Introduction
Several data assimilation methods have been used in the field of atmospheric
chemistry and air quality in many studies as exemplified in the
reviews of. Yet, how
efficiently data assimilation schemes operate in high-dimensional and
heterogeneous models such as those used in the field remains largely unclear.
Indeed, atmospheric chemistry models are becoming increasingly complex, with
multiphasic chemistry, size-resolved particulate matter, and possibly coupled
to numerical weather prediction models. In the meantime, data assimilation
methods have also become more sophisticated. Let us briefly and
non-exhaustively describe this evolution. Kalman filters have been used with
atmospheric chemistry models by . The numerical cost of
this algorithm was addressed by the use of the ensemble Kalman filter and
variants thereof in , , ,
, and . In order to address rank
deficiencies and sampling issues, localisation and inflation have been used
in this context
. Moreover, 3D-
and 4D-Var techniques have been applied to chemical transport models (CTMs)
in the wake of their success in operational meteorology
. These methods, however,
require the development of the adjoint models
e.g. and this has led to the implementation
of easier approximate adjoints
. Attention has been
paid to the construction of the background error covariance matrix
. Getting the best of the ensemble Kalman filter
and variational methods through an hybrid ensemble–variational approach is
a quest recently initiated in meteorology
that
could be applied to atmospheric chemistry models .
Finally, the recent development of coupled chemistry meteorology models
(CCMMs) opens the Pandora's box of data assimilation in coupled systems
characterised by heterogeneous dynamics with distinct timescales,
heterogeneous sources of uncertainty, and complex interactions.
Hence, it will become increasingly difficult to disentangle the merits of
data assimilation schemes, of models, and of their numerical implementation
in a successful high-dimensional data assimilation study. That is why we
believe that the increasing variety of problems encountered in the field of
atmospheric chemistry data assimilation puts forward the need for simple
low-order models, albeit complex enough to capture the relevant dynamics,
physics, and chemistry that could impact the performance of data assimilation
schemes. Low-order models, also called toy models, are models of reduced
dimension meant to capture the prominent characteristics of the dynamics of
larger models, but at a much lower computational cost. They are not meant to
be realistic, but their study provides insights into the larger models and
their dynamics. Their low numerical cost also comes with the ability to
compute reliable statistical scores in various regimes and hence to validate
methods with greater confidence. Moreover, they can be distributed and used
with the goal to benchmark data assimilation methods since their baseline
performance can easily be reproduced.
The Lorenz-95 and tracer model
The Lorenz-95 (L95) model is a very popular low-order meteorology model
. It is a one-dimensional model, whose M=40 state
variables extend over a mid-latitude circle. It is defined by the following
set of ordinary differential equations
dxmdt=(xm+1-xm-2)xm-1-xm+F,
for m=1,…,M. The domain is periodic (circle-like, i.e. x-1=x39, x0=x40, and x41=x1). F
is chosen to be 8 so that the dynamics is chaotic with a doubling time of about 0.42 Lorenz time units and with 13
positive Lyapunov exponents. A time step of Δt=0.05 is meant to represent a time interval of 6 h in the
real atmosphere. The L95 model has been extensively used as a test bed and benchmark for data assimilation experiments.
added a tracer field to the L95 model state vector.
The field is discretised into 40 additional variables meant to represent
40 tracer concentrations. The 40 scalar variables of L95 are considered
to be the magnitude of winds at 40 locations, their sign giving their
direction. The tracer field is advected by these winds with a simple
Godunov/upwind scheme. These 80 variables are defined on the circle using
an Arakawa C-grid as
shown below.
The equations of the model are those of Eq. () together with
dcm+12dt=Φm-Φm+1-λcm+12+Em+12,whereΦm=xmcm-12ifxm≥0,Φm=xmcm+12ifxm<0 .
This model will be called L95-T
in the following. Because the meteorological and tracer state vectors are
simulated together, it is an online model. The tracer is emitted
over the whole domain and the emission fluxes are denoted
Em+12. It is deposited over the whole domain, using a simple
scavenging scheme parametrised by a scavenging ratio λ. The reference
values for those parameters in are Em+12=1 and λ=0.1. Additional details and illustrations can be found in
. L95-T is a one-way coupled model in the sense
that there is no physical feedback from the transport part to the meteorology
part. However, when applying data assimilation to the model, information is
exchanged both ways through covariances between the meteorological
and transport subsystems. In particular, observations of tracer
concentrations can in principle improve the estimation of the meteorological
variables.
Hence, L95-T represents an instructive model for more ambitious CCMMs . From a dynamical perspective,
this model couples chaotic meteorology with non-chaotic transport, as any realistic online tracer model would
do. Uncertainty in the meteorology comes from errors on the initial conditions, which grow due to the chaotic
dynamics. Uncertainty in transport comes from the uncertainty in the emission field and from the wind uncertainty, but
the dynamics being stable, there is no exponential growth of the error in the transport subsystem. This is meant to
mimic CCMMs. Data assimilation techniques applied to the model, which are meant to reduce these uncertainties, should be
able to control the growing error modes as done in numerical weather prediction, and also estimate forcings, typically
emissions, as done with CTMs.
However, in order to develop a qualitatively representative low-order CCMM, nonlinear chemistry must be added. The
primary goal of this article is to extend the L95-T model with a simple photochemical kinetic mechanism. To that end, we
will use the generic reaction set GRS;. The resulting coupled model, which will be introduced in
detail in Sect. , will be called L95-GRS. In particular, it is meant to be useful to test
a state-of-the-art data assimilation and parameter estimation method with a significant potential for such coupled
models with heterogeneous observations and dynamics.
