68686540 storm surge prediction using kalman filtering

8/13/2019 68686540 Storm Surge Prediction Using Kalman Filtering

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rijkswaterstaat

communbatbnsstorm surge predictionusing kalman filtering

bydr ir a w heemink

no 46/1986ex


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storm surge prediction using kalman filtering

REF. NR.53S


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rijkswaterstaat Communications

storm surge predictionusing kalman filtering

bydr. ir. a.w. heemink

thesis Twente University of Technology 1986thehague 1986


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All correspondence and applications shouid be addressed to

rijkswaterstaatdienst getijdewaterenhooftskade 1postbus 209072500 ex the haguenetherlands

the views in this article are the author s own

recommended catalogue entry:Heemink A.W.

Storm surge prediction using kalman filtering / by A.W. Heemink; Rijkswaterstaat. - The Hague:Rijksw aterstaa t, 1986. - 194 p.: ill. ; 24 cm. - (Rijkswaterstaat Com mu nications; no. 46)With refer.: p. 185-187Thesis Twente University of Technology, 1986.With Dutch summary


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Contents

Page1 Introduction 81.1 Introduc tion 81.2 Tidal prediction methods 131.2.1 Introduction 131.2.2 Prediction of the astronom ical tide 141.2.3 O peration al metho ds used in the Netherlands 161.2.4 Recen t results in tidal prediction 181.3 Kalm an filter approach to tidal prediction 201.4 Scope of the investigation 23

2 The Shallow water equations 242.1 Introduction 242.2 The two dimensional equations 242.3 The one dimensional equations 27

3 Discretefilt ringtheory 303.1 Introduction 303.2 Linear filtering theory 303.2.1 The Kalman filter 303.2.2 Stability .. i 303.3 Non linear filtering theory 373.3.1 Introduction 373.3.2 Linearized and extended Kalman filters 383.3.3 Non linear filters 403.3.4 Para me ter estimation 44

4 Kalmanfilt rsfor the linear one dimensional shallow water equations 464.1 Introduction 464.2 Numerical approximation of differential equations 464.3 Discretisation of the one dimerisional shallow wa ter equations 494.4 On the choice of the system noise 55

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4.5 Kalman filtering in the wave number domain 574.6 General discrete system representation 744.7 G K observability 754.8 Numerical properties of some Kalman filter algorithms 794.9 Distributed parameter filters 86

5 A Kalman filter based on a one dimensional model 885.1 Introduction 885.2 The deterministic model 885.2.1 Introduction 885.2.2 The tidal movement in the southern North Sea 895.2.3 Meteorological effects in the North Sea 925.2.4 The mathematical model 945.3 The Kalman filter 985.3.1 Introduction 985.3.2 The stochastic model 995.3.3 The boundary treatment 1015.4 Experiments using simulated data 1025.4.1 Introduction 1025.4.2 The constant gain extended Kalman filter 1035.4.3 Parameter estimation 1075.5 A Kalman filter based on an Eastern Scheldt model 1115.6 Experiments using field data 1165.7 Discussion 126

6 AKalman filter based on a two dimensional model 1316.1 Introduction 1316.2 The deterministic model 1326.3 The Kalman filter 1396.3.1 Introduction 1396.3.2 The stochastic model 1396.3.3 The boundary treatment 1436.3.4 The steady state filter 1446.4 Implementation of the algorithm on a vector processor 1496.5 Experiments using simulated data 1526.5.1 Introduction 1526.5.2 Internal surge 1546.5.3 External surge 1636.6 Experiments using field data 1636.7 Discussion 1816


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onclusions tnd recommendations 182eferences J 185otation j 188Summary j 19Samenvatting 191


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1. Introduction

1 1 IntroductionThe delta area in the south-western part of the Netherlands lies below or justabove sea level see figure 1.1).Therefo re, special precautions have to be taken withrespect to th e dikes and the coast-lineingeneral. After the disaster of February 1953,when during a period of severe storm conditions the dikes in this area broke andapproximately 1800 people were drowned seefigure 1.2), the Dutch government

Noordzee - j -y

Figure 1.1: The area that would periodicallybeflooded if the Netherlandswould not be protected by dikes.


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Figure 1 2: The flooded area during the disaster of February 1953

SALTTIDALSALTNON-TIDALFRESH

Figure 1 3: The Delta Plan


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decided to solve the problem of the long and vu lnerable coast-line of the delta areaand planned to close all the estuaries except the free waterway to Antwerp, theWestern Scheldt. This massive construction program was called the D elta Plan seefigure 1.3 .

Before the start of the project in 1958, little attention was given to the fact thatimportant changes would occur in the estuaries. Since the closed estuaries wouldcease to be tidal and would be converted into fresh w ater lakes, the en tire flora andfauna of the delta area w ould change. By 1974, the D elta Plan was nearly completedexcept for the closing of the largest estuary, the Eastern Scheldt. At tha t time, workwas interrup ted as a result of pressure exerted by ecologists and nature p rotec tors,who wanted to preserve the tidal state of the Eastern Scheldt. The very specialecological characteristics of the Eas tern Scheldt, such as the grea t variety of types ofenvironment and the pure and relatively warm water, have brought about anexceptional abundance of flora and fauna Saeijs 1982). Closure would threaten notonly this rich and rare ecology but also the oyster and mussel industries located in theEastern Scheldt. In 1976, the Dutch government altered the original De lta Plan anddecided to insert a storm surge barrier containing many large gates that will be closed

Figure 1.4: The storm surge barrier.10


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Photo l.J : The storm surgc banier , February 1986.

only in case of extreme high water secfigure 1.4andphoto 1.1).in normal w eatherconditions, the gates will be opened to allow a reduccd tide to pass into the EasternScheldt.When the building of the storm surge ba nier is completed and the barnet is in

opcrational service, the prediction of water-levels in the Eastern Scheldt will be of


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3 30 3 45 4N

orth Sea

0 5 10 15 kmFigure 1.5: The Euro Channel and the Meuse Channel.

great importance. Since the barrier will have to be closed a few hours before theexpected occurence of extreme high water, accurate predictions are required toensure that the barrier will be closed in time.Accurate predictions for the entire Dutch coast, approximately six hours ahead,are also necessary to decide what precautionary actions have to be taken to protectthe d ikes. Furthe rm ore , with the increase in the size ofshipsusing the Eu ro C hanneland the Meuse Channel to Ro tterdam harbour (seefigure 1.5 ,this port has becomeinaccessible to many ships except during a short period at high water. Th erefo re, toallow safe passage, there is a demand for accurate tidal predictions.Previously, tidal prediction me thods have been either statistical or deterministic innature. Applying statistical methods - i.e. simple empirical or black-box models

derived from series of observations itispossible to use on-line measu rem ents of thewater-level. This feature is very importan t considering the f act that the num ber ofon-line measurement stations in the North Sea is increasing rapidly. Deterministicmethods, including the analytical or numerical solutions to the hydrodynamic equa-tions, do not have this possibility. However, they have a more physical basis andprovide a more realistic description of the w ater mov em ent. These facts suggest theuse ofamethod that is acombination of thestatistical and the deterministic approachin order to obtain the best features ofboth. Such a method is the Kalman filter. Thisstochastic filter is based on a deterministic model and has also the capability ofcorrecting model predictions using on-line information.12


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In this investigation Kalman filters have been developed for the p rediction of thewater-level during stormy periods. In the remaining pa rt ofthisintroductory chapterwe first summarize in Section 1.2 the tidal prediction methods used up to now.Section 1.3 describes the Kalman filter approach to the prediction of water-levelsand finally Section 1.4 gives an exposition of the scope of the entire investigation.

1 2 Tidal prediction methods1 2 1 Introduction

The astronomical tide of the oceans is created by forces from the sun and m oon .The attraction of the moon is approximately twice as strong as that of the sun.Because of their greater distance or smaller size other celestial bodies have anegligible effect on the tides. In the A tlantic Ocean tidal amplitude are rather small:less than 50 cm. In shallow w aters such as the North Sea the tide generating forceacting directly on the m ass of water in these areas can be neglected. However as theAtlantic tide progresses into the North Sea the tideisamplified and its am plitude canreach values of 1-2 meters along the Dutch coast. Furthe rm ore when the tidepen etrate s into shallow coastal waters or into an estuary its propagation is affectedby the coast and by the frictional effects caused by the reduced depth.

A surge is defined as the meteorological effect on the tidal propagation . A surgethat is generated by severe storms outside the North Sea area is called an externalsurge when it enters this area. This storm surge propagates through the North Seaapproximately like the tide. In the southern part of the North Sea external surgesusually have 50 cm or less amp litude. Interna l surges are caused by meteorologicalphenomena inside the North Sea area. During north-westerly storms owing to thefunnel shape of the North Sea - these storm surges can reach values of more than 2meters along the Dutch coast.

The m eteorological effect is created by two physical phen om ena. Firstly theatmospheric pressure profile of a storm affects the water-level lying directly under-neath . Secondly wind blowing over water c reates a frictional force on the surf ace ofthe water that sets up the water-level in the direction of the wind. In shallow waterssuch as the North Sea the wind effect is in general strongly predominent in relationto the pressure effect.Most tidal prediction techniques are based on the simple superposition of theastronomical tide and the meteorological effect. Since the astronomical tide can bepredicted relatively accurately attention has been concentrated on the prediction ofthe m eteorological effect. In this section we summ arize the existing tidal prediction

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m ethod s. In Subsection 1 .2 .2 the culmination metho d, the harmonie meth od and theresponse method for predicting the astronomical tide are briefly described. For athoroug h tre atm ent of these m ethod s, involving the descr iption of the mov em ents ofthe ear t h , m oon and su n, and the development of the t ide generat ing potential , thereader is referred to Hornman (1977) for the culmination method, to Dronkers(1964,1975) or Godin (1972) for the harmonie method, and to Munk and Cartwright(1966) for the respo nse m eth od . Subsection 1.2.3 is devo ted to the o pera tiona lmethods used in the Netherlands for predicting the meteorological effect along theDutch coast. Finally, some recent results in the development of tidal predictionmethods are discussed in Subsection 1 2 4

1.2.2 Prediction o f the astronom ical tideIn 1831 Lubbock published an analysis of the relationship between the tides andthe motion of the moon and sun in relation to the earth. It has been known for

centu ries th at, of all the factors that des cribe this m otion , the tim e of culm ination ofthe moon* has the most important influence on the high and low water-levels in theNorth Sea. To predict the times and heights of low and high water, Lubbockempirically determined the relationship between these tidal data and the time ofculmination using observations gathered over a few years. Corrections were derivedto take into account some other characteristics of the motions of moon, sun andearth . Lu bboc k s method w as cal led the culmination metho d and prov ed to be quitesuccessful in those days.

