We developed a carbon data assimilation system to estimate surface
carbon fluxes using the local ensemble transform Kalman filter (LETKF) and
atmospheric transport model GEOS-Chem driven by the MERRA-1 reanalysis of
the meteorological field based on the Goddard Earth Observing System model,
version 5 (GEOS-5). This assimilation system is inspired by the method of
Kang et al. (2011, 2012), who estimated the surface carbon fluxes in an
observing system simulation experiment (OSSE) as evolving parameters in the
assimilation of the atmospheric CO2, using a short assimilation window
of 6 h. They included the assimilation of the standard meteorological
variables, so that the ensemble provided a measure of the uncertainty in the
CO2 transport. After introducing new techniques such as “variable
localization”, and increased observation weights near the surface, they
obtained accurate surface carbon fluxes at grid-point resolution. We
developed a new version of the local ensemble transform Kalman
filter related to the “running-in-place”
(RIP) method used to accelerate the spin-up of ensemble Kalman filter (EnKF) data assimilation
(Kalnay and Yang, 2010; Wang et al., 2013; Yang et al., 2012). Like RIP, the
new assimilation system uses the “no cost smoothing” algorithm for the
LETKF (Kalnay et al., 2007b), which allows shifting the Kalman
filter solution forward or backward within an assimilation window at no cost. In the
new scheme a long “observation window” (e.g., 7 d or longer) is used to
create a LETKF ensemble at 7 d. Then, the RIP smoother is used to obtain
an accurate final analysis at 1 d. This new approach has the advantage of
being based on a short assimilation window, which makes it more accurate,
and of having been exposed to the future 7 d observations, which improves
the analysis and accelerates the spin-up. The assimilation and observation
windows are then shifted forward by 1 d, and the process is repeated.
This reduces significantly the analysis error, suggesting that the newly
developed assimilation method can be used with other Earth system models,
especially in order to make greater use of observations in conjunction with
models.
Introduction
The exchange of carbon among the atmosphere, land, and ocean contributes to
changes in the Earth's climate and is also sensitive to climate conditions.
The CO2 concentration in the atmosphere is affected by both the natural
variability of the Earth's planetary system and anthropogenic emissions.
The terrestrial and oceanic ecosystems absorb more than one-half of
anthropogenic CO2 emissions (Le Quéré et al., 2016). One major scientific
question is whether this rate of removal of CO2 from atmosphere will
continue in future and if it can be enhanced. It is thus essential to better
quantify the dynamics of Earth surface carbon fluxes (SCFs) and the
variations in carbon sources and sinks and their associated uncertainties.
A common approach for estimating SCF from atmospheric CO2 measurements and
atmospheric transport models is referred to as a “top-down” approach. The top-down methods estimate SCF through techniques such as Bayesian
synthesis approach (Rödenbeck et al., 2003;
Gurney et al., 2004; Enting,
2002; Bousquet et al., 1999), different types of ensemble Kalman filters
(EnKF) (e.g., Peters et al., 2005, 2007; Feng et al., 2009; Zupanski et al.,
2007; Lokupitiya et al., 2008), or variational data assimilation methods
(e.g., Baker et al., 2006, 2010; Chevallier et al., 2009).
Kang et al. (2011, 2012) developed a top-down carbon data assimilation
system by coupling an atmospheric general circulation model (AGCM),
including atmospheric CO2 concentrations, with the local ensemble transform
Kalman filter (LETKF) (Hunt et al., 2007). The meteorological variables
(wind, temperature, humidity, surface pressure) and CO2 concentrations were
assimilated simultaneously in order to account for the uncertainties of the
meteorological field and their impact on the transport of atmospheric CO2.
They carried out observing system simulation experiments (OSSEs), and their
carbon assimilation system achieved an accurate
estimation of the evolving SCF at the model grid resolution for the first time, without
requiring any a priori information. The surface carbon fluxes were considered
“unobserved evolving parameters” by augmenting the state vector at each
column with a surface carbon flux (SCF). The local ensemble transform Kalman
filter (LETKF) then estimated this evolving parameter from the error
covariance between the low-level atmospheric CO2 and the estimated SCF, and,
after a spin-up of about 1 month, the LETKF accurately recovered the
“nature” run seasonal surface carbon fluxes.
Kang et al. (2011, 2012) used a short 6 h assimilation window for both
atmospheric and CO2 observations because atmospheric observations are
usually assimilated at this frequency and because most ensemble Kalman
filter methods require short windows to ensure that the forecast
perturbation growth remains linear. Such a short data assimilation window,
required by the LETKF, also protects the system from becoming ill
conditioned (Enting, 2002, Fig. 1.3), and as a result it does not require
additional a priori information. We note further that the use of such a short
assimilation window differs very much from most other top-down
approaches for estimating SCFs that use long assimilation windows varying
from a few weeks to months or even years (e.g., Baker et al., 2006, 2010;
Peters et al., 2005, 2007; Michalak, 2008; Feng et al., 2009; Liu et al., 2016).
Although the Kang et al. (2011, 2012) methodology was successful, it is computationally
expensive, requiring ensemble forecasts and data assimilation, not only for
the carbon variables but also for the standard atmospheric variables, in
order to estimate the uncertainties of the CO2 atmospheric transport
process. In this study, we used an improved version of LETKF data
assimilation system with a state-of-the-art atmospheric transport model, the
GEOS-Chem (Bey et al., 2001; Nassar et al., 2013), which is driven by the
MERRA-1 reanalysis of the Goddard Earth Observing System model, version 5
(GEOS5). The improved data assimilation system, unlike Kang et al. (2011,
2012), does not include an estimation of transport uncertainties related to
the meteorological field.
