Forecasts of CO2 levels in the atmosphere and predictions of future climate depend on our scientific understanding of the natural carbon cycle (IPCC, 2007; Peters et al., 2007). To estimate the spatial and temporal distribution of carbon sources and sinks, inverse methods are used to infer carbon fluxes from geographically sparse observations of the atmospheric CO2 mixing ratio (Tans et al., 1989). The first comprehensive efforts in atmospheric CO2 inversions date back to the late 1980s and early 1990s (Enting and Mansbridge, 1989; Tans et al., 1989). With the increase in spatial coverage of CO2 observations and the development of three-dimensional (3-D) tracer transport models, a variety of numerical experiments and projects have been performed by members of the so-called “TransCom” community of inverse modelers (e.g., Law et al., 1996, 2008; Denning et al., 1999; Gurney et al., 2002, 2004; Baker et al., 2006; Patra et al., 2011). A number of studies have proposed improvements to the inverse methods of atmospheric transport – i.e., the efficient computation of the transport matrix by the model adjoint proposed by Kaminski et al. (1999b), use of monthly mean GLOBALVIEW-CO2 ground-based data (current version is for 2014) by Rödenbeck et al. (2003), development of an ensemble data assimilation method by Peters et al. (2005), flux inversion at high temporal (daily) and spatial (model grid) resolution for the first time using continuous CO2 measurements over Europe by Peylin et al. (2005), using satellite data to constrain the inversion of CO2 by Chevallier et al. (2005), and development of a new observational screening technique by Maki et al. (2010). Despite progress in atmospheric CO2 inversions, a recent intercomparison (Peylin et al., 2013) demonstrated the need for further refinement.

In recent decades, the density of the observational network established to monitor greenhouse gases in the atmosphere has been increased, and more measurements taken onboard ships and aircraft are becoming available (Karion et al., 2013; Tohjima et al., 2015). However, on a global scale CO2 observations do not exist for many remote regions not covered by networks. This lack of data is one of the main limitations of atmospheric inversions, which can be filled by monitoring from space (Rayner and O'Brien, 2001). The satellite observation data from current (GOSAT, Kuze et al., 2009; Yokota et al., 2009; OCO-2, Crisp et al., 2004) and future missions (CarbonSat/CarbonSat Constellation; Bovensmann et al., 2010; Buchwitz et al., 2013) offer enormous potential for CO2 inverse modeling. Optimal application of large observed data sets requires expanding the inverse analysis of CO2 to finer resolution, higher precision and faster performance.

To link surface fluxes of CO2 to observed atmospheric concentrations, an accurate model of atmospheric transport and an inverse modeling technique are needed. Generally, the atmospheric constituents transport may be described in two different ways: the Lagrangian and the Eulerian approaches. The Eulerian method treats the atmospheric tracers as a continuum on a control volume basis, so it is more effective at reproducing long-term patterns – i.e., the seasonal cycle or the interhemispheric gradient. Lagrangian Particle Dispersion Models (LPDMs) consider atmospheric tracer as a discrete phase and tracks each individual particle, therefore LPDMs are better for resolving synoptic and hourly variations.

To relate fluxes and concentrations of long-lived species like CO2, a transport model must cover a long simulation period (e.g., Bruhwiler et al., 2005). Therefore, computing time is a critical issue and minimization of the computational cost is essential. For chemically inert tracers, the transport can be represented by a model's Jacobian matrix, because the simulated concentration at observational sites is a linear function of the flux sets. Theoretically, to compute such matrix the transport model is run multiple times with a set of prescribed surface fluxes. However, this would require an extremely large number of forward model evaluations. The adjoint of the transport model is an efficient way to accelerate calculation of concentration gradients of the simulated tracer at observational locations (Kaminski et al., 1999a). Marchuk (1974) first applied the adjoint approach in atmospheric science. After that, this method became widely used in meteorology. In the 1990s, the use of this approach was expanded to the field of tracer transport modeling (Elbern et al., 1997; Kaminski et al., 1999b).

Adjoint models have numerous applications, including the assimilation of concentrations, inverse modeling of chemical source strengths, sensitivity analysis, and parameter sensitivity estimation (Enting, 2002; Haines et al., 2014). Recent studies have used this method to constrain estimates of the emissions of CO2 using retrieved column integrals from the GOSAT satellite (Basu et al., 2013; Deng et al., 2014; Liu et al., 2015).

Using the adjoint model speeds up the process of high dimensional inverse modeling. However, high CPU and memory demands prevent us from using Eulerian chemical transport models (CTMs) with high-resolution grids in inversions. It would be beneficial to increase the model resolution close to observation points, where the strong observation constraint can significantly improve the optimization of the resulting emission fluxes.