Another approach to t idal predict ion was developed by Keivin and Darwin,following the su ggestion by Lap lace th at as a result of perio dic variatio ns in the forceexerted on the mass of water the water will move with the same periodicity. Thisapp roac h to the analysis of tides was called the harm onie m etho d an d trea ts the tidalelevation at a single location as the summation of a large number of independentsinusoidal motions:

Nh(t) = h0 + 2 A scos (2 ia f. 0,) (1.1)i = lwhere :

h(t) = water-levelh0 = m ean sea levelAj = am plitude of the i-th harm onie constitue nt0; = ph ase of the i-th con stitu ent = astronomical frequency of the i-th constituent.

* The time at which the path of the moon crosses the meridian of the observer.14


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In principle, the number of astronomical frequenciesisinfinite. Using the gravita-tional tidal potehtial due to the moon and sun, the most important harmonieconstituents can bje selected. However, if shallow water effects become significant,constituents that ^re not found or have been neglected in the original tidal po tentialcan become very impo rtant. These frequencies have to be selected empirically foreach location. Sinpe the harmonie analysisisusually applied to time series no longerthan one year, the estimated amplitude A, and phases {depend on tidal motionswith longer periods* . In order to take these effects into account, correction factorsfj t) and u^ t) are Jntroduced by rewriting 1.1) as:

Nh t) = h0 + 2 f, t)Ajcos 2 Dj+ 0, + u, t)) 1.2)i= lwhere:fj t) = correc tion for the amplitude of the i-th constituent

U| t) = correc tion for the phase of the i-th constituent.The correction factors are computed using the tidal potential. The unknow n am plitude Aj and phase {are determined by least squares estimation.

The harmonie rnethod is usually applied using time series of approximately oneyear. Since the shallow water effects along the Dutch coast are very strong, manytidal constituents; 100-150) are required to describe the tide. Most of these constituents have am plitude smaller than3cm . To give an example the harmonie analysishas been applied to hourly observations of the water-level registered during 1983 atthe measurement station OS IV, located in the mouth of the Eastern Scheldt. Theanalysis was carried out by Voogt 1984) using107constituents. The most importantharmonie constituents are summarized in table 1.1.nameSAoN2M2S2M4MS4M62MS6

w degrees/hour)0.041013.943028.439728.984130.000057.968258.9841


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Another approach to tidal predictions was introduced by Munk and Cartwright1966).Their response m ethodisphysically more realistic than the harmonie m ethodandisbased on the response of the sea surface to the tidal potential. T he results haveproved to be comparable with the harmonie analysis Cartwright 1978). However,the response method requires approximately19years of observations to estimate theresponse w eights. In most locations observations recorded during such alongperiodare not available and, moreover, at many locations the conditions do not remainconstant over a period of 19 years owing to sedimentation, doek construction orchanging salinity. Since in addition, harmonie analysis is easier to use and interp ret,the response method has seldom been applied to predict the astronomical tide.

In the Netherlands, until 1986 the tide tables* were produced by means of amodified version of the culmination method using five years of observations. In thisrespect the Netherlands were fairly unique, since almost every other country usedharmonie tide analysis. Furthermore, most textbooks on tides or tidal predictiononly mention the culmination method as an historical curiosity or do not refer to it atall. Nevertheless, due to the very strong shallow w ater effects at most locations alongthe Dutch coast, the harmonie analysis is, surprisingly, not significantly moreaccurate than the culmination method. However, since the latter only predicts thetimes and heights of high and low water, it has been replaced by the harmonieanalysis in 1986.

1.2.3 Operationalm ethods used in the etherlandsThe operational tidal prediction techniques in the Netherlands are based on thesuperposition of the astronomical tide and the meteorological effect. While theastronomical tide can be determined by the culmination method or by harmonieanalysis, the meteorological effect is predic ted on an operational basis by using twoapproaches simultaneously.In the first approach predictions are accomplished using a numerical model of theNorth Sea and parts of the adjoining wa ters. This model is based on the linearized

shallow water equations** :u h u V2cos il 1 pa+ g f v + ^ --y-JL _ -pL = 0 1.3dt dx D D QW dx

* Tide tables consist of the predicted water-levels times and heights of high and low water) over aperiod of one year. These tables are published every year.* *The description of the general non-linear shallow water equationsaswell as the linearization of theseequations can be found in Chapter 2.16


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3v dh , v V2sin\l) 1 p,^ + ^ + f u + ^ D - Y - D ^ + o : ^ = 0 L 4 )ah + p u i + a p v I = o l 5 )dt dx dy

where:h = wa ter-levelu,v = w ater velocities in respectively the x and y-directionsD = de pth of the wa terf = Coriolis pa ram eterX, = linea r bo tto m friction cofficintY = wind friction co fficintV = wind speed\|J = wind direction= density of waterp a = a tmo spher ic pressureg = acce leration of gravity.Eq ua tion s 1.3)- 1.5) are discretised on a staggered grid using the Fischer schem e

Van der Ho uw en 1968). The mesh length in the southern part of the North Sea hasbee n chosen at4 km , while for reasons of s tabi li ty in more re m ote pa rts of the N orthSea and of the C hann el i t has been increased to 84 km in on e of the direction s. T hetime step has been chosen at 7.5 min. At the boundaries the meteorological effect isneglected and assumed to be zero.

Based on the work of Lauw erier and Dam st 1963) and of Va n der Ho uw en1966), the model was implemented by Timmerman on the computer at the

K .N .M .I. Royal Netherlan ds Meteorological Inst i tute) and has been used s ince1971. A detai led description of the model can be found in Tim me rma n 1975 ,1977).The model provides predictions of the meteorological effect along the Dutch coastover a 24 hour period. Unfortunately, i t takes approximately s ix hours to computethese predictions. This delay is mainly due to the time required to collect themeteorological data and to run the atmospheric model for predict ing themeteorological conditions which are the inputs for the North Sea model.

The second approach to s torm surge predict ions has been developed part icularlyfor predicting the m eteoro logical effect at the tim e of the next high wa ter . This is ofgreat importance for taking precaut ionary act ions to protect the dikes. The methodis base d on the early work of Schalkwijk 1947). On the basis of theo retica l con-siderat ions, Schalkwijk assumed a quadrat ic relat ionship between the wind speedand the wind effect during stationary weather conditions. He divided the North Sea

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and the C han nel into thre e area s and used a stationa ry mod el to write the wind effectat Hook of Holland as the sum of the contributions of the districts North, South andChannel, assuming for each of these districts a homogeneous wind field to occur:

s = aN V?, cos( -xN) + as O s ) V* + a c V2Ccos (i|>c-Xc) (1.6)where:

s = wind effect or set upV N , V s , V c = wind speed in the district N ort h, Sou th and Ch ann elXN, Xe = direction of the m axim um w ind effect in the districts No rthand Channel

' ^ s ' ^ c = w md direction in the districts North, South and ChannelaN, a c , as(x|Js) = coefficients.

The coefficients aN, a c , as(i|> s), xNand xc were computed with the aid of 14 stormsurges that occurred in the period 1920-1940.Weenink (1958) improved the method by subdividing the district South into three

areas. Furthermore, to give the method a more physical basis he determined thewind effect of these three areas by solving the shallow water equations (1.3)-(1.5).Since We enin k in tho se days was obliged to use analytical solutions it was necessaryto introduc simplifications and to consider the stationary equations. He compiledthe results into tables of the wind effect for various wind speeds and directions. Forthe districts North and Channel Weenink accepted Schalkwijk's results. In 1971,Timmerman again improved the method by dividing the North Sea and the Channelinto six areas. The stationary wind effect of each area was computed using the justdescribed numerical model of the North Sea and the resul ts were compiled intotables , l ike Weenink did.

The main advantage of using the tables instead of the North Sea model is that i ttakes less than three hours for the predictions to be available. The reason for this isdue to th e fact tha t the wind velocities, used as input for the tables , are predicte d bythe meteorologist and not by running the atmo spheric m odel . I t should be noted thatwhen using the tables, the external surges and the internal pressure effect are nottaken into acco unt . Th e No rth Sea model only neglects surges gene rated ou tside themodel .

