Storm surge forecasting using Kalman filtering (a review) - CiteSeerX

Storm surge forecasting using Kalman filtering (a review) Report 95-99

A.W. Heemink K. Bolding M. Verlaan

Faculteit der Technische Wiskunde en Informatica Faculty of Technical Mathematics and Informatics Technische Universiteit Delft Delft University of Technology

ISSN 0922-5641

Copyright c 1995 by the Faculty of Technical Mathematics and Informatics, Delft, The Netherlands. No part of this Journal may be reproduced in any form, by print, photoprint, microfilm, or any other means without permission from the Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands. Copies of these reports may be obtained from the bureau of the Faculty of Technical Mathematics and Informatics, Julianalaan 132, 2628 BL Delft, phone +31152784568. A selection of these reports is available in PostScript form at the Faculty’s anonymous ftp-site. They are located in the directory /pub/publications/tech-reports at ftp.twi.tudelft.nl

Storm surge forecasting using Kalman ltering A.W. Heemink, Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 356, 2628 AJ Delft, The Netherlands K. Bolding, Danish Meteorological Institute, Lyngbyvej 100, DK 2100 Copenhagen, Denmark M. Verlaan, Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 356, 2628 AJ Delft, The Netherlands

Abstract

Two data assimilation procedures for real time storm surge forecasting based on Kalman ltering are described. To reduce the computational burden of the Kalman lter, a time invariant lter approximation is suggested rst. This lter is computed using the Chandrasekhar-type algorithm. The resulting data assimilation procedure has been used for storm surge forecasting on a routine basis for a number of years in The Netherlands and in Denmark. The results of these operational systems are discussed in detail. Finally also a new ecient algorithm for time varying Kalman ltering problems is introduced and applied to storm surge forecasting.

1

1 Introduction In the countries around the North Sea, accurate forecasting of storm surges is of great importance. Many people in The Netherland have still vivid memories of the storm surge disaster of Februari 1, 1953, when the dikes in the southern part of the Netherlands broke and more then 1800 people lost their lives in a single night. Advance forecasting of high water levels ensure that dikes can be manned in time to prevent them breaking. Forecasting of high water levels is also necessary to allow decisions to be taken on the closure of a number of storm surge barriers. Where closure seems to be necessary, this has to be done at the right moment, usually fairly shortly after low water. Water also plays a major role in the transportation of goods by ships. Accurate forecasting of water levels is also vital as service to shipping. This is especially true in relation to the channels which provide access to the ports of the North Sea. During low water, some of them are not navigatable for vessels with a deep draught. To prevent such vessels getting into trouble, it is clearly essential to know exactly when the next period of low water is due to occur. To forecast storm surges one often employs large scale numerical models that are based on the shallow water equations (Abbott, 1979). However, during extreme conditions these shallow water ow models are far from perfect. Errors are introduced by uctuations in the meteorological input or by poorly known parameters in the model. Furthermore, considerable uncertainty is associated with the open boundary conditions. Therefore the use of a data assimilation technique to incorporate water level measurements into the numerical model to improve the forecasts is desirable. However, real-time assimilation of data into a numerical model for storm surge forecasting is far from trivial.

2

Existing data assimilation schemes were developed mainly for numerical weather prediction (Bengtsson et al., 1981). The most commonly used data assimilation technique in numerical weather prediction is optimal interpolation. However, optimal interpolation schemes cannot be used for data assimilation in coastal shallow water

