River, Coastal and Estuarine Morphodynamics: RCEM 2005 – Parker & García (eds)© 2006 Taylor & Francis Group, London, ISBN 0 415 39270 5

Using a sand wave model for optimal monitoring of a navigation depth

Michiel A.F. Knaapen & Suzanne J.M.H. HulscherWater Engineering & Management, University of Twente, Enschede, The Netherlands.

Meinard C.H. TiessenSchool of Civil Engineering, University of Nottingham, United Kingdom

Joris van den BergNumerical Analysis & Computational Mechanics, University of Twente, Enschede, The Netherlands.

ABSTRACT: In the Euro Channel to Rotterdam Harbor, sand waves reduce the navigable depth to an unac-ceptable level. To avoid the risk of grounding, the navigation depth is monitored and sand waves that reducethe navigation depth unacceptably are dredged. After the dredging, the sand waves slowly regain their originalheight. To reduce the high costs of surveying and dredging, the North Sea Service of the Department of Trans-port, Public Works and Water Management, is implementing a Decision Support System to reduce the requiredamount of surveys and provide optimal information on the necessity to dredge. Currently, the system predictsthe growth of sand waves using a linear trend. The trend is determined from observations using a Kalman-filterincluding geo-statistical components to incorporate spatial dependencies. This works well for sand waves thatare close to their maximum height. After dredging however, the sand wave height is far from its equilibrium andthe growth rate is much higher, making the linear prediction worthless. Here we show that replacing the lineartrend with a landau equation improves the predictions of the regeneration. Comparison shows that the landauequation predicts the crest evolution better than the linear equation for both undisturbed sand waves and dredgedsand waves, with an root mean square error that is 25% less.

1 INTRODUCTION

With the increased navigation depth, required for mod-ern ships, more and more harbor authorities are forcedto dredge the entrance channels to their harbors. In theNorth Sea sand waves limit the depth of the navigationchannel to Rotterdam Harbor, which makes dredgingnecessary. The guaranteed depth in this Eurochannelis 24 meter to allow for ships with a draught of 22.5 mor 75 feet. However, using the high tide, ships witha draught up to 26 meter can enter the harbor (VanDe Gouwe 2003). Therefore the guaranteed naviga-tion depth in the approach to the Eurochannel is 28.5meter. The costs of repeated dredging are high and theauthorities want to minimize these costs. To minimizethe dredging costs, it is crucial to know the rate atwhich the sand waves regain their original shape.

For this purpose, the North Sea Service of theDepartment of Transport, Public Works and WaterManagement, is implementing a Decision SupportSystem (DSS) to reduce the required amount of sur-veys and provide optimal information on the necessityto dredge. Currently, the system predicts the growth ofsand waves using a linear trend (Wüst 2004).The trend

is determined from observations using a Kalman-filterincluding geo-statistical components to incorporatespatial dependencies. This works well for sand wavesthat are close to their maximum height. Given the natu-ral variations, the linearization is sufficiently accurate.After dredging however, the sand wave height is farfrom its equilibrium and the growth rate is muchhigher, making the linear prediction worthless.

Here, we test if the DSS to manage the navi-gation channel sea bed can be improved followingthe approach proposed by Knaapen (2005). In theBisanseto Sea a field of sand waves has been topped off(Katoh et al. 1998) and there, the sand waves regen-erated in several years time. Knaapen and Hulscher(2002) proved that this regeneration can be modelledusing a Landau equation. In the North Sea navigationchannel to Rotterdam Harbor, individual sand waveshave been lowered. Here, we will show that the Landaumodel is valid to estimate the regeneration of the sandwaves for such dredging strategy as well.

Offshore sand waves are elongated rhythmic bedfeatures spaced about 200 to 500 meters apart, andseveral meters high. Sand waves are found in shallowtidal seas all over the world (Ludwick 1972; Boggs

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Figure 1. The approach to Rotterdam Harbor. The studyarea is found in the western part of the Euromaas channel.The length indicator in the picture covers 60 km.

