Introduction
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 (CO2) and methane (CH4)). The new system is referred to as NICAM-TM 4D-Var, the 4D-Var inversion system based on the Transport Model version of the Nonhydrostatic ICosahedral Atmospheric Model. It consists mainly of forward and adjoint
transport models and an optimization scheme. This paper presents derivation
of the transport models and evaluate their performances. The accompanying
paper describes the optimization scheme and demonstrates the
application of the new system to an atmospheric CO2 inversion problem.
The 4D-Var inversion method has evolved over the years to achieve higher
spatiotemporal resolution in inverse calculations of various atmospheric
trace gas measurements
that
include continuous measurements at the surface, as well as aircraft
and satellite observations
e.g.,. The 4D-Var method is an iterative method requiring multiple model simulations, not only forward but also backward using an
adjoint model. Moreover, a global inversion calculation of an atmospheric
greenhouse gas requires analysis over a long period ∼ 20 years;
e.g., to figure out interannual variations of surface fluxes,
resulting in at least hundreds of years of model simulations in total. This
provides strong motivation for us to develop ways of making the computations more efficient.
For GHG simulations, there are two types of atmospheric transport models: one
is online e.g., and the other is offline
e.g.,. Online models include atmospheric general
circulation models (AGCMs) which incorporate passive tracers of GHGs and
simulate their movements. Offline models are those that simulate transport
of tracer gases using archived meteorological data (e.g., temperature, wind
velocity, and humidity). Therefore, an offline model is computationally much
more efficient than an online model, and hence is favored for the 4D-Var
calculation. However, archived meteorological data usually consist of
reanalysis data with limited spatiotemporal resolution. Furthermore, temporal
snapshots of reanalysis data are not physically consistent with each other
. Therefore, in offline transport calculation, reanalysis
wind data should be modified in advance to restore the dynamical consistency
with pressure tendencies; otherwise the tracer mass cannot be conserved
.
An adjoint model integrates variables backward in time to calculate
sensitivities of a certain scalar variable against model parameters
, with applications for data assimilation and inversion
analyses. Furthermore, the adjoint sensitivity is a powerful tool to diagnose
tracer transport mechanisms e.g.,. For GHG inverse
analyses, the atmospheric processes are considered to be all linear, with
CO2 and CH4 transported as passive tracers and CH4 losses calculated
by simple linear equations with prescribed hydroxyl (OH) and chlorine (Cl)
radicals and O(1D) . However, in practice, nonlinearity is
introduced into the discretized model, which complicates adjoint model
formulation. One prominent example is the discretization of an advection
scheme. An advection scheme with higher-than-first-order accuracy must employ
a nonlinear algorithm to preserve tracer monotonicity .
Therefore, an advection scheme often uses a nonlinear flux limiter or fixer
that depends on tracer quantities, introducing nonlinearity and
discontinuity. However, the direct adjoint of such a nonlinear code is
computationally inefficient for a long simulation, because it requires
several checkpoints from which time forward simulations are restarted to
restore tracer quantities at every model time step. Furthermore, it has been
found that such an adjoint model is ill-behaved due to discontinuities
. Therefore, alternative approaches have been
proposed at the expense of linearity or the accuracy of numerical scheme
.
Most studies use either the “discrete adjoint” or “continuous adjoint”.
However, which approach performs better is still controversial. The discrete
adjoint is linear but reduces the accuracy of the numerical scheme, while the
continuous adjoint is nonlinear but maintains the monotonicity.
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
NICAM:. In fact, the model can be
considered as an offline version of the online transport model of
NICAM-based Transport Model NICAM-TM:. In the
offline model, tracer transport is calculated in the same way as in the
online model, but driven by meteorological data provided from the AGCM run
of NICAM in which winds fields are nudged toward reanalysis data. Compared to
the reanalysis data, the physical and dynamical consistency in the nudged
AGCM data is maintained. Furthermore, the use of the AGCM enables us to
change the spatiotemporal resolution of the meteorological input data.