1 2 4 Recent results in tidal predictionTidal predict ion techniques can roughly be divided into determinist ic methods

which include the analyt ical or numerical solut ions to the hydrodynamic equat ions18


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and statistical methods which are based on empirical or black-box models. Theparameters of these models are estimated by using long series of observations. Aspointed out in Subsection1.2.3 Schalkwijk used a statistical approach to pred ict themeteorological effect at Hook of Holland.The most importan t characteristic of statistical methods is the possibility to makeuse of the on-line pieasurements of the water-level. Furtherm ore, these methods arevery easy to empl j>y and can be implemented on very small com puters. Simple linearregression models based on wind, pressure and water-level data) have been derivedby Rossiter 1959) among othe rs, to predict the meteorological effect a t variouslocations along the eastern coast of Engeland. Christianssen and Sieferd 1978)presented a similor approach to predict the meteorological effect at the GermanBight. To predict the water height in inland waters such as Lake St. Clair, where thetide can be neglected, Budgell and El-Shaarawi 1978) applied a Box and Jenkins

1970) transfer function model using measurem ents of the water height, atmosphericpressure, wind atjd temperature. Unfortunately, the applications of the statisticalmethods just desqribedwerenot always satisfactory. Thisismainly caused ythe factthat the param eters of the models were estimated by using long series of observations,assuming stationarity of the observed processes. During storm surge periodswhen conditions can change rapidly, this is not a very realistic assumption.Therefore, attenfion has been more and more concentrated on the deterministicmethods which have a more physical basis.The early detbrministic methods were based on analytical solutions of thehydrodynamic equations. In order to obtain these solutions simplifications had to beimposed, e.g. a linear bottom friction term and a constant or linearly varying dep th.As described in Subsection1.2.3 this approach was used by Weenink to predict themeteorological effect along the Dutch coast. Other analytical methods for stormsurge prediction are reviewed by Bretscheider 1966, 1967).Within the last decennia, increasing computer capacity has permitted the com-putation of num^rical solutions to the hydrodynamic equations. In the countriesaround the North Sea two-dimensional numerical models of the North Sea and

adjacent waters have been developed. Linear models, neglecting the tide and onlydescribing the meteorological effect, were implem ented by Heaps 1969) and Timmerm an 1975, 1977) among others . Non-linear tidal models were developed forexample yFlatcher 1976) and recently yVoogt 1985). Unfortuna tely, non-linearmodels tend to be too large to be used on an operational basis. This is caused by thefact tha t at the open boundaries of the m odels the wind effectisnot known and has tobe neglected. T)iis is only a reasonable assumption when these boundaries arelocated on the Atlantic Ocean. In addition, for a sufficint description of the tidalmovement in th North Sea the grid size has to be rather small, yielding very large19


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models. Therefore, as described in Subsection 1.2.3, the operational model used inthe Netherlands does not consider the t idal movement and describes themeteorological effect only. As a consequence, the grid size can be larger yielding amuch smal ler model . However, in this case the interact ion between t ide andmeteorological effect is neglected.

In seeking prediction techniques that are not based on the simple superposit ion ofthe astronomical t ide and meteorological effect , small non-linear models have beendeveloped. To avoid that the model becomes too large, a rather coarse grid size hasto be used. Since the t ide cannot a ccurately be rep rese nted on this grid, the mode l isused to predict the meteorological effect by carrying out two computations, one fortide and surge together, the other with meteorological input removed and for thetide only. The difference of the two computations gives the required meteorologicaleffect including the important non-linear interaction between tide and surge, assum-ing that the errors caused by the crude discretisation can be eliminated to a largeexten t in this way. In Eng land Flatch er and Pro ctor 1983), G erm an y Soetje andBro ckm ann 1983) and Denm ark Duu n-Chris tensen 1983), this approach has beenin rout ine operat ion s ince the late sevent ies . In the Netherlands, plans have beenmade to replace the l inear North Sea model described in Subsection 1.2.3 by a non-linear model within a year or two.

During the last few years, at a number of research insti tutes three-dimensionalnume rical mod els have been develo ped. Rece nt ly , Davies of isnumerou s publ ica-tions we m ention D avies 1980)) has imp lem ente d a three-dim ension al mode l of theNorth Sea. Although the results look very encouraging, especially in the case ofstorm surges, three-dimensional models are not l ikely to be used on an operationalbasis in the near future. T hese m odels are very large and, furthermore , the dev elop-m ent of these mo dels is f ar from com plete .

1 3 Kalm an filter approach to tidal predictionAs described in Subsect ion 1.2.4 t idal predict ion techniques have been ei ther

statist ical or determ inistic in na tur e. E mp loying a statistical me thod it is possible touse on-l ine measurements of the water- level , whi le the determinis t ic approachprov ides a mor e realistic picture of the t idal dynam ics. W hen a num erical t idal mod elis com bined with a Ka lman fil ter on e obtains the best features of both the d eterm inist ic and the statis tical m ethod s. Fur ther m ore , in model ling the water movem ent thevarious par am eters tha t represen t the influence of the physical phen om ena on thesem ov em en ts, such as the bo ttom friction cofficint and wind friction cofficint, areinserted in deterministic models as constants. However, since the knowledge avail-able about the se phen om ena is far from co mplete and they are model led by empiri -2


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cal param etrizatiohs, th e param eters are sensitive to changing conditions. Em ploy-ing a Kalman filter lapproach itispossible to correct the predictions ofthewater-leveland to adapt the model to changing physical circumstances using on-line information. This propertjy of the Kalman filter becomes increasingly important duringstorm surge periois, when conditions usually change rapidly.Considering theifact that the num ber of on-line measuremen t stationsisincreasingrapidly and that in future it will be possible to obtain information of the watermovement in thejNorth Sea by means of satellites, the Kalman filter approachpromises to become very important.Since the original work of Kalman and Bucy 1960, 1961), Kalman filters havebeen successfullyi used in numerous applications. Most of these filters weredeveloped for the determination of satellite orbits and for the navigation of sub

marines, aircraft and spaceships. In the last decennium Kalman filter techniqueshave also gained acceptance in meteorology Ghil et al 1981), oceanography Miller1986) and in several areas of hydraulics and water resources Chao-lin Chiu 1978).Desalu , Gould and Schweppe 1974), Koda and Seinfield 1978) and Fronza, Spiritoand Tonielli 1979) have all developed Kalman filters for the prediction of airpollu tion, while Chao-lin Chiu and Isu 1978) have employed a Kalman filter toestimate the friction cofficint in the shallow water equations. However, thesetechniques have ^eldom been applied to tidal prediction problems. Budgell andUnny 1980, 1981) have developed a Kalman filter to predict tides in branchedestua ries. This filterisbased on the one-dimensional shallow water equations and isused to estimate and predict water-levels at spacially distributed measurementlocations as well as between these locations. Although this application is a valuablecontribution to the use of Kalman filters in tidal prediction p roblem s, the practicalusefulness of the filter is limited because the time interval over which predictions a reproduced,isonly3 minu tes. Fu rtherm ore, the filterhasonly been employed duringa period when the meteorological effect was very small and could be neglected.Therefore , th e Kalman filter approach has yet to be applied to storm surge prediction problems.The combinatiQn of the Kalman filter with a non-linear tidal model of the entireNorth Sea is, from a computational point of view, not yet) feasible. Therefore, inthis investigationjtwodifferent approaches have been developed. The first is basedon the approxim^tion of the tidal movement in the Dutch coastal area by a one-dimensional model. The two-dimensional effects due to the wind and the Coriolisforce a re taken ihto account by introducing some add itional, empirical, equations.Water-levels andj velocities as well as the parameters in the model are estimatedon-line by the K alman filter. Since the model is continuously being adapted to thechanging conditions even this simple conceptual model gives satisfactory predictions

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at a low com puta tiona l cost. Ho we ver the time interval over which accu ratepredict ions can be produced is l imited because the one-dimensional approxim ationis only realistic for a small part of the southern North Sea.

To increase the prediction interval the second Kalman filter approach that isdeveloped in this investigation is based on a two-dimensional model of the entireNo rth Sea and th e Ch ann el . The extension of the one-dim ensional f il ter to two spacedim ension s does no t give rise to conce ptual prob lem s but as noted befo r e imposean unacceptably greater com putat ional burd en. In order to obtain a computat ional lyefficint Ka lm an filter the filter is ap pro xim ated by a time -inv ariant o ne . In this casethe t ime-consum ing fi lter eq uat ions d o not have to be com puted o ver again as newm easure me nts becom e avai lable but need only be solved once. As a conseq uencethese compu tat ions can be carried out off-l ine on a large com puter. Fu rthe rm ore forthe com pu tation of a time-inv ariant filter special algorithms have been de velop ed toredu ce the am ou nt of com puta tions drastically. Unfo rtun ately in the case of a time-invariant fil ter i t is not possible to estimate the parameters in the model on-line.Ho we ver using a large two-dim ensional m odel this feature is less imp ortan t than inthe one-dim ensional appro ach. M oreov er the am ount of data avai lable for theNo rth Sea is too small to pro du ce reliable estimates of both the tidal m ove m ent andthe large number of parameters .

Sum m arizing the first K alma n filter ap pro ach is based on the simplification of themodel while employing the second approach the fil ter equations are simplified toobtain a com putatio nally efficint filter. B y using a small concep tual m od el it isposs ible to exploit all the capab ilities of Ka lm an filtering such as the possibility toest imate uncerta in param eters in the model . How ever by employing the secondapproach it becomes possible to use a detailed two-dimensional model.

Although in this investigation attention is concentrated on the application ofKalm an filters to pred ict storm surge s these filters can also be used to design andoptimize water-level monitoring networks. Since the fil ter provides estimates ofwater-levels and velocities based on the measured water-levels as well as the ac-curacy of these es tim ates it is possible to optimize the location of the m easu rem entstations.

Finally Ka lm an filters can be used to imp rov e empirical para m etrizatio ns in themathematical model . Employing the fi l ter based on the one-dimensional modelest imates of the uncerta in parameters in the model are produced. Studying theseestimates may increase the insight into the performance of the model and suggestimprovements of the underlying determinist ic model .