1974; Field et al. 1981; Aliotta and Perillo 1987;Harris 1988; Huntley et al. 1982; Ikehara andKinoshita 1994; Katoh et al. 1998). Sand waves onlyoccur in seas, where the tidal motion has to bestrong, the maximum tidal velocity should be between0.4–1 m/s (Stride 1982; Amos and King 1984). Fur-thermore, the bed should consist of non-cohesive sand(percentage silt less than 15%, gravel less than 5%(Terwindt 1971; Langhorne 1973; Bokuniewicz et al.1977; Hulscher and Van Den Brink 2001).

It is assumed, that the sand waves are free insta-bilities of the system of an erodible bed and the tidalflow over it (Hulscher 1996; Komarova and Hulscher2000; Gerkema 2000). It is shown that a stability anal-ysis helps to explain the generation of sand waves.Furthermore, this approach can be used to estimatein which regions sand waves can be found (Hulscherand Van Den Brink 2001). The consequence of thesand waves being free instabilities is that they tendto recover after dredging. Németh (2003) shows thatthe stability models can simulate the growth to finiteamplitude.

This paper has the following outline. First, the studyarea is described in section 2. In section 3 the approachto predict the regeneration of sand waves is explained.The testing data are described in section 4. In section 5the result of this approach are shown. Finally theseresults are discussed in section 6.

2 STUDY AREA

The study area is located in the North Sea at latitude51 N and longitude 3 E (5750 km northing and 480km easting in UTM ED50) at the entrance to the EuroChannel. The Euro Channel is the sea channel con-necting the harbor of Rotterdam with the deeper parts

Figure 2. The results of the data reduction.The smooth grid-ded data are shown as a map, while the resulting wave crestsand troughs are plotted as respectively black and white dots.The trench from North (top of figure) to South is the resultof the recent dredging of a pipeline.

of the North Sea. The channel is about 60 km long andis orientated from east to west (see figure 2). In the off-shore part crest of the sand waves have to be dredged tomaintain the guaranteed navigation depth. In this area,two bed patterns are dominant: sand waves and mega-ripples. The sand waves are approximately stationary.The mean depth relative to mean sea level is 32 meter,but sand waves reduce the navigation depth to about 28meter, which is the local guaranteed depth. The tidalrange is 0.8 meter and the average tide has a maxi-mum depth-averaged tidal velocity of about 0.8 m/s,but current velocities of over one meter can occur. Theprincipal tidal axis has an approximately ca. 15 degreesclockwise angle with the north. The waves are mod-erate with a mean wave height of 1.3 meter, and only10% of the waves exceed 2.5 meter (De Ronde et al.1995).

3 APPROACH

To predict the growth of individual sand waves anumber of techniques are combined. The basis of theapproach is the work of Wüst (2004), who developeda Bayesian modelling approach for the sand waves.In the Bayesian approach, the height of individualsand waves is assumed to grow with constant speed.The growth speed is determined from the availablemeasurements using a Kalman Filter. A geo-statisticalrelationship helps to reduce the impact of noise. Thisapproach works fine for undisturbed sand waves. Inthe case of dredging, however, the sand wave dynam-ics change significantly. Therefore, a new model isproposed based on the Landau model of Knaapenand Hulscher (2002). As this model only describesthe amplitude of the sand waves, the bathymetric data

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needs to be reduced to the crest positions. This has anadditional advantage that it reduces the computationaltime considerably. For this purpose, the approach ofKnaapen (2005) is followed.

3.1 Data reduction

The approach of Knaapen (2005) is based on the classi-cal way to analyze rhythmic bed patterns.The locationsof the crest and trough are compared to determinethe length, height and other characteristics (O’Conner1992). This approach has been proven to be robust andreliable. An automated procedure determines the crestand trough positions in both one-dimensional and two-dimensional bathymetric data. For this purpose thedata is filtered using a low-pass Bartlett window. Thisremoves all noise related to mega-ripples and ripples.The crests can than be identified as the local maximain the direction of the principle tidal axis, respectively.Due to the filtering procedure, this procedure resultsin a slight underestimation of the sand wave height(figure 3). This error is marginal however, comparedto the noise of mega-ripples and can be corrected usinga correction factor.