ow models. Here, the main problem is that some estimates of the error statistics of the numerical model have to be available to correct the result of the model using the measurements. However, since these errors statistics have to be determined by adopting some ad-hoc statistical assumptions, the correction produced by optimal interpolation is not consistent with the underlying numerical model. As a consequence, in case of a shallow water ow model describing the very complicated ow pattern due to an irregular coastline with estuaries and islands, the use of optimal interpolation will easily introduce numerical instabilities. Due to the numerical problems with optimal interpolation, data assimilation had not been used for operational storm surge forecasting until a few years ago. Here it should be noted that also without data assimilation, storm surge forecasting systems produces reasonable results. Data assimilation schemes can also be developed by employing Kalman ltering (Ghil et al., 1981). In order to use a Kalman lter for assimilating data into a numerical shallow water ow model, this model is embedded into a stochastic environment by introducing a system noise process. In this way it is possible to take into account the inaccuracies of the underlying deterministic system. By using a Kalman lter, the information provided by the resulting stochastic-dynamic model and the noisy measurements taken from the actual system are combined to obtain an optimal (least squares) estimate of the state of the system. With a Kalman lter the error statistics of the numerical model are, unlike with optimal interpolation determined by using the stochastic extension of the model. Therefore, the correction produced by this lter is, unlike in the case of optimal interpolation, guaranteed to be consistent with 3

this stochastic model even in the case of very irregular ow patterns. As a result, numerical instabilities can now be avoided. In the last decennium Kalman ltering has gained acceptance as a powerful framework for data assimilation in storm surge forecasting systems. However, the standard Kalman lter implementation would imposed un unacceptable computational burden. In order to obtain a computationally ecient lter, simpli cations have to be introduced. Budgell and Unny (1981) and Budgell (1986) have developed a Kalman lter to estimate tides in branched estuaries. This lter was based on a one dimensional shallow water ow model. To reduce the computational eort the Chandrasekhartype lter algorithm was used. Heemink (1988) and Heemink and Kloosterhuis (1990) developed a constant gain extended Kalman lter approach for weakly nonlinear twodimensional shallow water ow models. By again using the Chandrasekhar-type lter algorithm, the special structure of the lter problem could be exploited to obtain an ecient implementation. This lter is now used for the operational storm surge forecasting system in The Netherlands since 1992 (Heemink et al., 1991). A modi ed implementation based on a linearized storm surge model has been used on a routine basis in Denmark from 1994 (Bolding, 1995). Another approach has recently been introduced by Curi et al. (1995). They derived a lter algorithm using in nite dimensional ltering theory (Curtain and Pritchard, 1978) and used the algorithm for storm surge forecasting in the North Sea with the same data as used in Heemink (1988). Both the performance of the two lters as the computational eort required for the two dierent lter implementations seemed to be very simular. Another approach to data assimilation which possesses many of the desirable features of Kalman ltering is the adjoint method and is based on optimal control theory (Chavent, 1979). Here an unknown control function is introduced into the 4

numerical model. Using the available data, this control Here an unknown control function is introduced into function is identi ed by minimizing a certain cost function that measures the dierence between the model results and the data. In this way the model results can be corrected using the data available. In order to obtain a computationally ecient procedure, the minimization is performed by using a gradient based algorithm where the gradient is determined by solving the adjoint problem. The adjoint method has also been used for data assimilation in two dimensional shallow water ow models. Since this approach is more suitable for nonlinear estimation problems then Kalman ltering, Ten Brummelhuis et al. (1993) used the adjoint method mainly for the o-line estimation of model parameters. Here, a space varying bottom friction coecient, the wind stress coecient, the depth and the harmonic constants in the open boundary conditions were estimated using long series of water level observations. Mouthaan et al. (1995) used a simular approach to estimate these parameters from ERS-1 altimeter data. A data assimilation scheme for real time storm surge forecasting using the adjoint method was developed by Heemink and Metzelaar (1995). Here, the meteorological input was de ned as the unknown control function. Although this approach is more general than the time invariant Kalman lter, it turned out to be very time consuming. In Section 2 of this paper rst a brief introduction to shallow water ow models is given. Section 3 reviews the Chandrasekhar-type lter algorithm for time invariant systems and the application of this algorithm to storm surge models. The results of the operational storm surge forecasting systems in The Netherlands and Denmark are discussed in detail in Section 4. Finally, in Section 5 a new ecient lter algorithm for time varying Kalman ltering is introduced and applied to storm surge forecasting problems. 5