Similar AGCM-based offline models have been developed by previous studies
. In fact, the offline NICAM-TM has already
been used in a CO2 inversion studies using the conventional matrix
calculation method . In this study, we examine the relative
impact of the meteorological driver data with different temporal resolutions
in each of the transport processes (advection, vertical diffusion, and cumulus
convection) on model accuracies. Maintaining the same degree of flexibility
in the time resolution of the offline forward model, we develop a new
adjoint model. The new adjoint model can be run in discrete or continuous
mode. In order to achieve the exact adjoint relationship with its
corresponding forward model, the discrete adjoint method switches off the
nonlinear flux limiter in the advection scheme, while the continuous adjoint
utilizes the flux limiter to give preference to monotonicity over the adjoint
exactness.
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 CO2 concentrations simulated by the offline model with those
by the online model. In that evaluation, we test various temporal
resolutions of the meteorological data, which are separately determined for
each transport process. Also, we validate fundamental properties of the
adjoint model and demonstrate the utility of the adjoint sensitivity in a
back-trajectory analysis.
Methods
NICAM
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 . This avoids the pole problem inherent in
latitude–longitude grids and facilitates global high-resolution simulations.
Due to the feasibility of high-resolution simulations, the dynamical core
of NICAM is constructed with nonhydrostatic equations .
Furthermore, the model program is designed for an efficient parallel
computation with Message Passing Interface (MPI) libraries
. In fact, NICAM has been used for global
nonhydrostatic high-resolution simulations with 14 km to 850 m grid
resolutions . Nonetheless, in this
study, we use a moderate resolution to reduce the high computational cost
associated with the GHG inversion that requires repeated long-term
simulations.
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 , whose control volume is a shaped pentagon at
twelve vertices of the original icosahedron and hexagon at other grids
(Fig. 1b). Those control volumes are constructed by connecting mass centers
of the triangular elements that are produced by the recursive division of the
icosahedron (Fig. 1a). The mean grid interval of glevel-5 is approximately
240 km. Although this horizontal resolution is much coarser than the
high-resolutions that NICAM mainly targets, it is still comparable to the
resolutions used in previous GHG inversion studies
e.g.,.
The grid distribution of NICAM glevel-5. Triangular elements produced by dividing an icosahedron five times (a) and control
volumes constructed by connecting the mass centers of the triangular
elements (b).
Because the dynamical core is constructed with the finite-volume method,
NICAM achieves the consistency with continuity CWC: for
tracer transport , which cannot be achieved in
spectral AGCMs . Due to this CWC characteristic, tracer mass is
perfectly conserved without any numerical mass fixer. Indeed, thanks to this property of CWC, atmospheric transport studies have been conducted using
NICAM-TM with glevel-5. The model reproduces reasonably well the synoptic
scale and vertical variations of radon (222Rn) and the inter-hemispheric
gradients of sulfur hexafluoride (SF6) at the surface and in the upper
troposphere .
The model configuration in this study is essentially the same as the one
described in , except for the cumulus parameterization. The
cumulus parameterization scheme is changed from to
. The number of vertical model layers is 40, 12 layers of
which exist below about 3 km. The top of the model domain is at about
45 km. The tracer advection process is calculated with the scheme of
Miura (2007), and the vertical turbulent mixing is calculated with the MYNN
Level 2 scheme . The model time step of
glevel-5 is 20 min, both for the online and offline calculations. For the
nudging used in the online calculation, we use the 6-hourly horizontal wind
velocities of the Japan Meteorological Agency Climate Data Assimilation
System (JCDAS) reanalysis data .
Offline NICAM-TM
As is the case in the online model, the offline model integrates tracer
mass ρq (ρ is air mass density and q is tracer mixing ratio) as
∂ρq∂t=∇⋅ρvq+∂∂zρKv∂q∂z+fcρ,qw,T,MB,q,
where ∇ and v are the 3D
divergence operator and wind vector, respectively, and Kv is the
vertical diffusion coefficient. On the right hand side of the equation, the
first and second terms represent the grid-scale tendency of advection and the
sub-grid-scale tendency of vertical diffusion, respectively. The third term
fc denotes the sub-grid-scale tendency of cumulus convection,
determined by ρ and the mixing ratios of water substances
(qw), temperature (T), and cumulus base mass flux
(MB).