The shape characteristics of the sand waves can bededuced from the crest and trough positions of a sin-gle survey. The length of a sand wave is defined asthe distance between the trough positions on oppositesites of the crest. The height is defined as the differ-ence between the crest level and the mean of the twotrough levels. Finally, the lee-stoss asymmetry is char-acterized as the difference of the distance between thecrest and the lee trough and the distance between thecrest and the stoss trough divided by the sand wavelength.

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Figure 4. The sand wave growth according to the Landaumodel using several parameter combinations that are withinthe range of realistic values for North Sea conditions.

3.2 Landau equation

The Landau equation has been developed to describethe development of unstable waves in a variety of phys-ical systems in which turbulence occurs. This equationdescribes the development of patterns from their initialstages until their equilibrium state (Doelman 1990).Knaapen and Hulscher (2002) shows that the growthof sand waves can also be described by a normalizedform of the Landau equation:

This equation (1) describes the development of theamplitude of the sand wave (A) in time (t). Theterm (αA) describes the initial exponential growth forpositive α. The second term on the right hand sidedescribes the non-linear growth. For small values ofthe amplitude, the non-linear term is small comparedto the linear growth. With increasing amplitude, thenon-linear term becomes more dominant, resultingin a dampening of the exponential growth towards

an equilibrium value A∞ =√

− αβ

, provided that β is

negative.To translate the amplitude to a water depth zb

the sand wave height is added to an undisturbeddepth D:

Figure 4 shows the Landau curve for some values ofthe linear growth rate α and the damping parameter β.The initial state vector x0 were chosen to be α = 0.125and β = −0.0083, whereas D0 an A0 are determinedfrom the first measurement survey.

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3.3 Extended Kalman filter

A Kalman filter is a mathematical procedure toimprove predicting qualities of a model. Based on thedifference between the model prediction of a systemand some measurements, it determines the most proba-ble state of that system (Jazwinski 1970). The Kalmanfilter is based on a predictor-corrector algorithm. Tra-ditionally, Kalman filters are linear (Kalman 1960). Inorder to be able to use the non-linear Landau equation,an extended Kalman filter is used (Jazwinski 1966).

Given the state estimate xi, which in our case isdefined as:

the first step is a model prediction following the non-linear function f (xi):

which assumes that only the sand wave amplitudechanges, following the Landau equation, with an addednoise wi, which has a normal distribution with zeromean and covariance Wi. It is then assumed that themodel error changes linearly due to this prediction:

in which Pi is the estimated covariance of the state esti-mate xi. Fi is the update of that covariance, which isequal to f (xi) if the latter is a linear function. In our casepart of the prediction is non-linear: the change in theamplitude. In this case the changes in the error statis-tics can only be estimated. Therefore, in the extendedKalman filter, the real changes in the covariance areapproximated using the first order linearization off (xi) to the elements of the state estimate xi:

At some point in time tn a measurement is available:

in which M is the function that describes the measuringsystem leading to the measurement dn and rn is themeasurement noise with zero mean and covariance Rn.At that moment the model prediction is compared tothe measurements. In general, there will be a difference

between the two. This difference is used to correct thestate estimate based on the information on the modelerror and the measurement error:

The Kalman gain Ki compares the accuracy of themodel prediction with the accuracy of the measure-ments and is determined based on the covariances ofthe measurement and the predicted value:

The final calculation in the corrector phase is theupdate of the covariance of the state estimate, whichwill be reduced due to the additional information:

4 TUNING AND TESTING OF THE KALMANFILTER

Before, the Kalman filter can be used to predict the bedchanges in the Euro channel, it’s covariance matriceshave to be determined. Furthermore, we have to verifythat the Kalman filter can be used to tune a Landauequation.

