR has been advocated for onitoring CO 2 fro space, but even the best ground-based trace gas LIDARS have accuracies around 5%. Fig. 4: A coparison of the averaging kernels for colun CO 2 soundings using NIR absorption of reflected sunlight and theral IR eission. NIR easureents are uch ore sensitive to surface phenoena. 10

14 The CO 2 colun abundance will be retrieved fro the absorption band centred at 1.61 µ (see Fig. 5). This spectral region (called the weak CO 2 band in this report) is virtually free of absorption by other gases, siplifying CO 2 retrievals. Also, ost of the spectral lines are not saturated, such that their absorption increases alost linearly with the CO 2 abundance and path length. However, the absorption also depends on factors unrelated to the CO 2 abundance (e.g. clouds, aerosols, atospheric teperature profile, surface pressure). Measureents of a reference gas, whose concentration is unifor, constant, and well known, are needed to convert the observed CO 2 colun abundance to the dry air ole fraction, X CO2. Molecular oxygen is the best available candidate. X CO2 can be derived fro soundings of the total colun abundances of CO 2 and O 2 as follows: colun CO2 X CO = (8) colun O2 where is the O 2 ole fraction. Measureent biases produce systeatic offsets in X CO2. These biases can be introduced by the instruent (e.g. uncertainties in the zero offset and instruent lineshape, stray light containation, detector nonlinearity), surface and atospheric properties (uncertainties in cloud and aerosol opacity, teperature, surface pressure, surface reflectance, topography, etc.), and/or incopleteness and errors in the existing line list 11

15 database. While the instruent errors can be iniised by proper calibration, the other systeatic errors need to be accounted for. Fortunately, X CO2 can be easured to uch higher precision and accuracy than colun CO 2 alone, because any of the systeatic errors affect the CO 2 and O 2 coluns siilarly (e.g., cloud, aerosol, surface pressure), and therefore cancel in the ratio of these quantities. Aircraft easureents [O'Brien, 1992] have shown that O 2 A-Band easureents can provide colun O 2 (i.e., surface pressure) estiates with accuracies of 1 bar (0.1%). Airborne particles can absorb or scatter sunlight back to space before it traverses the full atospheric colun, precluding true colun CO 2 easureents in regions occupied by opaque clouds. The O 2 A-band is sensitive to clouds and sall aerosol particles [Heidinger, 2000; Stephens, 2000]. The fact that the band contains both weak and strong lines (see Fig. 5) provides additional inforation on the vertical distribution of clouds and aerosols. O 2 A-band easureents are therefore used to characterize the clouds and aerosols in each sounding, so that those with too uch scattering can be rejected. For less opaque soundings (optical depths 0.03 < τ < 0.3), cloud and aerosol scattering can introduce errors in X CO2 by adding uncertainty to the photon path length. However, O 2 A- band observations alone are not adequate for characterizing the scattering by water ice clouds and aerosols in the weak CO 2 band because their optical properties (optical depth, single scattering albedo, and phase function) can vary substantially between the two bands. 12

16 Aerosol effects on the photon path length can be characterised by cobining siultaneous spectroscopic observations fro the O 2 A-band and the CO 2 band near 2.06 µ (called the strong CO 2 band in this report). Unlike the weak CO 2 band, the CO 2 absorption near 2.06 µ is produced by strongly saturated lines that have a relatively weak (square root) dependence on the CO 2 concentration and enhanced sensitivity to clouds and aerosols (see Fig. 5). Measureents of water vapour absorption lines and CO 2 hot bands within the 2.06 µ region also provide explicit observational constraints on these properties. Fig. 5: Siulated atospheric transission (solar zenith angle=40 ) for the three OCO spectroeters (instruent effects included), including all absorbing species in each interval (O 2 -green, CO 2 -black, H 2 O-blue) The spectral range for each band includes the coplete band as well as soe continuu at both ends. By using the entire band, biases due to uncertainties in atospheric teperatures are iniised. The continuu at the band edges provides additional inforation about the wavelength dependent optical properties of the surface reflectance and airborne particles. 13