Meteorological parameters used for the offline forward and adjoint
models.
Parameter
Symbol
Time type
Related process
Air mass density
ρ
Snapshot
Advection
Air mass flux
V (=ρv)
Averaged
Advection
Vertical diffusion coefficient
Kv
Snapshot
Vertical diffusion
Water substances
qw
Snapshot
Cumulus convection
Temperature
T
Snapshot
Cumulus convection
Cumulus base mass flux
MB
Snapshot
Cumulus convection
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 V (=ρv), while instantaneous values
are saved for other meteorological parameters. The averaging of V
is intended to preserve the CWC property. Originally, NICAM calculates the
tracer advection using time-averaged air mass fluxes that are derived from
air mass fluxes at a shorter time interval. The tracer advection is
calculated with the Euler scheme, while momentums are calculated at a shorter
time interval using the Runge–Kutta scheme. The time-averaged air mass flux
retains the CWC property . The offline model uses air
mass fluxes that are further averaged for the thinning interval as
V‾t=1N∑i=1NVt+iΔτ,
where Δt is the thinning interval, Δτ is the model time
step, and N is the integer defined as N=Δt/Δτ. The
offline calculation, whose time step is the same as that of the online,
uses the above repeatedly during N steps from t to t+Δt as
ρqt+(i+1)Δτ=ρqt+iΔτ+Δτ∇⋅V‾tqt+iΔτ.
In order to preserve CWC, ρ is simultaneously integrated with the same
time-averaged air mass flux as
ρt+(i+1)Δτ=ρt+iΔτ+Δτ∇⋅V‾t+α,
where α is the modification term. If α=0, the Lagrangian
conservation (dq/dt=∂q/∂t+v⋅∇q=0) is achieved, which can easily be shown by
substituting Eq. (4) to Eq. (3). In practice, α is nonzero and the
Lagrangian conservation is not strictly satisfied due to evaporation and
precipitation (note that ρ includes not only dry air but also water
substances). This α is calculated as
α=1Nρt+Δt-ρt-Δt∇⋅V‾t,
so that the integrated ρ with Eq. (4) after N steps coincides with
ρt+Δt that is provided from the online calculation. The other
meteorological parameters (Kv, qw, T and
MB) are linearly interpolated from the thinned interval steps to
the model time steps.
Adjoint NICAM-TM
When M represents a forward model matrix and a and
b are arbitrary vectors, an adjoint model matrix M*
satisfies 〈a,Mb〉=〈M*a,b〉, where 〈.,.〉 is an inner
product. In the usual case, the inner product is defined as 〈a,b〉=aTb; therefore,
M* is equivalent to MT. In practice,
b represents mixing ratio or surface flux and a is its
adjoint variable. An adjoint model integrates adjoint variables backward in
time to calculate sensitivities.
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),
ρ is simultaneously integrated with the reversed winds and -α in
place of α.
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 fact, this flux limiter
improves the model accuracy to some extent . All the
transport calculations other than the flux limiter are linear. Therefore, we
obtain a completely linear forward model by just switching off the flux
limiter. From that linear forward model, we construct the adjoint model by
transposing the linear codes. Because of the linearity, this adjoint model is
expected to have the exact adjoint relationship with the forward model (with
the flux limiter off), as proven in Sect. 3.4. The relationship is expressed
as
MxTy=xTMTy,
where x and y represent the model input parameter vector and
the observation vector, respectively. By giving up the monotonicity, however,
the discrete adjoint produces negative (or oscillatory) sensitivities.
In the second approach, a continuous adjoint model is developed by
discretizing the continuous adjoint equation .
