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**Atmospheric Measurement Techniques**
An interactive open-access journal of the European Geosciences Union

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- Abstract
- Introduction
- Biases in OCO-2 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ due to erroneous surface pressure estimates
- Evaluation of OCO-2's footprint geolocations
- The OCO-2 v9 data product
- Brief evaluation of OCO-2 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ data
- Conclusions
- Data availability
- Appendix A: Theoretical motivation of dP parameters in the v9 parametric bias correction
- Author contributions
- Competing interests
- Acknowledgements
- References

**Research article**
12 Apr 2019

**Research article** | 12 Apr 2019

How bias correction goes wrong: measurement of X_{CO2} affected by erroneous surface pressure estimates

^{1}Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, USA^{2}Division of Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA, USA^{3}Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA^{4}Colorado State University, Fort Collins, CO, USA^{5}Climate Research Division, Environment and Climate Change Canada, Toronto, ON, Canada^{6}Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada

^{1}Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, USA^{2}Division of Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA, USA^{3}Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA^{4}Colorado State University, Fort Collins, CO, USA^{5}Climate Research Division, Environment and Climate Change Canada, Toronto, ON, Canada^{6}Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada

**Correspondence**: Matthäus Kiel (mkiel@caltech.edu)

**Correspondence**: Matthäus Kiel (mkiel@caltech.edu)

Abstract

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All measurements of ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ from space have systematic errors. To reduce a large fraction of these errors, a bias correction is applied to ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ retrieved from GOSAT and OCO-2 spectra using the ACOS retrieval algorithm. The bias correction uses, among other parameters, the surface pressure difference between the retrieval and the meteorological reanalysis. Relative errors in the surface pressure estimates, however, propagate nearly 1:1 into relative errors in bias-corrected ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. For OCO-2, small errors in the knowledge of the pointing of the observatory (up to ∼130 arcsec) introduce a bias in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ in regions with rough topography. Erroneous surface pressure estimates are also caused by a coding error in ACOS version 8, sampling meteorological analyses at wrong times (up to 3 h after the overpass time). Here, we derive new geolocations for OCO-2's eight footprints and show how using improved knowledge of surface pressure estimates in the bias correction reduces errors in OCO-2's v9 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ data.

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Kiel, M., O'Dell, C. W., Fisher, B., Eldering, A., Nassar, R., MacDonald, C. G., and Wennberg, P. O.: How bias correction goes wrong: measurement of X_{CO2} affected by erroneous surface pressure estimates, Atmos. Meas. Tech., 12, 2241–2259, https://doi.org/10.5194/amt-12-2241-2019, 2019.

1 Introduction

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Atmospheric carbon dioxide (CO_{2}) is currently being
measured from space by, among other instruments, NASA's Orbiting Carbon
Observatory 2 (OCO-2) and JAXA's Greenhouse gases Observing SATellite
(GOSAT). Accurate and precise measurements of atmospheric CO_{2} can
identify and quantify its sources and sinks and, more generally, improve our
understanding of biosphere–atmosphere fluxes. To do so, these measurements
must be sufficiently accurate and precise to properly capture the small (<1 %) spatial and temporal gradients of CO_{2}. OCO-2 and
GOSAT ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ data have been widely used in studies to
characterize fluxes from different sources, e.g., emissions from power plants
(Nassar et al., 2017) or fires in Indonesia (Heymann et al., 2017). Other recent
studies analyzed flux anomalies during El Niño periods (Liu et al., 2017).

OCO-2 and GOSAT share a common observational approach: solar reflectance
spectra centered around 1.6 and 2.0 µm are used to determine the
CO_{2} optical depth. The O_{2} optical depth is observed in the
so-called “A band” centered around
0.76 µm. The column-averaged dry air mole fraction of
CO_{2} (${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$) is determined by combining the
information from these three spectral regions. The A band is used to
determine the amount of dry air along the O_{2} optical path from the
sun to the spectrometer (air mass). The two CO_{2} bands provide a
measure of how many CO_{2} molecules are in the similar paths.
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ is the ratio of CO_{2} to the dry surface pressure.
Any error that does not affect both the CO_{2} measurement and dry
surface pressure in the same way is expected to propagate into
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. A fundamental challenge for the retrieval is that
photons are scattered in the atmosphere, and the efficiency of the scattering
– primarily by clouds and aerosols – depends on wavelength. The
wavelength-dependent scattering is, in turn, estimated by the retrieval
algorithm using information from both the O_{2} spectra and the relative
CO_{2} optical depths determined from the two different CO_{2}
bands.

Early analysis of ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ from the initial GOSAT spectra had
global and regional systematic errors. Wunch et al. (2011b) demonstrated,
however, that a large fraction of the error in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ was
correlated with retrieved components of the state vector in the retrieval
algorithm. In particular, difference between the retrieval of surface
pressure and that from the meteorological reanalysis was shown to correlate
with error of similar magnitude in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ (e.g., when the surface
pressure retrieval was ∼1 % too large, the retrieved
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ was ∼1 % too small). There are several reasons
why surface pressure is not accurately retrieved from the spectra. First,
errors in the knowledge of the spectroscopy of oxygen can produce spurious
air mass dependencies and can affect the pressure retrieval
(e.g., Yang et al., 2005; Wunch et al., 2011a). Second, the algorithm is not adequately
able to distinguish pathlength errors due to scattering from those due to
surface pressure variation. For example, overestimates of the amount of
aerosol near the surface (which shortens the path) can be compensated for by an
overestimate of surface pressure. Because in the retrieval aerosols are
generally assumed to scatter less efficiently at longer wavelengths, error in
retrieved pathlength maps differently into O_{2} and CO_{2},
resulting in a bias in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. Pathlength errors also largely
depend on surface albedo. For example, if the surface albedo is high,
multiple reflections between the surface and the aerosol layer are efficient
and lengthen the path. Moreover, the spectral variation of surface albedo and
aerosol optical properties also change the radiative transfer between the
A band and CO_{2} bands. For example, differences in the absorption
optical thickness structure between the three bands induce band-dependent
height sensitivities to different types of aerosols in the retrieval.