2 System representation of a shallow water ow model The basis of storm surge models are the shallow water equations, stating the conservation of mass and momentum, see Dronkers ( 1964):

p @u + u @u + v @u ? 2!v + gu u + v + g @h + 1 @pa ? aCdVx Vx + Vy = 0 (1) @t @x @y C (D + h) @x w @x w (D + h) 2

q

2

2

2

2

p @v + u @v + v @v + 2!u + gv u + v + g @h + 1 @pa ? aCdVy Vx + Vy = 0 (2) @t @x @y C (D + h) @y w @y w (D + h) 2

q

2

2

2

2

@h + @ f(D + h)ug + @ f(D + h)vg = 0 @t @x @y with: t = time x; y = space dimensions h = water level elevation above reference level u; v = depth averaged velocities in the x and y direction C = Chezy coecient Cd = wind stress coecient D = depth below reference level g = gravitational acceleration pa = atmospheric pressure Vx; Vy = wind velocities in the x and y directions a = density of the air w = density of water ! = angular frequency of the earth

(3)

At closed boundaries the perpendicular velocity component is set equal to zero, and at open boundaries the water level is given as a known function of time. Due to the nonlinearity of the dynamics, an additional condition has to be imposed at open boundaries in case of in ow. This condition states that the parallel velocity component vanishes at these boundaries. 6

In order to discretize the partial dierential equations (1) - (3) we de ne a space staggered grid G1 (see Fig. 1) and employ a nite dierence scheme (Abbott, 1979; k ; hk ;


]T , Van der Houwen, 1977). De ning the state as the n-vector X f (tk ) = [


:; ukm;n; vm;n m;n the numerical model can formally be written as a discrete system:

X f (tk ) = M (tk ; tk )X f (tk ) + Bu(tk) +1

(4)

+1

Here u(tk ) represents the boundary forcing at time k. In modeling the uncertainty of the system, it is assumed that most errors are caused u and W v (tk ) by the meteorological input of the model. Therefore noise processes Vm;n m;n are added to the right hand side of the momentum equations (1) and (2). These processes are assumed be mutually independent and to have statistics: n

o

n

o

E Qum;n(tk ) = 0

(5)

v (t ) = 0 E Wm;n k

n

(6) o

u (t )W u )t ) =  e? E Wm;n k p;q k

E fg

2

p m?p (

n?q)2

)2 +(

(7) (8)

The parameters  and  have to be speci ed. 2

The processes W u and W v are introduced to model the uncertainty of the momentum equations. The continuity equation (3) is assumed to be perfect. The meteorological input of the storm surge model, and therefore also the noise processes W u and W v , 7

vary very slowly in space compared to the variability of the water ow. As a result the noise processes can be de ned on a coarser grid G2 then the grid G1 of the underlying shallow water ow model. The stochastic system representation of the model can now be written as:

X t(tk = M (tk ; tk )X t(tk ) + Bu(tk) + (tk ) +1

(9)

+1

where (tk ) is a p-vector consisting of the noise components at the grid points of G2. The covariance matrix Q of (tk ) can be determined easily from the equations (5)-(8).  represents a sequence of linear operations to interpolate the noise at the grid points of G1. G2 is usually chosen to coincide approximately with the grid of the atmospheric model, yielding p  n. The uncertainty in the open boundary conditions can be modeled in a simular way by adding a boundary noise process (Heemink, 1988). Most storm surge models of the North sea are based on the equations (1)-(3) (Flather 1976). However, since the North Sea is rather deep, the nonlinearities of the model are small and may also be neglected. This is especially true in case of extreme meteorological conditions when the meteorological input for the model is far from perfect. Therefore, storm surge models can also be based on the linearized equations: q

@u ? 2!v + u + g @h + 1 @pa ? aCdVx Vx + Vy = 0 @t D @x w @x w D q

2

2

(10)

@v + 2!u + v + g @h + 1 @pa ? aCdVy Vx + Vy = 0 @t D @y w @y w D

(11)

@h + @ fDug + @ fDvg = 0 @t @x @y

(12)