In this approach, the flux limiter can be employed not only in the forward
model, but also in the adjoint model, keeping the tracer concentrations or
sensitivities positive (or non-oscillatory). However, due to the
nonlinearity of the flux limiter, the adjoint relationship is no longer
exact. The continuous adjoint equation of advection is written as
-∂q*∂t=∇⋅vq*,
where q* is the adjoint variable for q. Equation (7) can be derived with
the method of Lagrange multipliers and partial integrals from
the advection part of Eq. (1) (a detailed derivation can be found in
). Let q̃*=q*/ρ, and we obtain
-∂ρq̃*∂t=∇⋅ρvq̃*.
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 q̃*.
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.
CO2 flux data
For the validation of the offline forward model, we simulate atmospheric
CO2 for the year 2010. For the surface boundary CO2 flux input to the
model, we use the inversion flux of that is optimized for
atmospheric CO2 concentrations for 2006–2008. The inversion (posterior)
flux consists of prior flux data sets and monthly flux adjustments derived
from the observations. In this study, we replace the prior flux data sets with
those for 2010 other than the climatological air–sea exchange data from
. We use the monthly data of fossil-fuel emission from the
Carbon Dioxide Information Analysis Center (CDIAC) , of
biomass burning emission from the Global Fire Emissions Database ver. 3.1
, and of terrestrial biosphere net ecosystem production
(NEP) from the Carnegie–Ames–Stanford Approach (CASA) model
. To represent the diurnal variation of the terrestrial
biosphere flux, we redistribute the monthly CASA NEP into 3-hourly fluxes using
the same method as using 2 m height air temperature and
downward shortwave radiation data of JCDAS for 2010. Although the integrated
surface CO2 flux input to the model does not necessarily represent the
actual flux variations in 2010, the overall resulting atmospheric CO2
concentration field is consistent with the actual observed CO2
concentrations, permitting an effective evaluation of the model transport
performance. The initial concentration field is also constructed by running
the model with the inversion flux for 2003–2009.
Results
Computational cost
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.
Evaluation of the data thinning error
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 CO2 concentration simulated by the
offline model against the online model. Time interval of the meteorological
driver data is changed for each transport process as shown in Table 2
(a–e) and the same time resolutions are used as (e) but
with the flux limiter switched off (f).
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 CO2
concentration variation simulated by the online model for 2010.
Notation
Temporal interval (h)
Relative error (%)
Advection
Vertical
Cumulus
Surface
300 hPa
diffusion
convection
A6V6C6
6
6
6
12.9
5.3
A3V6C6
3
6
6
12.7
4.8
A3V6C3
3
6
3
12.6
3.3
A3V3C3
3
3
3
6.3
2.2
A3V1C3
3
1
3
2.6
2.0
A3V1C3 w/o flux limiter
3
1
3
7.5
9.0
Figure 2 shows a zonal mean pressure–latitude cross-section of the
root-mean-square deviation (RMSD) in CO2 concentrations between the
offline and online models. The RMSD value is calculated from hourly model
output. The RMSD value represents the error induced only by the
data thinning. Generally, the RMSD values are small even in the coarsest
resolution case in which all the transport processes are calculated with
6-hourly data (A6V6C6). In most areas, the RMSDs are less than 1 ppm,
indicating that the atmospheric transport is generally well simulated by the
6-hourly resolution. The relative error, which is defined as RMSD divided by
the standard deviation of the concentration variation simulated by the
online model, is 12.9 and 5.3 % on average at the surface and 300 hPa,
respectively (Table 2).
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 CO2
concentration variations in the mid- to upper troposphere. Furthermore, when
increasing the temporal resolution of the vertical diffusion coefficient from
A3V6C3 to A3V3C3, and to A3V1C3, we find greater RMSD reductions near the
surface (Fig. 2d and e). The relative errors at the surface are reduced to
6.3 and 2.6 % for A3V3C3 and A3V1C3, respectively. This is attributable to
the fact that vertical diffusion has a much higher temporal variability than
the other transport processes, especially near the surface.
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 CO2 concentration variations in the stratosphere.
