A
four-dimensional variational (4D-Var) method is a popular algorithm for
inverting atmospheric greenhouse gas (GHG) measurements. In order to meet the
computationally intense 4D-Var iterative calculation, offline forward and
adjoint transport models are developed based on the Nonhydrostatic
ICosahedral Atmospheric Model (NICAM). By introducing flexibility into the
temporal resolution of the input meteorological data, the forward model
developed in this study is not only computationally efficient, it is also
found to nearly match the transport performance of the online model. In a
transport simulation of atmospheric carbon dioxide (CO

We have developed a new four-dimensional variational (4D-Var) inversion system for estimating surface fluxes of greenhouse gases (GHGs; presently, primarily targets are carbon dioxide (CO

The 4D-Var inversion method has evolved over the years to achieve higher
spatiotemporal resolution in inverse calculations of various atmospheric
trace gas measurements

For GHG simulations, there are two types of atmospheric transport models: one
is online

An adjoint model integrates variables backward in time to calculate
sensitivities of a certain scalar variable against model parameters

In this study, we have achieved a level of computational efficiency to
conduct a 4D-Var inversion of atmospheric GHGs using offline forward and
adjoint models. The offline model is closely linked to the AGCM of
Nonhydrostatic ICosahedral Atmospheric Model

Because thinning (i.e., reducing time resolution, resulting in a decreased
number of data points) of the meteorological data might introduce some
additional model errors in the offline calculation, we evaluate those errors
by comparing CO

The horizontal grid of NICAM has a distinctive structure. Different from the
conventional latitude–longitude grid models, it has a quasi-homogenous grid
distribution produced from an icosahedron obtained by a recursive division
method

We set the horizontal resolution at “glevel-5” (Fig. 1). The “5” in glevel-5
denotes the number of division of the icosahedron. NICAM adopts the finite-volume method

The grid distribution of NICAM glevel-5. Triangular elements produced by dividing an icosahedron five times

Because the dynamical core is constructed with the finite-volume method,
NICAM achieves the consistency with continuity

The model configuration in this study is essentially the same as the one
described in

As is the case in the online model, the offline model integrates tracer
mass

Meteorological parameters used for the offline forward and adjoint models.

Table 1 shows the archived meteorological parameters that drive the offline model. Integrative time resolutions of these parameters are thinned out (i.e., reduced) from the model time step interval of 20 min to several hours. In this study, we examine the sensitivity of the model results to changes in the time resolution of each of the driving meteorological transport variables (advection, vertical diffusion, and cumulus convection; Sect. 3.2).

In the archiving of the meteorological data, averaged values are saved for
the air mass flux

When

An adjoint model is constructed based on the above offline forward model.
The adjoint model reads the archived meteorological data in the same way as
the offline model, but in reverse. Furthermore, similar to Eq. (4),

For the vertical diffusion and cumulus convection processes, we use the
discrete adjoint approach in which linear program codes are transposed. For
the advection process, we employ both the discrete and continuous approaches.
In the discrete adjoint approach, we give up the monotonicity. In NICAM, the
tracer monotonicity is achieved by the use of the flux limiter of

In the second approach, a continuous adjoint model is developed by
discretizing the continuous adjoint equation

By comparing Eq. (8) with Eq. (1), we find that we can reuse the divergence operator
of the forward code by reversing the wind direction and integrating it
backward in time. Thus, we can employ the nonlinear flux limiter to maintain
the monotonicity of

All the adjoint codes are manually written, achieving numerical efficiency of the model. Some studies use an automatic differentiation tool to readily create the adjoint model, but this carries the risk of making the model numerically inefficient. Furthermore, we retain the parallel computational ability of NICAM, allowing for significant savings in computational time.

For the validation of the offline forward model, we simulate atmospheric
CO

All the simulations are performed on PRIMEHPC FX100 with MPI parallelization by 10 nodes (each node has 32 cores). For the 1-year-long sensitivity test simulation discussed below, the offline forward model requires only 7 min, while the online model requires about 70 min. Therefore, the offline model is 10 times faster computationally than the online model. The corresponding adjoint calculation also requires 7 min, therefore the 4D-Var calculation is demonstrated to be reasonably feasible. These computational costs are evaluated using the highest temporal resolution of the input meteorological data in the following sensitivity runs (A3V1C3, see below). However, we found that the computational costs are not significantly affected by the data thinning interval.

As described in Sect. 2.2, the offline model can use a different data-thinning interval for each transport process. In order to determine an appropriate data-thinning interval, we perform five sensitivity runs (A6V6C6, A3V6C6, A3V6C3, A3V3C3, A3V1C3), changing the interval from 6 to 1 h, as shown in Table 2. In addition, we test A3V1C3 with the flux limiter in the advection scheme switched off (which is the counterpart of the discrete adjoint).

Zonal-mean latitude–pressure cross-section of annual
root-mean-square deviation (RMSD) of CO

Temporal intervals for advection, vertical diffusion, and cumulus
convection processes in each sensitivity test and relative errors globally
averaged at the surface and 300 hPa. The relative error is calculated at
each model grid by dividing RMSD by the standard deviation of the CO

Figure 2 shows a zonal mean pressure–latitude cross-section of the
root-mean-square deviation (RMSD) in CO

A closer examination shows that the temporal resolution of each transport
process affects the spatial distribution of RMSD. As shown in Fig. 2a and b,
halving the interval of the advection data from A6V6C6 to A3V6C6 does not
significantly reduce RMSDs, with the relative errors at the surface and
300 hPa decreasing slightly to 12.7 (from 12.9) and 4.8 (from 5.3) %,
respectively. However, the RMSDs values are noticeably reduced in the mid- to
upper troposphere when halving the interval of the cumulus convection data from
A3V6C6 to A3V6C3, with the relative error in 300 hPa reduced to 3.3 %.
This indicates a significant role of cumulus convection in CO