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1 4 Scope of the investigationAs described in Section 1.3 in this investigation two Kalman filters have beendeveloped to predict the water-level in the mouth of the Eastern Scheldt. First inChapter 2 a short description of the shallow water equations is given. Chapter 3

contains an introduction to the theory of Kalman filters linear and non-linea r. InChapter weapply the theory to the linear one-dimensional shallow w ater equationsto study some theoretical aspects of the filtering problem . F urtherm ore we developanalytical methods to investigate the performance of the filter and to increase theinsight into the complex filtering prob lem. Finally in this chap ter the numericalaspects of some Kalman filter algorithms are discussed. Chap ter5deals with the on e-dimensional approach to predict water-levels in the Eastern Scheldt. Attention isconcentrated on the development of the model equations. To examine filter performance both simulated data and field data have been used. In C hapter6 using th einsight gained from the one-dimensional approach the filter based on a two-dimensional model is developed Again this filter has been tested using simulateddata and field data. The investigation is concluded in Chapter7with a discussion ofthe results and recomm endations for further study.

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2 The shallow water equations

2.1 IntroductionTo make this study more self-contained, in this chapter a short description of the

shallow water equ at ions is given. We do not intend to present a detai led t reatm ent ,but describe some aspects of the shallow water equations that are relevant to thefiltering pro blem s dealt with in this study. Fo r a mo re com plete trea tm en t the r ead eris referre d to A bb ott 1966, 1979) or Ge rritsen 1982) am ong other s.

In Section 2.2 the general non-linear two-dimensional equations are derived torecall the basic assumptions of the theory of long waves. Section 2.3 deals with theone-dimensional equations. For this case, the characterist ic formulation of theequations is given. This formulation increases the insight into the wave motion andcan be used to derive boundary conditions.

2.2 The two dimensional equationsCo nsider an E uleria n system of coo rdinate s x, y, z with the z-axis vertically

upw ard. Neg lecting effects of viscosity the gene ral dynam ic equation s describing themotion of a fluid may be written:

du du du 1 dp + u + v + w + - = Fx 2.1)3t x 3y dz Q dxdv dv dv dv 1 3p + u + v + w + - = Fv 2.2)dt dx dy 3z Q 3y y

dw dw dw dw 1 dn + u + v + w + - = - g + F7 2.3)3t dx dy dz e dz vwhere :

u, v, w = velocities in respe ctively the x, y an d z-directionsQ = den sity of the fluidp = pressu reF x , F , F z = com ponen ts of extraneou s forcesg = acc elera tion of gravity

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The general equation of continuity for an incompressible fluid is:d u d v d w n . .. + + = 0 2.4)dx dy dzThe equa tions (2.1) - (2.3) describe the conservation of momeritum, while equation (2.4) describes the conservation of mass. The derivation of these equa tions canbe found in most textbooks on fluid dynamics (Batchelor 1970).We now derive the shallow water equations describing the propagation of longwater w aves, i.e. w aves with a length that is very great com pared to the depth of thewater. In the classical theory of long waves, the vertical accelerations are neglectedin comparison with the acceleration of gravity. Fu rtherm ore, it is assumed that theexternal forces in the vertical direction can also be neglected with respect to gravity.

Consequently, it follows from equation (2.3) that the pressure is assumed to behydrostatic and is a linear function of the water-level:P(z) = eg(h-z) + pa (2-5)

where:h = water-level with respect to the undisturbed water surfacepa = atmospheric pressureQW = density of water.To obtain the equations for the depth averaged flow the vertically integratedvelocity components are introduced according to:

hD

= rJTirdz < 2 6 >v = _ ^ D T h d z

where D is the distance between the undisturbed water surface and the bottom.Substituting equation (2.5) into the equations (2.1) and (2.2) and integrating theresulting equations over the region z = - D to z=h yields:du _ d _ d dh 1 dp + u + v + g = + Fx (2.8)d t d x d y d x ow d x vdv _ d v _ d v d h 1 dp + u + v + g =d t d x d y d y QW d y+ u ~ + v z: + ~ = - TT + Fy (2-9)

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O ther term s introduced by the integrat ion have been omitted. Ho we ver , in th is caseit is necessary that the velocity distributions over the vertical are fairly constant(Dronkers 1964, 1975).Including the effect of the earth s rotation and introducing em pirical formulasdescribing the bo ttom friction and the wind stress into the equa tions (2.8) and (2.9)results in:d d _ du ahhU hV + g = f V -dt dx dy dx

u V u 2 + v 2 V 2cosil) 1 dpu. + Y -=: ( 2 10)r D + h r D + h o w dxdv dv dv hh U h V h g = - f -at dx dy dyv V u 2 + v 2 V 2sinijJ 1 3p a^ D + h + Y D + h ~ 7 dy~ ( 2 H )

where :f = Coriolis para m eterH = bo tto m friction cofficintY = w ind friction cofficintV = wind ve locityij) = direc tion of the wind w ith respect to th e positive x-axis.In a similar way the equation of continuity (2.4) can be integrated over thevertical. The boundary conditions required can easily be established. If the equationof a bou nda ry surface is F (x ,y, z,t )= 0 th en , since a partiele that is on the surface willremain on it*, we have:dF 3F F F dF n = _ + U + V + W = 0 (2.12)dt at dx dy dx

At the free surface z=h:/ , x h h hw (h) = _ + u + v - (2.13)t x y

while a t the bo t tom z=-D:

* This is a fundamental consequence of the continum hypothesis (Batchelor 1970).26


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w (_ D ) + u : + v = 0 (2.14)dx dyWith these boundaries, integration of equation (2.4) over the region z=-D to

z= h yields: |ah 3((D + h)at x H L + ^ ^ L = (2.15)yThe m omentuirj e quations (2.10) and (2.11) and the continuity equation (2.15) arethe basis for the study of the tides in shallow water such as the North Sea.In practice the hon-linear shallow water equations are of en linearized, yieldingthe eq uation (1.3)-(1.5) that have been shown in Subsection1 2 3 As described in

this subsection, j:he complex astronomical tide can in this case be describedseparately by means of an harm onie analysis (Go din 1972). Consequently, a mo delbased on the linearized equations needs only be used to describe the meteorologicaleffects that are sperimposed on the astronomical tide.The wave motion is not completely described without boundary conditions. Ingeheral two types; of boundary conditions are to be distinguished: closed and openones.Closed bouttdaries are physical land-water bo und aries, whereas open boundaries are artificial pnes tha t have been chosen arbitrarily to restrict the dom ain of the

problem. Using the characteristic formulation of the equations (2.10), (2.11) and(2.15) itispossibl to determine the num ber of boundary conditions thatisnecessaryfor the problem to be well posed, i.e. the solution exists, is unique and dependscontinuously on i^iitial and boundary conditions. At a closed boundary where thenormal velocity is equal to zero, no additional condition is required. At an openboun dary on e coidition is required in case of outflow and two conditions in case ofinflow. Here, it is assumed that the flow is subcritical:2 + v2


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along the flow direction, the velocity com ponent v perpendicular to the flow can beneglected. In this case the equations of motion (2.10) and (2.11) become (leaving thebars off again):du du du uIuI V 2cosob dpn + u + g = -u . - J L + Y J f- (2.16)dt dx dx D + h D + h rjwdx

h V2sinop pqg = _ f u + Y - ^ ^ L _ _ ^ _ (2.i7)y D + h Qwy

and the continuity equation (2.15) becomes:ah a((D + h) u) + J- = 0 (2.18)

t dx y JIf these equa tions are used to describe the flow in a channel with very small wid th,the Coriolis force and the meteorological effect across the width can be neglectedand equ ation (2.17) can be left out of consideration.We now give the characteristic formulation of the equations (2.16) and (2.18).This formulation is important to gain insight into the tidal motion and to determ inewhether the num ber of boundary conditionsiscorrect and the problem iswell posed.Iftheequations (2.16) and (2.18) are generalizedbyintroducing the functions E , andE2:u u h + u - + g - = E, 2.19)at dx dx

ah a((D + h) u) + = E 2 ( 220)at dxthe characteristic form of these equations becomes:

( + (u + V g(D + h) - ) (u 2 V g ( D + h) ) = E , + V p r r T - E 2 + g ^ (2.21)at dx v D + h dxdD( - + ( u - V g ( D + h) - ) ( u - 2 V g ( D + h)) = E , - A / - J L - E , + g (2.22)at dx v D + h dx

These equations can be interpreted as equations that describe the propagation(left hand side terms) and the deformation (right hand side terms) of the (quasi)Riemann invariants

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u 2V g (D + h) (2.23)along the characteristic lines determined by:

dx- = u Vg (D + h) (2.24)Using this characteristic formulation, the num ber of boundary conditions requ ired,i.e. the number of characteristic lines pointing outside the dom ain of the prob lem,can be established easily.

On the basis of the physical interpreta tion of the equa tions (2.21)-(2.24) it is easyto formulate bouhdary conditions that correspond with a physical configuration ofthe boundary. Neglecting frictional forces and meteorological effects, we considerthe following boundary conditions:- total reflection at closed bou ndaries:

u - 2Vg (D + h) = -(u + 2V g(D + h))oru = 0 (2.25)

- f ree outflow:u - 2V g(D + h) 1is constantoru - V g (D j + h) + 2 VgD 1= 0 (2.26)

- partial reflection:u - V g (D + hj + 2 VgT? = - ] (u + 2 V g (D + h) - 2 VgD 1) (2.27)

where n is the reflection cofficint. Note that for TJ = 1 this boundary conditionreduces to relatiQn (2.25) while for y = 0 the condition (2.26) is obtained.Here we assumed that the flow is subcritical (u2< g(D +h )) and consequently the(quasi) Riemann invariants propagate in opposite directions. Introducing frictionand meteorological effects the boundary treatment becomes more complicated. Theinterested reader is referred to Verboom, Stelling and Officier (1982) or TenBrumm elhuis, De J ong and H eemink (1985).