The ultimate goal of our LETKF_C system is to estimate the
grid-point SCFs, which, as in Kang et al. (2011, 2012), are treated as
time-evolving parameters in the system. As mentioned before, an ensemble
Kalman filter requires a short assimilation window in order to have the
ensemble perturbations evolve linearly and remain Gaussian. On the other
hand, it is well known that the training needed to estimate evolving
parameters through data assimilation could be quite long, thus it
benefits from having many observations. Therefore, a short assimilation
window would shorten the training period needed for the estimation of the
SCF error covariance, and therefore lengthen the spin-up time.
To address this problem, we developed a new version of the LETKF using the
running-in-place (RIP) method to accelerate the spin-up of EnKF data
assimilation (Kalnay and Yang, 2010; Wang et al., 2013; Yang et al., 2012).
Like RIP, the new assimilation system uses the “no cost smoothing”
algorithm (Kalnay et al., 2007b) that allows shifting at a negligible cost
the Kalman filter solution forward or backward within a given assimilation
window. Briefly, the new scheme works as follows: a long “observation
window” (e.g., 7 d, containing all the observations within 7 d) is
used to create a temporary LETKF ensemble analysis at 7 d. Then the RIP
smoother is used to obtain a final analysis at 1 d. This analysis has the
advantage of being based on a short assimilation window, which makes it more
accurate, and of having been exposed to the 7 d of observations, which
accelerates the spin-up time. The assimilation and observation windows are
then shifted forward by 1 d, and the process is repeated. We have tested
this new method (short assimilation, long observation window), achieving a
significant reduction of analysis errors, and we believe that this method
could be useful in other data assimilation problems.
This paper is organized as follows: Sect. 2 briefly describes the new
system used for CO2 data assimilation (LETKF_C). Section 3
explores the effect of combining assimilation and observation windows in an
OSSE framework. Section 4 presents results of the proposed methodology
applied to CO2 data. A summary and discussion are presented in Sect. 5.
LETKF_C data assimilation system
A data assimilation system includes a forecast model, observations, and a
data assimilation method that optimally combines them. In the proposed
LETKF_C data assimilation system we use the GEOS-Chem as the
forecast model and LETKF as the data assimilation method. The
pseudo-observations for our OSSE experiments are created at the locations of
the real carbon observations from Orbiting Carbon Observatory-2 (OCO-2)
satellite (Crisp et al., 2004).
GEOS-Chem model and the “nature” run
GEOS-Chem is a global 3-D atmospheric chemical transport model driven by the
NASA reanalysis (MERRA-1) meteorological fields from the Goddard Earth
Observing System data assimilation, version 5, by the NASA Global Modeling
and Assimilation Office (Bosilovich et al., 2015). This model has been
applied worldwide to a wide range of atmospheric composition and transport
studies. The GEOS-Chem model used in this study is the version 10.01 with a
resolution of 4∘× 5∘ (latitude × longitude) and 47
hybrid pressure–sigma vertical levels for CO2 simulation (Nassar et al.,
2013). GEOS-Chem is driven by the MERRA-1 reanalysis with 72 hybrid vertical
levels, extending from the surface up to 0.01 hPa. The data used in this
study was provided by the GEOS-Chem support team, based at the Harvard and
Dalhousie Universities with support from the NASA Earth Science Division and
the Canadian National and Engineering Research Council, who re-gridded the
original data of spatial resolution of 0.25∘× 0.3125∘
into the resolution of 4∘× 5∘.
GEOS-Chem requires the SCFs as a set of parameters at each grid point in
order to simulate the CO2 concentration in the atmosphere. It is not
possible to observe the global SCFs directly. Therefore, the SCFs are
created from a “bottom-up” approach (considered “truth” in our
experiments) and used for the simulation of atmospheric CO2 concentration
with GEOS-Chem. The bottom-up SCFs used in this study include the three
components shown in Eq. (): (1) terrestrial carbon fluxes (FTA),
(2) air–sea carbon fluxes (FOA), and (3) anthropogenic fossil fuel emissions
(Ffe).
SCF=FTA+FOA+Ffe
The FTA values are derived from the VEgetation Global Atmosphere Soils
(VEGAS) model (Zeng et al., 2004, 2005), forced by the real
evolving weather, obtained from the GEOS-Chem. The FOA values are from
Takahashi et al. (2002), a climatological seasonal cycle estimated for the
1990s, and the Ffe values are from the Fossil Fuel Data Assimilation System
(FFDAS) for the year 2012 (Asefi-Najafabady et al., 2014). The air–sea
carbon flux and Ffe values were scaled using the global carbon budget
data of Le Quéré et al. (2015) in order to include interannual
variations. A nature run for atmospheric CO2 concentration simulation is
driven by the SCFs in units of (kgC (m2 yr)-1) based on all three
datasets.
In OSSEs, the nature run serves as the truth. We assume that the true
bottom-up carbon fluxes are not known in our data assimilation
experiments, and they will be estimated using the atmospheric
pseudo-observations derived from the truth, as described in more detail
below. The nature run obtained by coupling GEOS-Chem with VEGAS is fairly
realistic (figure not shown), so we use it to create the pseudo-OCO-2
observations for the period of January 2015–March 2016.
Pseudo-observations
The ultimate goal of this model–data assimilation system is to estimate the
SCFs at every grid point using real observations such as the conventional
surface CO2 measurements of GlobalViewplus (GV+) flask network provided by
Cooperative Global Atmospheric Data Integration Project (2016) and the
observations from satellites such as the Greenhouse Gases Observing
Satellite (GOSAT) (Yokota et al., 2004), and the Orbiting Carbon
Observatory-2 (OCO-2) (Crisp et al., 2004). Therefore, it is very beneficial
to choose a realistic observation network to generate the
pseudo-observations for testing the proposed data assimilation system. In
this study, we developed the pseudo-observations for the OSSE assimilation
experiments, based on a realistic OCO-2 observation product.
