We present an analysis of nadir observations in six cases, encompassing summer and winter at 3 sites, namely Park Falls, Wisconsin, USA (46°N) (“PF”), Darwin, Northern Territories, Australia (12°S) (“DA”), and Lauder, New Zealand (45°S) (“LA”). These are not intended to be comprehensive of all relevant conditions, but to be examples of the range of conditions expected in observations over land at low and middle latitudes. They are also 3 of the sites where high-precision CO2 column measurements are made, which will be used for OCO validation. The goal of OCO is to produce reliable CO2 values in cloud-free footprints with aerosol optical depth τ < 0.3. The footprint of OCO was made unusually small (3 km2) to increase the number of soundings which are cloud-free, and a cloud screening algorithm is under development to identify them. Aerosol opacities >0.3 are not considered here.
 This analysis includes our best estimates of noise errors and geophysical variability, and a treatment of spectroscopic error, and is meant to be representative of our current understanding. A full error analysis, including instrument error, is beyond the scope of this study, and awaits completion and characterization of the OCO instrument hardware. Tests of the complete instrument will be conducted at the Jet Propulsion Laboratory during 2007–2008, followed by a complete prospective error analysis before launch.
4.2.1. Assumptions for Noise, Albedo, and Operational Constraints
 An example of the three spectra to be measured by OCO in each sounding is shown in Figure 1. Operationally, we expect to use a measurement noise covariance, Sɛ, which is diagonal, with values derived for each sounding by the operational calibration algorithm. These values consist of a constant component plus one varying as the square root of incident intensity. Thus noise will vary with scene brightness, so for present purposes we use the best available estimate of noise for each wave number and each scene. These vary significantly with surface type and spectral region, as given in the Table 3.
Table 3. Continuum Signal-to-Noise Ratio (SNR) for Aerosol Optical Depth = 0.1 for O2 A-Band, Weak CO2, and Strong CO2
 The albedo of each scene varies in a similarly complex manner. We have assumed the values in Table 4 which are taken from the ASTER Spectral Library (Vol. 1.2, available through http://asterweb.jpl.nasa.gov).
Table 4. Mean Surface Albedo for O2 A-Band, Weak CO2, and Strong CO2
 The spectra and the Jacobians have been simulated using H2O and temperature profiles from the ECWMF ERA 40 data set, CO2 profiles and surface pressure from the MATCH/CASA model run [Olsen and Randerson, 2004], each interpolated in time and space. We have used an exponentially decreasing, tropospheric aerosol profile with a scale height of 2 km, which has been scaled to reproduce the different total aerosol optical depths. The aerosol optical properties are for a continental type.
 Increasing the aerosol optical depth from 0.01 to 0.3 mostly increases the signal observed by the satellite instrument, depending on surface albedo and solar zenith angle. For very bright surface, aerosol extinction can also result in a decrease of the signal. For the O2 A-band region, we observe an increase of the signal of ∼1% for the continuum and by up to 10% for strongly absorbing regions. For the weak CO2 band, the intensity increases relatively smoothly throughout the band by several percent for bright surface and up to 50% for snow. The largest effect of aerosol is observed for the strong CO2 band region with an intensity increase for a vegetated surface of 10% for the continuum and up to 30% for strongly absorbing lines. For snow surfaces, the signal in the strong CO2 band region is dominated by contributions from aerosol scattering and we observe an increase between 130 and 170%.
 In addition to tropospheric aerosol, we have also included a fixed stratospheric aerosol profile, which is based on SAGE 2 measurements. The surface albedo is from the ASTER database as described above.
 The a priori covariance matrix, Sa, has been constructed out of several submatrices arranged along its diagonal, one for each of the physically distinct components of the state vector as given in section 2.2, i.e., CO2, H2O, etc. These components are assumed independent of each other, so all elements not within the submatrices are set to zero.