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3 Discrete filtering theory

3.1 IntroductionSince the original pap ers by Kalm an and Bucy (1960,1961) a consid erable am ount

of literatu re has beco me available on the theory and applications of Kalm an filters.In this chap ter we give a short review of the filtering theo ry. We by no mean s atte m ptto give a complete treatment. We have concentrated our attention on the aspects ofdiscrete filtering theory that are of major relevance to the problems dealt with in thisstudy. Rather than strive for the mathematical precision of a theorem-proof struc-t u re , we merely recall the basic assumptions and characteristics of filtering theory.For a thorough treatment that is still accessible to most practical engineers, thereader is referred to the excellent textbooks of Jazwinski (1979), Maybeck (1979,1982) or Anderson and Moore (1979).

In Section 3.2 we introduc the Kalman filter for linear discrete systems. Conside rable a tten tio n is paid to the stability of the filter. S ection 3.3 is dev oted to so measpects of non-line ar filtering th eory . S ome n on-linear filters th at have proved to besuccessful in practical applications are summarized. In this chapter we do not payattention to the important numerical aspects of the filtering problem. These arediscussed in detail in Chapter 4 in connection with the problems dealt with in thisstudy.

3.2 Linear filtering theory3 2 1 The Kalman filter

Assume that modelling techniques have produced an adequate description in theform of a linear stoc hastic system to describ e the propa gat ion in time of state vectorvX t k = * ( tk , tk_,) X t k x + B ( tk ) u t k + G ( tk ) W V k = 1, 2, . . . (3.1)X t 0 = X0

H ere X t is an n-vector state proce ss, (tk , tk_,) is the non-singular n-by-n systemdynamics matrix, B(tk) is an n-by-r input matrix, uj k is an r-vector deterministic3


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input, G tk) is an n-by-p noise input matrix and W t is a p-vector white Gaussiannoise process. The statistics of this noise process are assumed to be:E {W t k } = 0 3.2)E f W ^ W I ^ R C k ) , k = l0 , k4 1

with Q k) being a p-by-p symmetrie positive-semidefinite matrix. The system noiseW t includes the effects of variability in the natural systemaswell as model struc tureerrors.The initial condition X0is also assumed to be Gaussian with statistics:E { X 0 } = X 0 3.3)E { [ X 0 - X 0 ] [ X 0 - X 0 H = P 0

where P0 is an n-by-n symmetrie positive-definite matrix.Measurements are available at discrete time points t,, t2, ... and are modelled bythe relation:Z t k = M tk )X t k + V t k 3.4)

Here Z t is the m-vector measurement process, M tk) is the m-by-n measurementmatrix and V t is an m-vector white Gaussian noise process with statistics:E { V t k V t i } = 0 3.5)E { Y t k V , } = R k ) , k = l0 , k 1

The measurement noise t represe nts the uncertainty associated with the measurement process. Itisfurther assumed tha t R k) is a symmetrie positive-definite matrixand that the initial state X, the system noise W t and the measurement noise V tare mutually independent.Itisdesired to combine the measurements Z t , taken from the actual system, withthe information provided by the system model in order to obtain an estimate of thesystem state Xt . To solve this filtering problem we adopt the Bayesian approachand determine the conditional probability density of the state X t k , conditioned onthe entire history of the measurements taken: Z t , Z t , ..., Z t , l ^ k . Once thisdensity is explicitly described an optimal estimate of the state Xt k can be defined.

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U nd er th e assump tions of the mo del described abov e it can easily be proved th at theconditiona l d ensity is Gaussian . As a result i t is comp letely characterized by its mea nand covariance matrix. Therefore, the mean, mode, median or any other logicalchoice of estim ate of X t based on th e condition al den sity will result in the s am eest imated value X(k11 and the same covariance matrix of the estimation errorP(k 11). Recursive fil ter equations to obtain these quantities can be summarized asfollows. The optimal state estimate is propagated from measurement t ime tk_j tomeasurement t ime tk by the equations:

S ( k |k-1 ) = $ ( tk , V , ) X (k-11 k - l ) + B( t k) utk (3.6)p (k |k - i ) =


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X (k-11 k-1)^

DELAY1

^ X t k t k - ^

X ( k | k )

* ,B (tk )

+ (k k-1) *-

f

M ( t k )

K(k )

2 t k

> +

^

Figure 3 .1 : Block diagram represen tation of the Kalman filter.

The Kalman filter has a predictor-corrector structure. Based on all previousinformation, a prediction of the state vector at time tk is made by means of theequa tions 3.6) and 3. 7). Once this prediction is known it is possible to predict thenext measurement by means of the equation 3.4). When this measurement hasbecome available the difference between this measurem ent and tspredicted value isused to upd ate the; prediction of the state vector by means of the equ ations 3.8)- 3.10). igure3 1 is a block diagram representation of the algorithm. Note that thefilter gain K k) does not depe nd on the measurem ents and therefore may beprecomputed.

The performance of the filter can be judged by monitoring the residuals R t ,defined as the difference between the measurements and the prediction of thesemeasurements bas^d on all previous information:R t k = Z t k - M t k ) ^ k | k - l ) 3.12)

It is easy to verify that:E { R t k } = 0 :E { R t R L}= M tk)P k| k- l )M tk)T + R k)

3.13)3.14)

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Since the theoretical statistics of the residuals R t are known, the actual residuals kcan be monitored and compared with this description. By checking whether theresiduals indeed possess their theoretical statistical properties we are able to judgewhether or not the mathematical model satisfactorily describes the real system

behaviour.The model described in this section is not the most general on e. It is possible toallow both P0 and R(k) to be positive-semidefinite instead of positive-definite.Furthermore, the system and measurement noise may be correlated. However,these generalizations have been omitted here since they are not relevant to theproblem considered in this study. The interested reader is referred to Jazwinski(1970) or Maybeck (1979).Finally, we note that if the initial condition X 0, the system noise W t and the

measurement noise t are not assumed to be Gaussian but only described by theirmean and covariance, the Kalman filter equations (3.6) - (3.10) are still optimal inleast squares sense. Howeverinthis case, unlike the Gaussian one, knowledge of themean X(k|k) and covariance P(k|k) does not provide complete information aboutthe probability density function of X t , conditioned on the measurem ent: Z j , Z t ,

3.2.2 StabilityOptimality of the filter does not imply stability. In order to define the stability ofthe filter it is useful to rewrite the Kalman filter as:X(k|k) = W tk, tk_ t)X(k-l| k-1) + B(tk)u t k + K(k)Zt k (3.15)

where K(k) is determined by the equations (3.7), (3.9) and (3.10) andV k, K-i)=[I - K(k)M(tk)] (tk , tM ) (3.16)

is the state transition matrix ofthefilter. The filter just describedissaid to be stableifthere exists a constant c,>0 so that:l l ^ g l l ^ . f o r a l l t ^ t , , * (3.17)

Here || . || denotes a matrix norm.* This stability definition is not the only possible one. For the definitions of various types of stability thereader is referred to Hahn (1963) among others.4


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The filter is uniformly exponentially stable if there exist constantsc2> andc3>so that:II^(tk.to)||


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As can be seen from equation (3.22) this implies that the system noise sequenceW t M, W t w ,. , . . . , W t, , affects all the com pon ents of the state X t . Fu rth erm or e,K iN 1 K ~ I N l i l K l tthis effect is bounded from below as well as from above. Controllabili ty preventscertain components of the state to be determined almost exactly in which case newobse rvation s would hav e very lit t le effect on the estimates of these com po nen ts andthe estim ates and m easu rem ents could easily diverge. This pro blem is kno wn as filterdivergence. If a filter is not controllable numerical difficulties also are very likely tooccur. Since in case some state components (or l inear combination of state components) can be estimated very accurately, some eigenvalues of the covariance matrixbecome almost zero. Due to the f ini te wordlength on the computer, these e igenvalues can easily become negative, a condition that is theoretically impossible andusually leads to a total failure of the recursion. This important aspect of Kalmanfiltering as well as the problem of filter divergence are discussed in Chapter 4.

A concept dual to that of controllabili ty is observability. Suppose the systemmodel (3.1) is noise-free, i.e. W t = 0 for all k. In tha t case , applying som e cleverma trix algebra (M aybeck 1979), the equa tions to obtain the covarian ce matrix can berewritten as:

P ( k | k ) - ' = * ( tk,t ())-TP 0-1cI(tk,t 0)-i +. ^ J K V ^ M ^ R O ) ->M(t.) >(tk,ti)-> (3.24)

Defining the observability gramian as:0 ( k , k - N 2 ) = . 2 0 ( tk , t i ) - ' r M ( t i ) T R ( i ) - ' M ( t s ) * ( t k , t i ) - ' , O ^ N 2 < k (3.25)i = k - N 2

equation (3.24) can be rewritten as:P (k |k)-> = ( tk , t k _ N r l)~ T P ( k - N 2 -l | k-N 2-l)-> >( tk ,V N r l) - > + 0 ( k , k - N 2 ) (3.26)

Th e system m odel (3.1) and (3.4) is now said to be uniformly co mp letely o bserv ableif there exist an N 2 and positive constants c6 and c7so tha t:c6I N 2 (3.27)

As can easily be deduced from equation (3.26) this implies that incorporating themeasurem ents Z tk N2 , Z tk , . . . , Z tk , improve the estimates of all the componentsof the state X_tk*. Fu rthe rm ore , this impro vem ent is bou nded from below and from* In fact this property is defined as reconstructibility (Kwakernaak and Sivan 1972). Observability isdefined such that incorporating the m easurements Z tk_N , Ztk_N,+j, Z ttimprove the estimates of allthe components of the state Xtk_N. However, for the problem described in Subsection 3.2.1 observability and reconstructibility are not significantly distinct issues.36


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above. Observabili ty guarantees that the entire state can be determined from theda ta and pre ven ts that certain eigenvalues of the covariance matrix can grow w ithoutbo un ds. It also implies that the effects of changes in any com pon ent of the state canbe observed in the measurements .

