Rapid Estimation of Column-Averaged CO2 Concentration Using a Correlation Algorithm

Igor Polonsky Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

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D. M. O’Brien Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

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Abstract

Measurement of XCO2, the column-averaged mole fraction of CO2, using reflected sunlight in the near-infrared bands of CO2, is strongly influenced by photons that are scattered in the atmosphere because scattering can either decrease or increase the mean pathlength compared with the direct path from the sun to the surface to the satellite. A very simple algorithm that can be used to compensate for the errors introduced by scattering is presented. The algorithm is based on the observation that the apparent optical path differences in selected pairs of channels in the weak CO2 band at 1.6 μm and the O2 A band at 0.76 μm are tightly correlated for large ensembles of scattering atmospheres. The number of tightly correlated pairs of channels is many hundreds for the bands measured by NASA’s Orbiting Carbon Observatory (OCO). The physical reasons for the correlation are that the mean photon pathlengths are comparable for the members of each pair of channels, and that the extinction profiles vary similarly with height. For atmospheres with modest scattering optical thickness (less than 0.3), the slope and the intercept of the linear correlation for any pair depends weakly on the surface reflectance, the surface pressure, and the viewing geometry. Through numerical simulations the slope and intercept may be parameterized simply in terms of these variables. Thereafter, the task of retrieving XCO2 from measured spectra may be reduced to linear interpolation in precomputed tables of slopes and intercepts. Results with simulated data for NASA’s OCO satellite are presented, and random errors and biases are investigated. Although OCO did not reach orbit, the method is applicable to any instrument that operates using similar principles [such as those on the Greenhouse Gases Observing Satellite (GOSAT) and the replacement satellite OCO-2].

Corresponding author address: Denis O’Brien, 1375 Campus Delivery, Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO 80523-1375. Email: obrien@atmos.colostate.edu

Abstract

Measurement of XCO2, the column-averaged mole fraction of CO2, using reflected sunlight in the near-infrared bands of CO2, is strongly influenced by photons that are scattered in the atmosphere because scattering can either decrease or increase the mean pathlength compared with the direct path from the sun to the surface to the satellite. A very simple algorithm that can be used to compensate for the errors introduced by scattering is presented. The algorithm is based on the observation that the apparent optical path differences in selected pairs of channels in the weak CO2 band at 1.6 μm and the O2 A band at 0.76 μm are tightly correlated for large ensembles of scattering atmospheres. The number of tightly correlated pairs of channels is many hundreds for the bands measured by NASA’s Orbiting Carbon Observatory (OCO). The physical reasons for the correlation are that the mean photon pathlengths are comparable for the members of each pair of channels, and that the extinction profiles vary similarly with height. For atmospheres with modest scattering optical thickness (less than 0.3), the slope and the intercept of the linear correlation for any pair depends weakly on the surface reflectance, the surface pressure, and the viewing geometry. Through numerical simulations the slope and intercept may be parameterized simply in terms of these variables. Thereafter, the task of retrieving XCO2 from measured spectra may be reduced to linear interpolation in precomputed tables of slopes and intercepts. Results with simulated data for NASA’s OCO satellite are presented, and random errors and biases are investigated. Although OCO did not reach orbit, the method is applicable to any instrument that operates using similar principles [such as those on the Greenhouse Gases Observing Satellite (GOSAT) and the replacement satellite OCO-2].

Corresponding author address: Denis O’Brien, 1375 Campus Delivery, Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO 80523-1375. Email: obrien@atmos.colostate.edu

1. Introduction

There is strong evidence that measurements of the column-averaged mole fraction of CO2 (denoted herein by XCO2) would enable the more accurate identification of regional sources and sinks of CO2 (Rayner and O’Brien 2001; Rayner et al. 2002). There also is evidence that measurements of XCO2 from space are feasible; options thereof include the thermal infrared (Engelen et al. 2001; Engelen and Stephens 2004; Engelen et al. 2004; Chahine et al. 2008; Crevoisier et al. 2009), the near-infrared (NIR; O’Brien and Rayner 2002; Kuang et al. 2002; Mao and Kawa 2004), and combinations of the two (Christi and Stephens 2004). The National Aeronautics and Space Administration’s (NASA’s) ill-fated Orbiting Carbon Observatory (OCO) and Japan Aerospace Exploration Agency’s (JAXA’s) Greenhouse Gases Observing Satellite (GOSAT) measure high-resolution spectra of reflected sunlight in the NIR to measure XCO2.

It was argued by O’Brien and Rayner (2002), and subsequently by Kuang et al. (2002), Dufour and Bréon (2003), and Mao and Kawa (2004), that simple differential absorption spectroscopy will not yield XCO2 with sufficient accuracy unless corrections are made to compensate for scattering in the atmosphere, whether by clouds or aerosols. O’Brien and Rayner (2002) proposed a correction procedure, in the context of an etalon spectrometer, that relies upon the correlation between the apparent optical path difference (AOPD)1 of the atmosphere in an O2 channel with that in a CO2 channel. Provided that the channels are selected so that their optical thicknesses are comparable and the extinction profiles vary similarly with height, the correlation is tight and it provides a way to link the column-averaged mole fraction of CO2 to that for O2. Kuang et al. (2002) and Bösch et al. (2006) use two CO2 bands and the O2 A band to estimate not only XCO2 but also a comprehensive representation of the atmospheric state. For both OCO and GOSAT, the operational requirement is that accurate retrievals of XCO2 should be made whenever the combined optical thickness of cloud water, cloud ice, and aerosol is less than 0.3 at the frequency of the O2 A band.

Figure 1 illustrates the tightness of the correlation between the apparent optical path differences for two selected channels in the O2 A band and weak CO2 band of OCO for two different values of XCO2. Each dot represents the AOPDs in the O2 and CO2 channels for one atmosphere. The ensemble of dots arises by sampling over an ensemble of atmospheres, varying not only the vertical profiles of temperature and water vapor but also the distributions of water cloud, ice cloud, and aerosol. There are hundreds of pairs of channels with similarly tight correlations in the OCO bands.

The procedure proposed by O’Brien and Rayner (2002) was only tested in an idealized setting, in which scattering by both optically thin clouds and aerosols was approximated by single scattering. The purpose of this paper is to show how the ideas proposed by O’Brien and Rayner (2002) may be developed more rigorously. All of the simulations will use a full multiple-scattering radiative transfer code embedded in the OCO simulator (O’Brien et al. 2009), which employs spectroscopy for the O2 A band and the weak CO2 band derived from the High Resolution Transmission (HITRAN) database (Rothman et al. 2009), with additional corrections to account for line mixing in the weak CO2 band (Hartmann et al. 2009).