The iterative ensemble Kalman smoother
Hence, the secondary goal of this work is to illustrate the usefulness of
low-order CCMMs to better understand the application of specific data
assimilation techniques to CCMMs and CTMs. The data assimilation method we
shall use is the iterative ensemble Kalman smoother (IEnKS). It was
introduced and developed by . A short account of the
scheme can be found in , which we do not repeat here.
However, the main characteristics of the method are recalled in the following
and its algorithm is recalled in Appendix . The IEnKS is
an ensemble variational (EnVar) method. It solves a variational problem over
a data assimilation window (DAW), as 4D-Var would do. However, because the
variational problem is solved in the reduced space generated by an ensemble
of state vectors, the adjoint of the model can easily be estimated without
the burden of generating the full adjoint model. Indeed, one can for instance use the ensemble of
perturbations within the DAW to estimate the sensitivities using finite
differences. Because it can solve nonlinear variational problems, it has an
edge over a state-of-the-art ensemble Kalman filter. In particular, it is
known to handle parameter estimation very well, as well as nonlinear models
. As such, it is as good a candidate method as
4D-Var when accounting for nonlinear chemistry
such as in a photochemical model. Unlike 4D-Var, a posterior ensemble is
generated as the output of the analysis using techniques known in
deterministic ensemble Kalman filtering. The IEnKS then propagates the
updated ensemble, allowing a better transfer of the errors from an update to
the next.
It was shown with the L95 and L95-T models that the IEnKS outperforms the ensemble Kalman filter (EnKF) and 4D-Var for
filtering applications (i.e. present-time estimation and forecasting), especially in strongly nonlinear conditions. It
was also shown to outperform the 4D-Var and the standard ensemble Kalman smoother for smoothing
(i.e. reanalysis). As for any EnVar method, the toll for applying these methods to high-dimensional models is to use ad
hoc techniques to regularise the error statistics obtained by empirical ensemble statistics that are prone to
sampling errors. As a consequence, localisation and possibly inflation are required when implementing the IEnKS in
high-dimensional systems.
If Δt is the time interval between two batches of observations,
LΔt is the DAW length over which the IEnKS variational analysis is
performed. The simplest variant of the IEnKS algorithm is given in
Appendix . Note that the case L=0 corresponds to the
ensemble square root Kalman filter in its ensemble transform implementation
, while the case L=1 corresponds to the iterative ensemble
Kalman filter .
Outline
In Sect. , the L95-GRS model with a reduced photochemistry module is introduced and justified. This
entails numerical complications similar to what is experienced in larger numerical CTMs and CCMMs due to the numerical
stiffness of the chemical reactions. Its physical and chemical relevance is also discussed. The following sections
illustrate the usefulness of these models. In Sect. , additional experiments with the IEnKS on L95-T
are described, focusing on features not discussed in . In particular, data assimilation for the full
L95-T model is compared to an offline variant where the tracer data assimilation system is operated independently from
the L95 data assimilation system. We also demonstrate the importance of the emission regime for the efficiency of the
data assimilation scheme and we assess the impact of the tracer observation network density. In
Sect. , the EnKF and IEnKS are applied to the newly developed L95-GRS model, with emphasis on the
precision, localisation, and parameter estimation. Conclusions are given in Sect. .
A low-order photochemical and transport model
In this section, we substitute the tracer part of L95-T for a reduced-order
photochemical kinetic mechanism to form the low-order coupled chemistry
meteorology model L95-GRS. We will first describe the resulting model, then
we will evaluate its ability to reproduce major physical and chemical
characteristics of the processes considered. All the parameters and equations
described in the following, with additional details, are gathered in
Appendix .
Description of the model
The photochemistry module is based on the GRS of . GRS
consists of seven chemical species meant to represent the atmospheric
chemistry of ozone formation from VOC (volatile organic compound) and NOx
emissions. The chemical reactions are
ROC+hν⟶k1O2RP+ROCRP+NO⟶k2NO2NO2+hν⟶k3NO+O3NO+O3⟶k4O2NO2RP+RP⟶k5RPRP+NO2⟶k6SGNRP+NO2⟶k7SNGN
where ROC represents the reactive organic compounds, RP is the
radical pool, SGN is the stable gaseous nitrogen product, and
SNGN is the stable non-gaseous nitrogen product. The kinetic rate
constants, taken from , are as follows
k1=10 000×e-4710T×k3min-1,k2=5.482×e242Tppb-1min-1,k4=2.643×e-1370Tppb-1min-1,k5=10.2ppb-1min-1,k6=0.12ppb-1min-1,k7=0.12ppb-1min-1,
where T is the temperature, chosen to be constant equal to 300K
for the sake of simplicity and k3 is the photolysis rate for NO2
in min-1, which is a function of sunlight. Details are given in
Appendix .
Since k6=k7 and Reactions () and () are similar,
one can merge the last two species of the scheme into a lumped species and
use a kinetic rate of 0.24ppb-1min-1 that represents the
formation of all the stable nitrogen products, whether gaseous or
non-gaseous, i.e.
RP+NO2⟶k=2⋅k6S(N)GN.
To further reduce the GRS scheme and improve the efficiency of its numerical implementation, we use the quasi-steady-state approximation
(QSSA) for the radical pool species RP, which is highly reactive and has the shortest lifetime among all the GRS species. This means that the
radical pool is in a dynamic equilibrium, adjusting rapidly to the other species concentrations. Solving the algebraic second-order equation for the concentration of the radical pool, we
obtain
[RP]=k2[NO]+2k6[NO2]2k51+4k1k5[ROC](k2[NO]+2k6[NO2])2-1.
