An ordinary ensemble-based assimilation system (for example, CarbonTracker) usually begins with the preparation of an ensemble of NCFs Fi,g(i=1,…,N) based on the first-guess net CO2 surface exchange F∗(t) at the rth assimilation cycle: Fi,g(t)=λi,g,rFg∗(t), where λg,r represents a set of linear scaling factors (Peters et al., 2005) for each day and each grid (g) to be estimated and the subscript “r” denotes the rth assimilation cycle. Usually, the CTM would integrate and produce the 3-D CO2 concentration ensemble Um,i (i=1,…,N) N times derived by the ensemble of CFs Fi,g(t) from the same initial background CO2 concentration field. However, for Tan-Tracker, we seek a more innovative way to accomplish its implementation. Figure 1 shows the flowchart of the Tan-Tracker joint assimilation system: Tan-Tracker is initiated by two CTM runs – one is the background run (the blue part in Fig. 1) and the other is the sampling run (the red part in Fig. 1).

Figure 2 shows the makeup of the assimilation window (i.e., the optimized window + the lag window + the observational window; see Fig. 2) in Tan-Tracker. Fba(Fbs) denotes the prior CF series over the assimilation (sampling) window, and Fa∗(Fs∗) represents the first-guess CF series over the assimilation (sampling) window. In the background run, we integrate the CTM (GEOS-Chem) to produce the background CO2 concentration fields Ub forced by the prior CF series Fba at the rth assimilation cycle over the assimilation window Fba(t)=λb(t)Fa∗(t),(t=1,⋯,La), which is used to prepare the background joint state vector λb,UbT. Here La is the length of the assimilation window and λb,ris the prior scaling factor at the rth assimilation cycle. As mentioned, the assimilation window consists of an optimized window (1 week), a lag window (5 weeks) and an observational window (1 week). In each assimilation cycle, the observations in the observational window will be used to update the joint prognostic variables λ,UT in the optimized window.

Correspondingly, in the sampling run, we run the CTM from the background CO2 concentration field Ubs at the beginning of the sampling window (i.e., the Pre-Assim window + the Assimilation window + the Post-Assim window) (Fig. 2) driven by the prior CF series in the same (rth) assimilation cycle Fbs(t)=λb,rFs∗(t), where t=1,…,Ls; Ls (= LPre + La + LPos) is the length of the sampling window; and LPre and LPos are the lengths of the Pre-Assim and Post-Assim windows, respectively (see Fig. 2), over the sampling window to yield the sampling CO2 concentration series Uis (i=1,⋯,Ls, and U1s=Ubs). Next, a 4-D moving sampling strategy (Fig. 2; Wang et al., 2010) is adopted to create the large-scale vector ensemble λm,i,Um,iT (i=1,…,N, N=Ls-La+1) as follows: λm,i,Um,iT=Xis=FbsiFa∗1⋮Fbsi+La-1Fa∗LaUis⋮Ui+La-1s. As a result the large-scale joint state vector λ,UT is viewed as the prognostic variable in Tan-Tracker, with the identity operator (4) chosen to be the CF dynamical sub-model to describe the CFs' evolution: MCF=I, where I is the identity matrix. This CF persistence forecasting model (4) follows Peters et al. (2005) and assumes that the prior (or background) scaling factors λb,r+1 for the next assimilation cycle [(r+1)th] are equal to the optimized scaling factors λa,r of the current (rth) assimilation cycle. In the actual implementations, the following dynamical model (5) is applied to the linear scaling factors, λ λb,r+1=1Lo∑j=1Loλa,rj, where Lo is the length of the optimized window (Fig. 2) and λa,rj are the daily optimized scaling factors λa,j(j=1,….,Lo). The CF dynamical sub-model MCF is thus utilized to constitute the augmented dynamical model M=ICTM for Tan-Tracker together with the CTM (GEOS-Chem) model. By applying the observation operator H to the modeled CO2 concentrations Um,i and the background CO2 concentrations Ub, we can obtain the ensemble simulated observations Umo and the background simulated observations Ubo as follows: Um,io=HUm,i and Ubo=HUb. So far, the background joint vector λb,UbT, the joint vector ensemble λm,i,Um,iT, Eqs. (8) and (9) and the real CO2 measurements Ubo would be input to the PODEn4DVar assimilation processor, which yields the assimilated λa,UaT and the optimized CFs Fa=λaF∗ as a result.