Whereas the benchmark retrieval algorithm for OCO is the “full physics” algorithm (Bösch et al. 2006), the approach advocated by O’Brien and Rayner (2002) might be described as “minimal physics,” meaning that the number of retrieved parameters is comparable to the information content of the spectra. The algorithm described in this paper adapts just the correlation component of the approach outlined above, and therefore might be termed the “zero physics” algorithm. It relies on parameterizations of empirical correlations, so it is very fast and very simple, but these gains are made at the expense of diagnostic ability and accuracy. The algorithm may be applied to both the weak and strong CO2 bands, but the performance is better in the weak CO2 band. The reason appears to be that it is easier with the weak CO2 band to find pairs of frequencies at which the vertical profiles of absorption for O2 and CO2 vary similarly with temperature and pressure. Furthermore, the strong CO2 band is contaminated with water vapor, which is highly variable in space and time, while the O2 A band and weak CO2 band are not. Consequently, this paper focuses on the weak CO2 band, with only occasional references to the strong band.

Section 3 describes how the pairs are identified using spectra simulated for ensembles of atmospheres whose thermodynamic and optical properties span the range of the almost clear atmospheres that are likely to be encountered by OCO. These simulations constitute the training runs for the algorithm. Section 3 also investigates how to account for the dependence of the slopes and the offsets of the linear relations between the O2 and CO2 AOPDs upon variables such as the surface reflectance in the O2 and CO2 bands, the surface pressure, and the viewing geometry. Ideally, the dependence on surface reflectance should be weak for atmospheres with low scattering optical thickness, but the dependence is not negligible.

Section 4 addresses several knotty issues: the treatment of solar lines, corrections for spectral variations of surface reflectance across the bands, and corrections for Doppler shifts caused by the relative motion of the sun, target, and satellite. Section 5 outlines the steps of the algorithm, while section 6 evaluates the vertical averaging kernel.

Finally, section 7 describes tests used to quantify the accuracy of the algorithm. The tests employed spectra simulated for the orbit of OCO using thermodynamic and scattering properties interpolated to the CloudSat vertical grid.

2. Training simulations

a. Atmospheres

Atmospheric profiles for training were selected from the European Centre for Medium-Range Weather Forecasts (ECMWF) database of 60-layer atmospheres prepared by Chevallier (2002). The selection criterion was that the combined scattering optical thickness of water and ice clouds should not exceed 0.4. As a rough guide, the column mass densities of cloud water w and cloud ice i should be less than 2 and 10 g m−2, respectively. There were 965 such profiles in the database from Chevallier (2002). Table 1 gives the breakdown of the cases into totally clear scenes, scenes free of cloud water, scenes free of cloud ice, and scenes with both water and ice.

Figures 2 and 3 present histograms of cloud water w, cloud ice i, and the mean pressure levels pw and pi of cloud water and ice, with the latter defined by
i1520-0426-27-12-2002-eq1
where the summation extends over the layers, wj and ij are the water and ice mass densities of layer j, and pj is the pressure of layer j. The histograms show that although many of the atmospheres have low amounts of cloud water and ice, there also are cases with significant amounts.
The histograms of column water vapor (kg m−2) and mean profile temperature are presented in Fig. 4. The latter is defined by
i1520-0426-27-12-2002-eq2
where Δpj and Tj are the pressure thickness and temperature of the jth layer. The histogram of water vapor shows that dry atmospheres were overrepresented in the collection of atmospheres used for training.

b. Cloud optical properties

Mie calculations were performed for the phase functions of water clouds, with the phase functions integrated over the assumed size distributions. The effective radii of the drop size distributions of water clouds were modeled on experimental data, as suggested by Bower et al. (1994). With this model the effective radius can vary with height in accordance with the density of cloud water.

Ice clouds used phase functions published by Baum (2007) and Baum et al. (2005a,b) but were originally derived from finite-difference time domain techniques performed by Yang et al. (2000). Baum (2007) tabulates the phase function for the frequencies of several satellite sensors; the frequencies closest to the OCO bands were chosen for this study.

Because ice crystals generally are large, the contribution of scattering to the forward diffraction peak of the phase matrix also is large. So-called δ transmission represents the direct transmission of rays through the ice crystals, and contributes up to 10% to the forward peak. Because expansion of the phase matrix in generalized spherical functions would require many thousands of terms, a δ correction was applied, not in the usual manner as described by Wiscombe (1977), but alternatively by truncating the forward peak of the phase matrix at a scattering angle of 10° and then renormalizing the phase matrix. This truncation was accompanied by a simultaneous δ correction of both the scattering and absorption coefficients, which generally reduces the apparent optical thickness, because forward-scattered photons are treated as unscattered.

Ice crystals were assumed to have a bimodal size distribution, with the size of the large mode depending on temperature, but not the small mode (Ivanova et al. 2001). For every layer, the large-mode effective radius was computed from the layer temperature. The large-mode effective radius was rounded to the nearest 5 μm, and moments for the phase matrix corresponding to that size were used for the layer.

c. Aerosol optical properties

Only a very simple aerosol model was used for training. Aerosol was considered to be in three layers: 0–2, 2–5, and 5–12 km. In the boundary layer (0–2 km), the aerosol was assumed to be either the standard oceanic or continental model, according to the surface type. In the 2–12-km range, the continental aerosol model was assumed. The option of a dust aerosol component between 2 and 5 km was included in the code to represent elevated dust clouds, such as Saharan dust, but this option was not used during training. The optical thickness of aerosol was varied randomly in the range from zero to 0.16 over land and 0.07 over ocean.

Figure 5 shows the histogram of the combined optical thickness of cloud water, cloud ice, and aerosol used in the training simulations. All of the optical thicknesses are specified at the frequency of the O2 A band. The scattering phase matrices for aerosol were calculated using Mie theory with particle size distributions from Shettle and Fenn (1979).

d. Surface

The surface was assumed to be Lambertian and nonpolarizing for the training runs. Simulations were carried out for only four values of surface albedo in each band—0.0, 0.1, 0.5, and 0.9. The radiance for each atmosphere and wavelength then was represented as a cubic polynomial in the surface albedo. Last, the polynomial representation was used to generate spectra on an albedo grid that varied from 0 to 1 in steps of 0.05. Pairs, slopes, and offsets were calculated on this finer albedo grid.

3. Training algorithm

a. Strategy

During training a collection of surfaces and atmospheres, denoted A1, A2, … , AN, was used with model variables x0, x1, … , xn−1 to compute spectra in the OCO bands. The atmospheres and the model variables uniquely define the system, and hence the spectra. Each of the model variables ranged over a set of predefined values ki in number for variable xi. Thus, with n model variables, the total number of combinations was the product
i1520-0426-27-12-2002-eq3
thus, training required at most KN simulations. In practice, the number could be smaller if one run of the OCO simulator (O’Brien et al. 2009) provided more than one of the combinations. For example, all of the combinations of M surface albedo values could be generated from the spectra of just M simulations with the same surface albedo assigned to each band.