This approximation has been validated a posteriori. There was little to no
impact on the simulated concentrations, while the mean adaptive time step of
the chemistry solver increased significantly. Further explanations on how
Eq. () is obtained are given in Appendix .
GRS is coupled to the L95 model. As for the L95-T model, the L95 variables
are seen as wind speeds that advect the GRS chemical species. The objective
is, therefore, to create a simplified model that is able to reproduce the
temporal variability of ozone chemistry at a regional to transcontinental
scale. There is a total of 40 wind variables and 200 concentration
variables, namely the ROC, NO, NO2, O3, and
S(N)GN concentrations at each of the 40 grid points defined on the
circle using the C-grid. Note that the RP concentrations are obtained
from Eq. ().
The transport equations for species [Ci] are consequently
d[Ci]m+12dt=ψmi-ψm+1i+Ri[Cj]j=1,…,6-λi[Ci]m+12+Em+12i,withψmi=xm[Ci]m-12ifxm≥0,ψmi=xm[Ci]m+12ifxm<0,
where λi is the scavenging ratio, Ei is the emission rate and
Ri[Cj] is the production term for Ci. There
is no such production term for ROC (RROC=0) so that
ROC behaves as the tracer of L95-T. The full equations are given in
Appendix for completeness.
When L95-GRS is seen as a global low-order model, the photolysis rate constant k3, which depends on sunlight, should
vary around the domain and with the season since it is directly linked to the solar zenith angle at a given grid
point. Hence, there are points on the grid where it is nighttime, with k3=0, and others where it is
daytime, with k3≠0. However, for the sake of simplicity, it has been chosen constant over the domain
and it varies according to a uniform daily cycle. This choice does not impact the order of magnitude of the simulated
concentrations. A test where the coefficient varies around the domain was performed and led to the same visual result as
in Fig. but with a delay around the domain: the black stripes of the figure that signal the time when
the NO concentrations reach 0 are slanted instead of straight.
Time evolution of the L95-GRS variables at
one grid point.
The L95
variables, flagged “Wind”, are shown with the original Lorenz unit, while the concentration unit is ppb (ppbC for ROC).
As ROC is not consumed in Reaction (), it will eventually
produce enough RP to consume all the NO, NO2, and
O3. Therefore, we have added emission fluxes for ROC and
NOx and a single scavenging ratio for all the species. The
emissions are considered constant over time and uniform over the domain, even
though a distinction between continent and ocean will also be made in the
following. These constants have been chosen using a genuine emission
inventory. Since the domain of our model is supposed to be a mid-latitude
circle discretised with 40 grid points, one cell of our domain is roughly
a few hundred kilometres in length. We used an emission rate for
NOx of 0.27 ppbday-1, where NO
accounts for 90% of these emissions and NO2 for
10%. This corresponds to an emission of
3 kgyear-1inhabitant-1 of NO for 60 million
inhabitants in a volume of 700km×700km×3km (typically France). We have fitted the values of the ROC
emission and scavenging coefficient in order to obtain concentrations of
O3 and NOx within the range of realistic
continental concentrations. Specifically, we used an emission rate of
0.0235 ppbCday-1 for the ROC species and a scavenging
ratio of 0.02 day-1. This ratio is the same as the reference
value of the L95-T model, since 1 Lorenz time unit corresponds to
5 days. All parameters are listed in Appendix .
Time integration of the model
The L95-T model is integrated in time using a fourth-order Runge–Kutta (RK4)
scheme with a time step of δt=0.05 in Lorenz time units, i.e.
6 h .
Similarly, the L95 subsystem of the L95-GRS model and the transport part are
integrated with the RK4 scheme. A first-order splitting of this integration
and the chemistry integration is performed, integrating first the L95 and
species transport part, followed by the GRS integration.
The chemical reactions of L95-GRS have a wide range of rates, which leads to
numerical stiffness. Hence, the RK4 scheme is an inadequate solver to
integrate the chemistry, even though it is more precise. An implicit or
semi-implicit scheme is required. That is why the GRS chemical scheme is
integrated with a second-order Rosenbrock method, following
. This method is costly since it is based on
a semi-implicit scheme that requires using the tangent linear model and
solving two linear systems. This is potentially the most time-consuming
operation of the whole model integration. Since the chemistry is local and
because of the splitting, the Rosenbrock scheme is actually implemented
block-wise, one block per grid point. The linear systems to be solved
point-wise have a size equal to the number of species. Because the
integration of the chemistry is block-wise, it can easily be parallelised.
The tangent linear model of GRS required by the Rosenbrock scheme is simple
to derive analytically and implement given the limited number of reactions.
Furthermore, an adaptive time stepping has been implemented that adjusts the
time step to the instantaneous stiffness of the reaction rates. However, it
has often been proven unnecessary in the free model run (i.e. without data
assimilation) in conjunction with the QSSA used for the radical pool.
The typical integration time step for the chemistry is δt=1h. The L95 and transport subsystem of L95-GRS is integrated with
δt=1h (δt=0.05/6 in Lorenz units).
Qualitative analysis of the L95-GRS model
Maximal ozone concentration (in ppb)
averaged over the domain depending on maximal averaged ROC (in
ppbC)
and NOx concentrations (in ppb). Each dot corresponds to a run with a different emission rate for ROC and
NOx, leading to different maximal averaged concentrations. The reference run is circled in black.
