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On the Robustness of Data
Assimilation Methods in Air
Pollution Modeling
X.F. Zhangl
A.W. ~ e e m i n k '
Abstract
The performance robustness properties of two data assimilation methods, Kriging ancl Kallnan filtering presented for the estimation of air
pollution are investigated under perturbations of noise uncertainty ancl
znodeP parameter uncertainty. The si~nulationresults are used to illustrate the advantages of Kalman filtering approach under various circumst ances.
INTRODUCTION
In Zhang et. a1 (1995) two d a t a assimilation methods, a Kriging approach
and Kalman filtering are developed for the estimation of two dimensional air
pollution problem. T h e deterministic system model and the statistics of system
noise and measurement noise are assumed to be known exactly in Zlmng et.
a1 (1 995). However, such an assutnption is often uilrealistic in applications,
and noticeable deterioration of system performance may be caused by model
errors and noise nncert ainty.
In this paper we investigate the performance robustness properties of the
two data assimilation methods proposed in Zhaug et. a1 (1995). T h e effects
of hot 11 noise uncertainty and parameter uncertainty are examined. In Section
2, the stochastic system model is described, and in Section 3 t h e two d a t a
assimilation methods are briefly reviewed. In Sec,tion 4, the influence of noise
uncertainty on the Kalmau filtering approach is analysed, and a strategy to
minimize the worst case performance is proposed. Then, in Section 5, s i n ulation results are presented, showing the advantages of t h e proposed robust
Kalman filter design method compared with the conventio~lalKalman filter
desigrr and the Kriging approach a t present of noise unc,rrtainty. Furthermore,
the performance robustness is checked with respect t o uncertain transport
speed of air pollution. Finally, Co~lclusionis given in Section 6.
Department of Technical Malhenlatics and I n f o r ~ n a t i c s ,Delft IJniversity of Technology,
Mekelweg 4, P. 0. Box 5031, 2600 CA Delft, The Netherlands
THE STOCHASTIC DYNAMIC AIR POLLUTION MODEL
The transport phenomena of air pollutant in the environment are usually
described by an advection-diffusion equation
where c is the concentration of the pollutant, t is time, u is the velocity field,
D is the dispersion tensor including ~noleculardiffusion and dispersion. By
introducing a numerical scheme (Heemink 1990, Van Eijkeren 1993) Eq. (1)
can be represented by a discrete time state space model.
where x(k) is an 11-vector state process taking corresponding values at model
grid point at time instant k , n is the total number of model grid points at the
is an r L x r~ dimension transient matrix
pollutant concentration field, and
oht ained by the numerical scheme discretizing the advection-diffi~sio~l
Eq. ( I ) ,
is an n x q noise input matrix that interpolates q dimensional system noise
process, with
w ( k ) at 11 model grid points, r ( k ) is an m-vector ~neasure~nent
and H are constant inatrices, w ( k )
measurement noise sequence v ( k ) . @,
and v(k) and the initial state x(0) are assumed to be multiply independent
and satisfied the following conditions:
r
where Q 2 0 is a positive semidefinite constant matrix, R > 0 is a positive
definite constant matrix, hjr,= 1 if j = k and hjk = 0 if j # k.
DATA ASSIMILATION METHODS
The Kriging Approach
Kriging is a method for the optimal estimation of a quantity y at an arbitrary location that has not been measured by using measurements taken at
surrou~idingpoints. By changing the position of the point of esti~nation,it is
possible to estimate the whole field of spatial variables. the estimation of the
pollutants concentration value x at it11 grid point of G,,, and time instant k is
given by:
where .c is the total nu~nberof measurement stations. To determine the optimal
choices of K ( i , I ) , K ( i , 2), ..., K ( i , s) of minimizing the esti~nationvariance for
an arbitrary ith grid point of G,,,. We get (Zhang et. a1 1995)
where yj,l denotes the value of the semi-variogram y(hj,l) of the deviation field.
Hj,l denotes t lie value of the known covariance of m e a s ~ ~ r e m esnnoise.
t
The
spatial correlation in the form of a semi-variogram is auto~naticallyestimated
based on the available ~neasurernentdata at each time instant. The set of
K ( i , j ) , j = 1, ..., s is obtained by Eq. (7) and then applied to Eq. (6). This
produces an unbiased esti~nationwith minimu~nvariance of estimation error.
The updated picture of the whole conc,entration field is obtained by adjusting
x; throughout every grid point of the concentration field using Eq. (6).
The Kalman Filter Approach
Since the linear discrete-time system Eq. (2) and Eq. (3) is time-invariant,
the Chandrasekhar-type algoritlml (Morf rt a1 1974) can be employed to get a
steady state Kahnan filter as follows:
where the gain matrix can be obtained by (Zhang et. a1 1995):
with initial condition:
The equations from Eq. (10) to Eq. (15) are iterated until:
where 0 is determined by prespecified accuracy. Under the assumption that the
dimension of system noise process q is much less than the dimension of system
, Chandrasekhar-type algorithm provides a significant
states 11, i.e, q << r ~ the
co~nputationreduction (Heernink 1988,1990).