In the training runs used for the results presented in this document, five model variables were used for both nadir and glint modes over land, as shown in Table 2. For glint over the ocean, the surface reflectances were replaced by the surface wind speed, leading to the list of variables in Table 3. Nadir mode was not subdivided into land and ocean cases because generally the radiance reflected by the ocean to the satellite in nadir mode was too low for reliable retrievals.

The model vector x = (x0, x1, … , xn−1) defines a point in an n-dimensional mesh. We define an upper boundary point of the mesh to be any point for which any of the coordinates is at its upper limit. For brevity, we refer to such points as “boundary points,” omitting the qualification that only upper boundary points are intended. Figure 6 represents the mesh in two dimensions, with the boundary points as squares, the vertices of a unit cell in the mesh as pentagons, and the lower corner of the unit cell as a circled pentagon. We specify a unit cell by its lower corner. By letting the lower corner roam over the nonboundary vertices, the corresponding unit cells will cover the mesh. Pairs will be selected at every nonboundary vertex x. At each of the 2n vertices of the unit cell whose lower corner is x, the slope and offset of the correlation between the O2 and CO2 channels are computed for each selected pair, and then written to the pair database. Thus, while the selected pairs may vary from vertex to vertex, the pairs selected at the lower corner x of a unit cell are used to interpolate the slopes and offsets to interior points of that unit cell.

For component xr of the state vector, we let
i1520-0426-27-12-2002-eq4
denote the values of xr defining the mesh. When
i1520-0426-27-12-2002-eq5
we define
i1520-0426-27-12-2002-eq6
Thus, every model state x can be specified in terms of the indices (s0, s1, … , sn−1) defining the lower corner and a vector ɛ = (ɛ0, ɛ1, … , ɛn−1), all of whose components lie between 0 and 1.

b. Surface albedo grid

Let denote the radiance computed for atmosphere Ai and state vector x at wavelength j in band r.2 For the purpose of training, the surface was assumed to be Lambertian, with albedo Ar in band r.

The training datasets were very large, even for the few values of albedo listed in Table 2. Furthermore, the correlations were found to depend weakly on the surface albedo, which implied that a much finer resolution of the surface albedo range was needed than that provided in Table 2. Finer resolution would lead to even larger datasets, and also would increase the cost of preparing the training data. Fortunately, it was found that the radiance for any pixel could be expressed with high accuracy as a polynomial in the surface albedo; in practice, a cubic polynomial sufficed. Therefore, the spectra computed for the four surface albedo values listed in Table 2 were used to compute the coefficients in the representation
i1520-0426-27-12-2002-e1
The accuracy of this representation was tested by computing spectra on a fine albedo grid, and then comparing the result with the spectra predicted by Eq. (1). Generally, the differences were smaller by an order of magnitude than the instrument noise expected for OCO.

c. Correlation calculation

Consider now any nonboundary vertex x. Let (r)(x) denote a matrix whose (i, j) element is . Let jr denote a reference wavelength selected for band r in a continuum region that is relatively free of water vapor and solar absorption lines. The AOPD is defined by
i1520-0426-27-12-2002-e2
where m(x) is the airmass factor. In its simplest form,
i1520-0426-27-12-2002-eq7
where θs and θυ are the solar and view zenith angles. The results described in this report used this simple form for the airmass factor, but a line-of-sight correction was to be added before analyzing the real data. Let Ij(r) denote the jth column vector of the matrix (r), and let Lj(r) similarly denote the jth column vector of the matrix (r).
The training algorithm computed the correlation between AOPDs in bands r and s,
i1520-0426-27-12-2002-eq8
where the covariances were evaluated over the ensemble of atmospheres, and the dependence upon x has been suppressed on the right-hand side for clarity. Thus,
i1520-0426-27-12-2002-eq9
where N is the number of atmospheres, and μj(r) is the mean of the jth column of the matrix (r),
i1520-0426-27-12-2002-eq10
Given that there were 1016 pixels per band, the correlation matrix (rs) for bands r and s consisted of 10162 elements. These were sorted into descending order of absolute value, and the first 400 were passed to the next stage. Typically, the maximum correlation coefficient exceeded 0.995, and in practice a lower threshold of 0.990 was set. These steps leading to the correlation matrix were carried out for every nonboundary vertex of the mesh.

d. Pair selection

As described in the previous subsection, the pairs of AOPDs at each vertex s were sorted according to the absolute value of the correlation coefficient. To refine the selection of pairs at s, the index representing XCO2(the last dimension of the state vector) was incremented by one, thus increasing XCO2 while the other variables were held fixed. Slopes and offsets were computed for each pair at both the original vertex and the vertex with increased XCO2. Because the slopes and intercepts were assumed to depend linearly on XCO2, they provided the basis of a simple retrieval algorithm, which is valid in the neighborhood of s. Noise was added to the AOPDs computed at s at a level representative of the expected instrument noise, and trial retrievals were performed for all atmospheres in the ensemble. Ideally the retrievals would return the value of XCO2 at s, but in practice there were errors. The pairs were ranked according to the mean value of the absolute errors, with the mean being taken over the ensemble of atmospheres. The top 200 pairs were selected for the vertex s.

4. Conditioning

The simple idea that the AOPD depends linearly on the number of CO2 molecules along the ray path through the atmosphere clearly fails in the vicinity of solar lines, so a correction to remove or reduce the effects of solar lines is required. The simple, but effective, procedure described by O’Brien et al. (2010) was applied.

Training for the algorithm was performed with Lambertian surfaces whose albedos were constant within each of the OCO bands. In practice, the world is more complex. The O2 A band sits on the “red edge” of vegetation, where the reflectance rises rapidly with wavelength from absorption by chlorophyll to strong reflection from cellular walls. Similarly, the reflectance of vegetation often is highly variable in the weak and strong CO2 bands.

To reduce the sensitivity to variations in surface albedo, the spectrum in each band after correction for solar lines and Doppler shifts was detrended, meaning that a linear slope was removed in a two-stage process, again following O’Brien et al. (2010).

In training the correlation algorithm, even though surface albedos were assumed to be constant within each band, the spectra still had a slope inherited from the solar spectrum and atmospheric features, such as Rayleigh scattering in the O2 A band and ice absorption features in the CO2 bands. Therefore, to ensure consistency between the training and the application of the algorithm, the same slope removal procedure was applied to all of the training data.

