We present the development of the Adjoint of the Global
Eulerian–Lagrangian Coupled Atmospheric (A-GELCA) model that consists of the
National Institute for Environmental Studies (NIES) model as an Eulerian
three-dimensional transport model (TM), and FLEXPART (FLEXible PARTicle
dispersion model) as the Lagrangian Particle Dispersion Model (LPDM). The
forward tangent linear and adjoint components of the Eulerian model were
constructed directly from the original NIES TM code using an automatic
differentiation tool known as TAF (Transformation of Algorithms in Fortran;

Forecasts of CO

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 CO

To link surface fluxes of CO

To relate fluxes and concentrations of long-lived species like CO

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 CO

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.

In this paper we use a global Eulerian–Lagrangian coupled model, the
principles of which are described by Ganshin et al. (2012). The coupled model
consists of FLEXPART (version 8.0; run in backward mode) as the Lagrangian
particle dispersion model, and NIES TM (version NIES-08.1i) as the Eulerian
off-line global transport model. For concentration

The computational scheme of the coupled model.

Since the first publication of the GELCA model in 2012, the NIES transport
model has undergone significant updates. We provide a brief outline of the
major features of the current model. NIES TM is a global three-dimensional
CTM that simulates the global distribution of atmospheric tracers between the
Earth's surface and a pressure level of 5 hPa. The model employs the
standard horizontal latitude–longitude grid with reduced number of meshes
towards the poles and a spatial resolution of
2.5

Inverse modeling assumes that the model reasonably well reproduces the
relationship between atmospheric mixing ratio and surface fluxes, assuming
that the biases between the simulated and observed concentrations are mostly
due to the emission inventories errors. To ensure that this is the case, the
NIES TM model has been evaluated extensively. Comparisons against SF

FLEXPART, like other LPDMs, considers atmospheric tracers as clouds of individual particles and tracks the pathway of each particle. The advantage of this approach is the direct estimation of the sensitivity of the measurements to the neighboring sinks and sources by tracking the particles backward in time. Usually it is sufficient to simulate for a limited number of days (2–10) to determine where particles intercept the surface layer before they spread vertically and horizontally.

To run both models we use a reanalysis data set combining the Japanese 25-year
Reanalysis (JRA-25) and the Japanese Meteorological Agency Climate Data
Assimilation System (JCDAS) data set (Onogi et al., 2007). The JRA-25/JCDAS
data set is distributed on a Gaussian T106 grid with horizontal resolution
1.25

Isolation of the transport equations is an effective way to save a significant amount of CPU time during tracer transport simulation. At the preprocessing stage, the NIES TM core produced a static archive of advective, diffusive, and convective mass fluxes with time step similar to the one of the original JRA-25/JCDAS data (6 h). After that the archive is used by an “offline” model specially designed only for passive transport of tracer. Intermediate fluxes are derived by interpolation.

Besides the mass fluxes, the static archives contain fields of temperature,
pressure, humidity, vertical grid parameters (variation of the
sigma-isentropic vertical coordinate over time), and others. The
pre-calculated and stored data field can be used directly for any of the
inert tracers. It is also possible to simulate chemically active tracers if
the chemical reaction can be written in the linear decay form; e.g., for

Originally, FLEXPART was driven by a ECMWF reanalysis data set distributed on a
grid with regular latitude–longitude horizontal structure and
sigma–pressure vertical coordinate. The current version of the model was
adapted to use JRA-25/JCDAS data, by horizontal bilinear interpolation of the
required parameters from a Gaussian grid to a regular
1.25

Given the large differences in structure, resolution, and parameter estimation methods used in different reanalysis data sets, the use of the same meteorology for both Eulerian and Lagrangian models provides significant benefit.

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

Map showing the location of the 19 WDCGG sites (red dots, blue labels) and six tower network sites in Siberia (magenta dots, green labels) for which we have performed comparison using forward GELCA simulation.

The minimization of the cost function (Eq. 2) has an analytic solution that
involves a matrix inversion. If the Jacobian

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

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 CO

We considered several cases with different model resolutions. For NIES TM we
tested grids at 10.0, 2.5, and 1.25

The coupled model setups analyzed in this study.

WDCGG continuous observation sites.

Here AEMET – Izana Atmospheric Research Center, Meteorological State Agency of Spain; CNR-ICES – International Center for Earth Sciences – CNR, Institute of Acoustics and Sensors; DNA-IAA – Direcion Nacional del Antartico – Istituto Antartico Argentino; EC – Environment Canada; HMS – Hungarian Meteorological Service; IAFMS – Italian Air Force Meteorological Service; ITM – Department of Applied Environmental Science, Stockholm University; JMA – Japan Meteorological Agency; KMA – Korea Meteorological Administration; LSCE – Laboratoire des Sciences du Climat et de l'Environnement; NOAA/ESRL – National Oceanic and Atmospheric Administration/Earth System Research Laboratory; RSE – Ricerca sul Sistema Energetico – RSE S.p.A.; FMI – Finnish Meteorological Institute; SAWS – South African Weather Service; UBA – Federal Environmental Agency, Germany.