A ROBUST FILTER DESIGN METHOD UNDER
PERTURBATIONS OF NOISE UNCERTAINTY
I11 general, it is a hard task to get suitable Q and R matrices which fit best
the stochastic system model Eq. ( 2 ) with various observations. We suppose
that the real steady state covariances of system noise and measurement noise
are uncertain, and they can be described by: Q = Qo AQ, R = Ro AR.
Further, we suppose that all information on AQ and A R is that the norm of
AQ and A R are bounded by: 11 AQll 5 a , IlARll 5 9 . AS the optimal gain
matrix I( of the Kalman filter is a function of Q, and R , the I( obtained
by using Qo and Ro is in general no longer optimal due to the existence of
uncertainties in Q and R matrices. First, we consider the prediction-type
Kalman filter with gain matrix K p . Let e ( k 1) = z ( k ) - x(klk - 1) the
actual estimation covariance denoted by NO(k 1lk) is determined by
+
+
+
+
Sangsuk-Iarn (1990) prove that as k -+ oo Eq. ( 1 8 ) always converges to some
constant No which satisfies the following algebraic Riccati equation
From Eq. ( 1 9 ) we notice that the actual estimation covariance No is a 1110110tonic increasing function of the actual noise covariances Qo AQ and Ro A R.
With Q := AQo aI, R := ARO+ P I where I is an identity matrix, we have
Q 2 Qo AQ ,R 2 Ro AR. It is a reasonable choice to take the worst case
covariance
and R to design the Kaltnan filter which minimizes the upper
bound of Kal~nanfilter performance, i.e
+
+
+
+
+
where I(,* is the optimal gain matrix of the Kal~nanfilter designed for the fixed
pair of worst case noise covariances Qo aI and Ro /?I, and I i p could he
any other gain matrix. We have the following saddle point inequality:
+
+
Now, we consider the current estimation-type Kalman filter with gain matrix
l i e . From Eq. (2), Eq. (3), and Eq. (8) we have the estimation error covariance
of the form:
Obviously, for any determined I<,, the actual estimation covariance P ( k ) is also
a monotonic increasing function of Q AQ, and R AR. We have following
saddle point inequality
+
+
where K: is the optimal gain matrix of the current estimation-type Kalman
filter designed using the fixed pair of worst case noise covariances Qo a I and
Ro [3 I , and I(, could be any other gain matrix. From above discussion we can
see that the selection of worst case noise covariances is a strategy to design a
robust Kal~nanfilter (either prediction- type or current estimation-type) which
mini~nizesthe upper bound of the Kalman filter performance.
+
+
SIMULATION STUDY
The simulation studies are aimed to examine the performance robustness of
the two data assimilation methods with respect to noise uncertainty and model
parameter uncertainty. The experimental region is a cycle area bounded in Q
(see figure 1) .
(a): True field
(b) 9 Sample Locations
2
3
4
south
(c) : Filtering results
2
3
south
4
4
2
3
south
(d): Kriging results
2
3
south
4
Figure 1: (a): The concentration field at t=64; (b): "*" Observations, "."
Model grid G,; ( c ) , (d): The eorespponding results of the Filtering and
the Kriging approaches without model parameter and noise uncentainty.
The concentration of the pollutant is denoted by z((,t ) where ( is the coy)' E 0. The true field is a stochastic process described by Eq.
ordinate t = (g,
(2). The deterministic part of this stochastic process is driven by the advectioudiffusion equation Eq (1) with a circular velocity u = u((;c,Y)') = 2n(- Y, g)' by
I
using the numerical approximation described in Heernink (1990) and V ~ I Eijkeren (1993). The initial condition of Eq. (1) consists of a constant background
field and a small cosine-squared shape cone interpreted as a high density concentration area of local pollution. In our experiments, the simulated true field
at t = 64 is shown in figure 1, (a). The regular model grid G,,, is shown in
figure 1, (b). The systern noise ut is generated by w = (wl , ..., w,)' with zero
mean and constant covariance Qo(i,j) = 2 x 10-'e
-~ . ' a[s ( z t - . ~ ) ~ + ( ~ ~ - g ~ ) ~ ]
measurement noise v is generated by Ro(i,j) = 1 x l ~ - ~ e - ~ [ ( " ' - " j ) ~ + ( ~ i - ~ ~ ) ~ ~ .
The location of 9 observation stations is also plotted in figure 1 (b). When
the deterministic model , the statistics of system noise and measurement noise
are exactly known the reasonable good results of both data assimilation approaches are plotted in figure 1 (c), (d).