Having defined the concepts of observabili ty and controllabil i ty, both introducedby Kalman, Lyapunov stabil i ty theory can be applied to establish exponentiallystabil i ty results. Through the explicit generation of an appropriate Lyapunov func-tion, i t can be proy ed th at if a system m odel is both uniformly com pletely observ ableand controlla ble, the Kalman fi l ter is uniformly expone ntially stable Kalm an 1963).

Another approach to the stability of the filter can be obtained from the filterequ ations 3.6)- 3.10 ). It is easy to show that for the model described in Subsection3.2 .1 :

|| I - K k)M t k )] || 1 , for all k 3.28)so that:

| | V t k , t , ) H | |< P t k , t 0 ) | | 3.29)Th is implies that if the original system 3.1) is exp one ntially) stable the Ka lm anfil ter is also expon entially) stable Kw ake rnaa k and Sivan 1972). From equa tion3.2 9) it can be seen th at th e filterisalways mo re stable than the original system .T hisstabil i ty improvement property of the fi l ter is a very favourable property. Note that

if the filter is uniformly completely observable and controllable the stability of theoriginal system is not required for filter stability. A system model can be unstablewhile the Kalman filter is stable.

3.3 Non linear filtering theory3 3 1 Introduction

Using the prob abilist ic approac h to non -linear fil tering pro blem s we seek, as in thelinear case, the conditional probabili ty density of the state conditioned on themeasurements taken. This density is , unlike that of the l inear case, usually non-Gau ssian an d, the refo re, generally cannot be characterized com pletely by a finite setof pa ram ete rs. Since prop agating and upd ating an entire density or an infinitenu m be r of describing par am ete rs for this density) is not imple m enta ble, simplifyingassumptions have to be imposed. In this section attention is concentrated on theparametrization of the conditional density via central moments, since the result ing

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app rox im ate non -linear fi lters - from the com putatio nal point of view - are veryattractive and have proved to be successful in numerous applications.

As in l inear fi l tering problems, the main interest is to compute the conditionalme an a nd covarian ce matrix since the me an is always the minimu m varianc e estima teand the covariance matr ix measures the uncertainty in the es t imate. Note that ,unl ike the l inear case, the mod e, m edian or an other es t imate of the s tate do not , ingeneral , result in the same estimated value of the mean. Propagating and updatingthe conditional mean and the covariance matrix in non-linear problems usuallyrequires all the central moments of the conditional probabili ty density. By makingcertain assumptions about these mom ents approxim ate es t imators can be gene rated.One might assume for instance that the conditional density is nearly symmetrie, sothat thi rd and higher-ord er odd ce ntral mom ents are negl igible, and tha t in add i t ionit is concentrated near i ts mean, so that fourth and higher-order even centralmoments can be neglected as well . The result ing approximate fi l ter is called thetruncated second-order f i l ter . In another approximat ion the fourth-order centralmoment is not neglected but, assuming that the conditional density is nearly Gaus-sian, expressed in terms of the covariance. This additional assumption gives rise tothe so-called Gaussian second-order fi l ter. Computational considerations lead tofirst-order fi l ters, such as the celebra ted ex tend ed K alma n fi lter.

In Subsection 3.3.2 we first derive the extended Kalman fi l ter as a ratherstraightforward extension of the linear filter since the results from the linear theorycan be exploited easily for this non-linea r prob lem . Fu rth er m or e, this subsection canserve as an introduction to the more general non-linear fi l ters. Subsection 3.3.3 isdevo ted to the n on-linear fi lters described in this introdu ction a nd to some fi l ters tha tare not based on the parametrization of the conditional probabili ty density viacentral moments. Finally, in Subsection 3.3.4 the use of the non-linear fi l ters toestimate uncertain parameters in a model is discussed.

3 3 2 Linearized and extended K alman filtersSup pose tha t a l inear mode l does not provide a valid description of the pro ble m .

Assu me th at the system state can be presented by the non -l inear s tochast ic system:X t k = 0 X t k , , V , , tk) -f B t k ) u t k + G tk ) W t k , k = l , 2 , 3 . . . 3.30)X t 0 = x 0

where


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Ztk =m ( X tk , t k ) + Vtk (3.31)wherem( X t , tk) is anm-vector describing the relation between the state andthemeasurements.Theother model assumptionsarecompletely similarto thelinearcase described in Subsection3.2.1.Note that the system as well as the measurem entnoise are still assumedtoenterin anadditive fashion. Therefore the linear theo ryeanbeextendedtp thenon-linear problem just described.

Suppose that it possible to gene rate a discrete reference state traj ectoryxtk. Thestate equation (3.30) mayberewrittenas:Xtk=[ * ( X v1 , t k _ 1 , t k ) - 0 ( x t k _ 1 , t k _ 1 , t k ) ] + 0 ( xt k i , t k _ 1 , t k ) +B (tk )u tk +

G tk)Wt k 3.32)and the observation equation (3.31)as:

Ztk = [ m ( X tk ) : - r n ( x t k ) tk ) ] +m ( xt k , tk) + Vtk (3.33)If the deviationXt-xt from thereference trajectory issmall,aTaylor's seriesexpansion yields:^ ( X t n ^ p g - ^ C x t ^ . t , . , ^ ) ( x t k ^ - I A ) (X t^ -x t , . , ) (3.34)r n ( X v t k ) - m ( x t k , t k ) - M ( x t k , t k ) ( X t k - l t k ) 3-35)where:( ( x t ^ . t M . O X j = ~l. (3-36) ( x t k i ) j/ . , / - ( m ( x t k , t k ) ) i(M(xtkg)ii= r }

are the m atrices of partial derivatives along the reference trajectory. The equations(3.34)and(3.35) c&nbeusedtoobtaintheapproximate linear system:X tk= V A P X ^ - * x t H > t H , t k ) x V i + (3.38)^ d t ^ . t k - a . + B (tk) tk + G(tk) W t k

and the approximate linear observation equation:Ztfc= M ( xt k , t k ) X t k - M ( x t k , t k ) x t k +m (xtk ,tk) + Vtk (3.39)

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Given the linearized model described ythe equations (3.38) and (3.39) the StandardKalman filter can be employed to obtain the estimate of the state Xt a n d i t scovariance matrix.

The remaining problem is the choice of the reference trajectory. An obviouschoice is to take:

I t k = * ( x t M , W t k ) + B( t k )u t k (3.40)

so that the reference trajectoryiscompletely determ ined by the prior estimate of thestate.This estimator is called the linearized Kalman filter.The basic idea of the extended Kalman filter is to relinearize about each estim ate ( k | k ) : t , = * ( I t . i V i tp ) + B ( tp)M t, P = k + 1, k+ 2, . .. (3.41)x t k = ( k | k )

As soon as a new measurement is available and a new state estimate has beenobtained, a new and better reference trajectory is incorporated into the estimationprocess. W ith this choice of reference trajectory large initial estimation errors arenot allowed to propagate through time and therefore , the linearity assumptionislesslikely to be violated. Note that the extended Kalman filtergain, unlike the linearizedKalman filter gain, depends on the measurements and therefore, cannot beprecomputed.

3.3.3 Non linear filtersConsider a model described by the non-linear stochastic system equation:X tk = * ( X t k xA-v\ + B (t k)u tk + G(X t k , t k ) W t k , k = l ,2 ,3 , ... (3.42)X t = X

where G(Xt ,tk) is an n-by-p noise input m atrix. The m easurem ent relation is givenby equation (3.31). The other assumptions are completely similar to the modeldescribed in Subsection 3.3.2. As we have done in the linear case we seek theconditional mean and covariance matrix. Propagating and updating these quan tities4


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generally require th e knowledge oftheentire conditional density. Take for exam plethe propagation of the conditional mean:X ( k | k - 1 ) = E f X t J Z v Z t 2 , . . . , Z t H } ( 3 4 3 )

= _ / > ^ ( x , t k _ 1 , t k ) P x t k _ ] | Z t 1 , Z t 2 , . . . ) Z t k _ 1 ( x | z t l ) 2 t 2 , . . . , z t k i )dxwhere:

P X t k _ 1 |Z t l ,Z t 2 , . . . , Z t k _ ] ( x | ? t 1 > z t 2 , - . ? t k _ 1 )is the conditional probability density of the state X t based on the measu rementsZt ,Zt , . . . , Z t |Inthis subsection atten tionisconcentrated on parametrization ofthis density via central m oments.

We first consider the truncated second-order filter. In deriving this filter the thirdand higher-order central moments are neglected. This is appropriate if the conditional density is almost symmetrie and concentrated near its mean. This filter hasbeen derived by Henriksen (1980), correcting an erro r made in previous derivationsand yielding the following filter equations:X (k| k-1) = 0 (X k-11k-1), tM , tk ) +

v 2p(k-i |k - i ) jx ( k -i | k - i j . t ^ . gwith:

(P(k-11 k-1) 0 x x ( X ( k - l |k - l ) , tw , t k ) ) i =a ^ ^ C k - i l k - i ) , ^ , ^ ) ) ,

3.44)

1 (P(k-l kl))0j i i x v ' jl a ( x ( k - i | k - i ) ) j a ( ^ ( k - i | k - i ,

P(k|k -1 ) = $ (X (k - l |k-l) ,tk ,j ,tk)P(k- l |k -1 ) $ ( ^ ( k - l |k-l),tk_1;tk)T +G ( X ( k - l |k - l ) , tk_ 1)Q (k - l )G (^ (k - l |k-l),tk_x)T +P(k-11 k- l )Q (k- l )G x (X (k -l | k-l) ,tk_,) 2 +V2P(k-l| k - l )Q(k- l )Gx x (X (k - l |k- l ) , tk_ 1)G(^(k- l |k - l ) , tw ) +V2(P(k-l| k - l ) Q ( k - l ) G ( * ( k - l |k-l),tk_,)G(X(k-l |k-l),tk_ ())T (3.45)