LPDM running in the backward mode can explicitly estimate a source–receptor sensitivity matrix by solving the adjoint equations of atmospheric transport (Stohl et al., 2009), which is mathematically presented by a Jacobian expressing the sensitivity of concentration at the observational locations. Marchuk (1995), and Hourdin and Talagrand (2006) provided derivations proving equivalence of the adjoint of forward transport models to backward transport models.

In order to exploit the advantages of both methods, Lagrangian and Eulerian chemical transport models can be coupled to develop an adjoint that is suitable for the simultaneous simulation of contributions from global and regional emissions. Coupling can be performed in several ways; e.g., a regional-scale LPDM can be coupled to a global Eulerian model at a regional domain boundary (Rödenbeck et al., 2009; Rigby et al., 2011), or a global-scale LPDM can be coupled to an Eulerian model at the time boundary (Koyama et al., 2011; Thompson and Stohl, 2014).

The goal of this study is to present the development and evaluation of an Adjoint of the Global Eulerian–Lagrangian Coupled Atmospheric model (A-GELCA), which consists of an Eulerian National Institute for Environmental Studies global Transport Model (NIES-TM; Maksyutov et al., 2008; Belikov et al., 2011, 2013a, b) and a Lagrangian particle dispersion model (FLEXPART; Stohl et al., 2005). This approach utilizes the accurate transport of the LPDM to calculate the signal near to the receptors, and efficient calculation of background responses using the adjoint of the Eulerian global transport model. In contrast to previous works (Rödenbeck et al., 2009; Rigby et al., 2011; Thompson and Stohl, 2014), in which the regional models were coupled at the spatial boundary of the domain, we implemented a coupling at a time boundary in the global model domain (as described in Sect. 2.1). A-GELCA can be integrated into a variational inverse modeling system designed to optimize surface fluxes.

The remainder of this paper is organized as follows. An overview of the coupled model is provided in Sect. 2. In Sect. 3 we describe the variational inversion scheme. In Sect. 4 we address several problems regarding the coupled model that have not been covered previously (Ganshin et al., 2012). In Sect. 5 we describe the formulation and evaluation of the adjoint model. The computational efficiency of the adjoint model is analyzed in Sect. 6, and the conclusions are presented in Sect. 7.

Although the variational inversion method for minimizing the discrepancy between modeled and observed mixing ratios has been well described and published (i.e., Chevallier et al., 2005), we summarize it here.

The aim of the inversion problem is to find the value of a state vector x with n elements that minimizes the cost function J(x): Jx=12x-xbTB-1x-xb+12Hx-yTR-1Hx-y, where y is a vector of observations with m elements, and the matrix H represents the forward model simulation mapping the state vector x to the observation space. Here, R is the covariance matrix (size m × m) for observational error, which includes instrument and representation errors. The matrix R also includes errors of the forward model H. B is the covariance matrix (size n × n) of error for prior information of the state vector xb. The use of the cost function in the form of Eq. (2) assumes that all errors have Gaussian statistics and are unbiased (Rodgers, 2000).

The minimization of the cost function (Eq. 2) has an analytic solution that involves a matrix inversion. If the Jacobian H is available this analytic solution can implemented, unless the matrix sizes are too large for the available computing resources. Alternatively, Eq. (2) can be solved through an iterative minimization algorithm. In this case, the existence of the gradient of J(x) with respect to x allows the use of powerful gradient algorithms for minimization. This gradient is efficiently provided by the adjoint (Giering and Kaminski, 1998; Kaminski et al., 1999a; Chevallier et al., 2005).

The effect of different horizontal resolutions on Eulerian models is discussed in detail by Patra et al. (2008). In general, higher resolution helps to resolve a more detailed distribution of the tracer. However, the use of a higher-resolution grid leads to additional computational cost, which is not always justified by the resulting model output. Higher resolution does not produce better results largely due to the limited availability of high-resolution meteorology and tracer emission data sets.

The paper by Ganshin et al. (2012) describing the development of the GELCA model provides a model testing report. The advantage of GELCA in reproducing the high-concentration spikes and short-term variations caused mainly by anthropogenic emissions is more vivid when using high-resolution (1 km × 1 km) surface fluxes compared to standard-resolution (1∘ × 1∘) fluxes. However, those tests considered only short 4-month simulations for a limited number of locations.

We expanded the comparison undertaken by Ganshin et al. (2012) to a 2-year period using an updated set of prescribed fluxes, which combines four components similar to the analysis performed by Takagi et al. (2011) and Maksyutov et al. (2013): (a) anthropogenic fluxes from the Open source Data Inventory of Anthropogenic CO2 (ODIAC; Oda and Maksyutov, 2011) and the Carbon Dioxide Information Analysis Center's (CDIAC; Andres et al., 2009, 2011) data sets; (b) biosphere fluxes simulated by the Vegetation Integrative SImulator for Trace gases (VISIT) terrestrial biosphere model (Ito, 2010; Saito et al., 2011, 2014); (c) oceanic fluxes predicted by a data assimilation system based on the Offline ocean Tracer Transport Model (OTTM; Valsala and Maksyutov, 2010); and (d) biomass burning emissions from the Global Fire Emissions Database (GFED) version 3.1 (van der Werf et al., 2010). Biosphere fluxes have daily time step, while the others are monthly. The initial global CO2 distribution was obtained from GLOBALVIEW-CO2 (2014).