Experiment 1
We assume that the second moments of statistics of the system noise and measure~nentnoise are uncertain. Let nominal filter use the optimal gain matrix
corresponding to the nominal system noise covariance Qo and measurement
noise covariance Ro, and robust filter use the robust gain matrix which min= Ro P I with a = 1 x
imize the worst case of Q = Qo a1 and
and /3 = 5 x
First Q = Qo and R = Ro: The Root Mean Square (RMS)
of the nominal filter and robust filter are plotted in (a) of figure 2. It can be
seen that the RMS corresponding to robust filter is only slight higher than
that of no~ninalfilter which is the optimal gain matrix in this case. S F C O T L ~ ~ T J
Q = Qo cuI and R = Ro /?I: The RMS of the nominal filter and robust
filter are plotted in (h) of figure 2. We can see that the RMS of robust filter
is now 111uch lower than that of nominal filter. It is obvious from above simulation results the robust filter which minimize the worst case covariance of
the estimation error makes the performance of the Kal~nanfilter much better
from the point of view of robustness. To compare the two data assi~nilatiou
methods the corresponding RMS of the Kriging approach are also plotted in
figure 2 (a), (b). It can be seen that Kalman filter approach can provide more
accurate result then The Kriging approach.
+
+
x
+
+
Experiment 2
We suppose that the Kalrnan filter is perturbed by uncertain deterministic
system model. More exactly, it is assumed that the transport speed u which
is a key model parameter is uncertain. Let the Kalman filter be designed by
uo = (u5, uy) = 2 ~ ( - y , x ) , and let the true concentration field is run with
+ nu.
u = uo
Au = - 0 . 1 ~i.r.
~ 10% deviation. The true co~lcentratio~l
field
at t = 64 is plotted in (c) of figure 2. The correspo~ldingestimated field of the
Kalman filter approach and the Kriging approach are plotted in (d) and (e) of
figure 2,respectively. The RMS of the data assimilation procedures are plotted
in (f) of figure 2. It is clear in this case that the performance robustness of
the Kalman filter approach to tolerate model parameter uncertainty is much
better than that of the Kriging approach.
(a): RMS
0
(b): RMS
20
40
t
(c): True field
60
0
20
40
60
t
(d): Filtering results
4
2
south
(e): Kriging results
3
south
(1): RMS
4
40
t
60
10
1.5
-
.-
&
1
'
C
4
0.5
2
2
3
south
4
0
0
20
80
Figure 2: (a) and (b): The RMS of three methods with noise uncertainty.
(c): Truth field at t=64; (d) and (e): The corresponding results of Filtering and Kriging approaches with 10% model parameter uncertainty;
(f): RMS of both two approaches.
CONCLUSION
In this paper, the performance robustness of two data assimilation met hods
under the perturbations of noise uncertainty and parameter uncertainty is investigated in two dimensional air pollution problem. The experimental results
show the advantage of a robust Kalman filter design method with respect to
noise uncertainty. Moreover, it is demonstrated in the simulation studies that
the Kalman filter approach has much stronger performance robustness to tolerate model parameter uncertainty then the Kriging approach. Our current
research focuses on applying Kalman filtering to " the evaluation of CH4 budget in Europe " in cooperation with RIVM air quality laboratory.
ACKNOWLEDGMENTS
This work has been carried out in cooperation with and with financial support
from the RIVM.
REFERENCES
Heemink, A.W., 1988. Two-Dimensional Shallow Water Flow Identification.
Applied Mat. Mod., 109-118.
Heemink, A.W., 1990. Identification of Wind Stress on Shallow Water Flow
Surfaces by Optimal Smoothing. Stochastic Hydrol. Hydraul. Vol. 4,
105-119.
Morf, M., Sidhu, S.S., Kailath, T., 1974. Some New Algorithms for Recursive
Estimation in Constant, Linear, Discrete-Time system. Trans. Autom.
Control, Vol. AC-19, NO. 4, 315-323.
Sangsuk-Iam, S., 1990. Analysis of discrete-time Kalman filtering under incorrect noise covariance, IEEE Transactions on Automatic Control, Vol. 35,
1304-1309.
Van Eij keren, J.C.H., 1993. Backward Semi-Lagrangian Methods:an Adjoint
Equation Method. In: Numerical Methods for Advection - Diffusion Problems, Vreugde hi1 C.B and Kora B Eds. Vieweg, Vol, 45, 215-241.
Zhang, X.F., Van Eijkeren, J.C.H. and Heemink, A.W. 1995 Data Assi~nilation
in Dynamic Environmental Pollution Modeling. RIVM report, No. 421503
006, Bilthoven, The Netherlands.
BIOGRAPHICAL SKETCH
Zhang X.F. is doing a Ph.D. project on the development of data assimilation
methods for the estimation of air pollution.
Heemink A.W. is a professor of faculty of technical mathematics and infor~natics,TU Delft.