Several retrieval codes that have been used to analyze GOSAT and OCO-2 spectra treat this problem differently. For example, the RemoTeC algorithm does not retrieve the surface pressure from the spectra. It uses the surface pressure from the meteorological reanalysis (Butz et al., 2011; Wu et al., 2018). Others, such as the University of Leicester Full Physics algorithm (UoL-FP) first normalizes the retrieved ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ by the ratio of the retrieved surface pressure from the spectra and the surface pressure from the meteorological reanalysis. Then it uses the difference between the retrieved surface pressure and that from the meteorological reanalysis to “bias correct” the ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ product (Cogan et al., 2012). To date, all versions of the Atmospheric Carbon Observations from Space (ACOS) retrieval algorithm (O'Dell et al., 2012, 2018; Crisp et al., 2012), used for both OCO-2 and GOSAT spectra, have also used the surface pressure difference between the retrieval and that diagnosed from the meteorological reanalysis to bias correct the ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ product. This bias correction demonstrably improves the data set (Wunch et al., 2011b, 2017b; O'Dell et al., 2018). It also, however, places new demands on the accuracy of the meteorological analysis – demands that had not been considered at the time the OCO-2 mission was conceived. Error in the assumed pressure from the meteorological reanalyses at the field of view of the spectrometers will propagate nearly 1:1 into bias-corrected ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. Over land, for example, small errors in the knowledge of the pointing of the observatory can yield significant errors in estimates of surface pressure in regions with rough topography. This is illustrated in Wunch et al. (2017b), in which ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ variations near Lauder, New Zealand, showed strong sensitivity to (different) estimates of the pointing of OCO-2, introducing an apparent topography-related bias in the data. Finally, due to atmospheric tides, the estimate of the surface pressure is sensitive to when the meteorological reanalysis is sampled. Given the precision we need to achieve in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ measurements, seemly insignificant issues can not necessarily be ignored. For example, the mean canopy height of the Amazon rain forest is ∼25 m (Benson et al., 2016) and might vary temporally due to fires or deforestation. Furthermore, the usual tidal range in the open ocean is ∼0.5 m, but coastal tidal ranges can reach up to 12 m (NOAA, 2018). At sea level, altitude variations of ∼8 m correspond to changes in surface pressure of ∼1 hPa. This might introduce errors in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ on the order of ∼0.4 ppm.

In this analysis, we address two issues with the OCO-2 v8 estimate of surface pressure: erroneous surface pressure values from the meteorological reanalysis due to small miss-specifications of the geolocations of OCO-2's eight footprints in the instrument-to-spacecraft pointing offsets and erroneous surface pressure estimates due to sampling the meteorological reanalysis at incorrect times. We illustrate how, using improved knowledge of the surface pressure, we can improve the bias correction and reduce errors in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. The resulting hybrid product which uses version 8 (v8) retrieval results with a revised bias correction using updated surface pressure estimates is labeled as version 9 (v9). This paper is structured as follows: Sect. 2 describes the impact of erroneous surface pressure estimates in the bias correction on ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ estimates. New footprint geolocations for OCO-2 are derived in Sect. 3. Section 4 introduces the revised parametric bias correction in v9 and discusses changes in the v9 filtration scheme. Section 5 gives a brief evaluation of the OCO-2 v9 data product and illustrates changes and improvements of v9 over v8 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ on regional and global scales.

2 Biases in OCO-2 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ due to erroneous surface pressure estimates

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OCO-2 v8 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ estimates are derived using the ACOS retrieval
algorithm. The algorithm uses optimal estimation to solve for parameters of
the state vector to obtain the best match to spectra recorded in OCO-2's
three spectral bands. The state vector includes, among other parameters, the
surface pressure which is primarily derived from information retrieved from
the O_{2} A band. The prior surface pressure is taken from the GEOS-5
Forward Processing for Instrument Teams Atmospheric Data Assimilation System
(GEOS5-FP-IT; Suarez et al., 2008; Lucchesi, 2013) and is sampled at the
geolocation of each OCO-2 sounding. Surface pressure and prior surface
pressure are used in the bias correction of ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. The OCO-2
bias correction addresses three types of biases: footprint-dependent biases,
parameter-dependent biases, and a global scaling of ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ to
the World Meteorological Organization (WMO) trace-gas standard scale using
comparisons to the Total Carbon Column Observing Network
(TCCON; Wunch et al., 2011a). An overview of the three different bias
correction terms is given in Mandrake et al. (2015), Wunch et al. (2017b), and
O'Dell et al. (2018).

Biases in OCO-2 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ due to erroneous surface pressure
estimates were initially illustrated in OCO-2 observations over Lauder, New
Zealand (Fig. 10 in Wunch et al., 2017b). The Lauder TCCON site is situated in
a remote area with no urban sources of ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ nearby
(Pollard et al., 2017). The area is dominated by rolling hills, with mountain
ridges spanning from southwest to northeast, almost perpendicular to the
ground track of the observatory (southeast to northwest). The terrain changes
up to ±200 m in altitude over small distances (see
Fig. 1a). Figure 1b shows
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ enhancements retrieved by the ACOS algorithm (v8) over
Lauder for a target observation on 17 February 2015. No bias correction is
applied here. ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ estimates are uniformly distributed over
the observed scene with a mean value of 393.58 ppm and a standard
deviation of 0.92 ppm. Figure 1c shows OCO-2
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ estimates after the v8 bias correction is applied. The
bias correction changes the mean value to 395.95 ppm and increases
the standard deviation to 1.35 ppm. Bias-corrected
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ enhancements vary up to ±3 ppm over the
observed scene. The bias is spatially correlated with the underlying
topography, more precisely, with the topographic slopes. The observed bias is
introduced by erroneous values of the prior surface pressure in the
d*P* term (the difference between the retrieved surface pressure and
the prior surface pressure) in the parametric bias correction. The parametric
bias correction accounts for spurious variability in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$
which correlates with retrieval parameters like albedo, retrieval aerosol
quantities, or surface pressure. A multivariate regression is performed
between spurious ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ variability and the parameters that
account for the largest variance in the data to correct for these errors
(Wunch et al., 2011b; Mandrake et al., 2015; O'Dell et al., 2018). The erroneous values of the
prior surface pressure are caused by small misspecifications in the
geolocations of OCO-2's eight footprints in the specified
instrument-to-spacecraft pointing. As stated previously, at sea level, a
surface pressure difference of 1 hPa corresponds to an altitude
difference of ∼8 m. Therefore, in areas like Lauder with steep
topography, misspecifications in the pointing of the observatory of a few
arcseconds can cause the prior surface pressure to be substantially different
from the retrieved surface pressure. This introduces errors in bias-corrected
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$, typically observed on local scales in areas with highly
varying topography.