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
CO

Siberia is assumed to be a substantial source and sink of CO

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 CO

Tower network sites in Siberia (JR-STATION).

The second group includes sites with domination of long-term variability of
CO

The sites selected to the third group are strongly influenced by local
emissions and global transport at the same time. Therefore the CO

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 CO

CO

However, improvements in CO

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

In this section, we present the development of the adjoint of the coupled model. The incorporation of the Lagrangian component does not require any modification to the code, as LPDMs are self-adjoint. The development of the adjoint of the Eulerian part is more complicated. We decided to develop a discrete adjoint of NIES TM in order to make it consistent with the forward model. An alternative approach is the construction of a continuous adjoint derived from the leading equations of the forward model (Giles and Pierce, 2000). The main advantage of the discrete adjoint model is that the resulting gradients of the numerical cost function are exact, even for nonlinear or iterative algorithms, and this makes it easier to validate the adjoint model, which is an essential and complicated task.

The adjoint model for NIES TM was created manually to achieve maximum
computational efficiency, while the adjoint of NIES TM to FLEXPART coupler
was created using the Transformation of Algorithms in Fortran (TAF) software
(

The advantages of our coupled adjoint model are as follows.

Simple incorporation of the Lagrangian part, since no modification of the LPDM is required. Potentially, NIES TM can be coupled to any Lagrangian model.

Minimization of the simulation time can be obtained, as once calculated the output from the Lagrangian model is applicable for different long-lived tracers.

Reduction of aggregation errors can be achieved, as the sensitivity for small regions and even individual model cells near to observation sites is estimated using the LPDM part, while the sensitivity for large regions remote from the monitoring sites is derived using the Eulerian part (Kaminski et al., 2001).

Minimization of the computational cost can be obtained, as high-resolution simulations are performed over a limited number of regions nearby to the observational sites using the LPDM part, while for the rest of the globe the coarse-resolution results are calculated by the Eulerian part.

High consistency of the tracer fields calculated by the Lagrangian and the Eulerian models due to the fact that both models use the same input meteorology.

An essential stage of the adjoint model construction is its validation. A lack of accuracy in the adjoint model will likely degrade the performance of the cost function minimization (Eq. 2). Several different tests were carried out to evaluate the accuracy and precision of the constructed adjoint model. Considering the simple formulation of the Lagrangian part, we focused on testing the NIES TM adjoint.

The discrete adjoint obtained through automatic differentiation can be easily validated by comparing the adjoint sensitivities with forward model gradients calculated using the finite difference approximation (Henze at al., 2007).

The forward model sensitivity,

In the first test, adjoint simulations were carried out using an initial
CO

To quantify the difference between the two calculations of the sensitivity

Comparison of sensitivities of CO

Comparison of sensitivities of CO

As for Fig. 8, but for day 4.

The definition of the adjoint of the tangent linear forward model

The next series of calculations was made for real measurements. We used data
from the Siberian observation network (Table 3) for the period
1–4 January 2010. CO

The sensitivities of CO

Above, we have already stated that the Eulerian part of the coupled model is
more effective in reproducing of long-term patterns, while the Lagrangian
part is better for resolving synoptic and hourly variations. This follows
from the fact that the A-GELCA components have different footprints. The
Eulerian adjoint has a wider footprint, with the greatest values in an area
where the effect of all stations is summed. The Euler model monitors global
and large-scale changes, although some stations can be outside this zone
(i.e., YAK in Fig. 8a, g or NOY in Fig. 9a, b). These figures illustrate why
the Eulerian model, even with a sufficiently detailed grid, fails to
reproduce CO

The FLEXPART model sensitivity shows more irregular distributions, and higher values closer to the observational sites, thereby reflecting the model's ability to monitor small-scale changes (Figs. 8 and 9, panels c, d).

During coupling, the sensitivity is aligned due to the cross-linking of
components (Figs. 8–9 panels e, f). Thus, the intensity has maximum near the
stations and smoothly decreases when distance increases. The Eulerian and
Lagrangian models employ different approaches and grid resolutions for the
modeling of atmospheric tracers, and can thus resolve processes with
different time and spatial scales, and underlying physics. By changing the
Eulerian model resolution, it is possible to change size of the footprint.
This system can utilize responses calculated at higher resolutions, such as
0.5

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 (

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 CO

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 CH

All code in the current version of the NIES forward model is available on
request. Any potential user interested in these modules should contact
D. Belikov (dmitry.belikov@nies.go.jp) or S. Maksyutov (shamil@nies.go.jp),
and any feedback on the modules is welcome. Note that potential users
may need help using the forward and adjoint model effectively, but open
support for the model is not available due to lack of resources. The code of
the adjoint part of the current NIES model is unavailable for distribution,
as it was generated using the commercial tool TAF
(

The FLEXPART code was taken from the official web site

The authors thank A. Stohl for providing the FLEXPART model. We also thank
T. Machida for Siberian observation data (downloaded from