17 2. Retrieval Strategy The principal characteristics and flow of the X CO2 retrieval algorith are presented scheatically in Fig. 6. For each sounding, the retrieval process begins with an assued environental state, defined by the surface pressure p s, surface reflectance a, vertical teperature profile T(z), ixing ratios of CO 2, water vapor, and other trace gases, [X(z)], and cloud and aerosol optical depth distributions (τ c and τ a, respectively). These paraeters can be initialized fro known cliatology, or fro adjacent retrievals. This inforation is cobined with pre-tabulated, wavelength-dependent gas, aerosol, and cloud optical properties. The gas absorption cross-sections, σ λ (p,t), in the three spectral regions are derived and tabulated as functions of p and T using a line-by-line odel [Meadows, 1996] and spectral line databases such as HITRAN. For clouds and aerosols, the wavelength-dependent optical properties (absorption and scattering cross sections, and phase functions) for liquid water, water ice crystals, and a wide variety of coon aerosol types have been derived for a range of coon size distributions using Mie scattering (liquid water, sulfates), T-Matrix (sall dust), and geoetric optics (cirrus cloud) codes. This tabulated data is cobined with the atospheric state paraeters and inforation about the viewing geoetry and solar zenith angle, and used in a ulti-layer, spectru resolving (line-by-line), ultiple-scattering odel to generate angle-dependent radiance spectra for the three bands [Crisp, 1997]. These synthetic spectra are then processed with a odel that siulates the instruent s spectral response to the incident radiation, and produces results that can be copared directly to the calibrated spectra. 14

18 The X CO2 retrieval odel copares these synthetic radiance spectra to the observations and uses an inverse ethod [16] to odify the assued atospheric state paraeters (p s, T(z), [X(z)], τ c (z) and τ a (z)), and surface reflectances to produce an iproved atch to the easured spectra in all three spectral regions. These revised paraeters are then reinserted into the radiative transfer odel to generate a new synthetic spectru, and this process is repeated until the observed and synthetic spectra achieve the best possible atch. Finally, the X CO2 is coputed fro the best-atch atospheric state. Figure 6: Scheatic of the X CO2 retrieval algorith, showing the ajor coponents (input spectroscopic tables and atospheric state, radiative transfer odel, retrieval odel) and products (final atospheric/surface state) 15

19 2.1 Discrete Ordinate Method The nuerical schee used to solve for the radiance is the Discrete Ordinate Radiative Transfer ethod (DISORT) [Stanes, 1988]. A detailed discussion can be found in the above reference but a brief description will be given here. The phase function P can be expanded in a series of 2M Legendre polynoials P l. The notation is consistent with that in equation (3). 2M 1 l= 0 P( τ,cosθ) = (2l + 1) g l ( τ ) P (cosθ) (9) l where, by virtue of the orthogonality of Legendre polynoials, the expansion coefficients are given by 2 1 g ( τ ) = 1 P (cosθ) P( τ,cosθ) d(cos Θ ) (10) l 1 l Note that g 0 = 1 because the phase function is noralised to unity, and g 1 is the asyetry factor. The addition theore of spherical haronics can be invoked to obtain P( τ,cosθ) = 2M 1 l= 0 (2l + 1) g l ( τ ){ P ( µ ) P ( µ ') + 2 l l l = 1 Λ l ( µ ) Λ l ( µ ')cos ( φ φ')} (11) 16

20 Here, Λ l is a noralised associated Legendre Polynoial defined by ( l )! Λ l = Pl ( µ ) (12) ( l + )! where P (µ) is the associated Legendre polynoial. l The radiance can be expanded in a Fourier cosine series 2M 1 ( τ, µ, φ) = I I ( τ, µ ) cos ( φ0 φ) (13) = 0 where φ 0 is the aziuthal angle of the incident solar bea at the top of the atosphere. Substitution of equations (13) and (11) into the fundaental radiative transfer equation (2) gives 2M independent integro-differential equations, one for each aziuthal intensity coponent: di ( τ, µ ) µ = I ( τ, µ ) J ( τ, µ ) ( = 0,1,...,2M 1) (14) dτ where the source function is given by 17

