unit; 1m The ONJUKU COAST Total number of nodes; 600 Total nimber of elements; 1097 Onjuku Port 5 Iwawada Port No.5 No.2 No.3 No.4 No.

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1 Estimation of Tidal Currents by Kalman Filter with FEM Using Parallel Computing Naeko TAKAHASHI Abstract The purpose of this research is to estimate of tidal currents using Kalman Filter combined with finite element method. And in recent years, parallel finite element computations have been successfully applied to several large-scale problems. The stochastic model combined with the shallow water equations and the Kalman filtering technique is presented with parallel computing. INTRODUCTION Finite element computation of the shallow water flows can be applied to many practical problems. In the coastal area, currents and water level vary widely with the tidal force and wind. In general currents and water level are estimated using deterministic model, which is based on the shallow water equations with suitable boundary conditions. Sometimes currents and water level are observed and the data collected is very effective and useful for there estimation. In the field of civil engineering, the kalman filter is applied to various problems, for example, parameter identification of mechanical constants of rock mass, prediction of hydroscience system, and estimation of dispersion coefficient of water pollution, and the initial value of stress. The filter is one of probabilistic approach using measurement data. And the filter is based on a set of difference equations. The one is system equation which can express states, the other is observation equation which can express measurements. Using FEM in the system equation, states can be estimated at the whole computational area as well as measurement points. But the calculation of Kalman filter needs a large computational time. So in this paper, a purpose of this research is to reduce computational time using parallel computing more effective. 2 FINITE ELEMENT METHOD 2. Basic Equation Linear shallow water equation which consists of the momentum equation and continuity equation are shown as follows ; < momentum equation > u_ i + g ;i = () < continuity equation > _ + hu i;i = (2) where u i are velocities, is water elevation and g is the gravity acceleration. 2.2 Boundary Conditions The boundary condition is given as, ρ =^ on d u n = u i n i =^u n on n (3) where^denotes the given value. 24

2 2.3 Finite Element Equation The discretization in space can be made by applying the Galerkin method. The velocity and water elevation on a triangle element interpolated using liner interpolation function. For the weighting function the same function as the interpolation function is employed. The discretization in time can be made by the explicit Euler method as follows, where ~ M is represented as follows, μmu n+ = ~ Mu n tgh x n (4) μmv n+ = ~ Mv n tgh y n (5) μm n+ = ~ M n th(h x u n + H y v n ) (6) ~M = e μ M +( e)m (7) where e is a lumping parameter which depends on a criterion of computational stability. 3 KALMAN FILTER The Kalman filter can be applied widely, its state vector formulation allows consideration of both process and observation noise, and it is easily implemented with recursive algorithms. These are the main reasons for the popularity of the Kalman filter. 3. Derivation of the Kalman Filter Consider a discrete, liner, stochastic, time-invariant dynamic system whose behavior is described by the state equation < System Model > x(t +)=ffi:x(t)+w(t) (8) where x(t) isann-vector of state variables, ffi is a transition matrix of size n n, w(t) is an n-vector of Gaussian white noise with statistics w(t) ο N(;Q). Also consider a liner measurement (or output) equation of the form < Measurements Model > z(t) =H:x(t)+v(t) (9) where z(t) is an n-vector of measured output vector, H is an m n measurement matrix, v(t) is an m- vector of measurement errors with statistics v(t) ο N(;R) ; the covariance matrix R is positive definite. It is also assumed that the noise processes are uncorrelated with one another, that is E[v(t) T.w(fi)]= for all t and fi where ^ denotes the given value. and both processes are independent of the state which has the initial statistics x()=n(^x(),p ()). Let ^x(tjt) denote the best liner unbiased estimator of x(t) based on measurements Z(t)=fz(),z(2),...,z(t)g. ^x(tjt) is the filtered estimator of x(t) since it is the estimate of the state vector at current time based upon all measurements including the current one. It will be shown that the optimal estimator for the filtering problem will provide us with the optimal estimator for the prediction problem. Kalman rigorously derived the form of a recursive filter for the above system-that is, a system with known structure, parameter, and error statistics. 25