However, in the troposphere, the numerical oscillations are not so large compared
to the CO2 concentration variations. Consequently, the relative error is
7.5 % at the surface, which is larger than A3V1C3 but less than the
6-hourly vertical diffusion cases (A6V6C6, A3V6C6, A3V6C3), and 9.0 % at
300 hPa, which is the highest number in all the sensitivity tests (Table 2).
Annual mean difference of CO2 concentration at the surface model
layer between the offline and online models (offline minus online) for
each sensitivity test: A6V6C6 (a), A3V3C3 (b),
A3V1C3 (c), and A3V1C3 without the flux limiter (d). White
colored areas signify absolute values less than 0.2 ppm. The geographical
locations of Minamitorishima (MNM), Karasevoe (KRS), and Narita (NRT) are
also indicated in (a).
Figure 3 shows the annual mean difference of CO2 concentration at the
surface between the offline and online models, for A6V6C6, A3V3C3, A3V1C3,
and A3V1C3 without the flux limiter. In fact, these differences represent
biases from the online calculation induced by the data thinning. Figure 3a
shows that even the lowest temporal resolution of 6-hourly data input
(A6V6C6) reproduces the CO2 concentrations over the oceans well, where the
bias is quite small (< 0.2 ppm). Meanwhile over the terrestrial areas, we
see significantly larger biases. Specifically, they are more than 4 ppm over
the tropical regions of the Amazon and Africa; these values are all negative due
to systematically smaller nighttime accumulation, i.e., larger mixing, than
the online model. Furthermore, since the results of A3V6C6 and A3V6C3 are
very similar to that of A6V6C6, the resolution of the vertical diffusion
coefficient data is a major factor contributing to the data-thinning error,
particularly over the terrestrial biosphere in the summertime when strong
diurnal variations exist. These biases are reduced but still larger than
1 ppm even when halving the temporal interval (A3V3C3; Fig. 3b). However, by
increasing the temporal resolution of the vertical diffusion coefficient data
to hourly, the biases become nearly indiscernible (A3V1C3; Fig. 3c). Since we
still use the moderate resolution of a 3-hourly time step for the advection and
cumulus convection processes, the necessary disk storage of A3V1C3 for 1 year
is not extremely large (approximately 50 GB, after partially performing
2-byte data compression). Therefore, we set the A3V1C3 configuration to be
the default for the glevel-5 simulations. When the flux limiter is switched
off in A3V1C3, the bias increases slightly over the terrestrial areas but
remains mostly less than 1 ppm (Fig. 3d). This bias is relatively small
compared to the RMSD shown in Fig. 2f. Therefore, A3V1C3 without the flux
limiter would be permissible only if the focus is on the concentration in
the troposphere. This model configuration should be used when the model
linearity is stringently required, such as in the use with the discrete
adjoint.
Comparison with observations
In order to assess the magnitude of the offline model error, we compare the
simulated CO2 concentrations with the observed measurements at
Minamitorishima, located in the western North Pacific , at
Karasevoe, located in west Siberia , and with the
continuous aircraft CONTRAIL Comprehensive Observation Network for
Trace gases by Airliner: measurements obtained at 8–10 km
altitude over Narita, Japan (each observation location is shown in Fig. 3a),
representing marine background, continental, and upper-troposphere
conditions, respectively.
Time series of CO2 concentration for 2010 at
Minamitorishima (a), Karasevoe (b), and 8–10 km altitude
over Narita (c). Each upper panel shows the time series observed
(black) and simulated by the online model (red). Each lower panel shows
differences of CO2 concentrations between the online model and the
observation (gray), and between the offline model (green for A6V6C6, magenta
for A3V1C3, and blue for A3V1C3 without the flux limiter) and the online model.
The number in the parenthesis gives the RMSD value for each sensitivity case.
Note that only tropospheric data (determined by the dynamical tropopause;
) are used for the comparison over Narita.
Figure 4 shows the observed and simulated CO2 concentrations at each site.