In conclusion, Tan-Tracker works as follows: two CTM runs forced by the background CFs' series are firstly achieved over the assimilation window and the sampling window, respectively: the background run is used to prepare the background joint vector, and the sampling run is used to produce the joint vector ensemble by applying a 4-D moving strategy (Wang et al., 2010) to the sampling simulations throughout the sampling window. The background joint vector and the joint vector ensemble are then input into the PODEn4DVar processor, in which the usual observation operator (e.g., the interpolation function to interpolate the model gridded variables to the in situ observations) compares the simulated CO2 concentrations with the observed according to the 4D-Var cost function: the CO2 concentrations are assimilated to initialize the next assimilation cycle. Meanwhile, the scaling factors λ in the optimized window are also optimized and used for the next assimilation cycle through Eq. (5).

The PODEn4DVar approach is born out of the incremental format of the 4D-Var cost function J(x′)=12x′B-1x′+12y′(x′)-yobs′TR-1y′(x′)-yobs′, where x′=x-xb is the perturbation of the background field xb at the initial time t0, yobs′=yobs,1′yobs,2′⋮yobs,S′,y′=y′(x′)=(y1)′(y2)′⋮(yS)′,(yk)′=yk(xb+x′)-yk(xb),yobs,k′=yobs,k-yk(xb),yk=Hk(Mt0→tk(x)), and R=R10⋯00R2⋯0⋮⋮⋱⋮00⋯RS. Here index k denotes the observation time; the superscript T stands for a transpose; b represents background values; S is the total observational time steps in the observational window; Hk acts as the observation operator; and matrices Rk and B are the observational and background error covariances, respectively.

With the prepared background field xb, the initial model perturbations (MPs) x′(x1′,x2′,⋯,xN′), the simulated observation perturbations y′(y1′,y2′,⋯,yN′), the observational increments yobs,k′, and the background and observational error covariances B and Rk, the final PODEn4DVar analysis solution xa without localization of its analysis error covariance Pa is formulated through some necessary calculations (see Tian et al., 2010, 2011, for more details) as xa=xb+x′V(N-1)I+PyTR-1Py-1PyTR-1yobs′ and Pa=PxPa∗PxT, where Pa∗=(N-1)I+PyTR-1Py-1 and V is derivable from y′Ty′=VΛ2VT and Py=y′V. To clarify, the background covariance B is approximately estimated by B=PxPxTN-1(Px=x′V) in formulating PODEn4DVar.

In particular, in Tan-Tracker, yobs,k′=Uo,k-Ubo and y′=Umo-Ubo, where Ubo=HUb. Here we mark H=H10⋯00H2⋯0⋮⋮⋱⋮00⋯HS. As mentioned, the model state to be optimized is the joint vector (λ,U)T, which indicates xb=λb,UbT and x′=λm,UmT-λb,UbT in Tan-Tracker.

We have realized the coupling between the joint assimilation framework with the PODEn4DVar assimilation processor through Eqs. (18–22) (see the green part of Fig. 1).

As an ensemble-based assimilation system, Tan-Tracker also utilizes the covariance localization techniques to ameliorate the contaminations resulting from the spurious long-range correlations (Houtekamer and Mitchell, 2001). It uses the following exponential decay of the covariance structure with distance between state and observational variables (Gaspari and Cohn, 1999), ρh[i,j]=e-di,j/d0, to calculate the elements ρh[i,j] of the matrix ρh[Lx×Ly], where Lx and Ly are the lengths of the state vector x and the observational vector y, respectively; di,j is the distance between the ith state and the jth observation locations and d0 is the horizontal covariance localization Schur radius.