We considered several cases with different model resolutions. For NIES TM we tested grids at 10.0, 2.5, and 1.25∘ resolutions, with FLEXPART running at 1.0∘ (Table 1). The resolution of the input fluxes was matched to that of FLEXPART. Model results were compared with observations from the World Data Centre for Greenhouse Gases (WDCGG, 2015) and the Siberian observations obtained by the Center for Global Environmental Research (CGER) of the National Institute for Environmental Studies (NIES) and the Russian Academy of Science (RAS), from six tower sites (JR-STATION) as described by Sasakawa et al. (2010). The selected site locations are shown in Fig. 2.

Although the total number of observational stations contributing to the WDCGG is about several hundreds, the set of sites conducting continuous (high temporal resolution is needed for the coupled model) observations is much smaller. We selected 19 sites (Table 2). Most of them are concentrated in the temperate latitudes of the Northern Hemisphere, where the variations in CO2 concentration are most noticeable.

Siberia is assumed to be a substantial source and sink of CO2, with high uncertainties in the fluxes describing them (McGuire et al., 2009; Hayes et al., 2011; Saeki et al., 2013). As a result, CTMs tend to reproduce the interannual variability of CO2 quite poorly. We selected six tower JR-STATION sites to check the model performance in the Siberian region (Table 3).

The analyzed sites are divided into three groups. The first group includes remote and marine sites (ALT, AMS, BRW, CPT, IZO, JBN, MLO, MNM, ZEP) with very weak influence of local sources, so the seasonal variation of CO2 is controlled by global, large-scale variations. For these sites contribution by using the Lagrangian component is negligible (see Figs. 3–5 panel b to analyze the difference between the coupled and the Eulerian models).

The second group includes sites with domination of long-term variability of CO2 and relatively smooth and weak short-term variations. Typically, these sites are located on the border of two regions with very different fluxes (AMY, CMN, MHD, PAL, PRS, YON).

The sites selected to the third group are strongly influenced by local emissions and global transport at the same time. Therefore the CO2 concentration variation is controlled by the strength and direction of wind, the depth of the boundary layer and other factors. Such sites are mainly in the northern mid-latitudes (HUN, PUY, SSL, WSA) including all Siberian towers (DEM, IGR, KRS, NOY, VGN, YAK). For these locations contributions of the Eulerian and Lagrangian components are comparable. Therefore, the simulation of CO2 for these sites shows the full potential of the coupled model.

Figure 6 compares the coupled and Eulerian model results with observations from the Igrim and Vaganovo towers. The recent modifications indicated in Sect. 2.2 have significantly improved the performance of NIES TM compared with the results reported by Ganshin et al. (2012). However, compared to the updated NIES TM the coupled model is better at reproducing short-term peaks of concentration. This explains the observed reduction of the mean bias and SD (up to 1.5 ppm), and the better simulation of the seasonal variation (in phase and amplitude). Generally, the improvements in the CO2 simulations due to the addition of the Lagrangian component to the Eulerian model are higher than those obtained by increasing the resolution of the Eulerian NIES transport model, as seen for the third group of sites (Figs. 3–5).

However, improvements in CO2 simulation due to the implementation of the GELCA model were not obtained for all the considered sites. There are several factors that limit the coupled model performance improvement. First, no significant improvement can be expected for the remote and marine sites since they are influenced by very distant emissions and/or nearby homogeneous emissions that are managed appropriately by the Eulerian model. The Lagrangian model introduces very significant improvements for sites influenced by relatively nearby inhomogeneous sources. Second, the use of the very rough Eulerian grid (10.0∘) causes a wrong reproduction of the CO2 seasonal cycle due to the large aggregation error – e.g., this happens for ALT and BRW. However, note that such low resolution is used in a rather synthetic case, which is unlikely to be used for actual simulations. Third, temporal irregularities in the observations and noise in the meteorological data bring erroneous signal to the Lagrangian model, causing spurious short-term peaks of tracers, which cause degraded results at some locations (e.g., PRS, YAK). This shows that further modification of the setup (i.e., more detailed meteorological data, switch to higher resolution) is necessary. Fourth, the Lagrangian part is very sensitive to the local flux quality. Thus, it is quite problematic to use the highly uncertain surface fluxes to simulate the tracer concentrations and use these concentrations for estimating the quality of different model configurations. However, we cannot improve our analysis, because we do not have concentration measurements for tracers whose surface fluxes are more accurately known, like SF6.