21 18 ' ) ', ( '),, ( ), ( 1 1 µ µ τ µ µ τ µ τ d I D J = (15) and ') ( ) ( ) ( 1) (2 2 ) ( '),, ( µ µ τ τ ω µ µ τ l M l l l g l D Λ Λ + = = (16) The discrete ordinate approxiation to equation (14) is obtained by approxiating the integral in equation (15) by a quadrature su and thus transforing it into a set of ordinary differential equations: ) 2,..., 1, ( ), ( ), ( ), ( N i J I d di i i i i ± ± = ± = µ τ µ τ τ µ τ µ (17) Each µ i is called a strea, and this is called a 2N-strea approxiation. A double Gaussian quadrature schee is eployed for the streas, with weights w i. Equation (15) now becoes ), ( ),, ( ), ( 0 j j i N j N j j i I D w J µ τ µ µ τ µ τ = = (18) DISORT assues that the ediu consists of L adjacent hoogeneous layers, with the single scattering albedo and phase function constant in each layer. The boundary

22 conditions are the solar flux at the top of the atosphere and the surface reflectance at the botto. 2.2 Convolution The resolution at which we obtain the absorption spectru is deterined by the spectroeter characteristics. The effect of the instruent is to sear the spectru over a finite range of wavelengths. In other words, it converts a spectru fro infinite resolution to a finite resolution. For a grating spectroeter, as used in this project, the radiances coing out of the instruent are related to those entering it by a convolution with the instruent response function. ν + ν / I '( ν ) = I( ν ) F( ν, ν ) dν dν (19) 0 ν ν / where I and I refer to the radiances before and after passing through the instruent respectively, F is the instruent lineshape function (ILS), ν 0 is the centre frequency and ν 0 is the frequency grid spacing. In practice, we cut off the inner integral at a finite distance fro the centre frequency, since the ILS decays as we ove away fro the centre. Typically, the cutoff distance is expressed in ters of a paraeter called the full width at half axiu (FWHM) of the ILS. The FWHM is given by the distance between points on the ILS at which the function reaches half its axiu value. In probles involving ultiple scattering, the ILS used is the lorentzian (characterised by a central peak with syetrical long tails) or soe cobination of powers of lorentzians. 19

23 2.3 Inverse Method An inverse ethod based on optial estiation theory [16] has been used to retrieve X CO2 fro the observed spectra. This odel siultaneously retrieves several properties of the atospheric and surface state x, such as teperature, huidity, surface pressure, surface reflectance, cloud and aerosol optical depths in addition to the CO 2 volue ixing ratio. The function f(x) represents the forward odel, taking into account the radiative transfer and the instruent response. The easureent vector is denoted by y, and the easureent can be described as follows: y = f (x) + ε (20) where ε is the easureent error. Fitting the observed spectra using the odel involves iniising a cost function 2 T 1 T 1 χ = [ y f ( x)] S [ y f ( x)] + ( x x ) S ( x x ) (21) ε a a a where x a is the a priori state vector, S a is the a priori covariance and S ε is the easureent error covariance. The easureent errors are assued to have no correlation between different pixels, i.e. S ε is diagonal. 20

24 Crucial to optial estiation theory is the use of a priori constraints, which represent the expected range of values for each eleent of the state vector and the correlations between eleents. These constraints can be estiated fro cliatological data (teperature and huidity profiles, surface pressure), elaborate easureents (CO 2 profile) or Markov descriptions (cloud and aerosol profiles) [16]. Since the aerosol optical depth in the weak CO 2 band was obtained by interpolation between the other two bands, an interpolation error was also included in the state vector. Care ust be exercised when eploying a priori constraints. While good a priori data are needed for fast convergence in a highly nonlinear proble, too uch inforation in the a priori can bias the retrieval. The challenge is to find the right balance. The Levenberg-Marquardt ethod [16] is used for the iniisation of the cost function. The Gauss-Newton ethod is a specific case of this technique. The state vector update dx i+1 is given by 1 T 1 1 T 1 1 dxi = [(1 + γ i ) S a + K i S K i ] [ K i S ( y f ( xi )) S a ( xi x )] (22) + 1 ε ε a where K i is the weighting function, or Jacobian, given by K i = f ( x ) x i i (23) 21