4.1 Covariance parameters in the Kalman filter

The Kalman filter uses covariance matrices to deter-mine the accuracy of the model predictions and themeasurements. The covariances of the measurementsR and of the predictions Q need to be prescribed, aswell as the covariance P0 describing the accuracy ofthe initial estimate of the state vector. Here we describethe settings that are used for these covariances.

Within the prediction system, all errors sourcesare assumed to be independent. In that situation theprescribed covariance matrices reduce to diagonalmatrices with the variances on the diagonal. This doesnot imply that the prediction errors are independent.Due to the covariance prediction (equation 5) andthe covariance measurement update (equation 10) theerrors interact, which is represented by the elementsoff the diagonal in the prediction covariance matrix.

The measurement noise R is determined from thebathymetric surveys. The difference between the fil-tered data and the raw data shows that the noise due toripples and mega-ripples has a standard deviation ofis 0.55 meter. Furthermore, the measurement system(echosounder and GPS) has an error with a 0.25 meterstandard deviation (Knaapen et al. 2005). This valueis confirmed by the variation of the mean water depth.The total variance is set at the sum of the variances ofthese errors: 0.37 m2.

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The exact values of the variance of the initial stateestimate are relatively unimportant. As long as onechooses them large enough, the Kalman filter willadjust the state covariances to acceptable levels. Ifthe initial variances are chosen too small e.g. muchsmaller than the variances of the measurement, theKalman filter will ignore the measurements and relyon the model predictions. The variances of the initialparameters estimates were set at:

in which the variances of the amplitude and the undis-turbed water depth are determined from the errors inthe measurement system. The variances of the modelparameters agree with a standard deviation that islarger than the initial parameter value (α) and about25% of the initial parameter value (β).

Finally, the model error is defined as being a frac-tion of the covariance matrix of the state estimatesfollowing (West):

in which Wdf is a diagonal matrix with on the axis 0.98,except for the forth diagonal element, which is set at0.99.This implies that the prediction of the mean depthis more accurate than the predictions of the amplitudeand the parameters.

4.2 Testing the Kalman filter

The Kalman filter is tested using an artificial data set.With a given parameter set, the Landau model is ranand its output stored. Next, this output is given to theKalman filter as ‘measurements’. If the Kalman filteris working properly, it should be able to recover theparameter settings from these ‘measurements’. Fig-ure 5 shows that the Kalman filter works perfectly. Theparameter estimates smoothly converge to the param-eter values used in the original Landau model thatgenerated the ‘measurements’. Figure 6 shows that,although the Kalman filter is sensitive to the noise,but still manages to retrieve the original parametersettings. For this test, a random noise has been addedto the measurements produced by the landau model.This noise has zero mean and standard deviation of0.5 meter. This is comparable with the noise result-ing from mega-ripples. Initially, the Kalman filter hasdifficulty with the noise. The confidence in the predic-tions is less than the reliability of the measurements.However, as time progresses, the estimates becomemore accurate, giving more weight to the model runs,

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Figure 5. Testing the Kalman filter. top: the predicted waterdepth compared with the original model results. Bottom: theparameter estimates (αkf and βkf ) converge to their originalvalue (α and β).

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Figure 6. Testing the Kalman filter with noise (standarddeviation 0.5 m). Top: the predicted water depth comparedwith the original model results. Bottom: after some initialproblems, the parameter estimates (αkf and βkf ) converge totheir original setting (α and β).

and the Kalman filter does find the correct parametervalues. These results show, that it is possible to tune aLandau model using a Kalman filter algorithm.

4.3 Test data

The proposed approach is tested against several sandwaves, both undisturbed and dredged. From these sandwaves, two are selected to visualize the quality of thecombination of the Kalman filter and the Landau equa-tion. The first one is a large sand wave that is justbelow the critical depth, but has not been dredged.This will test the predictions on the natural dynamics

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Figure 7. profile of 7(a) an undisturbed and 7(b) a dredgedsand wave used for testing.

of the sand waves. Figure 7(a) shows the cross-profilein two years. The second data set crosses the criticaldepth and has been dredged twice in 1992 and 1995.Figure 7(b) shows the cross-section of this sand wave.