5. Application

The algorithm attempts to estimate XCO2 using linear relations between AOPDs in the O2 and CO2 channels. As explained above, the slope and offset are determined by finding correlations that persist over ensembles of atmospheres. However, the slope and offset of the correlations depend weakly on other variables, such as the surface reflectance over land, the near-surface wind speed over ocean, the surface pressure, and the viewing geometry. To apply the algorithm, these components of the state vector must be fixed, so that XCO2 remains the only unknown. Some variables, such as the solar zenith angle and the view zenith angle, will be known accurately from the satellite navigation and pointing systems. Others, such as the surface pressure, may be derived from meteorology or ancillary datasets, and will be sources of error.

a. Surface reflectance

The slopes and offsets of the linear correlations derived during training depend weakly upon the surface reflectance. In a clear atmosphere, the surface reflectance can be estimated reliably from the continuum radiance, but such is not the case when the atmosphere contains scattering material. Generally, a small amount of nonabsorbing aerosol over a dark surface will increase the apparent surface reflectance; conversely, aerosol over a bright surface will reduce the apparent reflectance. Fortunately, the accuracy of the AOPD correlation algorithm is not limited by the accuracy with which the surface reflectance is known, so we may adopt a simple model in which the surface reflectance r is estimated from the continuum radiance and the solar zenith angle,
i1520-0426-27-12-2002-eq11
Here I is the mean radiance, estimated over one or more continuum ranges of the spectrum. The coefficients ci(θs) were determined for
i1520-0426-27-12-2002-eq12
with separate parameterizations for nadir and glint modes over land. At other angles, ci(θs) is interpolated linearly from the coefficients for the adjacent tabulated values of the solar zenith angle. When θs lies outside the grid, the coefficients are extrapolated linearly from the last two points of the grid.

In operation, OCO would not have viewed the glint point exactly, so the view zenith angle would have been added to the state vector. In that case, parameterizations of surface reflectance would have been developed in terms of both the solar zenith angle and the view zenith angle.

b. Near-surface wind speed

In glint mode over the ocean, the principal variable controlling the brightness of the surface is the near-surface wind speed. Again, we developed parameterizations of the wind speed in terms of the radiance averaged over selected continuum pixels. In practice, the same ranges of pixels were used for the wind speed as for the surface reflectance, but the code did not impose this restriction. Unlike the model for the surface reflectance, the wind speed was represented as a polynomial in the reciprocal of the mean radiance,
i1520-0426-27-12-2002-eq13
The coefficients were interpolated in the manner described for the surface reflectance.

c. Estimation of XCO2

The algorithm to estimate XCO2 is very simple, given the infrastructure described above:

  1. Calculate the normalization factor to remove solar lines. This task requires the speeds of the sun and the satellite toward the target on the earth, a reference terrestrial reflectance spectrum, absorption coefficients of the gases, the instrument line shape function and the dispersion parameters, and the solar spectrum at the top of the atmosphere. The solar lines are removed by dividing the measured spectrum in the satellite reference frame by the solar line correction factor (O’Brien et al. 2010).

  2. Map the solar-corrected spectrum from the satellite reference frame to the terrestrial reference frame.

  3. Estimate the surface reflectance if the target is land, or the near-surface wind speed for glint over the ocean. This calculation uses the solar-corrected spectrum.

  4. Remove the slope from the solar-corrected spectrum.

  5. Calculate AOPDs in the terrestrial reference frame using the spectra from stage 4.

  6. Estimate the surface pressure at the target using meteorological data from ECMWF and a digital elevation model of the topography.

  7. Compute the remaining components of the state vector x, except for the last, which represents XCO2.

  8. Using the state vector x, identify the cell in the training mesh that contains x, and find the vertex of its lowest corner, denoted by
    i1520-0426-27-12-2002-eq14
    In the simplest case, the mesh will contain only two values of XCO2, so the last index will take only the values one of 0 and 1.
  9. Read from the pair database the list of pairs selected at the vertex s. For each pair, read the slope and offset at each of the 2n vertices of the cell with lower corner s.

  10. On the face of the cell defined by sn−1 = 0, interpolate the slope and offset from the vertices to the interior point defined by
    i1520-0426-27-12-2002-eq15
    Denote the interpolated slope and offset for pair k by ak0 and bk0. Repeat the process for the face defined by sn−1 = 1, denoting the interpolated slope and offset by ak1 and bk1.
  11. If (pk, qk) denote the pair of O2 and CO2 AOPDs, then define
    i1520-0426-27-12-2002-e3
    The corresponding value of XCO2 then is
    i1520-0426-27-12-2002-e4
    where xn−1,0 and xn−1,1 are the values of XCO2 at the lower and upper corners of the cell.
  12. Finally, compute the mean and variance of the estimates provided by the N pairs,
    i1520-0426-27-12-2002-eq16
    Alternatively, we assigned a weight wk to each pair that measured the separation of the lines of correlation with the values of XCO2 at the lower and upper corners of the cell, and then we computed the value of X that minimized the weighted sum of the squared residuals. However, these two methods give similar results.

6. Averaging kernel

It is important to understand the sensitivity of the algorithm to the vertical distribution of CO2. For this we calculate the averaging kernel, defined to be the functional derivative
i1520-0426-27-12-2002-eq17
where X denotes the estimated XCO2 and u(p) denotes the CO2 mole fraction at pressure level p. As described above, the estimate X is the average of values Xk for the different pairs, while
i1520-0426-27-12-2002-eq18
where the coefficients can be identified from Eqs. (3) and (4). The contribution to the functional derivative from Xk is
i1520-0426-27-12-2002-eq19
in which the expression of the coefficients Ck and Dk may be regarded as constants. The O2 AOPD pk does not depend upon u(p), apart from a small effect resulting from the variation of the composition of the air. This effect may be neglected.
The CO2 AOPD qk has the form
i1520-0426-27-12-2002-eq20
where I0 and Ik denote the radiances for the reference pixel and pixel k, respectively. Hence,
i1520-0426-27-12-2002-eq21
For the purpose of estimating the averaging kernel, we initially neglect the effects of scattering, so that
i1520-0426-27-12-2002-eq22
where ϕk(λ) is the instrument line shape function for channel k, F(λ) is the solar flux density at the top of the atmosphere, r(λ) is the surface reflectance [bidirectional reflectance distribution function (BRDF)], and τ(λ) is the optical thickness. The functional derivative is easily evaluated, leading after some straightforward manipulation to
i1520-0426-27-12-2002-eq23
where q(p) is the specific humidity at pressure level p and Md is the molecular weight of dry air. For this clear-sky calculation, the weighting function does not depend upon the surface reflectance. This assertion should remain approximately true even when the atmosphere contains a small amount of scattering material. Figure 7 shows the averaging kernel as a function of pressure level. It is approximately constant until a height of 200 hPa, and then smoothly rolls away toward zero. The units of the functional derivative Ik−1δIk/δu(p) and the averaging kernel are Pa−1.