 The CO2 submatrix naturally has the most impact on retrieval of XCO2. In this paper all results use a single CO2 covariance matrix. That covariance has been constructed by assuming a root-mean-square (rms) variability of XCO2 of 12 ppm, which is an estimate of global variability [Dufour and Breon, 2003]. Variability as a function of height is assumed to decrease rapidly, from ∼10% at the surface to ∼1% in the stratosphere. The covariance among altitudes in the troposphere is derived from aircraft observations at Carr, Colorado (P. Tans, private communication, 2003). The total variability embodied in this covariance is unrealistically large for most of the world (all relatively clean air sites). It is intended to be a minimal constraint on the retrieved XCO2; the use of a single covariance everywhere at all times eliminates the covariance matrix as a source for variation in retrieval characteristics.
 The a priori covariance for the other state vector components has been set as follows. In all cases, we have assumed variability as large as expected at any of the sites, to minimize dependence on the a priori state vector itself. For H2O, we have used the observed variability at Park Falls in July, 1.4 cm precipitable water. For temperature, we have similarly used the covariance calculated for Park Falls in July, but imposed variability increasing in the lower troposphere to 10 K at the surface. The surface pressure is assumed to have a standard deviation of 20 mbar. For aerosol, total optical depth varies by ±0.15; the variability decreases from 150% at the surface to 50% with altitude with a scale height of 2 km; it has a correlation length of 1 km. The uncertainty in mean albedo is formally set to the unphysical value of ±1 (which effectively nullifies any a priori influence), with a slope which implies a variation of ±0.5 at each end of the spectral range.
4.2.2. Ensemble Variability and Error Sources
 We have considered 3 classes of error source: random, systematically variable, and fixed. Random errors include noise and some portion of each type of geophysical error, for example surface pressure. The last two classes merit more extended description. Certain errors will vary systematically with time and place, producing a bias in comparative values of XCO2. For example, errors due to aerosol will vary systematically with optical depth and zenith angle. We assume that such errors will not be reduced by averaging, but will produce a consistent error in XCO2, and class them as “bias” errors. This is an inherently conservative assumption, likely to overestimate the effect of these errors.
 The class of fixed error sources consists of those inherently constant everywhere at all times. They are exemplified by error in spectroscopic line parameters, which will be discussed in the next section.
 The submatrices making up the ensemble covariance matrix, Sc, used for calculating smoothing and interference error, are in principle chosen to be the best estimate available of actual atmospheric variability at the time and site in question. For the CO2 submatrix, the a priori covariance has been scaled to correspond to XCO2 variability observed with solar viewing FTS instruments by the authors at the Park Falls and Lauder sites, namely 2.5 and 0.7 ppm, respectively. For Darwin, 0.7 ppm is used as a representative value for the Southern Hemisphere.
 It will be seen below that interference error, that is, between CO2 and other state vector components, is relatively small, and so does not depend critically on the actual ensemble variability. In the current work we have made the following assumptions.
 For H2O and for temperature we have calculated covariance matrices based on ECMWF H2O for each site. For surface pressure, a 5-mbar standard deviation is based on pressure measurements at Park Falls. The aerosol covariance is based on an ad hoc constraint using a Markov description with a scale length of 1 km. The aerosol covariance has been subsequently scaled to reproduce a standard deviation for the total aerosol optical depth of ±0.09 for the Northern Hemisphere, and values half that in the Southern Hemisphere, which is based loosely on MODIS data for nonpolluted conditions. For albedo and spectral dispersion we have assumed the ensemble variability equals the a priori variability.
4.2.3. Spectroscopic Errors
 Spectroscopic errors belong to the class of error sources which are truly fixed. Unfortunately, because of the varying amount of information in each measured spectrum relative to the a priori constraint, the resulting errors in retrieved XCO2 are not fixed. In fact, the errors vary systematically with the observing conditions, depending principally on ground albedo, aerosol optical depth, and, most importantly, solar zenith angle. Fixed errors in XCO2 would leave the gradients unchanged, and so not affect flux inversions. However, the implications of such systematic variations in the XCO2 errors are profound; uncorrected, they will introduce significant biases into the determination of relative changes in XCO2, and consequently into inference of surface flux. This malign influence would be somewhat ameliorated if the correlations of the systematic error among soundings were properly taken into account.