Generally, the online model reproduces the observed CO2 concentration
variations relatively well, partly due to the inversion flux we use. In the
inversion, we used the Minamitorishima and CONTRAIL data to constrain the
terrestrial biosphere and ocean fluxes . However, the high
reproducibility of the synoptic variations indicates reasonable transport
performance of NICAM-TM, given the fact that we used monthly mean
observations in the inversion. Table 3 shows correlation coefficients of the
synoptic variations between the simulated and observed concentrations at the
three sites, all of which are found to be statistically significant. Here, the
synoptic variations are defined by residual CO2 concentrations from a
smoothed curve represented by a linear trend and three harmonics, similarly
to .
Temporal correlation coefficients of simulated residual CO2
concentrations (see the text for details) with the observations at
Minamitorishima, Karasevoe, and 8–10 km altitude over Narita for the
online and offline (A6V6C6 and A3V1C3) simulations. The results of A3V1C3
without the flux limiter and the online model nudged by JRA-55 are also
shown.
Site
Online
A6V6C6
A3V1C3
A3V1C3
Online
w/o flux limiter
nudged by JRA-55
Minamitorishima
0.577
0.572
0.575
0.573
0.587
Karasevoe
0.610
0.579
0.613
0.607
0.609
Narita 8–10 km
0.323
0.318
0.324
0.311
0.302
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
JRA-55:, instead of JCDAS, are used for the
nudging. The RMSD values between the two online models (JCDAS versus JRA-55)
are 0.22 and 0.40 ppm, respectively, for Minamitorishima and over Narita,
which are larger than the RMSDs between the online and offline models.
Also, the correlation coefficient change from JCDAS to JRA-55 is larger than
the changes from the online to the offline (Table 3). In fact, these
correlation coefficients would change more distinctly with a different model,
given the fact that showed a large range of the correlation
coefficients (∼ 0.4–0.7) for Minamitorishima among multiple models.
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 CO2 values compared to those produced by the online
model during the summer. As stated earlier, this is likely due to the
systematically smaller nighttime accumulation and is the cause of the
negative bias against the online model shown in Fig. 3a. This lower CO2
of A6V6C6 does not necessarily cancel out the positive deviations of the
online model from the observation (Fig. 4b) and hence is not closer to the
observation than the online model. In fact, the correlation coefficient of
the synoptic variations diminishes from the online model to A6V6C6 (from 0.610
to 0.579), whose magnitude is relatively large compared to the other changes
(Table 3). By increasing the temporal resolution of the vertical diffusion
coefficient to hourly (A3V1C3), we obtain a sufficiently small RMSD value of
0.24 ppm compared to the online model.
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.
Validation of the adjoint model
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
. In Eq. (6), if x and
y are the basis unit vectors having 1 for ith and jth elements,
respectively, and 0 for all the others (i.e., x=(0⋯0,1,0⋯0)T and y=(0⋯0,1,0⋯0)T), a value sampled at j, which is simulated from x
with the forward model ((Mx)Ty=(Mx)j=Mj,i), should coincide with the value simulated
from y with the adjoint model and subsequently sampled at i
(xT(MTy)=(MTy)i=Mj,i).
Checking this reciprocity, we can verify the exactitude of the adjoint code.
To evaluate the reciprocity for both the discrete and continuous adjoint
models, we use the forward model without and with the flux limiter,
respectively. The former forward/adjoint model set is linear but does not
ensure monotonicity, while the latter set is nonlinear but ensures
monotonicity. Both in the forward and adjoint model simulations, we use the
configuration of A3V1C3 for the meteorological input.
For a case study, we examine an Asian outflow event, which is a typical
transport phenomenon in East Asia during the winter–spring season
e.g.,. We prescribe a surface flux at the model grid (X)
located on the coast of East Asia representing the basis unit vector
x (its location is denoted by the open cyan circle in Fig. 5b).