The outcome of a free run (after spin-up) at a grid point is shown in
Fig. . One notices the daily cycle induced onto the
NO, NO2, and O3 concentrations by the variation of the
photolysis coefficient k3, since they are directly related to the value of
that coefficient through Reactions () and (). The wind
speed and orientation variations are responsible for the wave with a period
of about 1 week. In real situations, O3 and NO
concentrations can drop close to zero at night . In our
simulation results, only NO reaches zero at night while O3
remains at high levels. However, if the ROC emissions are sufficiently
lowered or if the NOx emissions are sufficiently
increased, the opposite behaviour occurs.
This model, even if not chaotic, is highly nonlinear, exhibiting distinct
chemical regimes. This can be seen in Fig. , which
represents ozone isopleths for different mean ROC and
NOx concentrations. This feature is typical of lower
troposphere ozone chemistry and is commonly known as an Empirical
Kinetic Modeling Approach (EKMA) diagram . Two
different regimes are visible in this graph. The top part of the diagram,
where the isopleths are steep, corresponds to a ROC-limited regime. In
this regime, a reduction in the emissions of the ozone precursor
NOx leads to an increase of the ozone concentrations (as
long as the regime does not change), while a diminution of ROC
emissions reduces the ozone concentrations significantly. This is due to the
fact that in that ROC-poor regime, RP concentrations are low
and NO reacts preferentially with O3. On the contrary, the
bottom part of the diagram, where the isopleths are flat, corresponds to a
NOx-limited regime. In this regime, a reduction in the
emissions of the ozone precursor NOx leads to a strong
decrease in the ozone concentrations, whereas the ROC emissions have
little to no impact on the ozone concentrations. Since the black circle
corresponds to our reference case, it is located in the ROC-limited
regime.
In the NOx-limited regime, the low levels of NO concentrations reduce the amplitude of the
daily cycle of the ozone. Hence, the resulting concentrations rather correspond to a background ozone simulation, with
very low concentrations of NOx and little daily variability of ozone. Because GRS is meant to be used
with concentrations typical of urban areas, we chose to remain in a ROC-limited regime even though a global
simulation should be NOx-limited. Nevertheless, several free runs and data assimilation experiments
have been performed as well in a NOx-limited regime and lead to one noteworthy result: ROC
concentration estimations are worse than in our reference case, unlike NO2. This makes sense because the
NO2 concentrations mainly control the model and an error on the ROC concentrations has less impact on
other species in this context.
Time evolution of the L95-GRS variables over the whole
domain in the case of a continent/ocean division.
The L95 variables, flagged “Wind”, are shown with the original Lorenz unit, while the concentration unit is ppb (ppbC for ROC).
To emphasise the impact of the transport of the chemical species by the wind
in the model, an experiment was performed, where the domain was split into
a continental zone and an oceanic zone. In this experiment,
we set the ROC and NOx emissions on the
continent to Ei>0 for i in [1,20] and on the ocean
to Ei=0 for i in [21,40]. The results of this experiment, displayed in
Fig. , show that puffs of ozone and its precursors can cross
the ocean, similarly to what is witnessed over the Pacific
and the Atlantic . Moreover, ozone
concentrations are higher above the ocean in the absence of NO
emissions. Note also that the tracer plumes move eastward (increasing
indices), which is consistent with a positive group velocity for the L95
model, while the peaks and lows of the L95 field move westward according to
the L95 negative phase velocity .
So far, the wind kinetics (amplitude and variability) has been determined by the original L95 model characteristics. In
the reference experiment, the waves of the wind extend over several days. The concentrations are driven by this wind
kinetics but vary within those waves according to the photochemical daily cycles. However, other types of behaviour are
possible with L95-GRS by choosing differently the timescale of the L95 model. If time within the L95 model is rescaled
by α and the wind variables are rescaled by β,
dxmdt=αβ(xm+1-xm-2)xm-1-xm+Fβ,
it is possible to reduce the period of the wind wave and to match that of the
chemical daily cycles. This way, the species concentrations are significantly
modulated by the wind variations. In terms of spatial scale, this would
correspond to regional modelling rather than continental to global modelling
of the species fields. Figure illustrates this
timescale change with α=20 and β=1. The winds fluctuate at
a higher rate than the concentrations, quite differently from the reference
configuration of Fig. . Adjusting β, it is also
possible to rescale the amplitude of the winds in the L95 equations to match
a more regional/lower troposphere transport behaviour with weaker mean wind
magnitude. Note that α, β are only rescaling parameters that do
not fundamentally impact the nature of the model dynamics in contrast to,
e.g. .
Time evolution of the L95-GRS variables at
one grid point
with a time rescaling
of α=20 applied to the L95 model. The L95 variables, flagged “Wind”, are shown with the original Lorenz unit, while the concentration unit is ppb (ppbC for ROC).
Numerical experiments with the L95-T model
In this section, we experiment on the use of data assimilation techniques for
forecasting and reanalysis with the tracer model (L95-T) beyond the
preliminary results of . The aim is to demonstrate the
advantages brought by this model to study certain data assimilation
strategies. Several of the results and interpretations in this section will
also apply to data assimilation systems operating with the L95-GRS model.
Definition of online and offline data assimilation systems
A typical offline model is a CTM where the
meteorological fields have been generated externally and are given as an
input to the model. These fields usually stem from operational meteorological
prediction centres or from any independently run meteorological model. On the
other hand, online models consistently process meteorology, chemistry, and
transport of species all together, but at a higher numerical cost. The choice
of an offline or online approach is a crucial issue as far as modelling is
concerned . It is even more so when data assimilation
techniques are applied to offline or online models because of the fluxes of
information between the two subsystems: chemistry and transport on the one
hand and meteorology on the other hand .