25 and γ i is the Levenberg-Marquardt paraeter. If γ i = 0, the iteration becoes Gauss- Newton. γ i is chosen at each step to iniise the cost function. To test for convergence, we use a paraeter dσ i 2, which is effectively the square of the state vector update in units of the solution variance. dσ (24) 2 T ˆ 1 i = dx i+ 1S dx i+ 1 where the solution variance ˆ 1 S is given by ˆ 1 1 S T 1 dx i+ 1 = K i Sε ( y f ( xi )) S a ( xi x a ) (25) If dσ 2 i > dσ 2 i-1, there is divergence. In this case, γ i is increased and equation (22) is solved again. Convergence is reached if dσ 2 i << n (diension of the state vector). However, convergence does not ean that the right result has been obtained, since the nonlinearity of the proble eans that we could have hit a local iniu. A chi-square fit is done to confir that the true solution has indeed been achieved. χ 2 i = j 2 2 ( y j f j (ˆ)) x / σ ε j i (26) 22

26 where i refers to the spectral band, j is the pixel in the band under consideration, xˆ is the 2 final coputed state, σ is the variance of the easureent error for the jth pixel in the ε j 2 ith band and i is the nuber of pixels in the ith band. If χ i > 1 in any of the bands, then we do not have a good fit. X CO2 is obtained by averaging the CO 2 profile, weighted by the pressure weighting function, h, such that T X = h xˆ (27) CO2 The foral error variance in the retrieved X CO2 is therefore given by 2 T σ h Sh ˆ (28) X CO = 2 3. Case Study: Retrieval of Aircraft Measureents O Brien and co-workers [O'Brien, 1998; O'Brien, 1997] took aircraft easureents of the O 2 A-band spectru of sunlight reflected fro the sea surface. They deonstrated, using differential absorption techniques, that the surface pressure can be deterined with an error of less than 0.1% fro such easureents. The ideal test case for the retrieval algorith would be to retrieve the colun O 2 (equivalently, since oxygen is well ixed, the surface pressure) fro O Brien s easureents since the ixing ratio of O 2 is known and constant (so we already have the correct answer if the pressure profile is known). 23

27 Any differences between the retrieval and the truth ust then be due to inadequacies in the easureent or analysis ethods. The transfer function t(x) of the instruent was given by O Brien as the convolution of the entrance slit transfer function with that of the detector pixels. 0, 5a / 2 x 2 ( x 5a / 2) /(2a), 3a / 2 x 5a / 2 t ( x) = (29) 2a x, a / 2 x 3a / 2 2 7a / 4 x / a, 0 x a / 2 where x is the physical distance along the detector, and 2a is the distance between successive detector pixels (25 µ in this case). The distance can be converted to wavelength by a linear transforation. The transfer function is syetric, so the specification above is sufficient. An additional coplication that needs to be handled is the surface boundary condition. The easureents were taken over the sea surface, which is highly non-labertian (different reflectivities in different directions). For such surfaces, we need to specify the BRDF, which is a specification of reflectance in ters of both incident and reflectedbea geoetry; i.e., the ratio of the reflected radiance in the viewing direction to the irradiance in the incident direction. For the sea surface, the BRDF has been odelled by Cox and Munk [Cox, 1954]. 24

28 As a first cut, we attepted a straightforward retrieval using our algorith, assuing the instruent was well-calibrated. Fig. 7 shows the results. The black line refers to the coputed radiances and the green line denotes the observed radiances. The residuals (coputed observed) are represented by the black line at the botto. The radiances are noralised by the axiu value so that they range between 0 and 1. The root ean square (rs) residual is 8.8%, which is clearly not good enough. A closer look at the results reveals that there is a wavelength grid isatch between the observed and calculated radiances. This isatch occurs because the optical properties of the instruent change with tie due to teperature changes, thus changing the wavelengthdiode apping. This in turn is caused by the expansion with teperature which changes the orientation/distance between grating and slit, etc. Also evident is the fact that the continuu is not aligned. This arises fro iproper calibration of the spectroeters. To correct for these effects, we did a piecewise linear scaling of the wavelength grid and fitted the continuu using the following two-paraeter for: xobs in( xobs ) I = + corr I CL CT (30) ax( xobs ) in( xobs ) where I corr and I are the corrected and coputed radiances respectively at wavelength x obs, and CL and CT are the retrieved paraeters, called the continuu level and continuu tilt respectively. 25