3 3.2 Formulation The problem can be stated follows: given a measurement sequence Z(t) observed by the measurement equation (9), estimate the state of the dynamic system (8) such that the error of estimate, ~x(tjt)=x(t)- ^x(tjt), will minimize the quadratic performance function J = E[~x(tjt):L:~x T (tjt)] () where L is any positive semi definite matrix and, for simplicity, is chosen to be the identity matrix I. Assume that a prior estimate ^x(tjt ) of the system state x(t) at t is based upon previous measurements up to t. An updated estimate ^x(tjt) is desired which takes into account the new measurement z(t) at t. Consider this updated estimate as the liner combination of the prior estimate and the new (noisy) measurement; thus ^x(tjt) = ~ K(t):^x(tjt ) + K(t):z(t) () where ~ K(t) andk(t) are time-varying weighting matrices as yet unspecified. Introducing (9) into () and utilizing the statistics of the noise process, it can be seen that () is an unbiased estimate only if ~K(t)=I-K(t).H. Hence, the state estimator ^x(tjt), using the new measurement z(t), is of the form ^x(tjt) =^x(tjt ) + K(t)[z(t) H:^x(tjt )] (2) where K(t) is still unspecified, and the initial condition at t = for the state estimation is ^x(j)=^x(). A measure of accuracy for the estimate can be expressed by thecovariance matrix P (:) of the prediction error defined as P (tjt) =E[~x(tjt):~x T (tjt)] (3) the initial condition of which is P (j)=p (). Using () with the measurement noise statistics, it can be shown that the covariance matrix of ^x(tjt) can be projected from that of ^x(tjt ) as P (tjt) =(I K(t)H)P (tjt )(I K(t)H) T + K(t)R:K T (t) (4) Since the loss function () is the trace the error covariance matrix (3), the problem is to minimize the trace norm kp(t t)k of P (tjt), that is, the length of the estimation error vector. Using the properties of matrix derivatives, it can be seen that the weighting matrix K(t) can be obtained from kp (tjt)k K(t) =P (tjt ):H T :[H:P(tjt ):H T + R] (6) which is referred to as the Kalman gain matrix. Equation (6) can be used to simplify (4) into P (tjt) =(I K(t):H)P (tjt ) (7) The one-step-ahead prediction of the state vector, given observation up to t, is ^x(t +jt) =ffi:^x(tjt) (8) The propagation of prediction errors P (tjt)p (t +jt) can be determined by computing the predicted error covariance matrix, P (t +jt) =E[~x(t +jt):~x T (t +jt)]. Using (8), (3), and the independence of the error terms, we find that P (t +jt) =ffi:p (tjt):ffi + Q (9) It can be shown that the one-step-ahead estimate of (8) will minimize (9); thus by finding the optimal estimator for the filtering problem, the optimal estimator for the prediction problem is obtained by(2). The optimal filter is summarized in Table. 26

4 System model (Eq:9) x(t +)=ffi:x(t)+w(t) M easurements (Eq:) z(t) =H:x(t)+v(t) Initial condition and E[x()] = ^x();e[x():x T ()] = P () other assumptions w(t) N(;Q);v(t) N(;R) E[w(t)v T (fi)] = for all t,fi State estimate update (Eq:3) ^x(tjt) = ^x(tjt ) + K(t)[z(t) H:^x(tjt )] Error covariance update (Eq:8) P (tjt) =[I K(t)H]:P (tjt ) Kalman gain matrix (Eq:7) K(t) =P (tjt )H T [H:P(tjt ):H T + R] State estimate prediction (Eq:9) ^x(t +jt) =ffi:^x(tjt) Error covariance prediction (Eq:2) P (t +jt) =ffi:p (tjt):ffi T + Q Tab.. Summary of discrete Kalman filter equation 4 PARALLEL COMPUTING The calculation of the Kalman filter takes long time. So it is necessary to use parallel computing. Actually it is necessary to apply parallel computing to the part of calculation of Error covatiance matrix. Because it especially takes a long time to calculate this part. Fig.. Image of Parallel Computing Fig. is image of summary of parallel computing. There is a necessity to communicate between each processors. Because if the communication time is too long, we'll have no effect. 5 APPLICATION TO THE ONJUKU COAST In this research, the Onjuku coast is employed as a numerical example. The Onjuku coast is located at the south part of Bousou peninsula (Fig.2). In resent years, the water quality pollution moves ahead with the inflow of pollution material from the river to the Onjku coast. So, it is necessary to grasp the trend of pollution material in estimating the situation of tidal flow. The measurement of currents and water elevation were carried out during July 6, 997 and July 2, 997 in Onjuku coast. Fig.3 are finite element mesh and water depth. The data about currents and water elevation were obtained at 5 points as shown in Fig.3. In this research, I use each parameters, increment ffit is.2 (sec). covariance R is : 4. System error covariance Q is : 4. Lumping parameter e is.8. 27

5 The ONJUKU COAST Fig.2. Location of the Onjuku coast Iwawada Port Onjuku Port 5 No.5 No.3 No.2 No.4 5 No. Total number of nodes; 6 Total nimber of elements; 97 Fig.3. Finite element mesh and Water depth unit; m 5. Numerical Result From Fig.4 to Fig.8 show the estimation results of water elevation, comparing with with the measurement data. The Estimation at from No. to No.5 using the observation data at from No. to No.5 was carried out. And Fig.9 represents velocity distribution of residual flow from 6 July to 2 July. Position at No. Position at No Fig.4. Comparison of Water elevation at No Fig.5. Comparison of Water elevation at No.2. 28

6 Position at No.3 Position at No Fig.6. Comparison of Water elevation at No Fig.7. Comparison of Water elevation at No.4. Position at No Fig.9. Velocity distribution of Residual flow Fig.8. Comparison of Water elevation at No.5 Tab.2 and Fig. are result of each computational times. In this case, there is good effect of using parallel computing. Single processor (sec) 3 processors (sec) 9 processors (sec) Off line On line (applied element by element method) Total CPU Time Tab.2. Comparison of CPU Time 6 CONCLUSION In this paper, the Kalman filter with finite element method was presented. And it could be achieved to reduce computational time with parallel computing and element by element method. For this reason, parallel computing proved to be effective for large scale analysis. For the future work, it consider to add to the total number of processors, and to savecommunicative waste. Fig.. Comparison of CPU Time 29