5 RESULTS

5.1 Undisturbed sand wave

As a first test against field data, both the Landau modeland the linear model are applied to undisturbed sandwaves. Both models are applied to the data sets of sandwaves that have not been dredged yet. For compari-son, the error statistics of both models are calculated.The results in the prediction of the development of anundisturbed sand wave development is shown in Fig-ure 8. The accuracy of the earlier data is still limited.But even without this error the initial settings are not inagreement with the data.The model predicts a growing

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Figure 8. The development of the undisturbed sand wave intime and the estimates with both the Landau model and thelinear model.

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Figure 9. The development of the dredged sand wave in timeand the estimates with both the Landau model and the linearmodel.

sand wave in the first year, whereas the sand wave isfairly stable.

As the changes in the sand wave are small andmostly within the range of the measurement uncer-tainty, both models have difficulty adjusting to thesituation. The Kalman filter has plenty of freedomto adjust the sand wave height and the undisturbedwater depth D with little parameter change. The lan-dau model, however, does estimate the bed levels betterthan the linear model.

5.2 Dredged sand wave

The main goal of this project is to improve the DecisionSupport System of the navigation channel sea bed inthe predicting of sand wave development after dredg-ing. Figure 9 shows the model results of a sand wave,

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that has been dredged twice since 1990. In 1992 and in1995 the crest of the sand wave is lowered. This humaninterference cannot be described by both models, dueto a lack of information about the dredging operations.Therefore, the models restart their predictions from thefirst observed water level after the dredging operation.

After the dredging operation of 1992, the differ-ence between the functioning of the Landau equationand the linear equation is striking. The Landau modelcorrectly predicts a high increase in sand wave ampli-tude, directly after dredging, while the linear modelpredicts a steady development. This difference showsthat the Landau model does describe the recovery afterdredging operations better than the linear model.

One year after the dredging operation the sand waveis totally recovered and at the same height as beforethe dredging operation. Now the linear model predicta strong further growth, whereas the Landau modeldoes show a relaxation to a stable bed level. This levelis a bit conservative, but well within the range of themeasurement uncertainty related to the mega-ripples.

The linear model is completely messed up by thedredging operations.The linear trend is being tuned forthe situation after dredging operations. After the firstdredging, it found an acceptable growth rate withintwo years. The first test of this rates takes place imme-diately after the second dredging operation. As thisprediction is correct, the trust in the trend is very high.Consequently, the Kalman filter is unable to adapt tothe stable sand wave height in the remaining years.

Again, the initial parameter settings of Landaumodel are incorrect. This results in an incorrect pre-dicted initial decrease in sand wave height predicted.In this situation, the initial sand wave height is big-ger than the maximum height assumed by the Landaumodel. After adjustment to this error, the Landaumodel reproduces the bed level dynamics well.

5.3 Overview

To compare and quantify the Landau model capabili-ties with the linear model capabilities, the models havebeen applied to more locations. In total 30 locations ofundisturbed sand waves, and 24 locations of dredgedsand waves are tested. However, due to limited infor-mation on dredging operations in the past, these 24locations are located at only eight different sand waves.This does result in some correlation between these 24locations with dredged sand waves.

Table 1 shows the root mean square differencesbetween the predicted model bed levels, without themeasurement update and the measured values. Toremove the tuning problems from the error estimates,the first 6 measurements are excluded from the errorestimates.

For the situation of undisturbed sand waves,the Landau equation can predict the sand wave

Table 1. Root mean square prediction error for the Landauequation and the linear equation for both undisturbed anddredged sand waves, after 2000.

Model Undisturbed Dredged

Landau 0.26 m 0.37 mLinear 0.32 m 0.45 m

Table 2. The initial and final parameter setting as well asthe root mean square error (RMSE) of the initial guess.