The OCO-2 observations are the CO2 column-averaged dry air mole fractions
over the entire OCO-2 pixel (defined as XCO2). The synthetic observations
cover the entire globe once every 14 d with very high spatial resolution.
This includes 24 samples per second along the satellite track within
∼7 km span. The observations are expected to be highly
correlated over a short length scale. Furthermore, the observation quality
is greatly affected by conditions such as cloud cover, surface type, and the
solar zenith angle at the time of measurement. The OCO-2 retrieval algorithm
uses a warning level (WL) between 0 and 19 to indicate the quality of
measurements, where WL = 0 means “most likely good”, and WL = 19 means
“least likely good” observations. To avoid highly correlated measurements
being treated as independent measurements and to bring the spatial
resolution in line with the resolution of atmosphere transfer model, David
Baker provided an OCO-2 observation dataset which averaged the synthetic
XCO2 in 10 s time window using the “good-quality” observations
retrieval defined by WL<=15 (David Baker, personal communication, April 2017).
The OCO-2 retrievals used to obtain averages are based on the NASA
Atmospheric CO2 Observations from Space XCO2 retrieval Algorithm version 7r
(O'Dell et al., 2012), as archived
at https://disc.gsfc.nasa.gov/datasets/OCO2_L2_Lite_FP_7r/summary (last access:
23 March 2017). A two-step averaging method has been used in order to avoid
the final average being disproportionately weighted to one part of the
averaging bin (track) with more good-quality retrievals. In the first step,
the “good-quality” retrievals, defined as WL<=15 and
XCO2_quality_flag = 0 (another quality
indicator of the data), are averaged over 1 s bins, with weights
inversely proportional to the square of each retrieval's posterior
uncertainty. In the second step, all the 1 s bins with at least one
valid retrieval are averaged over a 10 s interval to create 10 s
averaged data. The OCO-2 averaging kernels are similarly averaged to create
10 s mean averaging kernels. This averaging method had been used for
similar purposes in the recent study by Basu et al. (2018). In this study,
we further aggregated the observations from David Baker at the nearest
GEOS-Chem output time of 00:00, 06:00, 12:00, and 18:00 UTC for each model day. The
typical 1 d coverage of observation of OCO-2 is shown in Fig. 1. The
values of XCO2 in the winter are significantly larger than those in summer
of the Northern Hemisphere and the OCO-2 observations are missing in the
winter for midlatitude and high-latitude regions (latitude >∼30).
We used the actual location, timescales, and error scales of
the OCO-2 observations to create the pseudo-observations for our experiment.
The pseudo-observations are created by obtaining the true CO2 from the
nature run using the location and time of the valid observation, then
adding random errors with due consideration to the scales of the
corresponding real observations. These derived pseudo-observations used in
this study are based on the real observations associated error scales; thus, they
are much more realistic than the GOSAT observations also used in Kang et al. (2012) because they are anchored on the real OCO-2 observations,
their quality, and their statistical representation.
The 10 s average of good-quality OCO-2 XCO2 observations
(warning level <= 15), obtained from David Baker for (a) 1 January 2015 and (b) 1 July 2015.
The LETKF data assimilation system
The ensemble Kalman filter (EnKF) is a powerful tool for data assimilation
that was first introduced by Evensen (1994). The key attribute of this
method is to derive the forecast uncertainties from an ensemble of
integrated model simulations. A variety of ensemble Kalman filter
assimilation methods have been proposed (Burgers et al., 1998; Houtekamer
and Mitchell, 1998; Anderson, 2001, 2003; Bishop et al., 2001; Whitaker and
Hamill, 2002; Tippett et al., 2003; Ott et al., 2004; Hunt et al., 2004).
The local ensemble transform Kalman filter (LETKF) introduced by Hunt et al. (2007) is chosen for this study.
The LETKF is an extension of the local ensemble Kalman filter (Ott et al.,
2004) with the implementation of the ensemble transform filter (Bishop et
al., 2001; Wang and Bishop, 2003). It is widely used for data assimilation,
including several operational centers, and was also used for carbon data
assimilations by Kang et al. (2011, 2012).
As discussed earlier, we follow Kang et al. (2011) in estimating the SCFs
as evolving parameters, augmenting the state vector C (the prognostic
variable of atmospheric CO2) with the parameter SCF, i.e.,
X=[C,SCF]T. The analysis mean
X‾a and its ensemble perturbations
Xa are determined by Eq. (2.1, 2.2) at every
grid point, and the ensemble analysis is used as the initial conditions for the
ensemble forecast in the next cycle.
X‾a=X‾b+XbK̃(yo-y‾b)Xa=Xb[(K-1)P̃a]1/2
Here, X‾b is the mean of the forecast (background) ensemble
members; Xb is a matrix, whose columns are the background
perturbations of Xkb-X‾bfor each ensemble member
Xkb (k=1,..., K), where K is the ensemble size; yo
is a vector of all the observations; y‾b is the background
ensemble mean in observation space (y‾b=H(X‾b)), where H is
the observation forward operator that transforms values in the model space
to those in the observation space; P̃a=YbTR-1Yb+K-1Ir-1 is the
analysis error covariance matrix in ensemble space, which is a
function of Yb=HXb ,
the matrix of background ensemble perturbations in the observation space,
R, the observation error covariance (e.g., measurement error, aggregation
error, representativeness error), and of r, a multiplicative inflation
parameter; and K̃=P̃aYbR-1. LETKF simultaneously assimilates
all observations within a certain distance at each analysis
grid point, which defines the localization scale. Hunt et al. (2004)
introduced a four-dimensional version, and Hunt et al. (2007) provide a
detailed documentation of the 4-D LETKF that we are using.
