We describe a new 4D-Var inversion framework for nitrous oxide (N

Nitrous oxide (N

Rates of microbial nitrification and denitrification in soils depend strongly
on environmental characteristics such as temperature (Potter et al., 1996),
moisture (Bouwman, 1998; Bouwman et al., 2013), availability of reactive
nitrogen substrate, and the make-up of the soil microbial community
(Butterbach-Bahl et al., 2013), and as a result large uncertainties exist in
the spatial and temporal distribution of global N

Some previous work has found that uncertainties in stratosphere–troposphere
exchange (STE) and the associated influx of N

The global observing network for atmospheric N

In this paper, we introduce a new simulation and inversion framework for
atmospheric N

In this work we implement an N

N

A priori N

Mean annual N

Stratospheric destruction of N

Because of the long atmospheric lifetime of N

We use a 4D-Var inversion framework to solve for spatially resolved, monthly
N

In this study,

Initial model N

We assume 100 % uncertainty in the a priori emissions (for any given grid
square and month) and in the stratospheric loss frequencies, and impose
horizontal correlation length scales for emissions of 500 km over land and
1000 km over ocean, following Thompson et al. (2011, 2014a). The
observational error covariance matrix contains contributions from the
measurement uncertainty (typically 0.4 ppb; see next section for details)
and from model transport errors. We estimate the latter from the variance in
modeled N

The adjoint modules for optimizing N

Sites of surface flask and in situ N

Continued.

Below, we apply GEOS-Chem and its adjoint to assess the N

Global observing network for atmospheric N

Extensive airborne measurements of N

High-frequency airborne N

The use of different calibration scales results in offsets between different
networks measuring N

In this section we perform a range of pseudo observation tests to determine
how well N

Figure 4 shows the results of synthetic inversions in which we optimize
emissions using surface-based pseudo observations as described above. Here we
impose a time invariant a priori emission bias of

Global annual N

Pseudo observation tests optimizing N

Overall, the solution is of comparable quality whether we start with a high or low a priori bias, with some minor distinctions: the test with the positive initial bias performs slightly better over oceans and in later months of the simulation, and also converges more quickly (5 iterations versus 10 for the test with a low initial bias). However, the situation is very different when no upper bound is imposed on the solution. In this case, when given a low initial bias the optimization tends to overshoot the truth in high-flux regions while underestimating the truth in low-flux regions. Imposing both lower and upper bounds on the inverse solution (in this case, 0 and 10) is thus important to ensure a consistent solution across high and low initial bias scenarios.

Figure 4 also indicates that during the last several months of the
optimization window there is inadequate forcing for the inversion to
completely correct for the initial emission biases, particularly over the
Southern Hemisphere. This is largely due to the timescale required to
transport N

Figure 5 shows zonally integrated, annual a posteriori emissions from
synthetic inversions using surface, CARIBIC, or HIPPO pseudo observations. In
each case the state vector for optimization includes monthly emission scale
factors on the model grid (but not stratospheric loss rates), and an initial
bias of

Pseudo observation tests optimizing N

Pseudo observation tests optimizing N

We see in Fig. 5 that inversions based on the HIPPO data are also able to
capture the zonal distribution of N

Based on these experiments, we conclude that relatively sparse observations
in the upper troposphere and lowermost stratosphere, such as those from
CARIBIC, are sufficient to correct a prior bias in the global annual N

The above OSSEs were performed based on an initial fractional emission bias
that is uniform in space and time (i.e., a priori emissions set everywhere to
0.5

An important finding from previous work is that N

Figure 6 shows the resulting a posteriori scaling factors for stratospheric
loss frequencies when N

For the inversion using surface data, the optimized annual global sink in the
first year of the simulation is very close to the true value (Table 3), but
the loss frequencies are only adjusted throughout the first year in the
tropics. In the extratropics, they adjust primarily during the summer months.
The extratropical timing corresponds to the observed seasonal minimum of
N

Also shown in Fig. 6 are the a posteriori scaling factors for stratospheric
loss frequencies of N

Thompson et al. (2011) also examined the feasibility of constraining
stratospheric loss rates of N

Along with the rate of N

Figure 7 shows the fractional perturbations to the stratospheric and
tropospheric burdens of N

Results from a two-box model illustrating the sensitivity of the
tropospheric N

The top panel of Fig. 7 shows that on long timescales a perturbation to

However, on short timescales (as is used for our inversions), the importance
of stratosphere–troposphere exchange versus chemistry for tropospheric
N

Overall, we can see that N

The OSSEs in Sect. 5.1 and 5.2 demonstrate that the inversion (and N

To explore the temporal aspect of this question, we performed a test in which
we assimilate pseudo observations generated with aseasonal (model truth)
emissions, while imposing a simple seasonal bias in the a priori emissions
from natural and agricultural soils (50 % higher than model truth from
March–August; 50 % lower from September–February). As before, we
assimilate surface, CARIBIC, or HIPPO observations, and retrieve monthly
scaling factors for terrestrial and oceanic N