Another source for erroneous surface pressure estimates in v8 is caused by a temporal sampling error of the surface pressure estimate from the meteorological reanalysis. The prior surface pressure is taken from the GEOS5-FP-IT 3-hourly output. A coding error in the meteorological sampling algorithm caused for some soundings the surface pressure estimate to be sampled as much as 3 h after the overpass time. This mostly affected soundings of orbits whose first and last soundings fully lie between synoptic GEOS5-FP-IT's 3-hourly outputs (00:00, 03:00, etc.); the soundings in such an orbit would be erroneously sampled at the upper bounding synoptic time for that orbit. For example, for an orbit whose soundings lie fully between 06:00 and 09:00 UTC, the OCO-2 meteorological sampling algorithm erroneously samples the GEOS5-FP-IT surface pressure field at 09:00 UTC for each sounding in that orbit. On average, this introduced a mean prior surface pressure error of about +0.5 hPa for affected soundings. In some cases, however, the prior surface pressure error reached up to ±20 hPa for individual soundings. The sampling error also affects temperature and water vapor. Soundings over land are affected more than over ocean since diurnal surface heating tends to be stronger over land and because the surface pressure bias correction term over land is nearly 50 % larger than over water. While the sampling error of the prior surface pressure is easy to correct for via the bias correction by fixing the coding error and rerunning the meteorological sampling algorithm, erroneous surface pressure estimates caused by misspecifications in the instrument pointing offsets need greater attention.

3 Evaluation of OCO-2's footprint geolocations

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The core of the OCO-2 instrument is a three-channel grating spectrometer that
records spectra of reflected sunlight in the O_{2} A band
(0.76 µm), the weak CO_{2} band (1.61 µm), and the
strong CO_{2} band (2.06 µm). The incoming light is guided
through a common optics assembly, but the light is sampled and focused
sequentially and independently onto three spectrometer slits, each
3 mm long and 28 µm wide (Haring et al., 2004; Crisp et al., 2017).
These long, narrow slits are aligned to produce nominally co-boresighted
fields of view. After passing the slit and being spectrally dispersed, the
light is focused on a two-dimensional focal plane array (FPA) with eight
independent readouts along the slits – the so-called footprints. Spectra for
the three spectral bands and each footprint are recorded simultaneously.

To obtain the best estimate for the geolocation of the eight footprints, the
following must be known: (1) the location of the spacecraft along the orbit
track, (2) the pointing of the instrument boresight relative to a local
coordinate system, and (3) the relative pointing of the fields of view (FOV)
of the eight footprints in the three spectrometers. A Global Positioning
System (GPS) sensor provides the location of the observatory along its orbit
track. The on-board star tracker determines the orientation of the
observatory relative to fixed stars. The relative alignment of the eight
footprints is characterized with respect to the spacecraft body axes. The
spatial FOV, defined along the long axis of the slit by the eight footprints,
is aligned parallel with the spacecraft *y* axis. The boresight of the
spectrometer points down the *x* axis. The spacecraft *z* axis points across
the narrow axis of the spectrometer slit, perpendicular to the *y* axis (see
Fig. 2). For nadir and glint measurements, the *z* axis is
rotated around the *x* axis so it is oriented 30^{∘} (clockwise from
above) from the principal plane (i.e., the plane that includes the sun, the
surface target, and the instrument aperture). To maintain this viewing
geometry, the spacecraft slowly rotates counterclockwise (from above) around
the *x* axis as it travels from the southern terminator, across the sub-solar
latitude, to the northern terminator. South of a latitude that is
∼30^{∘} north of the sub-solar latitude, footprint 1 (FP 1) is to
the west of footprint 8 (FP 8). North of this latitude, FP 1 is east of FP 8.
For target mode observations, the *z* axis is always pointed along the
spacecraft orbit track, so that FP 1 is always to the west of FP 8.
Pre-launch instrument ground tests were performed to characterize the spatial
FOV of each footprint, and correction factors – the so-called pointing
offsets – have been derived and integrated into the geometric calibration
algorithm (v0001 configuration; see Fig. 3). The pointing
offsets are on the order of hundreds of arcseconds. A change in the
pointing offsets of, for example, 25 arcsec corresponds to a shift of
the instrument FOV of ∼80 m at nadir. During the OCO-2 in-orbit
checkout (IOC) period in 2014, lunar measurements were performed, and in
combination with data from coastal crossings the alignment of the three
spectrometer slits was tested. The alignment of the instrument angular
footprints in the coordinate system defined by the star tracker was within
mission requirements (<720 arcsec). Updated pointing offsets were integrated into the geometric calibration algorithm in November 2014
(v0006 configuration; see Fig. 3). The findings in the
previous section, however, indicate that a reevaluation of the pointing
vector correction factors is desirable.

The analysis of the IOC lunar data exposed some deficiencies of its usage in
elaborating footprint geolocations. Lunar data are typically taken in
so-called single pixel mode when each pixel of the array is read out
individually. This is in contrast to normal operations where 20 spatial pixel
samples are co-added to form each footprint. In addition, the moon only
illuminates a fraction of the FPA. Furthermore, defocus compromises the
analysis of the strong CO_{2} band results, and the moon only provides
positive constraints for the *z* axis.