8

2

2

where  is the linearized bottom friction coecient. In the linear case only one condition is required at the open boundaries. Because of the linearity of the model, in practice, the complex astronomical tide is usually predicted separately by means of harmonic analysis (Godin, 1972). The shallow water ow model is in that case only used to describe the meteorological eects which are superimposed on the astronomical tide. In the linear case the discrete system representation of the model is: X t(tk ) = M X t (tk ) + Bu(tk) + (tk ) (13) +1

Observations Yko are assumed to be available according to:

yko = HX t(tk ) + 

(14)

where H is the measurement matrix and  is a measurement noise process with zero mean and covariance matrix R

3 Time invariant Kalman lter approach Using a Kalman lter, the observations taken from the actual system and modeled by relation (14) can be combined with the information provided by the system model, Eq.(13), in order to obtain an optimal estimate of the state of the system. Recursive lter equations to obtain these quantities can be summarized as follows. For the linear system the optimal state estimate is propagated from measurement time k ? 1 to measurement time k by the equations:

Xf (tk

+1

= Mxa(tk ) + Bu(tk )

Pf (tk ) = MPa(tk )MT + QT +1

9

(15) (16)

At time k, the observation Yko becomes available. The estimate is updated by the equations

Xa (tk ) = Xf (tk ) + Kk dk

(17)

Pa(tk ) = (I ? KiH)Pf (tk (

(18)

where dk = Yko ? H Xf (tk ) and

Kk + Pf (tk )HT [H Pf (tk )HT + R J?

1

(19)

is the lter gain. Since the model, Eqs.() - (), is time-invariant, this lter gain will become time-invariant as well (Anderson and Moore, 1979) High dimensionality of the Kalman lter equations can be avoided by using a discrete form of the Chandrasekhar-type algorithm. This algorithm was rst proposed by Morf, Sidhu and Kailath (1974) and has been used in numerous applications. It can be used for constant systems and exploits the fact the for certain initial conditions the incremental covariance has a rank p (p is the dimension of the system noise process) and can be factorized as follows:

Pa(tk ) ? Pa(tk ) = S(k)L(k)S(k)T

(20)

S (k) is an n  p matrix, and L(k) is a p  p matrix. For the constant model, Eqs.() - (), the recursive equations to obtain the steady-state lter gain are, see Heemink (1988):

Y(k + 1) = MS(k) M(k + 1) = M(k) + Y(k + 1)L(k)Y(k + 1)T HT 10

(21) (22)

R"(k + 1) Kk S(k + 1) L(k + 1)

= R" (k) + HY(k + 1)L(k)Y(k + 1)T HT

(23)

= G(k + 1)R"(k + 1)?

(24)

= Y(k + 1) ? K(k + 1)HY(k + 1)

(25)

1

= L(k) + L(k)Y(k + 1)T HT R" (k)? HY(k + 1)L(k) 1

(26)

where K is the lter gain at time k; Y(k); M(k) and R"(k) are factor matrices and the initial condition for the recursion is given by

Y (1) = G R"(0) = R

M = 0 L(0) = Q

(27)

kKk ? Kk k < "kKk k

(28)

Equations () - () are iterated until +1

where " is speci ed in advance. Since the underlying deterministic model is of the hyperbolic type, the number of iterations depends on the traveling times of the waves in the model and, therefore, on the size of the domain of the problem. Since for our data assimilation problem p  n, the computational eort that is required to solve the complete ltering problem is very modest. In case of the nonlinear system () with measurement equation () the well-known extended Kalman lter can be summarized as follows (Jazwinski, 1970):

X f (tk ) = M X f (tk) + B U (tk ) +1

X a(tk ) = X f (tk ) + Kk dk

(29) (30)

Using the extended Kalman lter, the time varying Kalman gain Kk is determined by rst linearizing the model dynamics about a reference trajectory and then by using the lter equations for linear time varying systems. For the storm surge model 11