 We have adopted a two-pronged approach to deal with spectroscopic errors. First, we will minimize them by a combination of laboratory measurements and the calibration of spectral measurements from upward looking spectrometers by in situ aircraft measurements. Washenfelder et al.  performed an extensive comparison of upward looking FTS data taken at Park Falls, Wisconsin to simultaneous aircraft measurements in situ. The FTS data were analyzed in one of the CO2 spectral bands to be used by OCO (λ ∼ 1.61 μm) as well as an adjacent CO2 band, and the 1.27 μm band of O2. They concluded that by appropriate adjustments to the band strength and air-broadened line width of these bands, that they could achieve an absolute accuracy of 0.25% in the O2 column, and 0.3% of the CO2 profile-weighted mean mixing ratio.
 The calibrated FTS column measurements are combined with laboratory measurements by the following procedure. It is observed that the retrieved column depends on assumed line width differently for weak and saturated lines, being proportional to width for saturated lines, and approximately proportional to the square root of width for weak lines. Thus we assume that the column measurements, C, are proportional to the product of the band intensity I and a single parameter characterizing line width, W:
where we expect β = 1 for the strong O2 A-band, and β = for the weaker CO2 bands.
 In effect, we then have two estimates of each parameter characterizing strength and width, one directly from laboratory measurement of that parameter, and the other by solving equation (22). These two estimates may be formally combined in a variance-weighted average, which statistically is the most likely value of the parameter, given the two independent estimates, and which itself has inverse variance given by the sum of the two inverse variances.
 In practice, we calculate the variance of this weighted average by performing a notional a posteriori retrieval of the intensity and width parameters, using the FTS column as the “measurement” and the laboratory values and uncertainties as “a priori information” on the intensity and width. For present purposes, it is a notional retrieval only because we are only interested in the net uncertainties in the strength and width parameters, not their values.
 There has been intensive laboratory work done recently on spectroscopy relevant to OCO. The near infrared bands of CO2 have been analyzed by Toth et al. [2006a, 2006b] and Devi et al. . On the basis of their work, we assume uncertainties in laboratory data of 0.5% for band strength and 1.0% for air-broadened line widths. The same investigators are now studying the O2 A-band. For present purposes, we will assume that uncertainties comparable to those for the CO2 bands will be achieved. We use the procedure described above with these values, and the FTS uncertainties given by Washenfelder et al.  for the measured columns. The resulting estimate of net uncertainties are given in Table 5.
Table 5. Spectroscopic Uncertainties Used for CO2 and O2
| ||Uncertainties, %|
|Column Measurement||Laboratory Strength||Laboratory Width||Net Strength||Net Width|
 Second, we will assess the residual errors in XCO2 and their variability by the techniques used in this paper, combined with the net width and strength errors shown in Table 5. To that end, we have analyzed twelve cases covering the realistically useful range of solar zenith angle, albedo, and aerosol optical depth, specifically zenith angles of 10° and 70°, albedo of 0.05, 0.2, and 0.5, and aerosol optical depth of 0.01 and 0.3.
 The critical issue is not the magnitude of the errors themselves, but rather how they vary with observing conditions, and thus spatially and temporally. Therefore we take the RMS variability of the errors to be indicative of typical bias errors in XCO2 gradients, and the maximum difference in the errors as the upper limit, for typical observing conditions, of XCO2 gradient error due to spectroscopic parameters. These values are shown in Table 6. They show that spectroscopic error, particularly CO2 air-broadened line width error, is a potentially important source of variable bias, but one which is usually smaller than 1 ppm, and always within our 1–2 ppm target. Nevertheless these results emphasize the need for both accurate laboratory data and for precise FTS column measurements at sites representing a broad range of observing conditions, well calibrated by aircraft overflights wherever possible. The OCO validation plan incorporates all of these elements in an effort to reduce such bias errors.