The L95-T model stands as a well-suited simple tool to experiment on this
issue. In the following, we apply the quasi-static IEnKS to L95-T using
either an offline or an online approach. A distinction is made between
The full online data assimilation system for the L95-T model. Even though the L95 subsystem of the model does not depend
on the tracer subsystem, it should be kept in mind that information propagates both ways in advanced data
assimilation methods, as long as the error covariance matrices are defined over both subsystems.
The offline data assimilation system for the L95-T model. The L95 subsystem
is run separately. The IEnKS is applied to L95 with a DAW length
Lw (in units of Δt). The IEnKS is applied separately to
the tracer subsystem (transport, deposition, and emission) with a DAW length
Lc (in units of Δt). For advection, the winds are
provided by the analyses of the independent L95 data assimilation system.
Therefore, no feedback from the tracer subsystem to the L95 subsystem is to
be expected. The information gained from the observations flows one way.
Moreover, for the tracer subsystem, the uncertainty of the wind field
constitutes a realistic and significant source of model error. We believe
that this is an elegant way to create consistent model error, beyond
stochastic noise or offset parameters.
We conduct synthetic data assimilation experiments, applying the IEnKS to the
L95-T model. A simulation of L95-T that represents the truth is generated,
with E=1, λ=0.1. Synthetic observations are generated from the
truth every Δt=0.05. The system is fully observed, on both wind and
concentration variables. An unbiased Gaussian white noise is used to perturb
the observations. At any observation time, the observation error covariance
matrix for the wind and the tracer is in both cases R=I
the identity matrix. The analysis, output of the data assimilation system, is
compared to the truth using a root mean square error (RMSE), for the
meteorological subsystem as well as for the tracer subsystem. Reliable
statistics are computed over runs of 105×Δt with a burn-in
period of 5×103×Δt.
The ensemble size of the IEnKS has been chosen to be N=20 in a dynamical
regime of L95-T where localisation is unnecessary. Yet, sampling errors due
to the finite-size of the ensemble would require the use of inflation of the
errors. To avoid this issue, we use the finite-size scheme of
, , and where no
inflation tuning is necessary. Practically, it means that whenever the IEnKS is mentioned, the
finite-size IEnKS (IENKS-N) has been used instead or,
equivalently, that we enforced optimal inflation meant to account for
sampling errors. However, note that the finite-size approach does not account
for model error but rather sampling
errors.
We consider several practical variants of the offline data assimilation
system for the tracer model. In the first offline system, called
Offline 1a in the following, the mean analysis wind is provided to
the IEnKS of the tracer subsystem, both for the forecast step and the
analysis step of the IEnKS. In this baseline case, we choose
Lw=Lc. In a second variant, the winds are obtained
through an EnKF; i.e. Lw=0 and Lc is varied
(experiment Offline 1b). In a third variant, the winds are obtained
from an IEnKS with a given Lw and an EnKF is run for the tracer
subsystem, i.e. Lc=0 (experiment Offline 1c).
Because the uncertain winds are a source of model error for the offline system, we also implement a multiplicative
inflation on top of the IEnKS-N. It is applied on the prior by a rescaling of the anomalies. We choose the inflation
that leads to the best RMSE.
In a last variant of the offline model, called Offline 2, the analysis mean wind is still provided for the
analysis step of the IEnKS applied on the tracer subsystem. Yet, the full analysis wind ensemble, rather than
the mean, is provided in the forecast step of the IEnKS applied on the tracer subsystem. If the wind ensemble
spread is representative of the wind uncertainty, it is hoped that the uncertainty in the winds will be
properly accounted for. As in the Offline 1 experiments, a multiplicative inflation is also applied for any residual
model error.
The full online data assimilation system is also run for comparison (experiment Online), without any multiplicative inflation.
Comparison of online and offline data assimilation systems
Average RMSEs of the L95-T data assimilation
system using the IEnKS, as a function of the DAW length (in units of Δt=0.05 of the L95 model). The two top panels show the filtering RMSEs,
while the two bottom panels show the smoothing RMSEs. The scores of the wind
variables are on the left, while scores of the concentration variables are on
the right.
The case L=0 corresponds to the ensemble transform EnKF.
The performance of these systems as a function of the DAW length is reported
in Fig. . The best estimate of the present-time wind and
concentration state vectors are compared to the truth leading to the
filtering RMSE, which is a good indication of the forecasting
quality in this context. Best estimates for the past state vectors
(reanalysis) are also compared to the truth, leading to the
smoothing RMSE. For a DAW of length L, a run of length Nt (both
in units of Δt), and a state vector of size M, the formulas of the
RMSEs are
RMSEfiltering=1Nt∑i=1Nt1M||ML←0(xai)-xti+L||2
and
RMSEsmoothing=1Nt∑i=1Nt1M||xai-xti||2,
where
xak is the state analysis at time k,
xtk is the truth at time k, and for a vector x of
size M, ||x||2=∑j=1Mxj2.
First of all, the online system has a very significant edge over the offline
systems because of the two-way information flows, both for the concentration
variables and for the wind variables. This shows that concentration
observations can significantly improve meteorological forecasts, in agreement
with the results of obtained when assimilating real
observations of lower stratospheric ozone. For all offline systems, the wind
variables cannot benefit from the assimilation of concentrations, but only
from the L95 observations. Therefore, from now on, we shall focus on the
tracer subsystem.
The extrinsic model error due to the uncertain winds must be accounted for in the tracer subsystems. Otherwise, the
ensemble of the tracer subsystem collapses (the ensemble method diverges). In the absence of any correction for model
error, we observe that the estimation is close to a free run, with an average filtering RMSE of about 0.65.
Average filtering analysis RMSEs of the wind variables
(left) and concentration variables
(right) of the L95-T, as a function of the scavenging ratio for the ensemble transform EnKF (IEnKS with L=0) and the IEnKS with L=15.