To overcome the aforementioned limitations for the v0006 configuration, the
IOC lunar data results were used to constrain the pointing vector for FP 6
and 7, whereas for the other FPs the ground test results were used. Here, we
follow a different approach to derive new pointing offsets. We shift from
estimating geolocations with lunar images, which are strictly geometric
measurements, to optimizing footprint geolocations with retrieval variables.
We utilize the ACOS Level 2 Full Physics (L2FP) algorithm and its associated
prescreeners, the A-band Preprocessor (ABP) and the IMAP-DOAS Preprocessor
(IDP) to estimate footprint geolocations. The ABP performs a fast retrieval
of surface pressure using the O_{2} A band and assumes that no clouds or
aerosols are present. The IDP performs clear-sky fits to the weak and strong
CO_{2} bands to derive CO_{2} columns (Taylor et al., 2016). Using the
preprocessors instead of the L2FP algorithm saves computational effort and
allows us to study pointing offsets for each spectral band individually. The
footprint geolocations for the O_{2} A band are derived by minimizing
the variation in the difference between the surface pressure retrieved from
the ABP and the meteorological analysis (d*P*_{ABP}). The
location of the CO_{2} band footprints is determined by minimizing the
variation in the CO_{2} columns divided by the dry air column determined
from the meteorological analysis (${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{met}}$). These
two metrics are systematically explored for a set of different pointing
offsets. The geolocations that provide the smallest standard deviation over a
given scene for d*P*_{ABP} are good estimates for the
location of the O_{2} A band. The same holds for the standard deviation
of ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{met}}$ regarding the weak and strong CO_{2}
band. The assumption here is that there are no significant variations in
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ over the field of analysis. This may not be true in
regions with large heterogeneous sources (e.g., urban areas) or sinks
(vegetated areas) of CO_{2}. It is only true for areas with a clean
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ background. Therefore, in our analysis we focus on remote
desert-like mountainous areas to study pointing offsets.

We identify two desert areas in the Northern and Southern Hemisphere with
topographic relief and frequent clear-sky conditions during nadir and glint
observations to derive new footprint geolocations: a remote area in the Death
Valley National Park, CA, USA, and an area in the Atacama Desert, Chile. The
Death Valley National Park area ranges from 35 to 37^{∘} N and from 118
to 115^{∘} W. The area in the Atacama Desert ranges from 18 to
19^{∘} S and from 69.8 to 69.25^{∘} W. Both areas are far from
anthropogenic CO_{2} sources. A topography-related bias in v8
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ is apparent in both areas (see
Fig. 4). Observations over the Death Valley National
Park include ∼1800 soundings from September 2014 to September 2017.
Observations over the Atacama Desert include ∼1000 soundings from
September 2014 to October 2017. All these soundings are aggregated into
0.02^{∘} × 0.02^{∘} latitude–longitude grids. To account
for the secular increase and seasonal cycle in CO_{2} and different air
mass values for different overpasses for each orbit, we normalize all
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{met}}$ soundings by the orbital mean. The orbital
mean is calculated by taking into account all soundings of a particular orbit
that are within the latitude and longitude limits of the analyzed scene. The
standard deviation of d*P*_{ABP} and
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{met}}$ is calculated by taking into account all
grid squares in the analyzed latitude and longitude limits. Analyzing data
from both hemispheres allows us to check for possible errors introduced by
the reversed orientation of the *z* and *y* axis in the Northern and Southern
Hemisphere in our pointing offset derivation (e.g., errors introduced by a
timing error).

We run the ABP and IDP for a set of different pointing offsets for which the
relative footprint positions of the v0006 configuration are preserved. If not
otherwise stated, in the following we refer to the pointing offset of FP 4 of
the O_{2} A band when we refer to pointing offset values. For example,
if the pointing offset of FP 4 of the O_{2} A band is shifted by
+25 arcsec along the *y* axis, then all other footprint
geolocations are also shifted in the same direction by +25 arcsec
along the *y* axis (even though their absolute positions differ from the FP 4
O_{2} A-band position). The same holds for the *z* axis. For the
*y* axis, we run both algorithms for four different pointing offsets ranging
from 175 to 250 arcsec in 25 arcsec steps. For each of these
shifts, we also run a set of different offsets for the *z* axis, ranging from
−250 to +100 arcsec, also in 25 arcsec steps. This leads
to a total of 60 different geolocation configurations.

Figure 5 shows the standard deviation of
d*P*_{ABP} and ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{met}}$ for FP 4 for
all 60 geolocation configurations for the Death Valley National Park. The
observed metrics are less sensitive to changes along the footprint axis than
along the *z* axis. Differences in the standard deviation between neighboring
pointing offsets are small, typically <0.5 hPa for the O_{2}
A band and <0.2 ppm for the two CO_{2} bands. This holds for
all footprints in the three spectral bands. For example, for FP 2 to 7, the
standard deviation of d*P*_{ABP} is minimized for a
pointing offset of 225 arcsec along the footprint axis. A pointing
offset of 200 arcsec minimizes the standard deviation of FP 1 and 8.
Similar results are derived for the Atacama Desert (not shown here). In
general, a pointing offset of 225 arcsec along the footprint axis
minimizes the standard deviation of d*P*_{ABP} and
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{met}}$ for the majority of the footprints. This
offset value is nearly identical to the v0006 configuration
(222.4 arcsec). Therefore, we adapt a pointing offset of
225 arcsec along the *y* axis for all footprints in the three
spectral bands. The absolute pointing offsets along the footprint axis are
summarized in Table 1.