7. Retrievals of XCO2 with simulated spectra

Training of the algorithm used a fixed ensemble of 60-layer atmospheres, drawn from the database of profiles from ECMWF prepared by Chevallier (2002), and uniform vertical profiles of CO2 mole fraction. In contrast, the testing described in this section used profiles of meteorology, cloud, and aerosol defined on the vertical grid employed by CloudSat, with uniform 240-m-thick layers from the surface to the stratosphere, and CO2 profiles from the parameterized chemical transport model (Kawa et al. 2004) interpolated to the CloudSat grid. Training used spectrally flat Lambertian albedos in each band over land, and the glint model with fixed wind speeds developed by Cox and Munk (1954a,b) over the ocean. In contrast, test spectra submitted to the algorithm were generated using the reflectances predicted by the OCO simulator (O’Brien et al. 2009) for the target surface type. The results shown are for nonpolarizing surfaces; initial testing with polarizing surfaces indicates comparable results, though a comprehensive study with multiple orbits has not been conducted.

Whenever possible, the optical thickness of cloud was calculated using the profiles of cloud water and ice and their effective radii observed by CloudSat. However, the sensitivity of CloudSat to thin clouds, the cases of most interest to OCO, is low, so that whenever CloudSat reported clear sky we resorted to almost clear ECMWF profiles from Chevallier (2002) for the vertical distribution of cloud water and cloud ice, which then were interpolated onto the CloudSat grid. However, the profiles were not necessarily those used in the training. This is a shortcoming of the OCO simulator, which should use Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) estimates for the distribution and optical properties of cloud whenever CloudSat reports clear sky. Second, the simple aerosol distribution described by O’Brien et al. (2009) was used in both training and testing, although the optical thickness always was chosen randomly. Again, preliminary testing with aerosol profiles derived from CALIPSO indicates comparable results, except when the aerosol particles are large and absorbing. Interestingly, the more complex vertical distributions of aerosol derived from CALIPSO appear to have little impact upon the accuracy. These limitations in the testing will be removed in later work.

Results are presented for 19 nadir orbits and 8 glint orbits. The processing time per sounding is approximately 1 s on a single central processing unit (CPU) core, with almost all of that time taken in computing transmittances convolved with the instrument line shape function, as required for the Doppler corrections. Screening for thick cloud is assumed to be provided reliably by the cloud-screening algorithm.

Filtering of the results is very simple; for the present, an estimate of XCO2 is accepted if the standard deviation of the estimates from individual pairs is less than a threshold, set to 2 ppmv for these results. Considerable improvement in filtering is possible, based on a comparison of the measured AOPDs with the AOPDs expected for a clear atmosphere. The information for this comparison is derived as a by-product of the transmittance calculations.

a. Nadir mode

Figure 8 shows the single-shot retrieval errors for the nineteen nadir orbits. The color bar range is from −5 to +5 ppmv, and the rainbow distribution of color allows easy discrimination between the error bands. There is a hint of a latitudinal bias in the figure, with low values of XCO2 at high latitudes.

If the XCO2 errors are averaged over 10° × 10° regions, the results are as presented in the left-hand panel of Fig. 9. The color bar now ranges from −2 to + 2 ppmv, and the tendency of lower values at high latitudes is more obvious. Furthermore, there is a global bias of −0.86 ppmv indicated in the histogram of errors from all retrievals on all orbits shown in Fig. 10. More work is needed to identify the cause of the global bias. One possibility lies in the uniform vertical profiles assumed for the CO2 mole fraction during the training of the algorithm. The value of XCO2 reported as truth during the training was obtained by integrating the uniform profile through the atmosphere. In contrast, the sensitivity of the retrieved XCO2 to the mole fraction of CO2 rolls off above 200 hPa, so the potential for a biased estimate is apparent. This deficiency probably could be remedied by using a more realistic model of stratospheric CO2 in the training phase. If the global bias is removed, then the spatial distribution of XCO2 errors is shown in the right-hand panel of Fig. 9. The errors over Europe, Russia, and most of China now appear to meet the OCO mission requirement of 1 ppmv. Errors elsewhere are slightly larger, but it is likely that they can be reduced, as will be discussed later.

There are clues to the origin of the bias in the upper panel of Fig. 11, which shows the latitudinal dependence of the errors for all retrievals and all orbits. There appears to be a negative bias at higher latitudes, which is possibly stronger in the Southern Hemisphere, although the statistics there are much poorer than in the north. Large errors occur in the far north, but these are an artifact of the training, which was limited to solar zenith angles less than 75°. Furthermore, the angular grid for large zenith angles was too coarse. The lower panel of Fig. 11 is similar, except that the horizontal axis is longitude. Apart from the cluster of outliers near −40°, most of which correspond to the high northern latitude measurements over Greenland, there does not appear to be any significant correlation with longitude. Thus, the bias, which is quite clear in the upper panel of Fig. 11, seems to be the same regardless of the longitude. This suggests that the bias is not related to the type or topography of the land surface, which certainly varies with longitude.

The variation of solar zenith angle could cause a bias at high latitude. Indeed, when the errors in retrieved XCO2 are plotted against solar zenith angle, as in the upper panel of Fig. 12, there does appear to be an initial downward trend between 27° and 37°, but thereafter the error remains flat until 70°, at which point the limitations of the training cause the errors to increase dramatically. Thus, the behavior is not entirely consistent with the hump in Fig. 11.

The dependence of the XCO2 error upon the total scattering optical thickness of aerosol and cloud is shown in the middle panel of Fig. 12. In the 0.0 ≤ τ ≤ 0.1 range there appears to be almost no correlation, while in the 0.1 ≤ τ ≤ 0.3 range there is a slight increase in XCO2 error with the scattering optical thickness τ. Because we do not expect a strong correlation between the optical thickness of noncloud aerosol and latitude, especially in the Southern Hemisphere, then thin cloud would have to be the controlling factor if scattering were to be the cause of the high-latitude bias. This seems to be inconsistent with our understanding of the global distribution of thin cloud (Bréon et al. 2005).

The dependence of the AOPD correlation algorithm upon surface pressure is relatively weak, and the correlation with surface pressure shown in the lower panel of Fig. 12 is almost flat, except over very high terrain where the surface pressure is less than 650 hPa, near the lower limit of the pressure range used in training.

The most revealing correlation is between the XCO2 error and water vapor, as shown in the upper panel of Fig. 13. There appears to be a clear trend with the column amount of water vapor. This is supported by the lower panel of Fig. 13, which suggests a very weak increase in the error of XCO2 with increasing mean temperature of the atmospheric profile. The water vapor column responds most strongly to the temperature of the boundary layer, which in turn affects the mean temperature of the profile. To a good approximation, the water vapor column over the oceans will follow the Clausius–Clapeyron saturation vapor pressure curve, driven by the surface temperature (Stephens and Greenwald 1991a,b). Over land the relation between temperature and water vapor is much more complicated, but generally we expect a peak of the water vapor column near the tropics and a decreasing trend toward the poles.3 Furthermore, the shape of the pole-to-pole variation of water vapor is approximately the same at all longitudes, and therefore the hypothesis that the biases are caused by water vapor is consistent with Fig. 11, which shows the weak dependence of XCO2 errors upon longitude.