Consequently, the covariance localization in Tan-Tracker can be implemented by calculating the Schur product ∘ (i.e., piecewise multiplication) as follows (Greybush et al., 2011): xa=xb+ρh∘x′V(N-1)I+PyTR-1Py-1PyTR-1yobs′.

We simulate atmospheric CO2 concentrations using the global three-dimensional chemical transport model GEOS-Chem (version 9-01-03, http://acmg.seas.harvard.edu/geos/) driven by the assimilated meteorological data from the Goddard Earth Observing System (GEOS) of the NASA Global Modeling and Assimilation Office. The version of the model we use is driven by the GEOS-5 meteorological fields with a horizontal resolution of 2∘ latitude by 2.5∘ longitude and 47 vertical layers up to 0.01hPa. The original GEOS-Chem CO2 simulation was described in Suntharalingam et al. (2004) and updated by Nassar et al. (2010). Our simulations include CO2 fluxes from monthly fossil fuel burning and cement production CO2 emissions from the Carbon Dioxide Information Analysis Center (CDIAC) inventory for year 2009 (Andres et al., 2010), monthly biomass burning from the third version of the Global Fire Emission Database (GFEDv3) for 2010 (van der Werf et al., 2010; Mu et al., 2011), climatological biofuel burning (Yevich and Logan, 2003), monthly ocean exchange (Takahashi et al., 2009), 3-hourly biospheric fluxes from the Carnegie–Ames–Stanford Approach (CASA) model for 2000 (Olsen and Randerson, 2004), annual climatology terrestrial biosphere exchange based on TransCom CO2 inversion results adjusted with GFEDv2 fire emissions (Baker et al., 2006b; van der Werf et al., 2006), the chemical production of CO2 from the atmospheric oxidation of other carbon species (Nassar et al., 2010), the monthly emissions from shipping (Olivier and Berdowski, 2001), and aviation CO2 emissions (Friedl, 1997; Sausen and Schuman, 2000; Kim et al., 2005, 2007; Wilkerson et al., 2010). For this work, our model simulation was initialized on 01 January 2008 with a globally uniform 3-D CO2 field of 383.76 ppm. According to the record of NOAA-ESRL Mauna Loa Observatory in Hawaii (http://www.esrl.noaa.gov/gmd/ccgg/), which is a marine surface site, the annual mean CO2 at Mauna Loa in 2007 was 383.76 ppm, with monthly means of 383.89 ppm in December 2007 and 385.44 ppm in January 2008. A 2-year spin-up simulation from this initialized state allows for model transport, sources and sinks to develop the global spatial patterns of CO2; this approach was evaluated in Nassar et al. (2010). After the spin-up run, the obtained CO2 fields were used to drive the observing system simulation experiments. In all the following OSSEs, we firstly assume the default surface CO2 fluxes released with the GEOS-Chem model as the true CF series FTrue.Then we run the GEOS-Chem model, driven by the true CF series FTrue, to obtain the true CO2 concentration results from 1 January 2010 to 31 December 2010 (i.e., the assimilation period). The artificial CO2 observations are thus generated every day by sampling the daily true CO2 concentrations through adding small random noise (whose error variance is 0.01 ppm2) through the 136 observational sites used in this study (Fig. 3). The first-guess CF series F∗ are set to 1.8FTrue, which drives the GEOS-Chem model at the same resolution (2∘ latitude × 2.5∘ longitude) to produce the background CO2 simulations from the spun-up equilibrium state.