Results of this test indicate that a seasonal, global, emission bias is much
more difficult to resolve than is a constant bias based on the current
network of surface observations. Zonally integrated emissions (Fig. S2) begin
to approach the aseasonal model truth in the Northern Hemisphere during the
beginning of the simulation (when the a priori emissions are biased high),
but there is almost no correction of the seasonal bias in the latter half of
the simulation (when a priori emissions are biased low). Due to the long
lifetime of N

In areas near measurement sites, however, some seasonal constraints are
afforded in the inversion. For example, Fig. 8 shows monthly fluxes at four locations: a site with continuous observations (KCMP tall tower, MN, USA; 44.68

Resolving seasonal emission biases. The panels show the results
from an OSSE in which a seasonally dependent a priori emission bias is
applied and we test the ability of the inversion to recover the true model
fluxes. Results are shown for a site with continuous observations (KCMP tall
tower), a site with

Based on the above test, we can conclude that flask and in situ observations provide valuable corrections to seasonal emission biases upwind and in the vicinity of the measurements, though not necessarily on a monthly timescale. However, any seasonal biases arising from errors in model STE may be difficult to separate from such seasonal emission errors. Furthermore, large parts of the world (illustrated by the DR Congo site in Fig. 8) lack any meaningful seasonal constraints on emissions.

The spatial resolution at which current measurements constrain global
N

Figure 9 shows the resulting percent error reduction achieved in each model
grid cell using surface, CARIBIC, or HIPPO observations for a given month of
our 2-year simulation. Results using surface observations are shown for
month 1 (April 2010), but are comparable for all subsequent months. We see
appreciable error reduction near sites with continuous observations in North
America and Europe, and more modest error reductions in surrounding grid
cells, at sites with flask observations, and in the northern Atlantic upwind
of Europe. There is little (

Error reduction (%) in N

Figure 9 also shows that the sparse, high altitude CARIBIC observations
provide limited information on the spatial distribution of N

The spatial information provided by HIPPO observations varies by month
according to the flight tracks, and is complementary to that achieved with
surface data. For example, during August 2011, we see large error reductions
over the central USA, as well as some improvement for grid cells in East Asia
that are upwind of the HIPPO flight track. Some error reduction is also
achieved in these locations for May 2011, despite the fact that no HIPPO
flights occurred during this month (the next flights occurred in June). Given
the long lifetime of N

Based on the same stochastic approach used above to calculate the inverse
Hessian, we can also calculate the averaging kernel of the inversion. The
averaging kernel measures the sensitivity of the inversion to emissions in
any given grid square; we can thus use it to determine how well emissions in
a given location can be independently resolved from emissions elsewhere. If
emissions in one location are completely resolved from those in other grid
boxes, the averaging kernel value will be 1.0 in that location and 0
everywhere else. Here, we calculate the averaging kernel rows (based on the
surface observations only) for a selected group of locations in key emission
regions that vary in their proximity to N

Figure 10 shows the results for the same four locations shown in Fig. 8: KCMP
tall tower (MN, USA), Hegyhátsál (Hungary), East China, and DR Congo.
KCMP features continuous observations, and we see that emissions in this
model grid cell can be constrained independently (averaging kernel value near
1.0, and near 0.0 elsewhere) from those in other places. Significant
constraints are achieved at Hegyhátsál (averaging kernel value

Averaging kernel values for the central African location are very low
(

Rows of the averaging kernel for the inversion of N

The implications of this current lack of constraints on tropical N

This severe sensitivity of the solution to the a priori error assumption
reflects the ill-posed nature of the problem. It also highlights the fact
that, because the global N

In this section, we apply the error reduction statistics derived above to
identify priority regions where new observations are likely to have high
value for improving present understanding of global N

Figure 12 shows the distribution of unconstrained N

Inversion of N

Distribution of unconstrained N

The maps in Fig. 12 rely by necessity on a particular a priori estimate of
N

We developed a new inversion framework based on the GEOS-Chem model and its
adjoint for estimating global N

Observing system simulation experiments (OSSEs) showed that the surface and
HIPPO observations can accurately resolve a uniform bias in N

The surface and airborne data sets are all able to resolve a global bias in
the stratospheric loss rate of N

We employed a stochastic estimate of the inverse Hessian to quantify the
spatial resolution of N

From the atmospheric distribution of “unconstrained N

From our analysis it is clear that additional measurements are crucial to
obtaining a more complete picture of global N

The N

This work was supported by NOAA (grant no. NA13OAR4310086) and the Minnesota Supercomputing Institute. We thank J. Muhle and C. Harth (UCSD-SIO), D. Young (U. Bristol), P. Fraser (CSIRO), R. Wang (GaTech), and other members of the AGAGE team for providing AGAGE data. The 6 AGAGE stations used here are supported principally by NASA (USA) grants to MIT and SIO, and also by DECC (UK) and NOAA (USA) grants to Bristol University, and by CSIRO and BoM (Australia). We thank Environment Canada for providing data from the Churchill, Estevan Point, East Trout Lake, Fraserdale, and Sable Island sites. We thank R. Martin and S. Nichol for providing data from the Arrival Heights NIWA station. Edited by: O. Morgenstern