(??) - (??) with time invariant measurement equation (??) the time variations of Kk are only caused by the nonlinearities of the model. Since these nonlinearities are relatively small, the time variations of Kk are small too. Therefore a constant gain extended Kalman lter (Safonov and Athans, 1978) produces estimates that are nearly as accurate as the extended Kalman lter. This lter is based on the linearization of the model about an equilibrium state and is, as a consequence, time invariant. As a result the Kalman gain K is determined eciently by solving the Chandrasekhar-type lter equations (?? )-(??) based on the linear system (). For a weakly nonlinear shallow water ow model, the constant gain extended Kalman lter is an accurate approximation of the extended Kalman lter. This has been veri ed in the one dimensional case (Heemink, 1986). The most severe restriction of the steady state approach is, however, the fact that the measurement relation has to be time invariant. This does, e.g., not allow for the ltering of altimeter data from satellites. Furthermore, by using a time invariant lter, the system noise statistics have to be time invariant too. This assumption may not be very realistic.

4 Operational storm surge forecasting systems

4.1 Implementation at DMI

The Danish Meteorological Institute (DMI) is responsible for the storm surge warning for the Danish coastal areas. The objective of the storm surge warning system is to issue warnings to the local authorities whenever a surge is forecasted. To ful l this responsibility DMI in 1990 implemented an operational system (DKSS90). DKSS90 makes use of a deterministic shallow-water nite-dierence model developed by the Danish Hydraulic Institute. The model area includes the North Sea, the Baltic Sea and the inner Danish waters. The model operates with computational grids of dierent scales, and the computations for these grids are dynamically coupled, Vested et 12

al. (1992) [?]. Performance of the system has been documented in Vested et al. (1995) [?]. The performance of the system is very satisfactory, [ref til Lynch and Davies], when the external forcing is correct (meteorological and boundary forcing). Performance deteriorates when the meteorological forcing is not correct or when a so-called external surge, i.e. surge generated outside the model area, is in question. To overcome these limitations a project with the aim of using data assimilation in an operational storm surge forecast system, to be described below, was initiated. The system uses the above described method to assimilate tide gauge observations into a hydrodynamical model. The main areas of interest are the Danish Wadden Sea coast (which we will focus on in this paper) and the western part of the Baltic Sea. Warning criteria have been de ned for a number of sites. When the forecasted water levels for a given site exceed the criterion a warning is issued from DMI to the proper local authorities.

4.2 Area The system is very exible and has been implemented for dierent areas. Here, focus is on the North Sea, Fig. ??. The setup consists of a 62x54 numerical grid de ned on the UTM-grid with a spatial resolution of 9 nautical miles. The number of water points for the area is 1700 and the length of the state vector x is 4911. The domain has three open boundaries. The open boundaries in the English channel and between Scotland and Norway are speci ed as tidal elevation boundaries. The boundary in the Dansish domestic waters are of a Riemann type.

13

Figure 1: Model domain. Location of the 14 tide gauge stations are marked on the map. Depth contours in meters.

4.3 Meteorological forcing (HIRLAM) Meteorological forcing for the hydrodynamical model is obtained from the operational weather forecast model (HIRLAM) running at DMI. HIRLAM is run four times every day at 0Z, 6Z, 12Z, and 18Z. The 0Z and 12Z runs are full model runs making 36 hour forecasts. At 6Z and 18Z, HIRLAM is only integrated for 6 hours to produce better rst guesses for the full model runs. The temporal and spatial resolution of the meteorological model is presently 1 hour and 0.21. The surface stress is calculated according to a quadratic friction law:

~ jW ~j ~w = CD aW

(31)

~ is where the drag coecient CD = 2:5 10? , density of air a = 1:25 kgm? and W the wind velocity at 10 m height. Calculation of the stress and interpolation from the meteorological model grid to the hydrodynamical UTM-grid is done as a pre-processing job. It is important to 3

14

3

Figure 2: HIRLAM DKV model area notice that the interpolation, both in space and time, from the meteorological elds is performed in stress rather than in 10 m wind. This is done to avoid smoothing of the elds.