The performance of our Tan-Tracker system is examined by comparison with the simplified version (referred to as TT-S), taking only CFs as the prognostic variables. TT-S is somewhat similar to CarbonTracker except that the ensemble square root filter (EnSRF) has been replaced by the PODEn4DVar approach and the GEOS-Chem model is used instead of the TM5 model. Similar to CarbonTracker, the GEOS-Chem model in TT-S is actually the observation operator linking the CFs with CO2 observations. In TT-S, since the CO2 concentrations are not assimilated together with the CFs, we first obtain the optimized scaling factors through assimilating CO2 observations, and thus the CO2 concentrations are also updated by the GEOS-Chem modeling forced by the optimized CFs. All the assimilation processes are initiated by the GEOS-Chem model with first-guess CF series F∗ (=1.8FTrue) and conducted continuously by assimilating the daily pseudo-observations throughout the assimilation period. In all the assimilation experiments, the scaling factors are initiated from λb,0(i,j)=1.0 (where i and j are the longitude and latitude indexes, respectively, and 0 denotes the rth (=0) assimilation cycle). In all the OSSEs, the default lag window is 5 weeks, and the observational window and optimized window are both 1 week. The reference ensemble size N is 106 and the standard localization radius d0 is 900 km. Changes in the assimilation parameters might influence the assimilation performance. We further investigate the effects of the length of the horizontal localization Schur radius and the ensemble size in Tan-Tracker by means of several sensitivity numerical experiments, the results of which are presented in Sect. 3.2. In all assimilation experiments, we use the adaptive inflation technique proposed by Li et al. (2009).

To evaluate Tan-Tracker's performance in a general view, time series of the daily global mean fluxes and CO2 concentrations from the background simulations, the TT-S and the TT (Tan-Tracker) assimilations are compared with the true simulations in Fig. 4. Not surprisingly, the background simulations (referred to as Sim) will inevitably deviate seriously from the “true” simulations due to the predetermined background CF series Fb (=1.8FTrue). Remarkably, since both the CO2 concentrations and CFs are simultaneously assimilated under the joint assimilation framework, it could largely eliminate the uncertainty of the initial CO2 concentrations on the CO2 evolution during the assimilation window and maximize the observations' potential. Probably for this reason, Fig. 4 shows that Tan-Tracker works very well throughout the whole assimilation period, especially after the first few months, which can be considered a spin-up period. However, the performance of TT-S is not very robust and its assimilated errors do not show a trend of becoming less even though its performance seems to be substantially better than the background simulation case: obviously, the impacts of the CO2 concentration have not been taken into full consideration in the TT-S system and there must be some non-negligible errors remaining in the TT-S-optimized CO2 concentrations (Fig. 4b). The resulting errors in the initial CO2 concentrations will in turn contaminate the TT-S assimilation of CO2 fluxes for the next assimilation cycle. In the following discussions, we focus on the results only during the latter half of the year 2010 and thus remove the spin-up period occurring in the first half of the year. Figure 5 also shows that the posterior uncertainties of the analyzed CFs are gradually decreased with assimilation of CO2 observations. Furthermore, Fig. 6 shows time series of the daily globally averaged scaling factor. The daily averaged scaling factor is also decreased and becomes close to ∼ 0.56 (i.e., 1/1.8) with small fluctuations during the latter half of the year 2010.