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Here (X(k-l kl),tk_t,tk) is defined by equation (3.36) and(P(k-11 k-l)Q(k-l)G x (X(k - l k- l ) , t k J 2 ) i q =

p n , a (G (X (k - l | k - l ) , t k l ) ) i ,1 2 (P(k-1 k-l))M (Q(k -l)) ~ " "j , l = l s , t = l v ' stV v " a ( X ( k - i | k - i ) ) ,a(G(X(k-i |k- i ) , tk_ 1))q l

a ( X ( k - i | k - i ) ) t(P(k-11 k-l )Q (k- l)G x x (X(k - l k - l ) , V , ) G ( X ( k - l k - l ) , ^ , ) ) ^ =

P n , d2(G(X (k-l | k-l), tk 1)) ii1 2 (P(k-1 k-l))s , (Q(k-l) ) i l A L _ ^ i ^j , l = l s,t=l v " d ( X ( k - i | k - l ) ) ,a ( X ( k - l | k - l ) ) t(G(X(k-l |k- l ) , t k_ 1) )q l

X(k k) = X k k- l )+K(k) [Z t k - m ( X ( k k-l) ,tk) -V 2 P ( k | k - l ) m ( * ( k | k - l ) , t k )] (3.46)

with:m

(P(k |k - l )m x x (X(k |k - l ) , t k ) ) i = 2 (P(k |k - l ) )j lJ>' ^^(m(^(k[ k- l ) , , ) ) ,

CXCklk-l)^* X k|k-1)),K(k) = P (k | k - l )M (X (k |k - l ) , t k ) T [ M ( X ( k | k - l ) , t k )P (k | k - l )

M(X(k |k - l ) , t k ) T + R(k ) ] " 1 (3.47)Here M(X(k kl),tk) is defined by equation (3.37).

P(k k) = [I-K(k)M(X(k k- l ) , tk )] P(k k-1) (3.48)The vectors

b 0 (k-1) = V2P(k-l k-1) ^ xx(X(k-11 k-l) ,tk .1 ;tk) (3.49)bm(k) = V2P(k k)m xx(X(k k-l) , tk) (3.50)

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respectively appearing in equation 3.44) and 3.46), are called the bias correctionterms. Note that when using the second-order truncated filter just described themodel srelinearizjed about each new estimate X k|k) when it becomes available, asis the case with the extended Kalman filter.Now consider the Gaussian second-order filter. Whereas the truncated second-order filter ignores all the central moments above second-order, the Gaussiansecond-order filter accounts for the fourth moment as well by assuming that theconditional density is nearly G aussian and by expressing the fourth central m omentin terms of the covariance. In addition, the third and fourth-order non-linearitiesthat in this case appear in the equations are also neglected in the Gaussian second-order filter approximation. The resulting equations of this filter may be found inJazwinski 1970). Com pared to the truncated second-order filter add itional termsappear in the equations for the conditional covariance, while the equations for the

conditional mean are exactly the same.In cases where the dimension of the state is large, the calculation of the second-order filter equatjons to obtain the covariance matrix are very time-consuming.Howev er, the primary benefit is usually due to the bias correction terms 3.49) and3.50). Therefore, in deriving a first-order filter with bias correction terms, thesecond-order nonUinearities appearing in the equations for the covariance matrixare neglected. The equations for the conditional mean are identical to the second-order filter equations 3.44) and 3.46) including the bias correction term s. Whenthese terms can be neglected too, the approximate filter is called a truncated first-order filter. Finally, if G is only a function of tkand not ofXt the truncated first-order filter reduces to the extended Kalman filter which has been derived inSubsection 3.3.2.The non-linear filters just described are all based on the parametrization of theconditional probability density via central moments. Representing the non-linearfunctions,


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describing the propagat ion of the condi t ional mean. AssumingPX t \Zt.Zt.... Zu i x | z t l , z t 2 , . . . , z t k _ 1 ) - k - l l r 1 2 L k - 1

to be Gaussian with mean X k-11 k-l) and covariance matrix P k-11 k-l) , equat ion3.43) can be integrated numerically. Note that this assumption implies that thesefilters, unlike the other fi l ters described in this section, do not neglect higher-order

cent ra l moments .For the special case that G is not a function of the state, a statistically linearized

filter may be em ploye d M aybec k 1982). Structurally the fil ter equatio ns are thesame as those for the extended Kalman fil ter, but instead of neglecting second andhigher-o rder non-l ineari t ies , the no n-l inear functions P and m are approximated asfollows:

^ X t ^ V p t J = 0o tk . i) + fc^JXt,.. , + e , 3.51)m X t k , t k ) = m 0 tk ) + M t k ) X t k + e2 3.52)

where the vectors 0o tk_,) and m 0 tk ) and the matrices * tk_,) and M tk) are deter-m ined by the m inimization of respectively e j and e 2 in gene ralized) least squa ressense. Similarly to the assumed density fil ters, the expectations involved are calcu-lated by numerical integration, assuming the conditional densities to be Gaussian.

In som e application s, assumed density filters M aybec k 1982) and statisticallylinearized filters Kikk aw a and Iwase 1980) have shown to be more accu rate than theextended Kalman fi l ter . However, this advantage was gained at the expense ofseverely t ime-consuming computat ions. Therefore, when the dimension of the s tateis very large the use of these filters from a computational point of view is notat t ract ive.

3 3 4 Param eter estimation

A particular application of non-linear fi l ters is the estimation of an uncertainpa ram eter P in the model Eykhoff 1974). This par am eter can be t reate d as addi-tional state variable with system equation:

P t = P t + WP 3.53)l k l k ~ l p t k v

By adding the system noise WP to this equat ion the rando m character of thepa ram eter can be taken into accoun t . No te that in general even a linear model in this


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4. Kalman filters for the linear one-dimensional shallow waterequations

4 1 IntroductionIn this chapter some aspects of Kalman fil ters for the shallow water equations are

discussed. T o gain insight into the filtering p rob lem , filters are derived on th e basis ofthe l inear one-dimensional equat ions. Moreover, using these l inear equat ions, i tbec om es possible to em ploy analytical metho ds to investigate the perf orm anc e of thefilters.

In Section 4.2 we recall some basic aspects of the numerical approximation ofdifferential equ ation s. Section 4.3 and 4.4 are respectively devote d to the discretisa-tion of the shallow water equations and to the choice of the noise statistics. To gaininsight into the c om plex filtering pro blem , a special prob lem is solved analytically inSection 4.5. For this case, the sensitivity of the fil ter performance is studied withrespe ct to the choice of the finite difference schem e and m odelling erro rs. In Se ction4.6 the discrete system represe ntat ion of the general m odel is derived. Observabi l i tyof the filter is discussed in Section 4.7, while Section 4.8 deals with the numericalpro per ties of various Kalm an fil ter algo rithms. Finally, Section 4.9 briefly discussesthe dis t r ibuted parameter f i l tering problem in a more general context .

4 2 Numerical approximation of differential equationsThis section recalls in a tutorial way the basic aspects of the num erical app roxim a

tion of differential equations, such as consistency, convergence and stabili ty. Thetreatment is based on the l inear homogeneous part ial -different ial equat ions. How-ever, to i l lustrate the relevant concepts this is not a real l imitation. For a thoroughtrea tme nt the reade r is referred to Van der Ho uw en 1968), amo ng others .

Th e partial-differential equ ation is written as:Lf x , t) = O , xeQ , t e [ t0 ,T] 4 .1)

with initial condition:f x,t0) = f0x) , xeQ 4.2)

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and boundary ccjndition:Lbf x,t) = 0 b x,t), xer, te[t0T] 4.3)

where L and Lb are linear differential operators andTis the boundary of Q.Th e problem jlescribed above is assumed to be well posed so that it has a uniquesolution f x,t). In order to approximate the differential equation a grid is defined: aset of points with coordinates iAx,kA t). It is assumed th at:At = At Ax),

lim At Ax ):=0Ax-^0 jlDe note byQ thfc set of grid points tha t result whenQ iscovered with the grid and letTA denote the boundary of Q A. Using a finite difference scheme equation s 4.1) - 4.3) can be converted into a set of difference equations:

LAf =0 4.4)fO = f0iAx) ; 4.5)LbA ff = 0b iAJc, kA t) 4 .6)

where ff is the approximation of f iAx,kAt) and LAand LbAare the finite differenceoperators approkimating L and Lbrespectively.The finite difference scheme 4.4) is a consistent approxim ation of orde rnof 4.1)if:| |L A f iAx,kAt)Nc 8 A x)n 4.7)

where c8is a constant. Consistency of the app roximation of the boundary conditioncan be defined jn a similar way. The quantity under the norm here is called thetruncation erro r and m easureshow accurately the solution of the original differentialequation satisfies the finite difference equations.To show the analogy with the discrete system theory described in Section 3.2, wedefine:x t = [ . . 4 ft+1...]T 4.8)

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and write the difference equations (4.4) - (4.6) as* :A ^ t k + 1 = B xtk + ut k + i (4.9)

where A andBare cofficint matrices andut depends on the boundary cond ition.Symbolically we write:

xt = x t + A _ 1ut (4.10)- l k + l - Lk - l k + l y^ ^Jwhere = A_1 B. Defining:

x(k) = [.. .f(iAx ,kAt) f((i+l) A t, kAt) . . . ]T (4.11)the finite difference scheme (4.10) is said to be convergent of order n if:

| |x t k - x ( k ) | | s = c9 ( A x ) n (4.12)where c9 is a constan t. If a scheme is convergent th e solution of the differenceequations converges for Ax>0 and thereby At0 to the solution of the originaldifferential equation.

The finite difference scheme (4.10) is said to be stab le if there exists ac1 > 0 suchthat:|| k|| ss c1 0 , k = 1,2, ..., T/A t, for all At (4.13)

where c10is a constant. Stability implies that numerical erro rs in f; are bounded ifAx and therebyAt>0,with T constant. No te tha t stability is a property of thefinite difference scheme and is not related to the original differential equations.