4.4 Filter settings The spatial covariance matrix for the observational noise, R, has to be speci ed. We simply use a diagonal matrix for this purpose and the elements are all set to 0:03 m, representing the measurement standard deviation. The decay lenght scale of the forcing (system) error, , in equation ??, has been speci ed to be comparable to the rst baroclinic Rossby radius of deformation for the atmosphere which is of the order of 10 m . The standard deviation of the error has been speci ed to 0:5 Nm? corresponding to a wind speed error of the order of 10ms? . This value might seem somewhat high but best performance of the setup has been achieved with this parameter. 6

2

1

15

1

4.5 Tide gauge observations Observations of sea level from 14 tide gauge stations surrounding the North Sea are assimilated into the hydrodynamical model, see Fig. for locations. The international data are transmitted to DMI through the GTS net and have a temporal resolution of 1 hour. At present DMI receives new international observations every 6 hours. Observations from the Danish tide gauge stations are transmitted to DMI through dedicated tele-comunication lines and are accesible at DMI with a 5 min. lag. The observation frequency for the Danish tide gauge stations is 15 minutes. Before observed data are used for assimilation they are visually inspected. Based on the experience gained an automatical quality control system will be implemented. The dierent countries have dierent datum for the sea level observations and before they are used for assimilation the data are transferred to mean water level.

4.6 Solution method and code The explicit Sielecki-scheme [?], is used for the numerical solution of equations (?-?). The scheme operates on a staggered grid of the Arakawa-C type. The programmes have been developed in C for easier portability, and uses dynamical memory allocation so that the same executable can be used for dierent setups. The Kalman lter is calculated on DMI's Convex 3880, whereas the actual storm surge forecast is made on a work-station. The overhead of using the Kalman lter in preparing forecasts compared to an ordinary deterministic model is minor, and a 60 hour model run requires about 60 seconds cpu-time.

4.7 Operational Use The time step is 60 seconds and the length of the integration is 60 hours. The rst 24 hours of the integration is based on analysed meteorological information and the last 36 hours is the actual forecast. Results are stored every 30 minutes. 16

The advantage of the ability of running the system frequently is that the new information provided by new tide gauge observations can be included in the preparation of a new forecast. In the case of a potential storm surge the system can be invoked via a graphical user interface by the meteorologist on duty, to produce new forecasts.

4.7.1 Scheduled runs The system is run on a routine basis twice every day. This is done partly to produce initial elds for the next model simulation and partly to make a forecast archive for veri cation purposes.

4.7.2 Requested runs During periods with potential storm surges new forecasts can be produced when ever new tide gauge data are available. The meteorologist on duty makes a request for a new forecast through a graphical user interface. This will start the script responsible for generating necessary input les, run the hydrodynamical model and return updated post-processing les. Based on these new areal maps and time series plots are made and the temporal evolution of the storm surge with the new information from the tide gauge data is easily inspected.

4.8 Some results An example of the performance of the system with data assimilation compared to a purely deterministic system will be described in the following. The operational storm surge forecasting system in the Netherlands is based on a nonlinear shallow water ow model of the entire Continental Shelf (CSM, Continental Shelf Model), transformed in spherical coordinates (Verboom et al., 1992). The spatial domain is covered with a numerical grid of 100 (western latitude, stepsize 1=4 deg.) by 85 (northern latitude, stepsize 1=6 deg.) grid cells (Figure ?). A great 17

number of these grid cells are located on the main land, so the number of active grid cells is reduced to approximately 5000, which makes the dimension of the state vector approximately 15000.