Similar to Peters et al. (2005), we also aggregated the daily, gridded (2∘ latitude × 2.5∘ longitude) simulation and assimilation results to 24 “super-regions” corresponding to the TransCom 3 regions given by Gurney et al. (2002). Figure 7 shows the 24 super-regions' aggregated mean CO2 concentrations and fluxes during the latter half of the year 2010. Generally, Tan-Tracker is able to reproduce the true fluxes well and its superiority dominates most of the 24 super-regions except for 3 – CT-09 (Tropical Asia), CT-12 (North Pacific Temperate) and CT-20 (Southern Ocean) – whose absolute values are very small (Fig. 7a). Furthermore, as far as the CO2 concentration is concerned, the superior performance of Tan-Tracker beyond TT-S is increasingly obvious (Fig. 7b): the differences between the “truth” and the TT-assimilated CO2 concentrations are much less than those between the TT-S-assimilated and the “truth” in the overwhelming majority of cases, which illustrates once more that the simultaneous assimilation of CO2 concentrations and CFs is indispensable. The time series of daily mean fluxes and CO2 concentrations from the four selected super-regions (Temperate North America, Europe, Boreal Eurasia, and Southern Ocean) are shown in Figs. 8 and 9. Similar to the global mean case shown in Fig. 3, the ability of our assimilation system to represent the variations of seasonal peak-to-trough amplitudes of CO2 concentrations and fluxes is expressed thoroughly and demonstrates its power to make full use of the observations. Comparatively speaking, the ability of the TT-S system is considerably inferior to Tan-Tracker, especially in the Southern Ocean super-region during October–December, 2010: here the TT-S-optimized CO2 concentrations are even worse than the background simulations (Fig. 9d).

To evaluate the performance of our Tan-Tracker data assimilations system comprehensively, we show the root-mean-square errors (RMSEs) for the daily, gridded (2∘ latitude × 2.5∘ longitude) TT- and TT-S-assimilated fluxes from 1 July to 31 December 2010 in Fig. 10. In addition, their corresponding RMSEs for the assimilated (optimized) CO2 concentrations are also shown in Fig. 11. Compared with the Tan-Tracker case, larger RMSEs (> 300 × 10-11 kg C m-2 s-1) for the TT-S-assimilated fluxes can be found in the central parts of South America, most of East Asia, and southern Africa (Fig. 10b). Encouragingly, the TT-assimilated flux RMSEs are largely kept at a very low level (≤ 80 × 10-11 kg C m-2 s-1), in which relatively larger RMSEs (but still much less than the TT-S-assimilated) appear only in a very small area in the central parts of South America (Fig. 10a). Naturally, a parallel circumstance is also replayed in the CO2 concentration case (Fig. 11). Evidently, a relatively definite conclusion can be drawn that the uncertainty of the initial CO2 concentrations cannot be ignored and the joint assimilation framework contributes a lot to the final Tan-Tracker performance. Moreover, the application of the advanced hybrid assimilation approach (i.e., PODEn4DVar) would definitely make a positive contribution to its excellent performance (Tian et al., 2011). Of course, the imbalance of CFs and CO2 concentrations in TT-S partly explains its inferior performance.

Another group of experiments using the Tan-Tracker system with different horizontal localization radii (d0=100, 900, 1450, 2000 and 5000 km) are also conducted to explore the sensitivity of our Tan-Tracker assimilation system to the variations of the horizontal radius. As suggested by Peters et al. (2005), we take 900km as the default or reference radius. Figure 12 shows time series of the daily global CO2 concentrations and fluxes from the “truth” as well as the TT assimilations using the three different horizontal localization radii (d0=900, 1450 and 2000 km). Therefore, we can roughly judge that the Tan-Tracker system could perform well with its horizontal localization radius around 900 km. Nevertheless, two extremely inappropriate localization radii (d0=100 and 5000 km) are also tested in our experiments (but not shown here), whose poor performance demonstrates that the choice of an appropriate covariance localization radius is essential to Tan-Tracker's successful implementation.

Finally, to investigate the impacts of sample sizes on Tan-Tracker's assimilation results, we also conduct another group of Tan-Tracker assimilation experiments with the ensemble numbers N=60, 106 and 150. Figure 13 shows that the differences between the two assimilation experiments with N=106 and 150 are very small. However, if we decrease the ensemble number to 60 (not shown), the assimilation results become divergent. Synthesizing the above results, we can conclude that giving a certain number of sample sizes (≥ 100) could generally guarantee the robust performance of our system.