This discussion begs the question as to why water vapor might affect the AOPD results presented here. We do not think that the effect is related to spectroscopy, because the O2 A band and the weak CO2 band have little contamination by water vapor. However, we suspect that the fault lies in the training philosophy, where the surface pressure was used as a predictor rather than the partial pressure of dry air. As a simple illustration, consider two idealized and extreme cases, as shown in Table 4. The two scenarios have the same total mass, and hence the same surface pressure. They have the same constant profiles of mole fraction for CO2 and O2, but they have different masses of CO2 and O2. Of course the ratio of the CO2 mass to the O2 mass is the same in both cases; 4/2000 = 4.02/2010. The training, by chance rather than design, used an ensemble of atmospheres that was quite dry. When a moist atmosphere is presented to the algorithm, the number of CO2 molecules will be lower than in a dry atmosphere with the same surface pressure. Exactly how this will play out in the retrieved XCO2 is unclear, but we know that errors in surface pressure are correlated with errors in XCO2; a pressure error of 10 hPa produces an error of approximately 3 ppmv in XCO2. In the case of the example above, if the training is performed with dry atmospheres, represented by the “pole” column of Table 4, then the moist atmosphere of the “equator” column will have a pressure error of approximately 5 hPa, which would cause an XCO2 error of 1.5 ppmv, which is roughly comparable with that seen in the latitude dependence of Fig. 11.

If this hypothesis is correct, then one way to address the problem would be to train the algorithm with dry atmospheres, and then to estimate XCO2 using the partial pressure of dry air. The latter could be derived easily from the surface pressure by using the water vapor profile from ECMWF analyses. While there would be considerable uncertainty in interpolating the water vapor from the ECMWF grid to the target location, the accuracy probably would be sufficient, because the water vapor column provides only a small correction to the total pressure.

Last, the accuracy of the algorithm appears to change little with the vertical distribution of CO2. On average, the error in the retrieved XCO2 is about the same for atmospheres with uniform vertical profiles of CO2 as for atmospheres simulated with the parameterized chemical transport model (Kawa et al. 2004).

b. Glint mode

The corresponding results for the glint mode exhibit nearly all of the features discussed above, but with slightly larger biases. Figure 14 shows the spatial distribution of the glint orbits. The only differences worthy of note in glint mode are the sign of the bias, which is positive rather than negative, as shown in the histogram Fig. 15, and the distribution of error, which is skewed with a longer tail on the underestimate side. The panels of Fig. 16 show the errors averaged over 10° × 10° regions, again before and after a global bias has been removed.

8. Conclusions

The tests of the correlation algorithm with simulated data are encouraging. Training was performed with a fixed ensemble of scattering atmospheres, each with a uniform profile of CO2 mole fraction, overlying surfaces with prescribed Lambertian albedos. In generating the reflected spectra, an advanced radiative transfer solver was used, which accurately accounted for the effects of multiple scattering and polarization. The spectra used for testing shared the same solar model and gas spectroscopy, but otherwise reasonable steps were taken to make the testing and training datasets independent. In the test data, land surface were represented by unpolarized BRDFs from the Moderate Resolution Imaging Spectroradiometer (MODIS). The vertical profiles of temperature and water vapor were taken from ECMWF analyses, interpolated to the times, locations, and vertical grids of CloudSat observations. Wherever possible, the water and ice cloud profiles were drawn from CloudSat. The vertical distributions of CO2 were predicted with the parameterized chemical transport model, and again were interpolated to the times and locations of the CloudSat observations. The geometry was that for the simulated OCO orbits. The histograms of errors for single-shot retrievals are tight, with standard deviations approximately equal to 2 ppmv and bias less than 1 ppmv. There are some systematic errors in the retrieved XCO2, which appear to be associated with the misrepresentation of water vapor and/or temperature during training. The histograms of averages along segments of orbit tracks over 10° latitude bands come close to meeting the mission requirement for OCO.

The most serious weakness of the tests described here is the use of the same solar model and spectroscopy for both the training and testing phases. Work is in progress to adapt the algorithm to the Greenhouse Gases Observation Satellite (GOSAT) launched in January 2009 by the Japan Aerospace Exploration Agency (JAXA). The algorithm will be truly tested there.

Acknowledgments

This research was funded by NASA Jet Propulsion Laboratory Contract 1380533.

REFERENCES

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    • Search Google Scholar
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  • Baum, B. A., Heymsfield A. J. , Yang P. , and Bedka S. T. , 2005a: Bulk scattering properties for the remote sensing of ice clouds. Part I: Microphysical data and models. J. Appl. Meteor., 44 , 18851895.

    • Crossref
    • Search Google Scholar
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  • Baum, B. A., Yang P. , Heymsfield A. J. , Platnick S. , King M. D. , Hu Y-X. , and Bedka S. T. , 2005b: Bulk scattering properties for the remote sensing of ice clouds. Part II: Narrowband models. J. Appl. Meteor., 44 , 18961911.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bösch, H., and Coauthors, 2006: Space-based near-infrared CO2 measurements: Testing the Orbiting Carbon Observatory retrieval algorithm and validation concept using SCIAMACHY observations over Park Falls, Wisconsin. J. Geophys. Res., 111 , D23302. doi:10.1029/2006JD007080.

    • Search Google Scholar
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  • Bower, K. N., Choularton T. W. , Latham J. , Baker M. B. , and Jensen J. , 1994: A parameterization of warm clouds for use in atmospheric general circulation models. J. Atmos. Sci., 51 , 27222732.

    • Crossref
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    • Export Citation
  • Bréon, F. M., O’Brien D. M. , and Spinhirne J. D. , 2005: Scattering layer statistics from space borne GLAS observations. Geophys. Res. Lett., 32 , L22802. doi:10.1029/2005GL023825.

    • Search Google Scholar
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  • Chahine, M. T., and Coauthors, 2008: Satellite remote sounding of mid-tropospheric CO2. Geophys. Res. Lett., 35 , L17807. doi:10.1029/2008GL035022.

  • Chevallier, F., 2002: Sampled databases of 60-level atmospheric profiles from the ECMWF analyses. EUMETSAT/ECMWF, SAF Programme Research Rep. NWPSAF-EC-TR-004, 27 pp.