Table 6. Difference in Bias Error in XCO2 Due to Spectroscopic Parametersa
|Error Source||RMS Difference||Maximum Difference|
|CO2 band strength||0.1||0.2|
|CO2 line width||0.4||1.3|
|O2 band strength||0.1||0.5|
|O2 line width||0.2||0.6|
|H2O band strength||<0.1||<0.1|
|H2O line width||<0.1||0.2|
 It is also interesting to examine the averaging kernels for these cases, to understand how variations in observing conditions result in bias error. Figure 2 shows the column averaging kernels for 4 cases, as detailed in the Figure 2 caption. Case 1 corresponds to low aerosol values, high albedo (bright surface), and high sun, while case 4 has high aerosol for the same albedo and zenith angle. Similarly, case 2 has high aerosol, low albedo (dark surface), and low sun, while case 3 has the same except for high sun. The difference in XCO2 error from spectroscopic parameters is largest between cases 2 and 3, which have very different averaging kernels, owing to a large change in zenith angle in the presence of a dark surface and high aerosol scattering. Conversely, cases 1 and 4 show that changing aerosol makes little difference given a bright surface and high sun.
Figure 2. Averaging kernels spanning a wide range of three key observing parameters, solar zenith angle θ, surface albedo α, and aerosol optical depth τ. For case 1, θ = 10°, α = 0.5, and τ = 0.01. For case 2, θ = 70°, α = 0.05, and τ = 0.3. For case 3, θ = 10°, α = 0.05, and τ = 0.3. For case 4, θ = 10°, α = 0.5, and τ = 0.3.
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 The fact that fixed spectroscopic errors produce variable bias errors in XCO2 at all is worthy of comment. The fundamental reason for this behavior is the realistic assumption made in the a priori covariance input to the retrieval algorithm, that CO2 is more variable near the surface than at higher altitudes. This assumption has greater or lesser effect on the result, depending on the signal-to-noise ratio of the measured spectrum. The alternative, to assume an uncertainty uniform with altitude, would inhibit the algorithm from responding to real CO2 changes in the lower troposphere, and produce a bias dependent on the degree of boundary layer enhancement. Clearly, such enhancement is less well known than aerosol optical depth or albedo, which are retrieved from the data, or solar zenith angle.
4.2.4. Case Study Results
 The essential results for the six benchmark cases are shown in Figures 3 and 4and in Table 7. The six cases fall naturally into 2 groups, corresponding to high and low solar elevation. The high sun group includes Park Falls and Lauder in summer (July and January, respectively) and Darwin in both seasons, while winter at Park Falls (January) and Lauder (July) are low sun. We have performed error analyses for three values of aerosol optical depth at 0.76 microns, namely τ = 0.01, 0.1, 0.3. These values are expected to span the useful, cloud-free observing range.
Figure 3. (a) Column averaging kernels for aerosol optical depth τ = 0.01. The four high sun cases are Lauder in January, Park Falls in July, and both months at Darwin. (b) Same as Figure 3a but for τ = 0.3.
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Figure 4. Degrees of freedom for the CO2 profile for the six standard cases (Park Falls, Darwin, and Lauder in January and July), for τ = 0.1.
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Table 7. Error Sources and Magnitudes for the Benchmark Casesa
|Error Source||XCO2 Errors in ppm|
|Single Sounding||Regional Average|
|Random||Random + Bias||Random||Random + Bias|
|Park Falls, Jan|| || || || |
|Park Falls, Jul|| || || || |
|Darwin, Jan|| || || || |
|Darwin, Jul|| || || || |
|Lauder, Jan|| || || || |
|Lauder, Jul|| || || || |
 Figure 3 shows the column averaging kernels for the six cases. In Figure 3a, τ = 0.01, while in Figure 3b, τ = 0.3. In the lower optical depth case, all six cases are very similar, with values near 1 at the surface, decreasing to ∼0.7 at the tropopause.