29 Fig. 8 shows the iproved fit, with an rs residual of 2.3%. Contributions fro the isotopes of oxygen (green and orange lines) as well as trace gases like water vapour (light blue line) have been included. The dark blue line shows the total coputed radiance. The top panel is the residual. It is to be noted that the O 2 isotopic coposition is well known and constant for the purpose of this study. The ILS fro O Brien s instruent was never characterised well enough for the precisions sought in this work. Further iproveents can hence be ade by considering the zero level offset (a consequence of excess dark current in the detector and/or electronic offset) and instruent lineshape fits. A wide lorentzian was added to the ILS provided by O'Brien, which assued zero response beyond 1-2 c -1. The best fit for the wide lorentzian had a width of ~ 10 c -1 and aplitude of 0.02 (the peak of O'Brien's ILS is ~1.75). The width of O'Brien's ILS was also fitted which give a nuber 7% larger than that given by hi. This deonstrates the ability of the algorith to copensate for iperfect ILS knowledge as part of the retrieval process. The rs residual is now reduced to 1.4% (see Fig. 9). The residues are ostly systeatic. The effect of line-ixing is apparent at the band head (close to 0.76 µ). Also, there is a huge residual at µ. This is a solar feature not accounted for by the odel. The colun O 2 was retrieved to ~ 1% precision. Clearly, by averaging sufficient soundings, the rando errors can be iniised and precisions of around 0.1% are 26

30 potentially achievable. This suggests that the technology and physical insight required to retrieve X CO2 with precisions better than 0.3% exist. A thought experient was conducted to siulate the effect of accounting for the solar feature and reoving line ixing. Fig. 10 indicates that the residuals can be decreased to 1.1%. It ight thus be worthwhile to explore ways to odel line ixing. Figure 7: Retrieval of O Brien s easureents of the O 2 A-band radiances over sea surface, assuing proper instruent calibration (black observed, green coputed, botto residual) 27

31 Figure 8: Retrieval of O Brien s easureents of the O 2 A-band radiances over sea surface, with wavelength scaling and continuu correction (blue diaonds observed, dark blue line coputed) 28

32 Figure 9: Retrieval of O Brien s easureents of the O 2 A-band radiances over sea surface, with wavelength scaling, continuu correction, zero offset and instruent lineshape fits (black observed, aroon coputed, botto residual) 29

33 Figure 10: Retrieval of O Brien s easureents of the O 2 A-band radiances over sea surface, with wavelength scaling, continuu correction, zero offset, instruent lineshape fits and reoval of line ixing and solar feature (black observed, aroon coputed, botto residual) 30

34 4. Future Work The issues that need to be addressed can be broadly classed as follows: Polarisation: If electroagnetic radiation were scalar like sound, the intensity would provide a full description. However, the transverse nature of light allows the phenoenon of polarisation which requires additional paraeters for a full description. It can been shown [Hansen, 1974] that the intensity can be fully described by four nubers, known as the Stokes paraeters, which are denoted by I, Q, U and V respectively. The significance of this is that the equation of radiative transfer is actually a vector equation with four quantities needed to be deterined at each point. The question to be answered is whether polarisation akes a difference to the observed radiances, and if so, how does it affect the X CO2 retrieval? Surface Types: We considered the sea surface in the retrieval of O Brien s easureents. However, the OCO footprint could have several different types of surfaces, such as forest, snow, sand, etc.; indeed, it could be a cobination of surfaces. The effects of different kinds of surfaces and the respective BDRFs need to be considered. Other Radiative Transfer Models: The discrete ordinate ethod is just one of several techniques that exist to solve the equation of radiative transfer. A distinction needs to be ade here between exact and approxiate ethods. An exaple of the forer is the doubling-adding ethod [van de Hulst, 1962], while the ethod of successive scattering [Irvine, 1975] is approxiate. The successive scattering ethod 31