Choosing the long observation window (OW) and the short assimilation
window (AW)
Like other data assimilation methods, LETKF proceeds in analysis cycles that
consist of two steps, a forecast step and an analysis step. In the analysis
step, the model forecast (also called prior or background) and the
observations are optimally combined to produce the analysis (also called the
posterior), which is the best estimate of the current state of the system
under study. In the forecast step, the model is then advanced in time with
the analysis as the initial condition and its result becomes the forecast
for the next analysis cycle. All observations within the assimilation time
window are used to constrain the state at the end of the assimilation
window.
The focus of this study is on the estimation of SCFs that are time-varying
parameters in GEOS-Chem. As mentioned earlier, a preliminary LETKF analysis,
which provides the weights for each ensemble perturbation, is performed over
a longer window (e.g., 7 d, with observations starting at time t). Then,
the “no cost” smoothing (Kalnay et al., 2007b; Kalnay and Yang, 2010) is
applied, using the same analysis weights obtained at the end of the long
observation window (e.g., 7 d) for each ensemble member but combining
the ensemble perturbations at the end of the corresponding short
assimilation window (e.g., 1 d). This creates the final 1 d analysis (at
time t+AW), which benefits from the information from all the observations
made throughout the long OW (7 d) and from the linearity of the
perturbations in the short AW of 1 d, which is required for accuracy. At
this time the procedure is repeated starting at t+AW, which is 1 d later.
In this new approach, we have the flexibility to combine a short
assimilation window (AW) of length m (e.g., m=1 d) with a long
observation window (OW) of length n (e.g., n=7 d) to improve the
estimation of SCF. In the forecast step, the model is integrated from t to
t+n to produce the forecast corresponding to the observations within the
OW. In the analysis step, the observations and corresponding forecasts
within the OW are used by the LETKF to estimate optimal weights for the
ensemble members. The no cost smoother applies these optimal weights to
determine the analysis of the model state and the SCF parameter at t+m.
The resulting analysis is then used as the initial conditions for the next
analysis cycle starting from time t+m.
Experimental setup
In our experiments we used an ensemble size of 20 members, which was
reasonable since the data assimilation only includes one state variable (CO2
concentration) and one parameter variable (SCF). A similar experiment but
with 80-member ensemble size showed only slight improvement of assimilation
quality (figure not shown) but dramatically increased the computational
cost. The initial ensemble is created by random selection of the state and
flux values from the model-based nature run for both SCF and atmospheric
CO2 concentration. Therefore, the initial uncertainties of fluxes
and CO2 values are equivalent to their “natural” variability. Based on a
sensitivity analysis, we found a horizontal localization radius of 15 000 km
is optimal for our system. Following Kang el al. (2012), a vertical
localization is also applied by assigning a larger weight to the
CO2-updating layers near the surface, to reflect the expected dominance of
layers near the ground in the change of the total column CO2 measured by OCO-2.
Additive inflation method
Inflation is very important for our LETKF_C data assimilation
system. The LETKF uses the forecast ensemble spread to represent forecast
uncertainties. All EnKFs tend to underestimate the uncertainty in their
state estimate because of nonlinearities and the limited number of ensemble
members (Whitaker and Hamill, 2002). Underestimating the uncertainty
(ensemble spread) leads to overconfidence in the background state estimate
and less confidence in the observations, which will eventually lead the EnKF
to ignore the observations and result in filter divergence. This is also
true for our carbon-LETKF data assimilation system. The ensemble spread of
CO2 in GEOS-Chem model decreases during model integration when the ensemble
members are using the same meteorological forcing and SCF values, which is
very different from the system with prognostic meteorological fields where
the ensemble spread of model state increases during model integration (not
shown). The ensemble spread of SCFs also does not increase during model
integration because the SCFs are predicted using persistence, and the LETKF
decreases the ensemble spreads for both SCFs and CO2 during analysis steps.
Therefore, without inflation, the ensemble spread of the CO2 and SCFs would
be continuously decreasing during data assimilation, and soon would become
too small for LETKF to accept any observations, causing filter
divergence.
There are different types of inflation methods that address the problem of
overconfidence, such as multiplicative inflation, relaxation to prior, and
additive inflation (e.g., Anderson and Anderson, 1999; Mitchell and
Houtekamer, 2000; Zhang et al., 2004; Whitaker et al., 2008; Miyoshi, 2011).
For this study, we chose additive inflation, which adds random fields to the
analysis before the ensemble forecast of the next analysis cycle. Additive
inflation has some advantages compared to multiplicative inflation because
it prevents the effective ensemble dimension from collapsing toward the
dominant directions of error growth (Whitaker et al., 2008; Kalnay et al.,
2007a). We applied additive inflation to the ensemble of atmospheric CO2 and
SCF to increase perturbations in the initial conditions for the next time
step. It is important for an additive inflation method to minimize the
impact of model imbalance and initial shocks generated by adding the random
fields into a model. Following Kang et al. (2012), the added fields are
selected randomly from the model nature run. Pairs of atmospheric CO2 and
surface CO2 flux fields are chosen randomly from the model nature run within 1 year before the analysis time; their ensemble mean is removed and their
differences are scaled to a magnitude corresponding to 30 % of model
seasonal variance to create the ensemble of random fields for additive
inflation. Therefore, each selected random field is balanced, and when it is
added into model, the balance will be essentially maintained.