Meanwhile, we prepare 160 observational basis unit vectors (y), whose
sampling points are regularly located at 3 km altitude over an area enclosed
by 14–32∘ N and 111–159∘ E (denoted by cyan dots in
Fig. 5a). The simulation period lasts for 7 days, starting on 1 January 2010. The
flux is time invariant, i.e., the vector x is a function of space
only. Figure 5a shows the concentration field at the end of the period, as
simulated by the forward model with the flux limiter. In addition,
Fig. 5b shows the sensitivities of the observation Y that is located at the
eastern edge of the range (denoted by the cyan triangle in Fig. 5a) against
the surface fluxes (i.e., footprint). This sensitivity is calculated by the
continuous adjoint. We find that, using the discrete adjoint, the calculated
footprint pattern is quite similar to that shown in Fig. 5b. This is not
surprising since the forward simulation without the flux limiter does not
introduce substantial errors in the troposphere, as previously shown. As the
concentration field shown Fig. 5a is simulated from the unit flux, it also
represents the degree of spatial sensitivity between the flux and the
observation. According to Eq. (6), the concentration value sampled at the
observation point Y should coincide with the footprint value located at the
flux point X. By performing adjoint simulations for the remaining 159
observation points, we can evaluate the overall reciprocity of the adjoint
model.
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 X (a) and the sensitivities of the observation
Y against the surface fluxes (footprint) simulated by the continuous
adjoint model (b). The observation points, from which the adjoint
sensitivities are calculated, are denoted as cyan dots and the location of
observation Y is indicated by the cyan triangle (a). The location
of the basis unit flux X is indicated by the open cyan circle
(b).
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 X. The open red circles denote
values from the linear model setup (the forward model without the flux
limiter and the discrete adjoint model), while the open blue circles denote
values from the nonlinear model setup (the forward model with the flux
limiter and the continuous adjoint model).
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 X. It can be seen in the figure that the footprint values simulated by the discrete adjoint completely correspond to the forward concentration values within computer machine accuracy. This demonstrates that the discrete adjoint has
the exact reciprocity against the forward model without the flux limiter. Conversely, the continuous adjoint does not have the exact reciprocity
but it is reasonably approximated. Figure 6 also demonstrates that the
continuous adjoint successfully avoids negative sensitivities because of the
flux limiter, while the discrete adjoint calculation does produce negative
sensitivities.
Adjoint trajectory analysis
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
J=∫tto∫Ωg(q)dΩdt′,g(q)=q(x,t′)δ(x-xo)δ(t′-to),
where δ is the delta function and xo and to represent the
observed location and time, respectively. Therefore, the value of the cost
functional corresponds to the observed concentration q(xo,to). According
to Appendix A, the change of the cost functional is given by
ΔJ=Δq(xo,to)=∫tto∫ΩF*q*(x,t′)Δs(x,t′)dωdt′+∫Ωq*(x,t)Δq(x,t)dω,
where F* represents the adjoint of the transferring operator
from flux to concentration, and Δs denotes the flux perturbation.
F*q* and q* denote the sensitivities of the observed
concentration with respect to the surface flux and concentration,
respectively. If considered processes are all linear (which is the case in
this study since we consider only atmospheric transport), we can omit
Δ. Then, the first and second terms represent respectively the actual
contributions of the surface flux from t to to and the concentration at
t to the observed concentration. For our analysis, we investigate the
spatial structures of these sensitivity quantities normalized by
q(xo,to), i.e., ∫ttoF*q*(x,t′)s(x,t′)dt′/q(xo,to) and q*(x,t)q(x,t)/q(xo,to).
These quantities derived by the adjoint model have been utilized in previous
studies for diagnosing trace gas transport in the atmosphere
and a pathway of a water mass in the ocean
.
The normalized flux contributions (gray shades) and the adjoint
trajectory volumes (color contours) are shown for the high CO2 concentration events
observed at Minamitorishima on 24 January (a), at Karasevoe on
27 December (b), and at 8 km over Narita on 12 January 2010
(c), which are denoted by the cyan arrows in Fig. 4. See the text
for the definitions of the flux contribution and the adjoint trajectory
volume. The upper and lower panels present the horizontal and vertical
structures of the adjoint trajectory volume, whose maximum values are
projected onto the horizontal and vertical planes, respectively. The adjoint
trajectory volumes are drawn with contours starting from a minimum value 0.02
at an interval of 0.6. The color of the contour represents how many days
prior to the observation time to which the adjoint trajectory volume is
associated.