The basic experimental designs (such as the GEOS-Chem model, ensemble size, assimilation window, localization radius, etc.) are exactly the same as those adopted in Sect. 3. Nevertheless, in this real-data experiment, we took the default surface CO2 fluxes released with the GEOS-Chem model as the first-guess CF series F∗ and used spaceborne CO2 dry-air mole fraction observations (XCO2) instead of artificial CO2 observations. The spaceborne observations used here are originated from the Japanese Greenhouse Gases Observing Satellite (GOSAT), which was launched into orbit in 2009. TANSO-FTS, onboard GOSAT, operates in the shortwave infrared band (SWIR) between 758 and 2080 nm and thermal infrared band (TIR) from 5.56 to 14.3 µm, providing information on CO2 and CH4 in the atmosphere. Level 2 data or the so-called the column-averaged CO2 dry-air mole fraction XCO2 is taken from version 3.3 atmospheric CO2 observations from space (ACOS) data product (O'Dell et al., 2012). Validation against ground-based TCCON data shows a mean bias less than 1.4 ppm; these biases can be further reduced by applying the recommended data screening criteria and bias correction technique (for more details please refer to the document “ACOS Level 2 Standard Product Data User's Guide”, http://disc.sci.gsfc.nasa.gov/acdisc/documentation/ACOS_v3.3_DataUsersGuide.pdf). Furthermore, to guarantee the high quality of the assimilated data as much as possible, we discarded the XCO2 data with observation errors ≥ 0.75 ppm.

In order to assimilate the spaceborne XCO2 directly, the following observation operator (Eq. 25) needs to be incorporated into Tan-Tracker to provide a link between the observational variable XCO2 and the GEOS-Chem-simulated CO2 concentrations (Feng et al., 2009): XCO2=XCO2,a+hTAum-ua, where h is the pressure weighting function; A is the full averaging kernel matrix; Ua and XCO2,a are the prior CO2 profile and the associated column amount, respectively; and um is the GEOS-Chem-produced CO2 profile. The experiment period is from 1 January 2010 to 31 March 2010. In particular, we chose one arbitrary day's (15 March 2010 in this experiment) XCO2 data as the evaluation data set, which are designedly not assimilated in the experiments to provide an independent evaluation for the Tan-Tracker system.

The lack of reliable independent CF estimates derived from GOSAT XCO2 retrievals (Chevallier et al. 2014) forces us to seek an indirect way to evaluate the Tan-Tracker assimilations. Here, we performed a parallel free run of GEOS-Chem forward simulation without any data assimilation. Then, to examine Tan-Tracker's performance quantitatively, the simulated and assimilated CO2 dry-air mole fraction observations of XCO2 on 15 March 2010 were compared with the corresponding (independent) GOSAT observations. After the data quality control (observation error < 0.75 ppm) implemented in this experiment, there are still 163 valid footprints left for system evaluation. Compared with the Sim case, the TT-assimilated XCO2 is improved considerably with higher correlation (0.83 vs. 0.77) and a smaller RMSE (1.38 ppm vs. 2.95 ppm). The GEOS-Chem model generally underestimates the XCO2 values by a substantial negative bias of -2.46 ppm, where the mean bias is given by err =1163∑i=1163 XCO2,is(a)-XCO2,io, with XCO2,is(a) and XCO2,io being the simulated (assimilated) and observed XCO2 values for each valid footprint, respectively. However, the TT-assimilated case only has a very small bias (err = -0.45 ppm). Obviously, the above discussions could only demonstrate that our Tan-Tracker system is capable of yielding fairly good CO2 concentration results. It is encouraging to find that the performance of the TT-Sim case is slightly inferior to the TT case (RMSE = 1.45 ppm and r=0.83), suggesting that Tan-Tracker does enhance the CO2 concentration and flux estimations . It provides a promising new tool for CO2 surface flux (CF) inversion. In addition, in Fig. 14, α (0.01) is the confidence coefficient. Certainly, extra efforts should be made to give a more detailed assessment for Tan-Tracker satellite data assimilation, which will be provided in another study.