We present an analysis of uncertainties in global measurements of the column
averaged dry-air mole fraction of CO

The Orbiting Carbon Observatory-2 (OCO-2) was launched on 2 July 2014 and
has been making global measurements of CO

In this paper we assess the uncertainty in a single XCO

Our methodology is described in detail in Sects. 2 and 3 below. In overview, simulations of OCO-2 spectra were run with the CSU orbit simulator (O'Brien et al., 2009), retrievals were performed with the operational Level-2 (L2) code, and a linear error analysis was performed using a dedicated offline code. This study used simulations to allow full control of the calculations and their inputs. The use of simulations for this analysis should not significantly affect its overall conclusions or their applicability to OCO-2 operational measurements, since the simulations are calculated from “true states” drawn from a realistic atmosphere.

Simulated spectra were calculated for 3 days in June and 3 in December. Only
nadir spectra were calculated over land, and only glint spectra over ocean.
Calculations were done for a single footprint (as opposed to eight footprints
for the flight data), with the sounding frequency set at 1 Hz (as compared
to 3 Hz in operation). The resulting reduction in data volume (factor of 24)
was done purely for convenience and should not affect conclusions from this
study. After cloud screening using the oxygen A-band preprocessor (ABP)
(Taylor et al., 2016), the operational L2 code was run on the simulations,
and the results were screened for convergence. A second screen was performed to
minimize the occurrence of outliers, by restricting the accepted range of
some sounding and retrieved parameters. More than 20 000 soundings
passed these screens, and those were run with the L2 code a second time,
using an extended state vector including a set of interfering aerosols. The
interfering aerosols are tightly constrained to very small values in this
step, but are included to force the L2 code to calculate their Jacobians.
The Jacobian matrix,

Linear error analysis (Rodgers, 2000; Rodgers and Connor, 2003) was performed on the extended L2 output, using an offline code developed for the purpose (Connor et al., 2008). We calculate actual uncertainties using estimates of true error in the measured spectrum, the variability of the atmospheric ensemble, and forward model errors.

These calculations are intended to apply to a comparison of L2 results to the true atmospheric values, without applying a “bias correction” (Wunch et al., 2011) to the L2 results.

The paper is organized as follows. In Sect. 2 we briefly discuss the L2 retrieval algorithm and then present details of the error analysis methodology. This is followed by an enumeration and discussion of the error sources to be considered in Sect. 3. Section 4 contains the results of the linear error analysis. Section 5 is a discussion of the results, and Sect. 6 identifies needs for future research.

The OCO-2 level 2 full physics retrieval algorithm (“L2”), consists of a
forward model and inverse method, described in full detail in JPL (2015).
The forward model is a radiative transfer model of the atmosphere coupled to
a model of the solar spectrum to calculate the monochromatic spectrum at the
top of the atmosphere, which is then convolved with the response function as
measured for the OCO-2 instrument. The inverse method is a maximum a
posteriori likelihood method of a type which has been widely used in the
community (Rodgers, 2000; Rodgers and Connor, 2003; Connor et al., 2008,
O'Dell et al., 2012). For comparison, the retrieval algorithm for the
spectrally similar measurements by the GOSAT satellite is described in
Yoshida et al. (2011). Uncertainty in the OCO-2 measurements of XCO

The error analysis algorithm performs a linear analysis using Jacobians calculated by the operational OCO-2 forward model. This section closely follows the discussion in Connor et al. (2008).

As defined in JPL (2015),

Equations (1)–(6) follow the definitions of Rodgers (2000) and Rodgers
and Connor (2003). Given

For each error in the list, we calculate the resulting covariance of the
retrieved state vector, as follows. For measurement error,

For forward model error,

For interference error, which refers to error in CO

Finally, the total covariance is

The discussion of the preceding paragraph makes two assumptions – one, that
the retrieval is approximately linear within the region bounded by its
uncertainty, and two, that the error sources considered are themselves
uncorrelated. Whenever error sources are correlated, the correlations must
be included in, e.g., Eqs. (4) or (6), and the net effect on XCO

Many of the error sources we will consider do not vary randomly, and some do
not vary at all. Spectroscopic errors belong to the class of error sources
that are truly fixed. Unfortunately, due to the varying amount of
information in each measured spectrum relative to the a priori constraint, embodied
in changes in the gain function,

We note that the gain function,

Sources and sinks of CO

We will refer to this differential uncertainty as “variable error”. By its nature, variable error has both random components and those which are systematic in the sense that they depend on conditions such as solar zenith angle, atmospheric temperature, pressure, and aerosol, and surface properties. In other words, the concept of variable error applies to the difference between any two soundings, but its magnitude will depend on the difference in conditions between them.

Our quantitative estimate of variable error is a composite error calculated
from a selection taken from all error sources described above. Variable
error will be calculated from all error sources, but will exclude the mean
error produced by fixed error sources as discussed in Sect. 2.2. Then a
first approximation to the predicted error in the difference of XCO

We will consider four types of error: measurement, smoothing, interference, and forward model.