Figure 6 shows the standard deviation of
d*P*_{ABP} and ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{met}}$ as a function
of the *z* axis pointing offsets for FP 4 for the Death Valley National Park
(for a pointing offset of 225 arcsec along the footprint axis). The
analyzed metrics are strongly sensitive to changes of the pointing offset
along this axis. We perform a quadratic regression to determine the best
estimate of the location of the minimum. We only take data points into
account that are distributed symmetrically around the minimum. For FP 4, our
analysis indicates a minimum at −124 arcsec for the O_{2} A
band, −71 arcsec for the weak CO_{2} band, and
−44 arcsec for the strong CO_{2} band. We derive pointing
offsets for all other footprints for all three bands in the same way.
Figure 7a summarizes the *z* axis pointing offsets for all
footprints for all three bands for the Death Valley National Park and Atacama
Desert. On average, the derived pointing offsets for the two areas differ by
13 arcsec for the weak CO_{2} band and by 25 arcsec for
the strong CO_{2} band. For the O_{2} A band the differences
between the two areas differ, on average, by 46 arcsec. Footprints 3
to 5 have the largest pointing offset values. This is in agreement with the
relative footprint geolocations in the v0006 configuration. We average the
derived pointing offsets for the CO_{2} bands from both hemispheres.
This provides the best estimate for the footprint geolocations globally and
takes into account that the *z* axis is rotated by nearly 180^{∘} (in
glint and nadir mode) when the observatory overpasses the Equator. However,
for the O_{2} A band, the difference between the pointing offsets for
both areas reaches up to 60 arcsec for FP 2. In addition, the Atacama
Desert analysis indicate larger relative pointing variations for neighboring
footprints. Therefore, for the O_{2} A band, we only take the derived
pointing offsets from the Death Valley National Park analysis into account.
Final pointing offsets for all three bands are derived by applying a
quadratic regression to the pointing offsets as a function of footprint. This
preserves the parabolic shape of the relative footprint positions, which is
supported by findings from the pre-launch and IOC lunar analysis. The updated
pointing offsets for the *z* axis for each spectral band are summarized in
Table 1.

To evaluate the impact of the updated footprint geolocations we sample the
surface pressure from GEOS5-FP-IT with the updated meteorological sampling
algorithm (that was corrected for the time sampling error) at the footprint
geolocations of the O_{2} A band. The surface pressure is mainly
retrieved from the O_{2} A band; therefore sampling the meteorological
reanalysis at the O_{2} footprint geolocation should yield the best surface
pressure estimates. Figure 8 shows the prior surface pressure
difference between v8 and sampled at the updated footprint geolocations. The
striping pattern effect is mainly introduced by the updated sampling
algorithm and follows orbital paths. As stated previously, the updated
sampling method also introduces a mean bias of +0.5 hPa between the
v8 and newly derived surface pressure estimates. Figure 9
shows the change between the standard deviation of the prior surface pressure
in each grid box for both sampling methods. The observed structures are
mainly driven by changes in the footprint geolocations. The largest changes
are over mountainous regions, e.g., the Tibetan Plateau, the Andes, or the
US West Coast. This will mostly manifest as local-scale changes in
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. As expected, there are no significant changes over ocean
due to the updated footprint geolocations.

4 The OCO-2 v9 data product

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Our improved knowledge of OCO-2's footprint geolocations and the update of the meteorological sampling algorithm reduce errors in bias-corrected ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ that were introduced through erroneous surface pressure estimates in the v8 bias correction. The OCO-2 v9 data product combines the v8 ACOS L2FP retrieval results with a revised bias correction using updated surface pressure estimates from GEOS5-FP-IT. Moreover, filter limits that define the ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ quality flag and warning levels are adjusted, leading to a larger number of soundings that pass the filtration. Finally, the global scaling factor that is derived from direct observations over TCCON stations is updated. This section highlights the major changes in OCO-2's v9 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. The techniques that are used in the next sections are those presented in O'Dell et al. (2018). The derived results, with exception of the revised parametric bias correction, represent updates of the findings in O'Dell et al. (2018).

The parametric bias correction accounts for spurious variability in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ that is correlated with parameters in the retrieval state vector (Wunch et al., 2017b; O'Dell et al., 2018). A multivariate regression is performed between spurious ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ variations and the parameters that account for the largest fraction of the spurious variability. For all ACOS versions for GOSAT and OCO-2 observations, the mode-dependent parametric bias $\left({\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{para}}\right)$ has the following form:

$$\begin{array}{}\text{(1)}& {\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{para}}=\sum _{i}{c}_{i}\left({p}_{i}-{p}_{i,\mathrm{ref}}\right).\end{array}$$

Here, *c*_{i} are regression coefficients which express the sensitivity of
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ from the L2FP retrieval $\left({\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}\right)$ to the selected parameter *p*_{i}, and *p*_{i,ref} are
the corresponding reference values. In order to obtain bias-corrected
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ (${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{bc}}$), Eq. (1) is
subtracted from the raw ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ retrieved by the L2FP algorithm:

$$\begin{array}{ll}{\displaystyle}{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{bc}}& {\displaystyle}={\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}-{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{para}}\\ \text{(2)}& {\displaystyle}& {\displaystyle}={\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}-\sum _{i}{c}_{i}\left({p}_{i}-{p}_{i,\mathrm{ref}}\right).\end{array}$$

Note that we only focus on the parametric bias correction here and neglect
the footprint-dependent bias correction and global scaling factor for now. To
select the parameters and derive the regression coefficients in
Eq. (1), different truth proxy training data sets were used
for v8: TCCON, small area analysis (SAA), and
multi-model median. These truth proxies represent an independent estimate of
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ to which we compare OCO-2 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. A detailed
description of the truth proxies is given in Sect. 4.1 in O'Dell et al. (2018).
For v8 land observations, three different parameters were identified that
account for the largest fraction of variability: co2_grad_del, DWS, and
d*P*. Over ocean, only co2_grad_del and d*P* contribute
to the parametric bias correction. co2_grad_del represents the tropospheric
lapse rate of the retrieved CO_{2} profile and is defined as the
difference in the retrieved CO_{2} between the surface and the retrieval
pressure level at 0.6 times the surface pressure, minus the same quantity for
the prior profile. DWS represents the combined retrieved optical depth of
large particles in the lower-to-middle troposphere in the retrieval, namely
dust, water cloud, and sea salt aerosol. In v8, d*P* is defined as
the difference between the retrieved surface pressure and the prior surface
pressure from GEOS5-FP-IT.