Sensitivity analysis for AW and OW length
We tested the new version of the LETKF with short AW and long OW, described
in previous sections by conducting two sets of experiments using the
LETKF_C system in an OSSE framework with OCO-2-like
observations. The first set of experiments used the regular 4-D LETKF
settings (with a single window length AW = OW) to investigate the effect of
the length of AW for estimating SCF. In the second set of experiments, we
investigated the optimal OW length after choosing the best AW from the first
set of experiments. The assimilation period for all experiments was 1 January 2015
to 1 March 2016. The annual mean RMSE differences are
calculated from the simulation results by removing the spin-up period of the
first 2 months (January and February 2015). The average period is from
1 March 2015 to the end of February 2016. The details of experimental
settings are shown in Table 1.
Lengths of assimilation windows (AWs) and observation window (OWs)
and the resulting time-averaged global mean RMSEs for different experiments.
The first four experiments use a regular 4-D LETKF, with AW = OW. The last four
experiments use AW = 1 d, found to be optimal, and different OWs.
EXP1EXP2EXP3EXP4EXP5EXP6EXP7EXP8AW6 h1 d3 d7 d1 d1 d1 d1 dOW6 h1 d3 d7 d2 d8 d15 d30 dRMSE (kgC (m2 yr)-1)0.0770.0590.0680.0740.0530.0410.0400.050Sensitivity analysis for different assimilation windows
The sensitivity of SCF estimates to the length of AW was investigated based
on the first set of experiments (EXP1–EXP4) with regular 4-D LETKF settings,
where the length of OW is the same as that of the AW. All experiments used
the same observations and initial conditions. Since the temporal coverage of
the OCO-2 observation network is too sparse for our LETKF_C
assimilation system to estimate the SCF signal over short timescales, we
focus on evaluating the estimation of SCF for seasonal and longer timescales.
Figure 2 shows the estimated global total surface fluxes from the first set
of experiments. The true global total surface fluxes show a clear
seasonal cycle with very large carbon uptake during the growing season of the
Northern Hemisphere (NH), from May to August, and carbon release during
other seasons, with the peak release during November. All experiments
reproduced the seasonal cycle of SCF fairly well.
(a) The global total SCF from the nature run (“truth”, black line) and from the estimations of the first set of experiments with different AW. (b) The difference of global total SCF between the estimations from the experiments with different AW and the nature run (truth). (c) The global average RMSE of the estimated SCFs from the experiments with different AW.
When the AW is very short (6 h), there is large-magnitude and high-frequency noise overlaying the seasonal cycle. The magnitude of high-frequency errors of SCF estimation in EXP1 is comparable with the seasonal
variability of SCF (Fig. 2a). When the AW = 7 d, the high-frequency
errors of estimation decay but the long assimilation window increases the
analysis RMSE (EXP4). The EXP2 with AW = 1 d produced the best estimation
of SCF among all four experiments with equal observation and assimilation
windows (Fig. 2).
The advantage of AW = 1 d (EXP2) is clearly seen from the smaller average
global root-mean-square error (RMSE) (Fig. 2c). The RMSE of surface carbon
flux is calculated as follows:
RMSEt=Ex(Fax,t-Fnx,t2),
where x and t are space and time location; Fa and Fn indicate
the analysis and the true SCF from the nature run, respectively. Ex is
spatial average. The estimations from experiments with long AW (3 and 7 d)
have a smaller RMSE for the first 3 months (January to March),
when the truth had very little variation because the long AWs enhance
the signal and smooth the high-frequency noise. However, the experiments
with long AW can miss the fine-scale signals of SCF variation and fail to
catch its variations with time. As a result, the estimations with long AW
showed large RMSE during the period when SCF had larger variations. The
estimation with an AW of 6 h also showed very large RMSE because of the
overwhelming high-frequency noise. Thus, the estimation with an AW of 1 d had
the smallest RMSE among all of the experiments with a regular 4-D LETKF.
The time-averaged RMSEs of SCFs is calculated as follows:
RMSEx=Et(Fax,t-Fnx,t2),
which shows very similar spatial patterns but different amplitudes for
different experiments (Fig. 3). The large RMSEs of SCF estimation located
in the southeastern USA and the southeast of both China and Russia, resembled that of the
SCF variance (not shown). The regions of higher variance indicate more
information is needed to resolve such large variance by observations, which
is hard to achieve. As expected, the SCF RMSE of 0.059 from EXP2 with an AW of
1 d is significantly smaller than the RMSE from EXP1 with a short AW of
6 h (0.077 kgC (m2 yr)-1) and EXP3 and EXP4 with longer AWs of
3 d (0.068kgC (m2 yr)-1) and 7 d (0.074 kgC (m2 yr)-1), respectively.
The spatial pattern of the annual mean RMSE of estimated SCF from
the experiments with different AW (EXP1–4) for the average period from
1 March 2015 to the end of February 2016. (January and February 2015 are
treated as a spin-up period for our experiments).
Our results suggest that the optimal AW for estimating SCF is about 1 d.
This is distinctly different from previously published studies that indicate
that either a very short AW (6 h) (Kang et al., 2011, 2012), or a very
long AW (longer than a few weeks) is optimal (e.g., Baker et al., 2006,
2010; Peters et al., 2005, 2007; Michalak, 2008; Feng et al., 2009). A short
AW can better constrain the model state and therefore produce a better
parameter estimation. However, a very short AW of 6 h can degrade the
SCF estimation with high-frequency noise in our LETKF-C system. We postulate
that the high-frequency noise is related to the sampling errors in the
CO2–SCF covariance that has a smaller signal-to-noise ratio compared to those
in experiments with longer AWs.