The first and most obvious error is random noise in the measured spectrum.
This is calculated based on the operational noise model (JPL, 2015), and its
direct effect on XCO

However, it is observed that spectral residuals do not decrease with averaging as would be expected for pure random noise, but instead have a systematic structure. Because of this it was decided to derive empirical orthogonal functions (EOFs) representing this systematic structure, and to retrieve scale factors for these functions at every sounding (see Sect. 3.3.2.6 of JPL, 2015). Uncertainties in this process are to be addressed as interference error, below.

This represents error due to the a priori constraint of the state vector. As
suggested by Rodgers and Connor (2003), we have separated this into two
components. The first, smoothing by the true CO

The error due to the true atmospheric CO

We apply the Modern Era Retrospective analysis for Research and Applications
(MERRA) aerosol reanalysis climatology for daytime (local time 10:00,
13:00, and 16:00) in June and December, to represent the aerosol-related
variability in the OCO-2 spectral measurements (Rienecker et al., 2011). The
MERRA aerosol data are the basis for the OCO-2 forward model's aerosol types,
as described in detail in JPL (2015, pages 28–31). MERRA aerosol data consisting
of five composite types, namely dust (DU), sea salt (SS), sulfate (SU),
black carbon (BC), and organic carbon (OC), have nearly zero bias and a
correlation coefficient of

The L2 calculations for linear error analysis are performed at each sounding
with the operational state vector and a priori uncertainties, augmented as follows.
Ten additional aerosol quantities are added to the state vector, namely the
AOD for each of the five composite MERRA aerosols, integrated over two layers.
Using the relative pressure scale

Subsequently, the linear error analysis combines the Jacobians for all of
the aerosol and cloud quantities (liquid water, ice, the two types retrieved,
and the 10 additional interfering aerosols) with estimates of the ensemble
variability of their total atmospheric AOD, to calculate the resulting error
in XCO

The two retrieved aerosol types are counted twice by this procedure, once in
the operational state vector and again in the part of the state vector as
augmented for the error analysis. To avoid an error due to “double
counting”, we set the ensemble variance for the aerosols in the operational
state vector to very small values, ensuring they produce negligible error in
retrieved XCO

Interference errors due to the scale factors applied to the operational EOFs
are calculated as part of the error analysis by including the actual EOF
scale factors in the state vector. The results show negligible effects on
XCO

Other non-CO

For all of these components, an effort has been made to include an estimate of global variability as the a priori uncertainty, and this has been used as the estimated ensemble variability in the error analysis. The net effect of these uncertainties is fairly small compared to aerosols and forward model errors, so refining this ensemble estimate has not been a high priority, but may be considered later.

Uncertainties in spectroscopic parameters used in the L2 algorithm.

Forward model errors that have been evaluated in this analysis include those due to a variety of spectroscopic and calibration parameters.

Table 1 shows the estimated uncertainties in spectroscopic parameters used
in the L2 algorithm. The parameters listed are those required for the
spectroscopic line shape models used within the OCO-2 v7 L2 algorithm. For
CO

The majority of the uncertainties listed in Table 1 are based on published
values. The notable exceptions are speed dependence in the CO

It is also worth noting that the exponent of the temperature dependence of
the pressure broadened line widths in the O

A discrepancy between recent measurements of the line
strength in the WCO2 band is also of note. The values used by the OCO-2 algorithm are based
on Devi et al. (2007, 2016). Values from Polyansky et al. (2015) differ from
those in Devi et al. (2016) by

Spectroscopic uncertainties in interfering gas species are not a significant
source of error in retrieved XCO

Uncertainties in the calibration parameters are shown in Table 2. These are based on pre-flight laboratory calibration of the instrument at the Jet Propulsion Laboratory. The parameters are defined as follows. The instrument line shape (ILS) in each band is assumed to have a single uncertainty, in its width. Its shape as measured in the laboratory before launch is assumed to be correct. Radiometric gain is the factor applied to the measured voltages to convert them to absolute physical units. Finally, OCO-2 is only sensitive to one polarization of the incoming radiation, whose angle of orientation is the “polarization angle”.

Uncertainties in calibration parameters used in the L2 algorithm.

In applying the uncertainties in polarization angle, we note that the
observed spectrum

Figures 1–6 and Tables 3 and 4, below, display the summary of
results for the offline error analysis. The data are gridded into 10

Measurement error. Top: June, land; second row: June, ocean; third row: Dec., land; bottom: Dec., ocean.

Error due to spectroscopy. Top: June, land; second row: June, ocean; third row: Dec., land; bottom: Dec., ocean.

Instrument error. Top: June, land; second row: June, ocean; third row: Dec., land; bottom: Dec., ocean.