For v9, we define two different d*P* parameters for observations
over land (d*P*_{frac}) and ocean
($\mathrm{d}{P}_{s{\mathrm{CO}}_{\mathrm{2}}}$) that are used in the parametric bias
correction. The revised d*P* parameters take into account two
problems: (1) the misspecifications in the geolocation calibration algorithm
for the overall pointing of the observatory and (2) the pointing offsets
between the three spectral bands. The first is characterized by the
difference between the retrieved surface pressure of the v8 L2FP algorithm
(*P*_{ret,v8}) and the prior surface pressure at the new
geolocation where the O_{2} A band is pointing (${P}_{\mathrm{ap},{\mathrm{O}}_{\mathrm{2}}}$). The second is characterized by the difference between the prior
surface pressure where the O_{2} A band is pointing and the prior
surface pressure where the strong CO_{2} band is pointing
(${P}_{\mathrm{ap},s{\mathrm{CO}}_{\mathrm{2}}}$). For ocean, the revised d*P*
parameter has the following form (given in hPa):

$$\begin{array}{ll}{\displaystyle}\mathrm{d}{P}_{s{\mathrm{CO}}_{\mathrm{2}}}& {\displaystyle}=\left({P}_{\mathrm{ret},\mathrm{v}\mathrm{8}}-{P}_{\mathrm{ap},{\mathrm{O}}_{\mathrm{2}}}\right)+\left({P}_{\mathrm{ap},{\mathrm{O}}_{\mathrm{2}}}-{P}_{\mathrm{ap},{\mathrm{CO}}_{\mathrm{2}}}\right)\\ \text{(3)}& {\displaystyle}& {\displaystyle}={P}_{\mathrm{ret},\mathrm{v}\mathrm{8}}-{P}_{\mathrm{ap},s{\mathrm{CO}}_{\mathrm{2}}}.\end{array}$$

This approach allows us to reduce variations in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ due to
differences between the retrieved and estimated surface pressure without
rerunning the L2FP algorithm. Only the prior surface pressure sampled at the
geolocation where the CO_{2} bands are pointing is needed. Tests have
shown that the best results are achieved when the prior surface pressure is
sampled at the geolocation of the strong CO_{2} band. Over land, the
revised d*P* parameter accounts for the fractional change in
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ when error is present in surface pressure estimates
(given in ppm):

$$\begin{array}{}\text{(4)}& \mathrm{d}{P}_{\mathrm{frac}}={\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}\left(\mathrm{1}-{\displaystyle \frac{{P}_{\mathrm{ap},s{\mathrm{CO}}_{\mathrm{2}}}}{{P}_{\mathrm{ret},\mathrm{v}\mathrm{8}}}}\right).\end{array}$$

Here, ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}$ represents the v8 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$
from the L2FP run when no bias correction is applied. A theoretical
motivation for our choice of the d*P* parameters over land and ocean
is given in Appendix A. The definitions of co2_grad_del and DWS
remains the same in v9.

Similar to v8, we use three truth proxies to derive the parametric bias
correction coefficients for co_grad_del, DWS and the revised d*P*
parameters (see Table 2). Compared to v8, the truth proxy data
sets are extended in time to cover the longer OCO-2 data record. For the
multi-model median, nine models from the OCO-2 model intercomparison project
(MIP) are used (see Table 3). For all data sets a correction
was applied using the OCO-2 averaging kernels based on Connor et al. (2008). We
convolve the CO_{2} profiles from the truth proxies with the OCO-2
column averaging kernel before we compare it to OCO-2 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$.
The parametric bias correction coefficients for v9 are derived from the
average of all coefficients derived from the different truth proxies. The
adapted coefficients and reference values for land and ocean glint data are
summarized in Table 4. The d*P*_{frac}
coefficient over land is −0.9. This is in agreement with the theoretical
value since a change in surface pressure by ∼1 % also changes
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ by ∼1 % and seems to indicate that the
retrieved surface pressure is still not sufficiently accurate to yield the
best estimate of ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$; indeed, as shown in
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$, the coefficient implies that the optimal surface
pressure is a weighted average of the retrieved and prior surface pressure,
with the prior surface pressure weight being about 0.9. Figure 10 shows the different contributions of the v9 parametric
bias correction to the raw ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$.

Bad soundings (e.g., those affected by clouds and low continuum level
signal-to-noise ratio) are mostly screened out by the ABP and IDP
before the ACOS L2FP algorithm performs retrievals. Some soundings that pass
the prescreening criteria, however, show errors in raw ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$
when compared to the truth proxy training data sets that are too large to
provide reliable constraints on CO_{2} fluxes. Therefore, threshold
limits are defined for several variables to filter out these soundings. A
detailed description on quality filtering is given in Mandrake et al. (2015),
Eldering et al. (2017), and O'Dell et al. (2018). We apply slight changes to the
v9 filtration.