35 (or Neuann series solution) starts by calculating the distribution of all light that has been scattered once: this is the first-order scattered light distribution. The first-order distribution is used to find the second-order light distribution. This process is repeated until the distribution of light scattered n ties is negligible. The distributions for all orders are sued to deterine the final distribution of light. This ethod has potential use since scenes with cloud optical depth larger than 0.3 will be discarded and so potentially only a few orders of scattering are expected to be significant. The principle behind the doubling-adding ethod is that if the reflection and transission is known for each of two layers, those for the cobined layer can be obtained by coputing the successive reflections back and forth between the two layers. If the two layers are chosen to be identical, the results for a thick hoogeneous layer can be built up rapidly in a geoetric (doubling) anner. Analytic Weighting Functions: Recently, a radiative transfer odel called LIDORT was developed [Spurr, 2001] that siultaneously coputes radiances and weighting functions for a ultiple-scattering proble. The coputation of weighting functions is based on an internal perturbation analysis of the discrete ordinate solution; the results are derived analytically without the need for finite difference approxiations based on repeated calls to an intensity-only odel. Sensitivity Tests: Siulation tests need to be run to quantify the effects of uncertainties in various paraeters (such as the instruent line shape) on the retrieval. Such tests would give invaluable inforation about how accurate the calibration should be, for exaple. 32

36 Speed Iproveents: OCO will generate 2.4 illion spectra per day, which need to be analysed in real-tie. On average, each retrieval has to be done in about two seconds. Achieving this without coproising on the stringent accuracy criteria is possibly the single biggest challenge. 5. Conclusion An algorith has been developed to retrieve X CO2 fro spectroscopic easureents of absorption in the weak CO 2 band at 1.61 µ, the O 2 A-band at 0.76 µ and the strong CO 2 band at 2.06 µ, where the latter two bands are needed to account for the effects of paraeters other than the CO 2 ixing ratio on the absorption. The discrete ordinate ethod was eployed to do the radiative transfer. Optial estiation theory was used to perfor the inversion to obtain atospheric and surface paraeters such as CO 2 volue ixing ratio, teperature, huidity, surface pressure and aerosol vertical profiles. To test the algorith, colun O 2 (surface pressure) was retrieved fro sunglint easureents over ocean of absorption in the O 2 A-band. It was observed that the instruent was iproperly calibrated. After correcting for the calibration effects, excellent fit was obtained between the coputed and easured spectra. The colun O 2 was retrieved with an error of around 1%. Spectral averaging using ultiple soundings would reduce rando errors; it is thus possible to reduce the error to 0.1%. This indicates that it is possible to retrieve X CO2 with precisions better than 0.3%. 33

37 Acknowledgents The project was funded by the grant JPL P fro the Jet Propulsion Laboratory. The author wishes to thank the following people: Yuk Yung for his otivational guidance; David Crisp, Charles Miller, Geoff Toon, Bhaswar Sen, Hartut Boesch and Dan Feldan for their support and collaboration; Zhonghua Yang or invaluable help with the retrieval of the A-band spectra; and Jack Margolis, Charles Miller and Xun Jiang for detailed and insightful coents on the anuscript. References CDIAC, Carbon Dioxide Inforation and Analysis Center, Oak Ridge National Laboratory, Chandrasekhar, S., Radiative Transfer, Oxford Univ. Press, London, Cicerone, R.J., E.J. Barron, R.E. Dickinson, I.Y. Fung, J.E. Hansen, T.R. Karl, R.S. Lindzen, J.C. McWillias, F.S. Rowland, E.S. Sarachik, and J.M. Wallace, Cliate Change Science: An Analysis of Soe Key Questions (Prepublication Copy), pp. 28, National Research Council, Washington, DC, Conway, T.J., P.P. Tans, L.S. Wateran, and K.W. Thoning, Evidence for Interannual Variability of the Carbon-Cycle fro the National-Oceanic-and-Atospheric- 34

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