    • Search Google Scholar
    • Export Citation
  • Christi, M. J., and Stephens G. L. , 2004: Retrieving profiles of atmospheric CO2 in clear sky and in the presence of thin cloud using spectroscopy from the near and thermal infrared: A preliminary case study. J. Geophys. Res., 109 , D04316. doi:10.1029/2003JD004058.

    • Search Google Scholar
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  • Cox, C., and Munk W. H. , 1954a: The measurement of the roughness of the sea surface from photographs of the sun’s glitter. J. Opt. Soc. Amer., 44 , 838850. doi:10.1364/JOSA.44.000838.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cox, C., and Munk W. H. , 1954b: Statistics of the sea surface derived from sun glitter. J. Mar. Res., 13 , 198227.

  • Crevoisier, C., Chédin A. , Matsueda H. , Machida T. , Armante R. , and Scott N. A. , 2009: First year of upper tropospheric integrated content of CO2 from IASI hyperspectral infrared observations. Atmos. Chem. Phys., 9 , 47974810. doi:10.5194/acp-9-4797-2009.

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    • Search Google Scholar
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  • Dufour, E., and Bréon F-M. , 2003: Spaceborne estimate of atmospheric CO2 column by use of the differential absorption method: Error analysis. Appl. Opt., 42 , 35953609. doi:10.1364/AO.42.003595.

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  • Engelen, R. J., and Stephens G. L. , 2004: Information content of infrared satellite sounding measurements with respect to CO2. J. Appl. Meteor., 43 , 373378.

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  • Engelen, R. J., Denning A. S. , Gurney K. R. , and Stephens G. L. , 2001: Global observations of the carbon budget 1. Expected satellite capabilities for emission spectroscopy in the EOS and NPOESS eras. J. Geophys. Res., 106 , 20 05520 068.

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  • Engelen, R. J., Andersson E. , Chevallier F. , Hollingsworth A. , Matricardi M. , McNally A. P. , Thépaut J-N. , and Watts P. D. , 2004: Estimating atmospheric CO2 from advanced infrared satellite radiances within an operational 4D-Var data assimilation system: Methodology and first results. J. Geophys. Res., 109 , D19309. doi:10.1029/2004JD004777.

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  • Hartmann, J-M., Tran H. , and Toon G. C. , 2009: Influence of line mixing on the retrievals of atmospheric CO2 from spectra in the 1.6 and 2.1 μ regions. Atmos. Chem. Phys., 9 , 73037312. doi:10.5194/acp-9-7303-2009.

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Fig. 1.
Fig. 1.

Correlation between the AOPDs for a pair of selected channels in the O2 A band and weak CO2 band of OCO. The correlation is tight over the ensemble of scattering atmospheres.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 2.
Fig. 2.

Histograms of (top) cloud water and (bottom) the effective pressure level of cloud water. The cases with zero for the pressure level of water correspond to atmospheres free of cloud water.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 3.
Fig. 3.

Histograms of (top) cloud ice and (bottom) the effective pressure level of cloud ice. The cases with zero for the pressure level of ice correspond to atmospheres free of cloud ice.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 4.
Fig. 4.

Histograms of column water vapor (kg m−2) and mean temperature (K).

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 5.
Fig. 5.

Histogram of the combined optical thickness of water cloud, ice cloud, and aerosol at the frequency of the O2 A band.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 6.
Fig. 6.

The squares are upper boundary points for the mesh. The pentagons mark a unit cell. The circled pentagon is the lower corner of the unit cell.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 7.
Fig. 7.

Averaging kernel for clear-sky conditions. The horizontal axis denotes the functional derivative δX/δu(p). The CO2 profile has been assumed uniform for this calculation.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 8.
Fig. 8.

Single-shot XCO2 errors along the target track in nadir mode. Observations over the sea were discarded.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 9.
Fig. 9.

(left) Nadir mode XCO2 errors averaged over soundings within 10° × 10° regions, and (right) the same errors after a global bias of −0.86 ppmv has been removed from the retrieved values of XCO2.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 10.
Fig. 10.

Histogram of errors in single-shot retrievals of XCO2 for all nadir orbits. The bias is −0.86 ppmv.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 11.
Fig. 11.

Dependence on (top) latitude and (bottom) longitude of the errors in single-shot retrievals of XCO2 for the ensemble of all nadir orbits.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 12.
Fig. 12.

(from top to bottom) Dependence on the solar zenith angle, the combined cloud and aerosol optical thickness, and the surface pressure of the errors in single-shot retrievals of XCO2 for the ensemble of all nadir orbits.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 13.
Fig. 13.

(top) Dependence on the column water vapor of the errors in single-shot retrievals of XCO2 for the ensemble of all nadir orbits. (bottom) Dependence on the mean temperature of the profile.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 14.
Fig. 14.

Single-shot XCO2 errors along the target track in glint mode.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 15.
Fig. 15.

Histogram of errors in single-shot retrievals of XCO2 for all glint orbits. The bias is +0.4 ppmv.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Fig. 16.
Fig. 16.

(left) Glint mode XCO2 errors averaged over soundings within 10° × 10° regions, and (right) the same errors after a global bias of 0.4 ppmv has been removed from the retrieved values of XCO2.

Citation: Journal of Atmospheric and Oceanic Technology 27, 12; 10.1175/2010JTECHA1487.1

Table 1.

Breakdown of the profiles according to the column densities of cloud water and cloud ice. The total number of cases is 965.

Table 1.
Table 2.

Model variables for both nadir and glint modes over land.

Table 2.
Table 3.

Model variables for glint mode over ocean.

Table 3.
Table 4.

Two idealized scenarios representing columns with the same total pressure at the equator and the pole.

Table 4.

1

The apparent optical path difference is defined formally later. Here it suffices to say that the optical path (or thickness) is apparent because it is calculated from radiances measured at the top of the atmosphere as if the atmosphere were nonscattering. The optical path is a difference because each AOPD is the difference between optical paths at two frequencies, with one being a fixed reference frequency where absorption is weak.

2

The wavelength is in the reference frame of an observer standing at the target on the surface of the earth.

3

Of course there are major exceptions to this rule; for example, over the Sahara the temperature usually will be high but the air will be dry.

Save
  • Baum, B. A., cited. 2007: The development of ice cloud scattering models for use in remote sensing applications. SSEC, University of Wisconsin—Madison. [Available online at http://www.ssec.wisc.edu/~baum/Cirrus/IceCloudModels.html].

    • Search Google Scholar
    • Export Citation
  • Baum, B. A., Heymsfield A. J. , Yang P. , and Bedka S. T. , 2005a: Bulk scattering properties for the remote sensing of ice clouds. Part I: Microphysical data and models. J. Appl. Meteor., 44 , 18851895.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Baum, B. A., Yang P. , Heymsfield A. J. , Platnick S. , King M. D. , Hu Y-X. , and Bedka S. T. , 2005b: Bulk scattering properties for the remote sensing of ice clouds. Part II: Narrowband models. J. Appl. Meteor., 44 , 18961911.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bösch, H., and Coauthors, 2006: Space-based near-infrared CO2 measurements: Testing the Orbiting Carbon Observatory retrieval algorithm and validation concept using SCIAMACHY observations over Park Falls, Wisconsin. J. Geophys. Res., 111 , D23302. doi:10.1029/2006JD007080.

