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Data Assimilation Algorithms for Numerical Models
2002). Atmospheric chemistry transport models have been developed to understand
the processes controlling the formation of ozone, to study the potential effects of
ozone on ecosystems and humans, and to assess the effects of emission reductions
on the ozone concentration. Atmospheric models and observations are usually ap-
plied separately to obtain information about ozone and air-pollutant concentrations
in the boundary layer. Assimilating ozone observations in the model can improve the
information on the state of the atmosphere. One method to assimilate measurement
data into a model simulation to get a better estimate of the real ozone concentration
is data assimilation.
6.1 Eulerian chemistry transport model EUROS
The Eulerian atmospheric chemistry transport model EUROS (EURopean Opera-
tional Smog, Rheineck et al., 1990, Pul et al. 1996, Matthijsen et al, 2001) was
developed at RIVM (National Institute of Public Health and the Environment, The
Netherlands). The model contains parameterizations of the various chemical, dy-
namical and radiative processes in the atmosphere, as well as information about
the emissions of ozone precursors, nitrogen oxides, VOCs, carbon monoxide and
methane. The model can be used to examine the time and spatial behavior of
SO
x
,NO
x
, O
3
, and VOCs in the lower troposphere over Europe. The model area
extends over a large part of Europe and uses a shifted pole coordinate system, i.e.,
the real North Pole is at 30
o
northern latitude and the equator has been shifted to 600
northern latitude in the new coordinate system. In these shifted pole coordinates the
model domain is [−8.25
o
,20.35
o
] ×[−23.1
o
,7.15
o
]. The grid consists of 52 ×55
grid cells with a longitude-latitude resolution of 0.55
o
×0.55
o
(about 60 ×60 km).
The vertical stratification of EUROS consists of four layers from the surface up to
3000 m.
6.2 State space representation
In order to derive a state space representation of the EUROS model, a state vector,
defined as the vector containing the ozone concentrations at each grid point of the
model, is necessary. The EUROS model can now be rewritten as a state space model
in the form:
c
t
k+1
= M(c
t
k
). (34)
The stochastical model required for the Kalman filter is based on a specification of
the errors of the deterministic model EUROS and of the observations. Knowledge
of the uncertainties in the EUROS model and in the observations is crucial for a suc-
cessful data assimilation. The description of the errors strongly affects the results of
the assimilation. One of the goals of this study is to gain insight in the uncertain pa-
133
A.W. Heemink et al.
rameters of the model and the impact of these parameters on the assimilation. One
method to accomplish this is to treat these parameters as model input and to define
them as stochastic parameters. A disadvantage of modeling uncertainties using a
white noise input w is that there is no memory in the system, i.e., the w is uncor-
related in time. At a certain hour t the noise parameter may indicate an emission
increase of 30% with respect to the original field, whereas it estimates a decrease
of 30% at t + 1. Such irregular and unrealistic behavior can be prevented by the use
of colored noise. The white noise input w forces a colored noise process λ , which
operates on the parameters in the model. The exact size of λ depends on which
parameter in the model is considered to be stochastic. Each element of λ allows
use of different values for the time correlation parameter a = exp(
−1
τ
) and standard
deviation σ
a
= σ ·
√
1 −a
2
. The parameter τ is the time correlation length in hours.
Five parameters are considered uncertain. In the implementation of the Kalman fil-
ter with the EUROS model, the noise parameters form a part of the model state
(Segers, 2002), therefore making the use of the augmented state vector necessary.
The general stochastical model (equation (2)) takes the following form:
c(t
k+1
)
λ
k+1
(t
k+1
)
λ
k
(t
k+1
)
=
M(c(t
k+1
),λ
k
(t
k
))
αλ
k
(t
k
)
λ
k
(t
k
)
+
0
σ
α
I
0
w(t
k
), (35)
or:
x
t
(t
k+1
) = M(x
k
(t
k
)) + Gw(t
k
), (36)
where the left-hand side of equation (35) is the state vector, c is the concentration
array of all species in the model, and the matrix G distributes the impact of the noise
values for all the parameters considered uncertain in the model in specific areas.