We introduce the new filter variables $\mathrm{d}{P}_{{\mathrm{O}}_{\mathrm{2}}}$ and
$\mathrm{d}{P}_{s{\mathrm{CO}}_{\mathrm{2}}}$, the difference between the retrieved
surface pressure, and the estimated surface pressure at the geolocations of
the O_{2} A band and $\mathrm{d}{P}_{s{\mathrm{CO}}_{\mathrm{2}}}$ as given in
Eq. (3). These variables replace the d*P*
filter variable in v8, which was defined as the difference between the
retrieved surface pressure and a mean surface pressure estimate at the
geolocation of all three spectral bands. The improved knowledge of the
estimated surface pressure values allows us to relax the filter limits for
the standard deviation of the surface elevation in the FOV.
Figure 9 shows the bias and scatter in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$
over land relative to the multi-model median truth proxy data set as a
function of the standard deviation of the surface elevation. In v9, the
scatter in the ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ difference starts to increase for standard
deviations of the surface elevation larger than 110 m, whereas in v8
the scatter already increases for standard deviations larger than
60 m. Therefore, we extend the rather strict upper filter limit of
60 m in v8 to 110 m. This leads to a larger throughput of
soundings in mountainous areas in v9. The parameters Max_Declocking_wco2
and Max_Declocking_sco2 are removed from the v9 filtration scheme over
land. Moreover, filter limits for several other variables changed, e.g.,
rms_rel_wco2, *τ*_{oc}, Band 3 albedo, and
d*P*_{ABP}. The revised filter limits for rms_rel_wco2,
*τ*_{oc}, and Band 3 albedo cause a larger throughput for regions
with boreal forests at high northern latitudes. The updated limits for
*τ*_{oc} and Band 3 albedo also increase the number of soundings
over rain forests. The updated filter limits for d*P*_{ABP}
cause a larger throughput in regions with bright surfaces, e.g., the Saharan
desert (see Fig. 12). Overall, 10–15 % additional
soundings pass the new filtration scheme compared to v8. All v9 filter
variables and limits for land and ocean observations are summarized in
Table 5. For soundings that pass filtration in both v8 and v9,
the quality flag did not change.

The global scaling factor corrects for an overall bias in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$
which still remains after filtration and application of the parametric bias
correction. The global scaling factor is derived by comparing the OCO-2 data
to TCCON measurements which are tied to the WMO scale
(e.g., Wunch et al., 2010; Messerschmidt et al., 2010; Geibel et al., 2012). Due to changes in
the data filtration and the revised parametric bias correction in v9, the
global scaling factor *C*_{0} needs to be updated, too. TCCON stations that are
used to derive the global scaling factor are listed in Table 6.

We use the same geographic and temporal co-location criteria for OCO-2 data from direct overpasses of TCCON stations as in O'Dell et al. (2018). We apply the OCO-2 averaging kernels to TCCON data as discussed in the derivation of the coefficients in the parametric bias correction. The slope of the best fit line (forced through a zero intercept) is calculated using the method described in York et al. (2004). The global scaling factor is roughly the same for the different observational modes over land and ocean. Ultimately, we adapt a value of 0.9954 over land and 0.9953 over ocean in v9 (compared with 0.9958 over land and 0.9955 over ocean in v8).

5 Brief evaluation of OCO-2 ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ data

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Here, we evaluate the impact of the changes made in v9 on bias-corrected ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. To explore changes on local scales, we revisit the target observation over Lauder, New Zealand, on 17 February 2015. Figure 13 shows both v8 and v9 bias-corrected ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$. The improved knowledge of the prior surface pressure with the revised parametric bias correction clearly reduces the correlation between ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ and the underlying topography in v9. ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ values are distributed more uniformly over the observed scene. The standard deviation is reduced from 1.35 ppm in v8 to 0.74 ppm in v9. A small topography-related bias is still apparent. However, compared to v8, it is a factor of 2 improvement in reducing biases caused by erroneous surface pressure estimates.

Figure 14 shows the absolute change in bias-corrected
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ between v8 and v9 globally. The observed changes are
mainly driven by three factors: the updated meteorological sampling
algorithm, improved knowledge of the footprint geolocations, and the revised
parametric bias correction. In analogy to Fig. 8, the
striping patterns follow orbital paths and are caused by the updated
meteorological sampling algorithm. Differences over mountainous regions like
the Tibetan Plateau or the Andes are driven by the improved knowledge of the
prior surface pressure due to the updated footprint geolocations. The revised
d*P*_{frac} parameter in the parametric bias correction
over land also introduces changes in regions at high altitudes but not
necessarily with highly variable topography (e.g., South Africa). In addition,
the v9 global scaling factor introduces a systematic difference of
approximately +0.15 ppm between v8 and v9.

6 Conclusions

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The update of the pointing vector that is used to derive the geolocation for OCO-2's eight footprints, together with an update of the meteorological sampling algorithm that corrects for a temporal sampling coding error, provides a better estimate for the surface pressure in OCO-2's v9 data product. Biases in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ due to erroneous surface pressure estimates are clearly reduced in regions with rough topography. For example, over Lauder, New Zealand, the standard deviation of bias-corrected ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ is reduced by almost a factor of 2 when the updated surface pressure estimates are used in the revised parametric bias correction that accounts for misspecifications in the instrument pointing offsets.

Accurate knowledge of the surface pressure and its estimate is crucial to retrieve ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ accurately, and many challenges remain. The OCO-2 retrieval, for example, still has a latitudinally dependent bias in surface pressure, with a maximum in the tropics of nearly 5 hPa (O'Dell et al., 2018). Currently, it is thought that this originates in errors in describing the temperature dependence of the oxygen absorption. Moreover, uncertainties in the underlying elevation map and the question what the source of the scattering is might have an impact on surface pressure estimates. This does not only affect ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ retrieved from GOSAT and OCO-2 but may also affect future sensors with similar observational approaches.

Data availability

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Data availability.

All of the OCO-2 data products are publicly available through the NASA Goddard Earth Science Data and Information Services Center (GES DISC) for distribution and archiving (http://disc.sci.gsfc.nasa.gov/OCO-2; last access: 31 March 2019). TCCON data were obtained from the TCCON data archive hosted by CaltechDATA and are available from https://tccondata.org/ (last access: 31 March 2019).