The same results can be obtained from the same experiments with different
initial times, indicating the robustness of our findings (figure not shown).
The convergence of estimated SCFs from the experiments starting from months
with big SCF variation, such as April, is slightly slower than the
experiments from the time with small SCF variation, such as January. While
the estimated SCFs converge in a few analysis cycles (a few days) in our
system (Fig. 2), the small difference of convergence rate does not make
any significant impact on the quality of estimated SCFs. Moreover, the
calculation of RMSE of estimated SCFs has excluded the spin-up period of the
first 2 months to remove the potential impact of the initial conditions and
initial time.
Sensitivity analysis for different observation windows (OW)
The results presented earlier and associated discussion suggest that
parameter estimation through data assimilation benefits from a long training
time and having a sufficient number of observations, implying that the length
of OW is critical for the estimation of desired parameter(s). We
investigated the effect of such sensitivity to find out the suitable length
of OW for estimating SCF in the second set of experiments (EXP5–EXP8), all
based on the optimum AW = 1 d that was identified from the first set of
experiments but using different OW lengths.
The estimated global total SCFs in the second set of experiments show a
clear seasonal cycle matching the truth (Fig. 4a). Compared with EXP2
(OW = 1), shown with the green line in Fig. 2a, EXP5 (OW = 2 d) reduced
the high-frequency noise significantly when the OW length was increased from
1 to 2 d. There is still some high-frequency noise in the SCF
estimation for EXP5 because the observations for 2 d are not sufficient
to smooth out the high-frequency noise introduced into the estimation
through data assimilation. The estimated global total SCFs for EXP6
(OW = 8 d), EXP7 (OW = 15), and EXP8 (OW = 30) are much smoother than that of
EXP5 (OW = 1 d) because of their longer OW. However, the estimation for OW of 30 d shows a clear time-shift compared with the truth, especially
during the transient period when the majority of ecosystems and plants are
switching from dormant phase in the winter to the growing phase in the
spring. The surface carbon fluxes change rapidly during this period. The
time-shift can also be seen in the estimations for these experiments with an OW
of 15 d, but it is less pronounced. In the proposed LETKF technique, most
of observations in a long OW are introduced at a time later than the
assimilation time. Since the SCFs are temporally evolving parameters, the
information (variation) of future surface fluxes is brought into the
estimation of current time when the future observations are included in the
OW. Therefore, the estimated SCFs with a very long OW tend to shift towards
its future value. The estimated SCFs with moderate OW = 8 and 15 d
(EXP6 and EXP7) are more accurate than those with a short OW of 2 d
(EXP5) and very long OW of 30 d (EXP8) by avoiding the significant high-frequency noise observed in EXP5 (OW = 2 d) and the significant time-shift
present in EXP8, with a very long observation window (OW = 30 d). The
global mean RMSEs of estimated SCF from OW = 8 and 15 d (EXP6 and EXP7)
are significantly smaller than those from OW = 2 and 30 d, i.e., EXP5 and EXP8 (Fig. 4c).
Same as Fig. 2, except for the second set of experiments with
different OW but the same AW of 1 d.
The spatial pattern of time-averaged RMSE of SCF for EXP5 (OW = 2 d;
Fig. 5) is similar to those in the first set of experiments, which had
short AW = OW (Fig. 3). The regions with large RMSE in EXP5 (OW = 2 d)
disappear with OW = 7 and 15 d in EXP6 and EXP7 because the long OWs
enhance the signals for SFC estimation. The large RMSE in SCF estimates for
EXP8 (OW = 30 d) are primarily in the Northern Hemisphere midlatitudes
because of the time-shift in estimations with OW = 30 d. The mean RMSEs
of experiments with moderate OWs of 8 and 15 d are 0.041 and 0.040kgC (m2 yr)-1, respectively, which
is significantly smaller than those from experiments with OWs of 2 d
(0.053 kgC (m2 yr)-1) and 30 d (0.050 kgC (m2 yr)-1).
Same as Fig. 3, except for the second set of experiments with
different OW but similar AW of 1 d.
However, a longer OW requires a longer forecast period for each forecast
step, which results in additional computational time and cost. For example, EXP7
with an OW of 8 d used 8 times more computational time compared to EXP2.
Furthermore, the length of the OW is also constrained by the timescale of
estimation parameters. A long OW tends to generate a time-shift for its
estimation. For seasonal and longer timescales, OW(s) in the moderate range of
8–15 d appear to be most suitable for the
LETKF_C estimates of the SCF. EXP6 and EXP7 show almost the
same quality of SCF estimation, but EXP6 has higher computational
efficiency. The best configuration thus appears to be EXP6 with an OW of 8 d
and AW of 1 d, referred as the “benchmark” experiment hereafter.
We note that the high-frequency noise in EXP1 with a short AW of 6 h can
be smoothed out by a long OW (i.e., 8–15 d). We postulate that an
experiment with an AW of 6 h and OW 8 d will produce similarly realistic
estimations as the benchmark experiment; however, it would require much
more computational time.
Evaluating estimated fluxes from the benchmark experiment
With the moderately long observation and short assimilation windows, we
obtained best estimates of surface carbon fluxes, and their seasonal cycle.
This section describes the SCF estimates from the benchmark experiment (AW = 1 d, OW = 8 d).
Figure 6 shows a comparison of surface carbon fluxes based on the
benchmark assimilation experiment and the nature (truth) run for
Northern Hemisphere summer (June, July, and August) and winter seasons
(December, January, and February). The bottom-up carbon fluxes used in
the nature run show a very strong seasonal cycle over all of the continents
except Antarctica. The Northern Hemisphere midlatitude areas are very large
carbon sinks in the summer and carbon sources in the winter, as expected.