When the flux limiter is switched off in the A3V1C3 case, RMSDs are increased
globally (Fig. 2f). The region where the RMSD has most pronouncedly increased
is the stratosphere. This is probably because the flux limiter no longer
suppresses the numerical oscillation near the top of the model domain, which
is much larger than the CO

Annual mean difference of CO

Figure 3 shows the annual mean difference of CO

In order to assess the magnitude of the offline model error, we compare the
simulated CO

Time series of CO

Figure 4 shows the observed and simulated CO

Temporal correlation coefficients of simulated residual CO

At Minamitorishima and over Narita, the RMSD values between the observation
and the model are quite small; 0.92 and 1.36 ppm, respectively. Compared to
those RMSDs, the RMSD between the offline and online models is negligibly
small, even for the lowest resolution of A6V6C6 (Fig. 4a and c). Furthermore,
changes in the correlation coefficients of the synoptic variations are also
quite small (Table 3). These negligible influences of the data thinning are
accentuated by comparing with an additional online simulation in which
different wind data from the Japanese 55-year Reanalysis

However, for A6V6C6 at the continental site, Karasevoe, we found a significant
influence of the data thinning. Here, the RMSD between the observation and
the online model is 5.72 ppm, probably due to the fact that the observation
is independent of the inversion and consequently the flux data have a large
error for this area. Comparably, the meteorological resolution of A6V6C6
results in an RMSD value of 2.93 ppm. As shown in Fig. 4b, the offline
model produces lower CO

Without the flux limiter, the RMSDs are modestly small (at most 0.86 ppm for Karasevoe) and the difference does not have any distinct positive or negative tendency (Fig. 4). Meanwhile, the correlation coefficients are reduced by switching off the flux limiter coherently at the three sites (Table 3). Although they are all minute changes, they suggest that the flux limiter has improved the model accuracy.

We now validate the exactitude of the adjoint model using the reciprocity
property with its corresponding forward models. A detailed description of the
reciprocity property can be found in the literature

For a case study, we examine an Asian outflow event, which is a typical
transport phenomenon in East Asia during the winter–spring season

The concentration field at 3 km altitude on 00:00 UTC
8 January 2010 simulated by the forward model (with the flux limiter) from
the basis unit flux

The scatter diagram showing 160 concentration values simulated by
the forward model at the observation points versus their corresponding
adjoint footprint values at the flux point

Figure 6 shows a scatter diagram of the 160 pairs of forward concentration
values at the observation points with their corresponding adjoint footprint
values at the flux point

Finally, we apply the adjoint sensitivities to a transport trajectory
analysis. Generally, the adjoint model provides sensitivities of a specified
scalar value with respect to concentrations and surface fluxes (Appendix A).
When the scalar value is set to an observed concentration, the cost
functional is defined as

The normalized flux contributions (gray shades) and the adjoint
trajectory volumes (color contours) are shown for the high CO

Using such adjoint-derived quantities, we analyze three high CO

Interestingly, the analysis indicates that three high concentration events
were produced by three distinctly different transport phenomena. The flux
contribution shows that the event observed at Minamitorishima originated from
the Korean Peninsula and eastern China (Fig. 7a). Furthermore, the sharp
adjoint trajectory volume indicates that the transport of the high CO

We have developed forward and adjoint models based on NICAM-TM, as part of the 4D-Var system for atmospheric GHGs inversions. Both of these models are offline. Therefore, the models are computationally efficient enough to make the 4D-Var iterative calculation feasible. The computational cost of the offline forward model is about 10 times less than that of the corresponding online model calculation, irrespective of the temporal resolution of the meteorological data input. Furthermore, the adjoint model computational cost is nearly the same as that of the forward model.

The archived meteorological data used in the forward and adjoint models were
prepared by the online AGCM calculation of NICAM in advance. In this study,
we have developed the variable temporal resolution capabilities for
individual meteorological transport data to minimize the offline model
errors due to the data thinning. Through sensitivity tests using CO

For the adjoint model, we have explored the relative impact of using discrete
adjoint or continuous adjoint on the advective transport process. Using an
Asian outflow case, we have demonstrated the perfect adjoint relationship of the
discrete adjoint with its corresponding forward model in which the flux
limiter is turned off. In the same analysis, the continuous adjoint has also
shown reasonable adjoint exactitude against the forward model with the flux
limiter turned on. Furthermore, we have found that the adjoint model can be
used in attribution studies in which surface flux contributions are diagnosed
as a function of air mass pathway when interpreting observed high CO

Based on the results of this study, we have developed a new 4D-Var system for
performing CO

Icosahedral grid models such as NICAM are a new model type and are
becoming popular in dynamical meteorology research fields as remarkable
innovations in supercomputers are made. However, there are still only a few
studies of its applications in atmospheric chemistry and
inversion/assimilation calculations

Development of NICAM-TM is being continued by the authors.
The source codes of NICAM-TM are available for those who are interested. The
source codes of NICAM-TM are included in the package of the parent model
NICAM, which can be obtained upon request under the general terms and
conditions (

Here we explain the theory of the adjoint sensitivity following the
description of

We characterize the solution of the tracer transport Eq. (A1) by the cost
functional

The authors declare that they have no conflict of interest.

Comments from Maarten Krol and Arne Babenhauserheide helped to improve the
manuscript. They were greatly appreciated. We also thank Kaz Higuchi of York
University, Canada, for his fruitful comments on the manuscript. The
Minamitorishima CO