Given the large difference in computational costs running the NIES TM model when using the lower- and the higher-resolution grids (e.g., the computational cost increases by a factor of ∼ 4 between Case 2 and 3), the coupled model is an effective way to improve the CO2 simulation without changing the Eulerian model resolution.

We tested several different methods to reduce the computational cost of the adjoint model. First, the Eulerian part of the adjoint model was driven by static archives of meteorological parameters, as described in Sect. 2.4.1. Second, the Lagrangian part of the adjoint model made use of pre-calculated response functions, as described in Sect. 2.4.2.

To run the adjoint model we used a Linux workstation with eight Intel(R) Xeon(R) E5-4650 2.70 GHz processors and 64 GB of RAM. The CPU time of the adjoint model (backward only) was almost equal to the CPU time required to run the forward model. It took about 1.3 min for a week-long iteration (forward and backward). The memory demand was about 1 GB. Henze et al. (2007) reports that the ratio between simulation time in backward and forward modes for adjoint models derived for other CTMs is as follows: GEOS-Chem: 1.5, STEM: 1.5, CHIMERE: 3–4, IM-AGES: 4, Polair: 4.5–7, and CIT: 11.75. Thus, the adjoint of the developed coupled model GELCA is quite efficient. To achieve this level of efficiency, a substantial amount of manual programming effort is required, despite the automatic code generated by TAF. The main disadvantage of TAF is that many redundant recomputations are often generated by the compiler. A crucial optimization of the adjoint code is required to eliminate these extra recomputations.

In this paper we have presented the construction and evaluation of an adjoint of the global Eulerian–Lagrangian coupled model GELCA that will be integrated into a variational inverse system designed to optimize surface fluxes. The coupled model combines the NIES three-dimensional transport model as its Eulerian part and the FLEXPART plume diffusion model as its Lagrangian component. The Eulerian and Lagrangian components are coupled at a time boundary in the global domain. The model was originally developed to study the carbon dioxide and methane atmospheric distributions.

The Lagrangian component did not require any code modification, as FLEXPART is a self-adjoint and tracks a significant number of particles backward in time in order to calculate the sensitivity of observations to the neighboring emissions areas.

For the Eulerian part, the discrete adjoint was constructed directly from the original NIES TM code, instead of contrasting a continuous adjoint derived from the forward model basic equations. The tangent linear and adjoint models of the NIES TM to FLEXPART coupler were derived using the automatic differentiation software TAF (http://www.FastOpt.com), which significantly accelerated the development. However, considerable manual processing of forward and adjoint model codes was necessary to improve the transparency and clarity of the model and to optimize the computational performance in relation to including MPI, as the TAF code used here (version 1.5) does not fully support MPI routines.

The main benefit of the developed discrete adjoint is accurate calculation of the numerical cost function gradients, even if the algorithms are nonlinear. The overall advantages of the developed model also include relatively simple incorporation of the Lagrangian part and fast computation using the Lagrangian component, scalability of sensitivity calculation depending on distance to monitoring sites, thereby reducing aggregation errors, and computational efficiency even for high-resolution simulations.

The transport scheme accuracy of the forward coupled model was investigated using the distribution of the atmospheric CO2. The GELCA components and the model itself had previously been validated using various tests and by comparison with measurements and with other transport models for CO2 and other tracers. The analyses in the present paper have shown that CELGA is effective in capturing the seasonal variability of atmospheric tracer at observation sites. Decreasing of the Eulerian model resolution does not significantly distort the transport model performance; however, running the coupled model using NIES TM with low-resolution grid can maximize simulation speed and use of data storage.

The Eulerian adjoint was validated using various tests in which the adjoint gradients were compared to gradients calculated with numerical finite difference. We evaluated each routine of the discrete adjoint of the Eulerian model and the adjoint gradients of the cost function. The precision obtained in the results of the considered numerical experiments demonstrates proper construction of the adjoint.

The CPU time needed by the adjoint model is comparable with those of other models, as we used several methods to reduce the computational cost. The forward NIES model was altered so that at each model time step it saved all variables that were also being needed by the adjoint model. These variables therefore did not have to be recalculated for use in the adjoint model. In addition, the adjoint simulation was isolated from the recalculation of NIES TM meteorological parameters and Lagrangian response functions. All supplementary parameters were pre-calculated before running the adjoint and were stored in static archives.

The developed A-GELCA model will be incorporated into a variational inversion system aimed at studying greenhouse gases (mainly CH4 and CO2), by assimilating tracer measurements from in situ, aircraft, and remote sensing observations. However, before performing real inverse modeling simulations it is necessary to select a proper minimization program and find the optimal values for the error covariance matrices R and B.