    • Search Google Scholar
    • Export Citation
  • Bower, K. N., Choularton T. W. , Latham J. , Baker M. B. , and Jensen J. , 1994: A parameterization of warm clouds for use in atmospheric general circulation models. J. Atmos. Sci., 51 , 27222732.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bréon, F. M., O’Brien D. M. , and Spinhirne J. D. , 2005: Scattering layer statistics from space borne GLAS observations. Geophys. Res. Lett., 32 , L22802. doi:10.1029/2005GL023825.

    • Search Google Scholar
    • Export Citation
  • Chahine, M. T., and Coauthors, 2008: Satellite remote sounding of mid-tropospheric CO2. Geophys. Res. Lett., 35 , L17807. doi:10.1029/2008GL035022.

  • Chevallier, F., 2002: Sampled databases of 60-level atmospheric profiles from the ECMWF analyses. EUMETSAT/ECMWF, SAF Programme Research Rep. NWPSAF-EC-TR-004, 27 pp.

    • Search Google Scholar
    • Export Citation
  • Christi, M. J., and Stephens G. L. , 2004: Retrieving profiles of atmospheric CO2 in clear sky and in the presence of thin cloud using spectroscopy from the near and thermal infrared: A preliminary case study. J. Geophys. Res., 109 , D04316. doi:10.1029/2003JD004058.

    • Search Google Scholar
    • Export Citation
  • Cox, C., and Munk W. H. , 1954a: The measurement of the roughness of the sea surface from photographs of the sun’s glitter. J. Opt. Soc. Amer., 44 , 838850. doi:10.1364/JOSA.44.000838.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cox, C., and Munk W. H. , 1954b: Statistics of the sea surface derived from sun glitter. J. Mar. Res., 13 , 198227.

  • Crevoisier, C., Chédin A. , Matsueda H. , Machida T. , Armante R. , and Scott N. A. , 2009: First year of upper tropospheric integrated content of CO2 from IASI hyperspectral infrared observations. Atmos. Chem. Phys., 9 , 47974810. doi:10.5194/acp-9-4797-2009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dufour, E., and Bréon F-M. , 2003: Spaceborne estimate of atmospheric CO2 column by use of the differential absorption method: Error analysis. Appl. Opt., 42 , 35953609. doi:10.1364/AO.42.003595.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Engelen, R. J., and Stephens G. L. , 2004: Information content of infrared satellite sounding measurements with respect to CO2. J. Appl. Meteor., 43 , 373378.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Engelen, R. J., Denning A. S. , Gurney K. R. , and Stephens G. L. , 2001: Global observations of the carbon budget 1. Expected satellite capabilities for emission spectroscopy in the EOS and NPOESS eras. J. Geophys. Res., 106 , 20 05520 068.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Engelen, R. J., Andersson E. , Chevallier F. , Hollingsworth A. , Matricardi M. , McNally A. P. , Thépaut J-N. , and Watts P. D. , 2004: Estimating atmospheric CO2 from advanced infrared satellite radiances within an operational 4D-Var data assimilation system: Methodology and first results. J. Geophys. Res., 109 , D19309. doi:10.1029/2004JD004777.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hartmann, J-M., Tran H. , and Toon G. C. , 2009: Influence of line mixing on the retrievals of atmospheric CO2 from spectra in the 1.6 and 2.1 μ regions. Atmos. Chem. Phys., 9 , 73037312. doi:10.5194/acp-9-7303-2009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ivanova, D., Mitchell D. L. , Arnott W. P. , and Poellot M. , 2001: A GCM parameterization for bimodal size spectra and ice mass removal rates in mid-latitude cirrus clouds. Atmos. Res., 59–60 , 89113. doi:10.1016/S0169-8095(01)00111-9.

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    • Export Citation
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  • Fig. 1.

    Correlation between the AOPDs for a pair of selected channels in the O2 A band and weak CO2 band of OCO. The correlation is tight over the ensemble of scattering atmospheres.

  • Fig. 2.

    Histograms of (top) cloud water and (bottom) the effective pressure level of cloud water. The cases with zero for the pressure level of water correspond to atmospheres free of cloud water.

  • Fig. 3.

    Histograms of (top) cloud ice and (bottom) the effective pressure level of cloud ice. The cases with zero for the pressure level of ice correspond to atmospheres free of cloud ice.

  • Fig. 4.

    Histograms of column water vapor (kg m−2) and mean temperature (K).

  • Fig. 5.

    Histogram of the combined optical thickness of water cloud, ice cloud, and aerosol at the frequency of the O2 A band.

  • Fig. 6.

    The squares are upper boundary points for the mesh. The pentagons mark a unit cell. The circled pentagon is the lower corner of the unit cell.

  • Fig. 7.

    Averaging kernel for clear-sky conditions. The horizontal axis denotes the functional derivative δX/δu(p). The CO2 profile has been assumed uniform for this calculation.

  • Fig. 8.

    Single-shot XCO2 errors along the target track in nadir mode. Observations over the sea were discarded.

  • Fig. 9.

    (left) Nadir mode XCO2 errors averaged over soundings within 10° × 10° regions, and (right) the same errors after a global bias of −0.86 ppmv has been removed from the retrieved values of XCO2.

  • Fig. 10.

    Histogram of errors in single-shot retrievals of XCO2 for all nadir orbits. The bias is −0.86 ppmv.

  • Fig. 11.

    Dependence on (top) latitude and (bottom) longitude of the errors in single-shot retrievals of XCO2 for the ensemble of all nadir orbits.

  • Fig. 12.

    (from top to bottom) Dependence on the solar zenith angle, the combined cloud and aerosol optical thickness, and the surface pressure of the errors in single-shot retrievals of XCO2 for the ensemble of all nadir orbits.

  • Fig. 13.

    (top) Dependence on the column water vapor of the errors in single-shot retrievals of XCO2 for the ensemble of all nadir orbits. (bottom) Dependence on the mean temperature of the profile.

  • Fig. 14.

    Single-shot XCO2 errors along the target track in glint mode.

  • Fig. 15.

    Histogram of errors in single-shot retrievals of XCO2 for all glint orbits. The bias is +0.4 ppmv.

  • Fig. 16.

    (left) Glint mode XCO2 errors averaged over soundings within 10° × 10° regions, and (right) the same errors after a global bias of 0.4 ppmv has been removed from the retrieved values of XCO2.

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