6.3 Ground based observations
To improve the estimation of the assimilated concentrations of ozone and to reduce
the uncertainties in the emissions the model is combined with available measure-
ments. Only ozone observations at background stations are used in the assimilation,
since the output of the EUROS model represents the ozone concentrations on the
same scale. For the period studied (summer 1996) data was available for in total 135
of those stations in Europe. (Figure 12). For each ozone measurement it is assumed
that the standard deviation of the error is equal to 10 µgm
−3
. This error accounts
for both the uncertainty in the actual concentration measured at the specific station
as well as for a representation error, reflecting the size of the grid cell.
134
Data Assimilation Algorithms for Numerical Models
Fig. 12 Stations with ozone observations in Europe in June 1996. Stations used (default) for as-
similation (set 1) are marked by filled diamonds, the stations used (default) for validation marked
with open diamonds (set 2) and open triangles (set 3). Part of the EUROS grid is shown as well.
If there is more than one station per grid cell, only one is used for the assimilation, the other is
indicated by an open triangle. For the year 1996 there is only one observation in France in the
databases.
6.4 Results
To quantify the effect of the various simulations, we calculated the residual between
the simulated and the measured ozone concentrations. We have introduced the term
Absolute Average Residual (AAR), defined as:
AAR =
1
N
N
∑
i=1
|c
observed
i
−c
simulated
i
|, (37)
where N is the number of valid measurements in the time series and the simulated
concentration can refer to the EUROS model alone or the KF results.
Figure 13 shows the Ensemble Kalman filter mean (KF mean) ozone concentra-
tion without (panel a) and with (panel b) assimilation. Also shown are the ozone
concentrations from the EUROS model and from the observations. Without data as-
similation (panel a), no observations are used in the simulation and the KF mean
is simply the statistical mean of the 20 separate model calculations. The improve-
ment in ozone concentration is clear in the case where observations are assimilated.
The assimilated time series for this station are in better agreement with the obser-
vations than the ones calculated by the model alone. This improvement cannot only
be observed at measurement sites the observations of which are used in the assim-
ilation procedure (i.e., so-called assimilation stations); it also holds for validation
stations the observations of which are not used in the assimilation. In panel c of Fig-
ure 13 the concentration for the validation station Hellendoorn is shown. A similar
135
A.W. Heemink et al.
a) Biddinghuizen - no assimilation
1 2 3 4 5 6 7 8
Day of the month
0
50
100
150
200
250
O3 conc. (µg m
-3
)
KF mean
Observations
Euros
b) Biddinghuizen - assimilation station
1 2 3 4 5 6 7 8
Day of the month
0
50
100
150
200
250
O3 conc. (µg m
-3
)
KF mean
Observations
Euros
c) Hellendoorn - validation station
1 2 3 4 5 6 7 8
Day of the month
0
50
100
150
200
250
O3 conc. (µg m
-3
)
KF mean
Observations
Euros
Fig. 13 Ozone concentrations (µgm
−3
) at station Biddinghuizen (the Netherlands) without (a) and
with (b)data assimilation for the Kalman filter mean (solid line) for June 1-7, 1996, calculated with
ENKF with 20 modes. Ozone concentrations at validation station Hellendoorn (the Netherlands),
of which the observations are not used in the assimilation process, are shown in panel (c) Data
from the EUROS model (dash-dot) and the observations (crosses) are also shown. The vertical bar
of the crosses represents the uncertainty in the observations given as input in the assimilation.
improvement of the concentration given by the Kalman filter mean in the direction
of the measurements is shown. In Figure 14 the absolute average residuals (AARs),
averaged over a month, are shown for all the validation stations in the Netherlands.
The first bar represents a run without assimilation, the second bar the EUROS run,
and the last one the KF mean. The difference between the first two bars is due
to the stochastic noise that was implemented in the model and the slightly better
performance of the first is a consequence of the averaging. In all the stations the
improvement given by the KF mean is evident. Since an improvement at the assimi-
lated stations is expected anyway, in most simulations the analysis was based on the
data for the validation stations.
136
Data Assimilation Algorithms for Numerical Models
0
5
10
15
20
25
30
35
Balk
Barsbeek
Houtakker
De Zilk
Eibergen
Hellendoorn
Huijbergen
Loenen
Sappemeer
Schipluiden
Volkel
Vredepeel
Westmaas
Zegveld
Zierikzee
no assimilation
Euros run
KF mean
Fig. 14 AARs (µgm
−3
) of an assimilation (ENKF, 20 modes) for all the validation stations in the
Netherlands for the EUROS model calculation, the Kalman filter and the mean without assimila-
tion. Simulation for June 1996.