Yet, as expected, accounting for model error offers better performance. Let
us first consider cases 1a, 1b, and 1c that use the best estimate of the mean
wind and apply multiplicative inflation to account for model error.
Configuration Offline 1a, i.e. when Lc=Lw, offers
a baseline performance for the filtering and smoothing RMSEs, which improves
as the joint DAW length increases. It remains quite far from the performance
of the online system since multiplicative inflation cannot compete with
a better wind estimate.
With configuration Offline 1b, where the mean wind estimate comes from an
EnKF (Lw=0), the average filtering RMSE does not benefit from an
extended DAW for the tracer, while the smoothing RMSE only marginally
benefits from short DAW (up to Lc=2) before degrading. Hence
a longer window for this CTM system is inefficient. We attribute this
important property to the stable and linear dynamics of transport.
With configuration Offline 1c, the tracer data assimilation system is based
on an EnKF, while the wind estimation gets better as Lw gets
larger. Therefore, the improvement that is observed for the filtering RMSE
comes from reduced model errors. In this configuration, the filtering and
smoothing RMSEs coincide since the concentrations are merely estimated by an
EnKF (Lc=0).
In the light of these results, we understand that the improvement that is
observed in configuration 1a comes from the reduced uncertainty in the wind
fields in the first place. Note that as Lc=Lw gets
larger, the tracer analysis within the DAW of length Lc uses wind
fields with lower error thanks to smoothing. This explains why the
improvement in the RMSEs is more pronounced than in configuration 1b, which
only benefits from the filtered winds at present time.
With configuration Offline 2, model error is addressed by not only the
multiplicative inflation but also the ensemble of winds in the forecast
steps. Each wind member is ascribed to a tracer member. This is similar to
stochastic parametrisation where one changes the model input parameters for
each member of the CTM . This shows much better performance. As
far as filtering is concerned, the optimal inflation is an increasing
function of Lc (with an inflation of 1.06 for anomalies at
Lc=25), whereas the absence of inflation is optimal for
smoothing.
One lesson is that a variational analysis over a long DAW is useless for the
offline transport subsystem, because of its linear dynamics; an
IEnKS, or a 4D-Var analysis does not perform better than an EnKF analysis in
this context. This conclusion will not necessarily hold with the L95-GRS
model because of the nonlinear chemistry.
Emission/deposition regime
The scavenging ratio λ and the emission rate E control the tracer
mass budget in the domain. Their ratio can lead to different regimes of the
physics of the model. It can impact the performance of data assimilation. In
the reference case (E=1, λ=0.1), parcels of tracer travel over
large distances before deposition. Hence, an observation of the tracer
concentration at a grid point gives information about the wind magnitude and
direction at other grid points several time steps in the past.
A synthetic experiment where the scavenging ratio λ is varied over
several orders of magnitude has been set up to highlight this point. The
emission flux E is tuned in order to keep the ratio E/λ constant.
Thus, the order of magnitude of the concentrations is unchanged so that the
relative precision of the concentration observations remains the same with an
unchanged error covariance matrix. The set-up of this experiment is the same
as in the previous section, but only the online data assimilation system
based on the L95-T model is used.
The average filtering RMSE of the concentration variables and of the wind
variables are plotted in Fig. as a function of the
scavenging ratio. The RMSE remains rather constant for both the
concentrations and the winds for small scavenging ratios, with the same
performance as in the reference case (E=1, λ=0.1). However, as soon
as λ>1, the behaviour changes. With such higher values of the
scavenging ratio, the wind does not have sufficient time to transport the
tracer over significant distances. The information about the wind embedded in
the observations of the concentrations diminishes and the wind RMSE
increases. On the contrary, the absence of transport of the tracer by the
wind reduces the detrimental impact of diffusion making the concentrations
easier to estimate (in the fully observed configuration at least). Moreover,
the benefit of using a larger DAW (L=15 here) is greater with lower
scavenging ratios because the tracer is advected farther in space. Indeed,
pieces of information contained in concentration observations help estimate
the wind back in time before the tracer was advected by that wind. These
results stress that variational schemes implemented over large DAW have an
advantage over the EnKF in that context.
Observation network
The performance of the EnKF and of the IEnKS is now studied with the L95-T model when the observations of the tracer
concentrations are sparser. The set-up of this synthetic experiment remains unchanged except for the density of the
observations. The wind variables are observed on all grid points while only some of the observations of the tracer
concentrations are assimilated. The observations of the concentrations are chosen to be evenly spread. The number of
observations is a divisor of 40 and belongs to {1,2,4,5,8,10,20,40}. The performance of the IEnKS is studied for
several DAW lengths L=0,1,2,3,4,5,10,20. The resulting average filtering RMSEs for the concentration variables and the
wind variables are shown in Fig. . In both cases, the gain derived from using a longer DAW is
undeniable: the gain in RMSE is greater for large L than for small L, but the marginal gain decreases with L.
Average filtering analysis RMSEs of the wind
variables (left) and concentration
variables (right) of the L95-T, as a function of the number of concentration observations for the IEnKS with
several DAW lengths. The case L=0 corresponds to the ensemble transform EnKF.
Applying the IEnKS to the L95-GRS model
The IEnKS is now applied to the L95-GRS model introduced in Sect. . We showed that the model could
reproduce the main physical and chemical processes of interest. The aim in this section is to show that it offers a rich
playground for testing data assimilation methods. The approach is similar to the one applied in Sect. .
Apart from an overall performance test, we will focus on specific aspects not addressed in Sect. of
relevance for this type of model.