 With increased aerosol optical depth (Figure 3b), winter at Park Falls and Lauder begin to stand out from the group, showing sensitivity slightly decreased at the ground but increased in the middle or upper troposphere. Both the winter results are driven by the same two factors: low sun and perturbed surface albedo, relative to the other cases. These factors act primarily to reduce the spectral signal-to-noise. In January at Park Falls, a snow covered surface is assumed, while at Lauder in July, “frost” is assumed. The snow surface is highly reflective in the O2 A-band, but very dark in the CO2 bands. The frost surface is intermediate, being similar to snow in the A-band, and similar to vegetation in the CO2 bands. The effect of low sun is for the contribution of light reflected from aerosol in the upper troposphere to grow relative to light reflected from the ground. This is exacerbated by lower albedo in the CO2 bands for snow covered surfaces.
 As illustrated in Figure 3, the effect of varying optical depth is modest for the conditions considered. For that reason, we have chosen to present all subsequent results for the intermediate case of τ = 0.1. The degrees of freedom for the CO2 profile are shown in Figure 4. There are approximately 1.4 to 1.5 degrees of freedom for the high sun cases, 1.2 for winter at Lauder, and 1.0 for winter at Park Falls.
 Table 7 shows detailed estimates of the contributions to XCO2 error for the six cases, both for a single sounding and for regional, semimonthly averages. The regional averages are taken to apply to an area 1000 × 1000 km. Because of the intensive computational demands of the OCO data, it is planned to process only 2% of the spectra with the algorithm described here, at least initially. Thus the regional averages are assumed to include typically 200 soundings. It is shown in Table 7 that the dominant random error, as expected, is noise. Also included in Table 7 are typical bias errors due to spectroscopic parameters, from Table 6. These are the dominant source of error in the semimonthly, regional averages.
 It is worth noting that, as a group interference errors (such as from the temperature profile) become both relatively and absolutely more important as sources of bias in the low signal-to-noise cases (Park Falls in January and Lauder in July). Even in those cases, noise is an insignificant contributor to regional average error. Thus uncertainties in the SNR values of Table 3 are unlikely to be important for regional averages.
 Concern is often expressed over the extent to which the a priori influences the retrieved result. Figure 5 shows the ratio of the retrieved to the a priori uncertainty for each state vector element, for the case of July at Park Falls. This ratio is a measure of the extent to which the measurement and retrieval process has reduced uncertainty. Values less than ∼0.5 indicate that the information in the measurement dominates the a priori information. Values near 1 show the a priori is dominant. It may be seen from Figure 5 that the measurement is overwhelmingly dominant for the albedo, spectral dispersion, and surface pressure. For CO2, the a priori plays a small role at each altitude, but for the column integral XCO2, the measurement dominates the a priori strongly. The temperature, water vapor, and aerosol profiles are fairly well determined by the measurement at lower altitudes, but strongly constrained by the a priori at higher altitudes.
Figure 5. Ratio of retrieved uncertainty to a priori uncertainty (“error reduction”) for Park Falls in July, τ = 0.1. CO2, H2O, temperature, and aerosol are each represented as profiles at 12 pressure levels from the ground to the stratopause. Within each profile, pressure decreases with increasing state vector index. Albedo and dispersion are described by six parameters each (see section 2.2).
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 Error in the retrieved values of XCO2 due to smoothing (CO2 profile variability) and interference is also a cause for concern. Figure 6 shows the quantity c (equation (16)), a direct estimate of this error due to each element in the state vector, again for the Park Falls July case. Note first that the errors are small for all state vector elements. The largest errors, 0.1–0.2 ppm, are due to smoothing by high-altitude CO2, and interference from tropospheric aerosol.
Figure 6. Smoothing and interference error in XCO2 due to each element of the state vector, for Park Falls, July, τ = 0.1. State vector index (y axis) as described for Figure 5.
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 Figure 7 summarizes the single sounding results from Table 7, separating error sources into noise, smoothing, interference, and forward model (the only forward model error considered here is spectroscopy). Noise is the largest error in all cases. Spectroscopic error is usually next, though interference error becomes larger in Park Falls winter. Note that smoothing is essentially negligible in all cases. Further note that interference error is near-negligible for the high sun cases. For the winter scenes with low sun, and snow covered scenes with low albedo in the CO2 bands, interference is significant, but noise becomes the dominant error.