Figure 1 shows measurement error due to random noise in the measured
spectra. It is typically

Variable error. Top: June, land; second row: June, ocean; third row: Dec. land; bottom: Dec., ocean.

Total error. Top: June, land; second row: June, ocean; third row: Dec. land; bottom: Dec., ocean.

Forward model error, divided into spectroscopic and instrument error, is
shown in Figs. 2 and 3, respectively. Spectroscopic and instrument error
make roughly equal contributions to forward model error. Spectroscopic error
in ocean glint observations shows little variation, and is

Maps of aerosol error are shown in Fig. 4a, and for comparison, the monthly
mean aerosol optical depth from MERRA is shown in Fig. 4b and its standard
deviation in Fig. 4c. The sensitivity of XCO

In most places, aerosol errors are surprisingly small, typically

Variable error is shown in Fig. 5. Comparison to Figs. 3 and 4 shows that
variable error over land is dominated by instrument error (due to instrument
line shape), but also by aerosol error over ocean. It is typically

Total error from all sources is shown in Fig. 6. It is

Inspection of the global mean and standard deviations in Tables 3 and 4
gives rise to some interesting observations. In general, the fixed, or
approximately fixed, error sources (spectroscopy and instrument
calibration) cause mean errors much larger than their standard deviations.
This implies that whatever the true value of the error in the relevant
forward model parameter, most of its effect can in principle be removed by
simple bias correction based on validation measurements. However, the
remaining, variable error, caused by the fixed error source, is of critical
importance. The error in the difference in XCO

As noted above, spectroscopic error varies little over the ocean, and
modestly over land. The main sources of this behavior can be traced to
WCO

Three components of instrument error were analyzed. Error due to uncertainty
in the polarization angle

Global mean errors in XCO

Global mean errors in XCO

* driven by Sahara dust and high-latitude outliers.

Uncertainty due to smoothing error is fairly small. It is typically

Tables 3 and 4 also emphasize that the dominant variable error is due to
aerosol. Although the absolute size of the aerosol error is fairly small, it
varies widely from place to place, with a standard deviation up to 195 %
of its mean value (the coefficients of variation are 134, 109,
195, and 132 % for June nadir – land, Dec. nadir – land, June glint –
water, and Dec. glint – water, respectively). Furthermore, it will depend on the
actual atmospheric aerosol distribution, which will vary in a complex
fashion with space and time. Correlation of the aerosol distribution is
likely to be a major source of correlation in XCO

We envisage a continual ongoing analysis to quantify uncertainties in the OCO-2 measurements. We believe such quantification is critical for using the data to constrain the geophysical carbon cycle. Linear error analysis as presented here will be a key part of that effort, and it is important to replicate the analysis when future versions of the L2 algorithm are released and mission data are reprocessed. The error analysis should be extended to further examine errors produced by the algorithm itself. This would include studying the effect of errors in algorithm inputs such as the a priori state vector. A more general subject for study is nonlinearity of the forward and inverse models. Both of these areas are foci of active research. In the particular case of nonlinearity, linear error analysis can be supplemented with Monte Carlo studies. The Monte Carlo approach can interrogate the probability distribution of retrieval errors under specified conditions and can characterize correlations between multiple error sources, such as interference and nonlinearity, for example. Monte Carlo studies require far more computational effort than the linear error analysis, so experiments should be designed for a carefully selected subset of conditions.

Specific recommendations for linear error analysis include the following. Linear error analysis, as applied here to simulations, will be used to estimate uncertainties in selected sets of actual OCO-2 measurements. This work will have two main goals. First, we will analyze sets of OCO-2 measurements, which have been used for top-down error estimates and validation, by comparison to data from TCCON (the Total Carbon Column Observing Network) and by examining observed scatter in uniform, local areas. The volume of OCO-2 data has provided a large collection of validation data sets for many regions, spanning all seasons. The results of these top-down estimates will be compared to the bottom-up estimates of linear error analysis. If these two types of estimates are consistent they will give us confidence in our overall understanding of measurement uncertainty. Any inconsistencies will require further investigation. One possible source of inconsistencies, already under investigation as described above, is nonlinearity of the forward model.

Second, the variability of the bottom-up estimates will be systematically compared to the variation in sounding geometry, atmospheric conditions, and surface type. This will improve insight into the causes of measurement uncertainty, and guide data users in quantitative applications.

The simulations and analyses reported in this paper are not a part of the OCO-2 data set or of the public record of the OCO-2 project. They are exploratory in nature and not publically accessible because it has not been feasible to select, catalog, and document the relevant material.

We thank the following members of the OCO-2 team for support and helpful discussions: Vijay Natraj, Linda Brown, Brian Drouin, Chris Benner, Malathy Devi, and Annmarie Eldering. Part of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The CSU contribution to this work was supported by JPL subcontract 1439002. The contribution by BC Scientific Consulting was supported by JPL subcontract 1518224. Edited by: I. Aben Reviewed by: two anonymous referees