5 Conclusions Acknowledgements The authors gratefully acknowledge the discussions with

References [1] Abbott, M.B. (1979), Computational Hydraulics, Pitman, London [2] Bolding, K. (1995), Using a Kalman lter in operational storm surge prediction, In: Second Int. Symp. on Assimilation of Observations in Meteorology and Oceanography, WMO, pp. 379-383 [3] Brummelhuis P.G.J. ten, A.W. Heemink and H.F.P. van den Boogaard (1993), Identi cation of shallow sea models, Int. Journ. on Num. Meth. in Fluids, 17, pp. 637-665 [4] Budgell, W.P and T.E. Unny (1980), A stochastic-deterministic model for predicting tides in branched estuaries, Proceedings of the 3th Int. Symp. on Stoch. Hydr., Tokyo [5] Budgell, W.P.(1986), Nonlinear data assimilation for shallow water equations in branched channels, J. Geophys. Res., 10, pp. 633-644 18

[6] Chavent G. (1979), Identi cation of distributed parameter systems : about the least square method, its implementation and identi ability , Identi cation and system parameter estimation, Proc. of the 5th IFAC symposium [7] Curi, R.C., T.E. Unny, K.W. Hipiel and K. Ponnambalam (1995), Application of the distributed parameter lter to predict simulated tidal induced shallow water ow, Stoch. Hydrology and Hydraulics, 9, pp. 13-32 [8] Curtain, R.F. and A.J. Pritchard, In nite dimensional linear system theory, Lec. Notes in Control and Inf. Sc., 8, Springer-Verlag, New York [9] Dronkers J.J. (1964), Tidal computations in rivers and coastal waters, North Holland Publishing Company, Amsterdam [10] Flatcher, R. (1976), Tidal model of the North-West European Shelf, Mem. Soc. R. Sci., Liege, 10 [11] Ghil, M., S.E. Cohn, J. Tavantzis, K. Bube and E. Isaacson (1981), Application of estimation theory to numerical whether prediction, In: Dynamic Meteorology: Data assimilation methods, Bengtsson, et al. (eds), Springer-Verlag, New York [12] Godin, G. (1972), The analysis of Tides, University of Toronto Press, Toronto [13] Heemink, A.W. (1986), Storm surge prediction using Kalman ltering, RWS Communications, 46 [14] Heemink, A.W. (1988), Two-dimensional shallow water ow identi cation, Appl. Mat. Mod., 12, pp. 109-118 [15] Heemink A.W. and H. Kloosterhuis (1990), Data assimilation in nonlinear lidal models, Journ. on Num. Meth. in Fluids 19

[16] Heemink, A.W and I.D.M. Metzelaar (1995), Data assimilation into a numerical shallow water ow model: a stochastic optimal contrlol approach, J. of Marine Systems, 6, pp. 145-158 [17] Van der Houwen, P.J. (1986), Finite dierence methods for solving partial dierential equations, Mat. Cen. Tracts, 20, Mat. Cen., Amsterdam [18] Jazwinski A.H. (1970), Stochastic processes and ltering theory, Academic Press, New York [19] Morf M., S.S. Sidhu and T. Kailath (1974), Some new algorithms for recursive estimation in constant, linear, discrete time systems, IEEE Trans. autom. control, AC 19 [20] Mouthaan, E.E.A., A.W. Heemink and K.B. Robaczewska (1994), Assimilation of ERS-1 altimeter data in a tidal model of the continental shelf, Deutsche Hyd. Z., 46, pp. 285-329 [21] Safonov, M.G. and M. Athans (1987), Robustness and computational aspects of nonlinear stochastic estimations and regulators, IEEE Int. Aut. Contr, 23, pp. 717-725 [22] Stelling G.S. (1984), On the construction of computational methods for shallow water ow problems, Rijkswaterstaat communications, The Hague, Vol 35 [23] Verboom, G.K., J.G. de Ronde and R.P. van Dijk (1992), A ne grid tidal ow and storm surge model of the North Sea, Continental Shelf Research, 12, pp. 213-233

20

Figure captions Figure 1: The computational grid Figure 2: The Continental Shelf Model Figure 3: Model results with (? ? ?) and without data assimilation (||{)

21

 6
y

n

6

?

-


x

ukm;n

hkm

;n+1

k vm;n

Dm;n

vmk

;n

hkm;n

ukm;n

hkm

;n

vmk

;n?1

+1

+1

+1

+1

Dm;n?

1

+1

- m Figure 3: The computational grid

22