Using such adjoint-derived quantities, we analyze three high CO2
concentration events, observed at Minamitorishima on 24 January, at
Karasevoe on 27 December, and at 8 km over Narita on 12 January 2010
(denoted by the cyan arrows in Fig. 4). These high-concentration events are
chosen because NICAM-TM captures this event well (see Fig. 4). Figure 7 shows
the normalized flux contribution and “adjoint trajectory volumes”
against each event. The adjoint trajectory volume is derived
by averaging q*(x,t)q(x,t)/q(xo,to) for each day prior to the
observation period. Overlaying the averaged q*(x,t)q(x,t)/q(xo,to) shows
the pathway of the air mass that caused each of the high CO2 events at the
observation location and time. This analysis approach resembles the one taken
by . The forward simulation that calculates q(x,t) and the
adjoint simulation that calculates q*(x,t) and
F*q*(x,t) are performed for 1 week prior to
each observation period. Here, we show the results calculated by the continuous
adjoint model, taking the advantage of its monotonicity property. However,
the discrete adjoint model produces similar sensitivity features (not shown).
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 CO2
plume was characterized by slow diffusion. For the event observed at
Karasevoe, the analysis indicates that the air mass that produced the high
CO2 concentration was advected from the west, but fluxes in the vicinity
of the observation site also contributed to the observed concentration
(Fig. 7b). This local flux contribution is a result of a very shallow mixed
layer, as indicated by the vertical structure of the trajectory volume that
is concentrated below 1 km. Figure 7c shows that the high concentration
event observed over Narita originated from southeast China. The adjoint
trajectory volume indicates that the horizontal transport of the air mass
from China to Japan was fast (taking only about 2 days) compared to the other
cases because of the strong westerlies in the free troposphere. Before this
fast eastward transport, the analysis also indicates the possibility of an
air mass propagation westward along the slope of the topography. Therefore,
this result suggests that the topographical uplifting may play a significant
role in high CO2 concentration events frequently observed over Narita
.
Conclusions
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 CO2 as
a tracer, we have determined that the temporal resolution of the vertical
diffusion coefficient should be high; otherwise, a significantly large
systematic bias is introduced near the surface due to the smaller CO2
accumulation during the nighttime. For the spatial resolution used in this
study (the horizontal grid interval is approximately 240 km), the use of a
1 h interval for the vertical diffusion coefficient and 3 h interval for
the other meteorological fields (A3V1C3) is enough to simulate CO2
concentrations that are reasonably consistent with the online calculation.
By comparing these with observations, we have found that the error from the data
thinning in A3V1C3 is negligible compared with the intrinsic model
performance. In a case without using the flux limiter, we have found
significant errors in the stratosphere, while the errors in the troposphere
were smaller and tolerable. Therefore, simulations without the flux limiter
can be carried out in studies focused only on the troposphere.
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 CO2
concentration events.
Based on the results of this study, we have developed a new 4D-Var system for
performing CO2 inversions. Application of the 4D-Var system and its
results are described in the accompanying paper by . In the
accompanying paper, the A3V1C3 configuration is used to judge which of the
adjoint calculation methods, discrete or continuous, is better suited for
global inversion studies.
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 e.g.,. One prominent
feature of the NICAM-TM 4D-Var system is the perfect mass conservation, as
described in Sect 2.1. Another advantage of the system is its computational
efficiency when applied to linear GHG inversion problems. If we limit the
analysis period to a short time, a global high-resolution inversion would be
feasible as long as sufficient data storage capacity is available.
Furthermore, regional high-resolution inversions would also be possible with
the grid stretching technique . It is expected that
the system developed in this study and in the accompanying paper can exploit
new observations and open up new avenues for GHG inversions.