Appendix A: Theoretical motivation of dP parameters in the v9 parametric bias correction

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Column-averaged dry air mole fractions of CO_{2} are defined as the
total column of CO_{2} (${C}_{{\mathrm{CO}}_{\mathrm{2}}}$) divided by the dry air column
(*C*_{dryair}):

$$\begin{array}{}\text{(A1)}& {\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}={\displaystyle \frac{{C}_{{\mathrm{CO}}_{\mathrm{2}}}}{{C}_{\mathrm{dryair}}}}.\end{array}$$

*C*_{dryair} is defined as

$$\begin{array}{}\text{(A2)}& {C}_{\mathrm{dryair}}={\displaystyle \frac{P}{{g}_{\mathrm{0}}\cdot {m}_{\mathrm{dryair}}}}-{\displaystyle \frac{{C}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\cdot {m}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{m}_{\mathrm{dryair}}}}.\end{array}$$

Here, *P* is the surface pressure, *g*_{0} the gravitational acceleration,
${C}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ the total column of water vapor, *m*_{dryair}
the mean molecular weight of dry air, and ${m}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ the molecular
weight of water vapor. The surface pressure *P*_{true} can be
written as

$$\begin{array}{}\text{(A3)}& {P}_{\mathrm{true}}=a\cdot {P}_{\mathrm{ap}}+(\mathrm{1}-a)\cdot {P}_{\mathrm{ret}}.\end{array}$$

*P*_{ap} and *P*_{ret} represent the
prior and retrieved surface pressure, respectively. The parameter *a* is the
fractional weight given to the prior in the assumed surface pressure. A value
of *a*=0 means that we completely trust the retrieval; *a*=1 means that we
completely trust the prior. Because of retrieval biases, the true surface
pressure is generally close to the prior surface pressure, such that
*a*≈0.9. For a start, we neglect the contribution of the total column
of water vapor. Then the dry air column is directly proportional to the
surface pressure and we can write

$$\begin{array}{}\text{(A4)}& {\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}\propto {\displaystyle \frac{{C}_{{\mathrm{CO}}_{\mathrm{2}}}}{{P}_{\mathrm{ret}}}}.\end{array}$$

For bias-corrected ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ we can write

$$\begin{array}{ll}{\displaystyle}{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{bc}}& {\displaystyle}\propto {\displaystyle \frac{{C}_{{\mathrm{CO}}_{\mathrm{2}}}}{a\cdot {P}_{\mathrm{ap}}+(\mathrm{1}-a)\cdot {P}_{\mathrm{ret}}}}\\ {\displaystyle}& {\displaystyle}={\displaystyle \frac{{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}\cdot {P}_{\mathrm{ret}}}{a\cdot {P}_{\mathrm{ap}}+(\mathrm{1}-a)\cdot {P}_{\mathrm{ret}}}}\\ {\displaystyle}& {\displaystyle}={\displaystyle \frac{{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}}{a\cdot ({P}_{\mathrm{ap}}/{P}_{\mathrm{ret}})+(\mathrm{1}-a)}}\\ \text{(A5)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}}{\mathrm{1}-a\left(\mathrm{1}-{P}_{\mathrm{ap}}/{P}_{\mathrm{ret}}\right)}}.\end{array}$$

Taylor expansion in
$x=a\left(\mathrm{1}-{P}_{\mathrm{ap}}/{P}_{\mathrm{ret}}\right)$
around *x*=0 leads to

$$\begin{array}{}\text{(A6)}& {\displaystyle}{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{bc}}{\displaystyle}\propto {\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}+a\cdot \underset{\mathrm{d}{P}_{\mathrm{frac}}}{\underbrace{{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}\cdot \left(\mathrm{1}-{\displaystyle \frac{{P}_{\mathrm{ap}}}{{P}_{\mathrm{ret}}}}\right)}}.\end{array}$$

The second term in Eq. (A6) is identical to the
d*P*_{frac} parameter that is used in the v9 parametric
bias correction over land (see Sect. 4.1). Here, *a*
represents the coefficient for the d*P*_{frac} parameter in
the parametric bias correction over land. Comparing Eq. (A6) to
Eq. (2), if *p*_{1}=d*P*_{frac}, then ${c}_{\mathrm{1}}=-a$. Further, if we assume that relative variations in
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}/{P}_{\mathrm{ret}}$ are small compared
to relative variations in (*P*_{ret}−*P*_{ap}), then we
can simplify to

$$\begin{array}{}\text{(A7)}& {\displaystyle}{\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{bc}}{\displaystyle}={\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}+a\cdot ({P}_{\mathrm{ret}}-{P}_{\mathrm{ap}}).\end{array}$$

Over ocean, ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ typically varies from
390 to 410 ppm and the surface pressure varies from
995 to 1025 hPa. The relative variations of
${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{raw}}/{P}_{\mathrm{ret}}$ are therefore on
the order of a few tenths of a percent on ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}},\mathrm{para}}$
(for d*P*), which itself is of the order of 1 ppm and
therefore negligible. The second term of Eq. (A7) has the form of
the $\mathrm{d}{P}_{s{\mathrm{CO}}_{\mathrm{2}}}$ parameter as defined in
Sect. 4.1. This form, however, does not account for the
fractional change in ${\mathrm{X}}_{{\mathrm{CO}}_{\mathrm{2}}}$ at higher elevations when error is
present in surface pressure estimates. Therefore, we use Eq. (A6)
over land and Eq. (A7) only over ocean. Note that the parametric
bias correction coefficient *a* in Eqs. (A6) and (A7) is
different for land and ocean observations (see Table 4).

Author contributions

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Author contributions.

MK performed substantial data analysis regarding the derivation of new pointing offsets, the revised bias correction, and the global scaling factor for v9. CO was involved in nearly all aspects of this work, in particular the revised bias correction, quality filtering, and the global scaling factor for v9. BF implemented many tests and performed data analysis. AE provided project leadership and algorithm guidance. CM and RN helped to understand the origin of the topography-related bias and contributed to the selection of the training data sets. PW provided critical guidance on nearly all aspects of the work, throughout all stages.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank David Crisp for helpful discussions on the viewing geometry of the
observatory. We thank Callum McCracken for contributing to
Fig. 4. This work was financially supported by NASA's
OCO-2 project (grant no. NNN12AA01C) and NASA's Carbon Cycle and
Ecosystems research program (grant no. NNX17AE15G).

Edited by: Christof Janssen

Reviewed by: François-Marie Bréon and one anonymous referee

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