Twin experiments are conducted where each chemical species is observed. The
observations are drawn from the truth every Δt=6h and
perturbed using a Gaussian white noise. The standard deviations of the error
for the concentrations have been chosen to correspond to about 10 %
of the average value of the concentration over the domain. Specifically, the
observation error covariance matrix of each species is of the form
R=σ2I, where σ2=1 for the wind in Lorenz
units, σ2=0.01ppbC2 for ROC, σ2=0.16ppb2 for NO, σ2=1ppb2 for
NO2, σ2=4ppb2 for O3, and σ2=0.01ppb2 for S(N)GN. All the RMSEs shown in this section
are normalised by the observational error standard deviation of the
corresponding species. All the data assimilation runs use the same set-up as
used with the L95-T model. The size of the ensemble is set again to N=20,
except when localisation is tested.
Performance
At first, the number and distribution of observations of the concentration
variables have been varied following the same set-up as in
Sect. . All the chemical species are observed
but only at selected grid points. The resulting average filtering RMSEs for
the concentration variables and the wind variables are shown in
Fig. . We found that with poor observability of the
concentrations, the system's state estimate can be imprecise. When the
concentrations are only observed at one point, the EnKF diverges from the
truth. The IEnKS with L=1 also fails to estimate the S(N)GN better
on average over the whole domain than the standard deviation of the single
observation.
Average filtering analysis RMSEs of the L95-GRS
variables, as a function of the number of concentration observations for the
IEnKS with three DAW lengths. The case L=0 corresponds
to the ensemble transform EnKF. The RMSEs are normalised by the standard deviations of the observation error.
To be more realistic, further experiments will assimilate sparse
concentration observations. We choose to keep eight observations in the
domain per species, that is to say at 1 every 5 grid
points. In this context, the DAW length has been varied over a wider range
of values. The time-averaged analysis filtering and smoothing RMSEs for the
wind and the concentrations are shown in Fig. . Even though
the model is strongly nonlinear, the IEnKS method can account for these
nonlinearities and, therefore, it performs well and improves with L for
both the filtering and smoothing RMSEs. The S(N)GN species is still
the one with the worst results, probably because it is little correlated with
the other species except for ROC. A misestimation of its concentration
has no consequences on the other species.
Average filtering and smoothing analysis RMSEs of the
L95-GRS variables, as a function of the DAW length (in units of Δt=6h). The case L=0 corresponds to the ensemble transform EnKF. The
RMSEs are normalised by the standard deviations of the observation error.
Parameter estimation
In atmospheric chemistry, there is a strong dependency of the model on the
values of the various forcings, such as the boundary conditions
, the meteorological fields or the
emission rates of the pollutants and their precursors . It
is therefore important to estimate these inputs and data assimilation can be
a powerful tool in this context. Here, we show how the L95-GRS model allows
us to test parameter estimation strategies, which is illustrated using the
IEnKS.
To estimate a set of model parameters θ∈RP along
with the state variables, the state vector is augmented from x∈RM to a vector
z=xθ∈RM+P,
in the joint state and parameter space. It is also necessary to define
a forward model for the parameters. The persistence model is chosen, i.e.
θk+1=θk.
The estimation of the main parameters of the L95-T model (forcing of the L95
and emission rate of the tracer) with various data assimilation methods,
including the IEnKS, has been experimented upon by .
Similarly, we conduct a twin experiment with the L95-GRS model where, in
addition to the state variables, the F forcing of the L95 model and the
emission rates of ROC and NOx are estimated
simultaneously. The state space is, therefore, augmented from a 240- to
a 240+3-vector of the joint state and parameter space. The parameter
variables in the ensemble are set as follows.
For the emission rates: the ensemble is initialised around the truth by
adding an unbiased Gaussian noise of standard deviation 10% of
the true value.
For F: the ensemble is initialised around the value F=7 by adding an
unbiased Gaussian noise of standard deviation 10% of the true
value.
Rather than the single data assimilation (SDA) version of the IEnKS presented
in Appendix , we use the multiple data assimilation (MDA)
version, which is very similar, less accurate for small L but more stable for
large L. The MDA IEnKS algorithm is described in detail in
.
Time evolution (days) of the parameter
variables for several DAW lengths without spin-up (main) or after a long time
(inset). The case L=0 corresponds to the ensemble transform EnKF.
F is shown with the original Lorenz unit, while the emission rate unit is ppbCday-1 or ppbday-1.
Let us first mention that the RMSEs of the state variables are barely changed
by the joint state and parameter estimation, as well as by the use of
a different variant of the IEnKS in this experiment. Hence, the results in
Fig. for state variables still hold. The parameter values
are plotted in Fig. , over time intervals of different
lengths. We observe that only a few days are required to converge to the
right value for the forcing parameter of the meteorology F, while a few
dozens of days are necessary for the emission rates to stabilise. This is due
to the high sensitivity of both wind and concentration variables to the
forcing F of the wind model, which controls its chaotic behaviour, as well
as to the fact that there is a bias on the initial value of F. At the end
of a long data assimilation run, the algorithm has converged to the right
values with a precision of less than a percent. The use of a long DAW
improves the estimation of the parameters and the smoothness of the results.
The case L=1 shows that the method can sometimes converge to a biased value
for a long period of time.
It could be possible to estimate chemical reaction rates, for instance, the ROC photolysis rate. However, our
experiments have shown that the filter diverges. It probably happens because this rate is equal to 0 at night. Hence,
it is imperative to set a prior distribution on this type of parameter to avoid divergence when the model becomes
insensitive to the parameter. But, this is outside the scope of this work.