The strong seasonal cycle of surface fluxes is mainly related to the
variability of terrestrial ecosystems that absorb a large amount of CO2
during the growing season (spring and summer) and release carbon back to the
atmosphere during dormant seasons (fall and winter). The estimated surface
fluxes in the seasonal timescale follow the truth closely. The
benchmark assimilation experiment closely reproduces the spatial pattern of
surface fluxes globally, for different seasons. The difference between the
benchmark estimation and truth shown in Fig. 6e, f are very
small. There are some positive carbon flux differences over Northern
Hemisphere midlatitudes in the winter, thus a positive bias in estimated
atmospheric CO2 concentration is expected.
The SCF of the “nature” run and an estimation from the benchmark experiment (AW = 1 d, OW = 8 d)
for Northern Hemisphere summer (a, c and e), and winter (b, d, and f). Panels (a)
and (b) are the “truth” from the nature run, panels (c) and (d) are the estimates from benchmark experiment, and panels (e) and (f) are the difference between estimation and truth.
The analysis of CO2 concentrations matches the nature run well. The
error pattern also matches the CO2 seasonal cycle and the error pattern of
estimated SCF. Figure 7 shows the comparison of surface atmospheric CO2
concentrations between the benchmark assimilation experiment and the nature
(truth) run for the Northern Hemisphere summer and winter. The spatial
pattern of assimilated CO2 matches the truth very well. The analysis
successfully reproduced the seasonal cycle of CO2 over Northern Hemisphere
midlatitudes, with low CO2 concentration in summer (Fig. 7a–c) and high
CO2 in winter (Fig. 7b–d), consistent with the seasonal cycle of CO2
absorption and release from terrestrial ecosystems. There are positive
CO2 concentrations located at high latitudes of the North American and East Asian
regions during winter 2016 (Fig. 7f), due to the positive bias in
estimated SCF (Fig. 6f).
Same as Fig. 6, except for surface concentrations of CO2. Where panels (a) and (c) share the upper left color bar; Panels (b) and (d) use the upper right
color bar.
The consistency of annual mean estimated SCF for both benchmark experiment
and truth is a very important feature for our LETKF_C
assimilation system (Fig. 8a). In EnKF assimilation the ensemble spread is
considered a good representation of uncertainties associated with both
parameters and model state (e.g., Evensen, 2007; Liu et al., 2014). The
surface carbon fluxes are special parameters that vary with time and it is
very hard to quantify their uncertainty during assimilation. When the
ensemble spread of parameters are too small to drive a model with a robust
response, the estimation fails. The additive inflation with 30 % of nature
variability is used to maintain the amplitude of parameter ensemble spread.
Although the ensemble spread of the global total surface flux, in our
experiments, is bigger than its error (Fig. 8a), we were still able to
estimate the global total surface CO2 fluxes (ensemble mean) and
their seasonal variability very well. This is consistent with findings of Liu el al. (2014)
that parameter estimation can tolerate some inconsistency between
parameter ensemble spread and parameter error.
(a) The global total SCF of “truth” and estimation from the benchmark experiment: the black line is the truth, the green line is the
ensemble mean of the estimation, and the yellow shading is the ensemble spread.
(b) The global mean RMSE of the estimated SCF from the benchmark experiment(AW = 1 d, OW = 8 d).
The global mean RMSE of SCF decreases from an initial value of
∼0.1 to ∼0.04 kg C m-2 yr-1
in just a few analysis cycles (Fig. 8b). It does not
further decrease during following assimilation cycles because the SCF values
vary temporally. The signals added by observations are mainly used to
reproduce the temporal variation in SCF.
It is very important for a SCF estimation to reproduce the spatial
distribution of the annual mean of the SCF, since it identifies the carbon
sources and sinks in the Earth system. Though the amplitude of annual mean
SCF is much smaller than the seasonal cycle of SCF, the estimated spatial
pattern of annual mean SCF in the benchmark experiment (Eq. 5) is generally
consistent with the truth (Fig. 9).
ΔFx=EtFax,t-EtFnx,t
In summary, we found that the OSSE experiments using long observation
windows and short assimilation windows resulted in the best estimates of SCF.
(a) The annual mean of SCF (with the Ffe removed) for the “nature” run, (b) the annual mean of estimated SCF (with the Ffe removed) from the benchmark experiment, and (c) their differences.
Same as Fig. 5, except for assimilating both OCO-2 and GV+
pseudo-observations. Panels (a), (b), (c), and (d) show the results with OWs of 2, 4, 8, and 15 d respectively.
Summary and discussion
We have developed a LETKF GEOS-Chem carbon data assimilation
(LETKF_C) system for estimating the surface carbon fluxes
(SCFs). The true GEOS-Chem atmospheric transport model is driven by the
single realization of meteorology fields from MERRA reanalysis. The proposed
data assimilation system captured the true SCF spatial and temporal
variability well. The system performed best with a choice of short assimilation
and long observation windows.
The LETKF requires a short assimilation window to avoid an ill-posed
condition caused by the nonlinear processes in the forecast model with a
long forecast time. The parameter estimation favors a long training period
and many observations. Based on these features, we developed a new method to
accurately estimate the SCF. The new scheme separates the original assimilation
time window into observation (OW) and assimilation (AW) windows, allowing for
the flexibility to apply an OW that is different to the AW. Like the running-in-place (RIP) method, the new technique takes advantage of the no cost smoothing algorithm developed for the LETKF by Kalnay et al. (2007b) that
allows the transportation of the Kalman filter solution forward or backward within
the observation window.