Direct proofs of convergence are of en hard to give. However, for a consistentfinite difference approximation to a linear problem, stability is a necessary andsufficint condition for convergence (Lax's equivalence theorem).Another important aspect of the behaviour of finite difference schemes isO'Brien-Hyman-K aplan stability, as treated by Van der H ouwen (1968). The finitedifference scheme (4.10) is said to be O'B-H-K stable if:| | * k | | ^ c l p fo ra l lk (4 . 14)

* Not all approximation problem s can be writteninthis form. How ever,aspointed outinSection4.6,thisis not a real limitation for the prob lems dealt with in this study.8


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where c u is a constant. O B-H-K stability impliesthat thenum erical errors inff arebounded ifT with xand At constant. Note that for the same problem O B-H-Kstability is equivalent w ith th e stability of a filter as defined in Subsection 3.2.2.From a practical point of view it is convenien t tha t a finite differential scheme is

stable in the sense of both stability definitions described in this section (Stelling1983).

4.3 Discretisation of the one dimensional shallow water equationsThe linear one-dimensional equations used throughout this chapter are (seeChapter 2):di di + C, + C 2 f = 0 (4.15)dt dx

where:f = [u h]T

Here we recall that:h = water-levlu = water velocityD = depth of the waterg = acceleration of gravityX = linear bottom friction cofficint (X = k/D)

The water movement is completely described by these equations, provided thatinitial and boundary conditions are given.To develop a discrete system representation for the description of the watermovement, the equation (4.15) is discretised. Defining a grid seefigure 4.1),weconsider two different finite difference schemes to accomplish this task. Firstly, weemploy the implicit four-point Preissmann scheme (Liggett and Cunge 1975):

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k+1 4

AtV

- - < Ax > -

k+1f .i < k * , 1- i + 1' r i i

kf.i O o

i - 1 i 1

Figure 4.1: The grid.

1 k + 1A t 1 + / 2 r i+ ' / , .

i r /f k + ec l t i + lAx f lk + e + c 2 f S v + ^ = o (4 .16)

where ff is the a ppr oxim ation of f(i A x, k A t) an d-f+y2= yaC ff+i + h

and 9 is a weighting fac tor. Fo r the special case X= 0 it is easy to show th at t he finitedifference scheme (4.16) is a consistent approximation of second-order of thepartial-differential equation:

3f 3f 2ft

+ C, - (9-0 .5) At C] = 0dx dx (4 .17)

Comparing equation (4.17) with (4.15) we see that only for 9 = 0.5 the originalequat ion (4 .15) i s solved wi th second-order accuracy . By choosing 0 . 5< 9s l .0 oneintroduces numerical diffusion.

Th e second finite difference sche m e that is con sidere d is the well-know n explicitLax-Wendroff scheme (Richtmyer and Morton 1967) :

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- f f a - f i c - i r ^ i + i ' - w ) U ( ^ ) 2 ( - i + i _ 2 i + ^ )-V2AtC2_ff_1 + fK +1 ) (4.18)

The scheme is of second-order accuracy.To analyse the stability and to gain insight into the dissipative properties of theschemes for initial value problems, a Von Neumann stability analysis can beemployed. Consider the behaviour of the Fourier integral:

l/2Axfk = ; k ( i ) e / 2 j t l A x dl (4.19)-l /2Axunder the operation given by equation (4.16). Here; is the imaginary unit, 1 is thewave number and 1/2Ax is the highest wave number that can be resolved. Substitut-ing the Fourier integral (4.19) into the Preissmann scheme (4.16) yields:

A ( l ) k + l ( 1 ) = B ( l ) | k ( l ) 4.20)where:

AtA(l) = I + l/2At C2 + 2/ 9 tan (JTIAX) C,xAtB(l) = I -V2At C2 - 2 ; (1-6 ) tan (jtIAx) C,Ax

Equation (4.20) can be rewritten as:|k + l ( l ) = G( l )Sk( i ) (4.21)

where G(l) = A(l) B(l) is the amplification matrix of the finite difference schem e.This matrix amplifies the value of 1)over a time step At. The eigenvalues of G(l)can be shown to be:

1-v2 8(1-0) V v 2 ->/4At2X2+ V2AtXv2((1-9) 2-0 2)7g ., ( l ) = - - (4.22) u U l + v 2 e 2 + AAfX2Lwhere:

, , Atv = VgT tan (JTIAX) = Cr tan (JTIAX) (4.23)Ax 51


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X = OX = 1.0 x l 0 -

0. - 1 1 1 1 1 1 12

P O I N T S P E R W V E L E N G T H L0610Figure 4.2: The modulus of the eigenvalues of the amplification matrix using the Preissmann scheme

X = 0X = i.oxio 610

I " 1 T i 1 1 1 r-1P O I N T S P E R W V E L E N G T H

Figure 4 . 3 : The celerity of the numerically computed wave using the Preissmann scheme.LOG10

with Cr a s the Courant number. nfigur s 4 2 and 4 3 respectively the modulus of theeigenvalues of the amplification matrix and the celerity of the numerically computedwave Ccgiven by* :cc 0 ) = -T T T a r c t a n f I m s u 0 ) /R e g u O ) ] (4.24)

* The derivation of the expression for the celerity can be found in Abbott (1979).52


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are shown for som values ofI incase Cr = 0.75. Her e, following Leend ertse (1967),we introduc the number of grid points per wave length N = 1/lAx as the independent variable. Furthermore we note that if:v 2 - V4At2X2 + V2A tX v 2((l-6)2-8 2)>0 (4.25)

or approximately ff the wave length 1/1 satisfies:l /K 2 j tV g /X ( 4 - 2 6 )

the eigenvaluesgl 1)have imaginary parts and| g , ( l ) = |g 2(l):| (4-27)

However, for verV long waves the eigenvalues of G(l) are real and positive withdifferent moduli and therefore these w aves do not advance. This is a physical effectand is no t caused cjy the finite difference approx imation . F or the very short waves asimilar behaviourl can be noticed. However, this effect is introduced by thediscretisation.The Von Neumann necessary condition for stability results from the fact that thematrix:G ( l ) k , k =1 2J... T/At (4.28)

has to be uniformly bound ed. He nce, we have the necessary condition:ga0 ) ^ 1 (4 .29)

where:g a( l ) = max{|g m (l)|} (4.30)mis the am plificatioh factor of the finite difference scheme. If the eigenvalues of G 1)

satisfy:I & CO I1for some1and consequently the schemeisunstable. In case* This condition has to be generalized in case k 1)is amplified by physical influences.

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0.5 < 9 =S 1 the amplification factor ga 1)


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V v A t < C r < V l + V2XAt' (4.33)Infigure 4 4the rhodulus of the eigenvalues of the amplification matrix for the Lax-Wendroff scheme are shown for some values of X in case Cr = 0.75. Th e celerity ofthe computed wave can be found infigure 4 5

4 4 On the choice of the system noiseThe tidal movm ent is not perfectly described by the finite difference equations.Therefore, we embed these equations into a stochastic environment by addingsystem noise. Fo)- the Preissmann scheme we obtain:

- (F i+v -F i + v ) + r -c ' (Fi+f - F i + e ) + c ^ + = ( 4 J 4 )A t 1 + / 2 - i + / 2 A x 1'+1 ' 1 + ' 2where:w H = [ W m f W c k ] T

is the system noise. Wmj' and Wcj' are the noise processes associated with theuncertainty of r4spectively the momentum equation and the continuity equation.The covariance q>f W_. is chosen to be:E j w f ; w f 2 J = Q i i ; i 2 k i , k l = k2

0 , k, 4 k2F ]1:is the stochastic generalization of the deterministic processf j By introducingthe system noisp as just described, we associate this random process with thediscretised momentum and continuity equations. However, when employing animplicit scheme jit is also possible to introduc the system noise process after thefinite difference;equations have been solved and F-' * 1is known explicitly. In thiscase the system (noise process is associated w ith the uncertainty of the com putedwater-levels and velocities. From the physical point of view we prefer the firstpossibility since the errors concerned with the continuity equation are usually verysmall and assunied to be zero*. This assumption also guarantees that if the finitedifference scheme conserves mass, the filter conserves mass too. A perfect con-

* Unfortunately, inisome cases this approach introduces numerical difficulties (see Section 4.5).55


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tinuity equation can only be modelled by associating the system noise with theuncertainty of the equations and not with the uncertainty of the components of thestate vector. Note that for most explicit schemes this discussion is not relevant.

For the Lax-Wendroff scheme we introduc the system noise as follows:

-VjAtQCFJLj + F k + 1 ) At W f (4 .35)The system noise represents the errors of the corresponding determinist ic model.

It includes the variability in the natural system, e.g. due to turbulent effects andmodel structure errors such as neglected non-lineari t ies or wrong parameter values,as well as err ors cause d by the discre tisation of the partial-differential eq ua tion s.Un for tun ate ly, in practic e very little is know n ab out the statistics of the system n oise.In some cases (Jazwinski 1970) it is possible to use a systematic approach todetermine the covariance Qj ,{ (k). However, usually i t has to be established bym ean s of trial and er ro r , i .e. the filter is em ploye d for various values of Qj ,j (k),until one gets satisfactory filter performance. In fact, the determination of thecovariance matrix of the system noise is the calibration of the filter.

In choo sing a suitable valu e for Qj ,j (k) , it has to be taken in to accou nt tha t finitedifference schem es are not able to accurately represe nt short wav es, i.e. waves with awave length of the orde r of 2Ax. The refo re, in ord er to obtain meaningful solutionswhen solving the t ime propagation of the covariance of the estimation error, theenergy of these short noise waves should be limited.

Assu ming the system noise is loc

68686540 storm surge prediction using kalman filtering

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