α β

Initial 0.125 −0.0083

UndisturbedFinal 0.054 −0.009RMSE 0.38 0.0128DredgedFinal 0.139 −0.009RMSE 0.64 0.0286

development about 23% better than the linear equa-tion. For the situation of dredged sand waves, theimprovement is about 22%.

The Kalman filter has adapted the initial parame-ter values to improve predicting capabilities for bothequations. The initial estimation of the parameter val-ues was expected to be close to the occurring values.In both examples displayed above, the figures sug-gested that the estimated parameter values were ratherinaccurate.

To analyze the quality of the initial parameter esti-mations, the initial and final parameter values arecompared. Table 2 shows the initial values and thefinal values as well as the root mean square differencebetween them. From these values it is clear that thedifferences in the initial value of the linear growth rate(α are large, whereas the value of the non-linear damp-ening term is small. The initial settings overestimatethe asymptotic sand wave height.

6 CONCLUSIONS AND DISCUSSION

The Landau equation is a clear improvement to thedecision support system for dredging and monitoringmanagement of the Approach to Rotterdam Harbor.Both the natural behavior and the regeneration of sandwaves after dredging are predicted more accurately.Moreover, the dynamics of the Landau equation are inline with the theoretical dynamics.

The non-linear model requires the use of theextended Kalman filter. The linear approximation of

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the error statistics in the extended Kalman filter resultsin a near optimal state, as long as the time intervalsare small enough for this linear approximation. Inour case, the time step between measurement surveysnever more than a couple of years. This period is smallenough for the extended Kalman filter to give reliableresults.

The data reduction procedure of Knaapen (2005)does reduce to amount of information enormously,without losing essential information on the navigationdepth. A negative side effect of the data filtering pro-cedure used in this project is that the average sandwave height at the crest is lowered with 0.61 meter.This could be repaired either, by adding a correctionterm on top of the crests, or by replacing the filteredcrest level by the absolute maximum bed level aroundthis crest position. This would result in a conservativeestimate of the navigation depth.

Without the geo-statistical information analysisused by (Wüst 2004), the noise in the results is unac-ceptably high. Therefore, this analysis should be com-bined with the Landau model. At first sight, the datareduction procedure compromises the geo-statisticsprocedure, as it removes a lot of spatial informa-tion. However, the most important information is stillpresent, being the level of the neighboring crest points.However, information on the mega-ripples might bereduced, which could be essential information toreduce the noise.

The sand wave regenerate much faster thanexpected. It never to more than a few years for thesand waves to recover. Occasionally, the sand wavesreturned to their natural state within a year. This ismuch faster than the theoretical growth rates (Hulscher1996; Németh 2003). It is also considerably faster thanthe regeneration observed by Katoh et al. (1998) inthe Bisanseto sea. One explanation may be that theremoval of a single crest does not effect the flow patternsignificantly.The bathymetry is out of balance with theflow field. As a result, increased sediment depositionin the dredging area is expected, from theory based ona growth of a group of sand waves.

The parameters that are based on local conditionsand existing theory are inaccurate. The Kalman filterhas to correct the initial parameters significantly.

The administration of dredging operations has neverintended to facilitate research. As a result it is diffi-cult to find information on the performed dredging. Itis especially difficult to determine exact location, thedredging method that is used and the dredging amountsfrom the archives. To get better insight in the regener-ation of the dredged sand waves, it would be favorableto archive future dredging operations on exact loca-tion, dredging quantities, and the location on whichthe material is dumped. A large step forward is thepost-dredging survey that is now the practice in theapproach channel to Rotterdam Harbor.

ACKNOWLEDGEMENTS

The research described in this paper, is funded by theTechnology Foundation STW, applied science divisionof NWO and the technology program of the Min-istry of EconomicAffairs (project number TWO.5805:Modelling of Spatial and Temporal Variations in Off-shore Sandwaves: Process-Oriented vs. StochasticApproach). We acknowledge the feedback of Dr. A.A.Németh and Ir. H. Wüst. Furthermore, we thank the theNorth Sea Directorate of the Netherlands Departmentof Waterways and Public Works for their cooperationand for making the measurements available.

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