The new method was applied to the LETKF_C system in the OSSE
mode using a dataset developed based on the OCO-2 observation
characteristics. The sensitivity experiments for this model assimilation
system demonstrated that the new technique, i.e., using a short AW and long
OW, significantly improves the SCF estimation as compared to a regular
4-D LETKF with identical observation and assimilation windows. The best AW
for SCF estimation is 1 d, which is different from the typical AW of 6 h
used in the meteorological assimilations. An OW in the range of 8–15 d
is required to estimate the surface carbon fluxes for seasonal and
longer timescales. The benchmark experiment with an AW of 1 d and the OW of
8 d successfully reproduced the mean seasonal and annual SCF.
Our working hypothesis was that the optimal OW for the estimation of
SCF could be reduced with more observations. We examined this hypothesis by
using simulated OCO-2 observations and GlobalViewPlus (GV+)
observations. Similar to the OCO-2 pseudo-observations, the GV+
pseudo-observations were also generated based on the actual location, time,
and corresponding error scale of the GV+ flask observations. The results
show that the AW and OW lengths of 1 d and 8 d, respectively, are also optimal using both the
OCO-2 and GV+ observation characteristics. We estimated the SCF using the
OCO-2 and GV+ pseudo-observations with the identical experiment settings
as the OCO-2 experiments, except we replace the experiment with very long OW
of 30 d with an experiment with a short OW of 4 d to better evaluate
the impact from short OWs. Thus, the current experiments settings are using
OW of 2, 4, 8, and15 d.
The results from these experiments show that the AW and OW lengths of 1 d and 8 d, respectively,
are still optimal for both the OCO-2 and GV+ observation
characteristics (Fig. 10). Generally, the time mean RMSE of estimated SCF
with OCO-2 and GV+ (Fig. 10) are smaller than the corresponding
estimates for OCO-2 only (Fig. 5). The short OW of 2 d performs worse
than the moderate OWs of 4, 8, and 15 d. The time-averaged
global mean RMSE is 0.046 kgC (m2 yr)-1 for experiments with an OW of 2 d (Fig. 10a). The time-averaged global mean RMSE is only 0.040, 0.037, and 0.039 kgC (m2 yr)-1 for experiments with OWs of 4, 8, and 30 d,
respectively (Fig. 10b, c and d). We only see a slight impact of
observation coverage on the optimal OW length. The best OW appears to be
8–15 d, which produces the smallest RMSE when only OCO-2
observations are assimilated. The smallest RMSE is
obtained in the experiment with the best OW of 8 d, when both OCO-2 and
GV+ observations are assimilated into the system.
Two different sets of experiments (OCO-2 vs. OCO-2 and GV+) suggesting the
same optimal OW of 8 d indicate that the observation coverage and
observation type are not the major factor in deciding the length of optimal
OW. We speculate that the optimal OW is mainly determined by the timescale
of model response to the SCF uncertainties because LETKF constrains
parameters (SCF) based on the mapping function of parameter-state
covariance; hence, only the model response to the parameter uncertainties
provide the signal for parameter estimation.
It is worth noting that our approach works best for estimating parameters
that vary slowly over moderate timescales. It may not be optimum for
estimating SCF variation for short timescales such as sub-daily to daily
because the variations shorter than the OWs are filtered out. Furthermore, we
used a coarse spatial resolution (4∘× 5∘) GEOS-Chem in
our study. We postulate that the optimal AW and OW could be different when a
higher spatial resolution version of GEOS-Chem is used with the proposed
assimilation system because models with different resolutions' responses to
the SCF may be different. This issue also merits further exploring in the
future.
Our newly developed short AW and long OW technique is different from both the
standard 4-D variational method and the 4-D LETKF. The 4-D Var (four-dimensional variational) and the 4-D LETKF methods
have been shown (Bonavita et al., 2015; Hamrud et al., 2015) to have an
essentially equivalent performance, and their hybrid Kalman Gain combination
(Penny, 2014) in a EnKF framework was comparable to the hybrid ensemble
data assimilation system currently operational at ECMWF but with a lower
computational cost. The hybrid ensemble data assimilation system at ECMWF
uses an ensemble of 4-D Var assimilations at reduced resolution to provide a
flow-dependent estimate of background errors for use in 4-D Var assimilation
(Bonavita et al., 2015). The short AW and long OW approach can be used with
other Earth system models for parameter estimation, when the parameters have
slow and smooth variations in time and space and the observations are too
limited to constrain the parameters well.
Code and data availability
This study focused on developing a new methodology for estimating carbon
flux based on a carbon cycle model–data assimilation system. It does not
generate any new datasets. The related code for GEOS-Chem and LETKF can be
accessed from
http://wiki.seas.harvard.edu/geos-chem/index.php/Downloading_GEOS-Chem_source_code (last access: 18 June 2019; GEOS-Chem, 2019) and
https://github.com/takemasa-miyoshi/letkf (last access: 18 June 2019; Miyoshi, 2019), respectively.
Author contributions
NZ, YL and EK developed the method. NZ, YL, BJ, ZC developed the model code, building on published work of Miyoshi et al and Kang et al. EK and YL designed the model experiments described in the paper and YL run all of them. YL, EK, NZ and GA wrote the paper. All contributed to the ideas and development of the model and methodology.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This research is partially supported by laboratory-directed research and development funding from the Pacific Northwest National Laboratory (PNNL), managed by the Battelle Memorial Institute for the US Department of Energy.
Financial support
This research has been supported by the NOAA OAR
(grant no. NA18OAR4310266 and NA10OAR4310248) NASA (grant no. 80NSSC18K0908 and NNX15AG95G).
Review statement
This paper was edited by Adrian Sandu and reviewed by three anonymous referees.
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