Localisation
Estimating covariances from a limited size ensemble of state vectors produces spurious long distance correlations
between variables. This degrades the estimation of the error statistics and can lead to divergence in ensemble data
assimilation methods. To address this issue, localisation is used in high-dimensional systems implementing ensemble
methods. There are two main localisation methods known as covariance localisation and local/domain analysis and references therein. The first method consists in tapering the empirical covariance
matrix using a Schur product with short-range correlation function. The second method performs the analysis in a local
domain around each grid point, using only nearby observations. Both localisation methods have been tested with success
with EnKF techniques applied to the L95-GRS model. Here, we provide an illustration of such experiments.
We tested covariance localisation on the L95-GRS model using the DEnKF data assimilation method from
. This method has the advantage of being computationally efficient while allowing a straightforward use
of Schur localisation. The RMSEs are shown in Fig. , as a function of the ensemble size, with
optimally tuned inflation and localisation radius. For comparison, the results without localisation are shown as
well. We see how localisation allows us to reduce the size of the ensemble below 18 without diverging, even though it
leads to a degradation of the scores for very small ensemble sizes (N<10).
Conclusions
Average filtering analysis RMSEs of the
L95-GRS variables, as a function of the ensemble size for the DEnKF without
localisation or with optimally tuned localisation radius. The L95
variables, flagged “Wind”, are shown with the original Lorenz unit, while the concentration unit is ppb (ppbC for ROC).
The aim of this article is to introduce low-order models on which to test
advanced data assimilation methods in order to gain insights on some of the many
difficulties encountered in data assimilation applied to meteorology and
atmospheric chemistry. Amongst them, the questions of inflation, localisation
for ensemble methods, model error, online and offline modelling, or
nonlinearities have been addressed.
Building on the L95-T model, where the transport of a tracer is coupled to
the L95 model, we introduced a new model, L95-GRS, where the tracer part is
replaced with a simplified ozone chemistry. The L95-GRS model shows important
peculiarities typical of tropospheric ozone chemistry. It has been adjusted
to simulate pollutant concentrations of realistic magnitude. Ozone precursors
can experience long-range transport by the meteorology and lead to ozone
episodes far from the pollutant sources. It is possible to tune the wind
magnitude in order to modify the time- and space scale of the model. Moreover,
it has stiff equations that require the use of the same numerical tools as
high-dimensional CTMs. Last but not least, it shows a nonlinear response to
the emission rates of the ozone precursors. It thus includes several of the
hardships of high-dimensional chemistry models without the high numerical
cost. As such, it can be used to experiment upon and validate new data
assimilation methods in the context of atmospheric chemistry modelling and
coupled chemistry meteorology modelling.
To illustrate the use of advanced data assimilation methods on these models,
and specifically ensemble variational methods, we first performed new
experiments on the L95-T with the iterative ensemble Kalman smoother
(including the ensemble Kalman filter). We showed that this model is suitable
to test online and offline strategies for data assimilation, as well as to
emulate model error stemming from a meteorological field, or an ensemble
forecast of meteorological fields.
More specifically, we experimented on the offline version of the L95-T model,
where the meteorology and the tracer subsystems are integrated and
assimilated separately. This decoupling introduces model error into the
tracer subsystem. In this context, having an ensemble of analyses from a data
assimilation on the meteorology as an input to the tracer subsystem gives us
a representative sample of this model error. By doing so, we have avoided the
use of inflation and obtained optimal performance. We noticed as well that,
for data assimilation purposes, the coupling of the two subsystems is only
relevant when they have similar evolution timescales. In the case where the
tracer subsystem evolves too quickly or too slowly compared to the
meteorology, the coupling of these two parts fails to improve the results of
the data assimilation compared to an offline case.
The use of data assimilation methods was also illustrated with the L95-GRS
model. The iterative ensemble Kalman smoother performs well despite the
nonlinearities of the model and even if the observation network is sparse. In
particular, the model can help testing parameter estimation techniques with
multiple parameters usually met in CCMMs and CTMs. The use of localisation
was also successfully tested with L95-GRS. By making this wide range of
experiments, we concluded that the L95-GRS model is suitable to test advanced
data assimilation schemes.
A broad class of models could be developed by exchanging the L95
meteorological part with another low-order model. The L95 model has
anti-correlations in space and time that are not observed in more realistic
models. It could be replaced by its continuous extension, the Lorenz 2005-II
model , which could offer a more complex set-up for testing
localisation, but still suffers from similar correlation patterns.
Alternatively, the L95 meteorological part could be exchanged with the
multiscale Lorenz 2005-III model to explore the impact of subgrid-scale model
error. To study space–time intermittent behaviours, the model of
Kuramoto–Sivashinsky and references
within could be implemented. Alternatively, the Burgers equation could be used to study the impact of
a front (concentration/rarefaction) on the chemistry. If such continuous
meteorological models are used, the choice of the advection scheme could be
revised as well, and a more accurate higher-order one compared to the upwind
scheme could be used, alongside with a flux limiter.
The
L95-GRS model depends on several key species-dependent chemical and physical
parameters that introduce many time- and space scales in the data
assimilation system and that impact its observability and controllability.
These parameters are likely to be representative of those of realistic CCMMs
and CTMs. We have only investigated the impact of a few of those parameters,
fixing the others. But a more general parameter-wise exploration of data
assimilation systems built on L95-GRS is desirable.
Finally, following this study, we are planning to test the IEnKS on the
Polair3D CTM of the research and operational Polyphemus modelling platform
building on the experience acquired with the
L95-T and L95-GRS low-order models.
Code availability
The code for the models L95-T and L95-GRS can be downloaded from the
following website: http://cerea.enpc.fr/l95-grs/