6.5 Performance of RRSQRT algorithm, ENKF algorithm and
COFFEE (Complementary Orthogonal subspace Filter For
Efficient Ensembles) algorithm
A parameter that is important for the assimilation in Kalman filter algorithms is the
number of modes used in assimilation. The AAR’s values for all three algorithms
are presented in Figure 15. The behavior of the algorithms with a real life large scale
model is consistent with the specific theoretical characteristics of these algorithms
discussed in Section 3, like the robustness of RRSQRT filter in comparison with the
ENKF for a small number of modes and the convergence of both of them when the
ensemble size (number of modes for RRSQRT) are increasing. This instability is
shown in Figure 15, where the RRSQRT algorithm has higher AAR values for 10
modes than for 3 and 5 modes.
The following filter set-ups were examined to gain insight into the efficiency dur-
ensemble members/modes and a COFFEE with q = 5, 10, 20, 40 modes and
N = 5, 10, 20, 40 ensemble members. The number of model evaluations required
comes to N for the ENKF, 1 + q for the RRSQRT filter and 1 + q + N for the COF-
FEE filter. The assimilation period was the month of July 2000. The results are
137
ing the assimilation process: ENKF and RRSQRT filter with N =5, 10, 20, 50, 100, 200
A.W. Heemink et al.
20
25
30
35
ENKF
RRSQRT
AAR (µg/m
3
)
EUROS
3 modes
5 modes
10 modes
20 modes
50 modes
80 modes
Fig. 15 Comparison of the performance of the ENKF algorithm and the RRSQRT algorithm using
different numbers of modes/ensembles. The AARs (µgm
−3
) are averaged over all the validation
stations in Europe. Noise was applied to all five parameters
shown in Figure 16. The COFFEE results fall somewhere between the ENKF and
RRSQRT results for less than about 30 model evaluations. For more than about 30
model evaluations, the COFFEE results are, in all cases, better than, or in a single
case, equal to, the ENKF and RRSQRT results. The results of the COFFEE simu-
lations with more than about 60 model evaluations turn out significantly better than
all the ENKF and RRSQRT simulations, also those with 100 and 200 modes or
ensemble members (not shown in the Figure).
7 Conclusions
In this Chapter we have introduced the main ideas of data assimilation. Using data
assimilation techniques, measurements and model results are both used to obtain an
optimal estimate of the state of the system. The two main directions for data assimi-
lation have been presented: variational and sequential methods. The basic theoretical
background was given and some new algorithms were introduced with their advan-
tages and disadvantages: variational methods (adjoint based approach) and sequen-
tial methods (Ensemble Kalman filter(EnKF), Reduced Rank Square Root Kalman
138
Data Assimilation Algorithms for Numerical Models
0
5
10
20
40
50
50
40
20
10
5
0
26
27
28
29
30
31
32
33
34
35
36
506HUURU JP
E
n
s
e
m
b
l
e
m
e
m
b
e
r
s
(
N
)
Mod
e
s (
q
)
Fig. 16 RMS errors (µg m
−3
), averaged over all validation stations in Europe, calculated with the
ENKF, RRSQRT, and COFFEE filters for various numbers of modes (q) and ensemble members
(N). The columns with q = 0 contain the ENKF results and the columns with N = 0, the RRSQRT
results. The RMS error of the EUROS model calculation (N = 0 and q = 0) is 42.4 µg m
−3
; it is
truncated in the figure.
filter(RRSQRT), and Complementary Orthogonal subspace Filter For Efficient En-
sembles(COFFEE).
We also presented a number of real life applications of these methods. Applica-
tion in coastal sea modeling (storm surge prediction) and coastal ocean modeling
(assimilation of high frequency radar data). The last application was in the field
of large scale atmospheric chemistry transport models. For each new application
one should decide first what kind of assimilation algorithm might work for that
specific problem. So, it would be very helpful if one can have the possibility to
easily switch between the data assimilation methods and find the one that it suits
best for the specific application under study. One solution is presented in Section 4:
the COSTA framework. It is a modular framework for data assimilation and model
calibration. Within the COSTA framework it is possible to combine models with
the available data assimilation and model calibration methods without the need of
139
A.W. Heemink et al.
additional software development. The usage of COSTA significantly reduces the de-
velopment costs for implementing data assimilation for simulation models.
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