International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

A Bayesian network approach to coastal storm impact modeling Jäger, Wiebke S.; den Heijer, C. (Kees); Bolle, Annelies; Hanea, Anca M. 2015-07

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata


53032-Paper_444_Jager_rev_2015_06_12.pdf [ 4.86MB ]
JSON: 53032-1.0076218.json
JSON-LD: 53032-1.0076218-ld.json
RDF/XML (Pretty): 53032-1.0076218-rdf.xml
RDF/JSON: 53032-1.0076218-rdf.json
Turtle: 53032-1.0076218-turtle.txt
N-Triples: 53032-1.0076218-rdf-ntriples.txt
Original Record: 53032-1.0076218-source.json
Full Text

Full Text

12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015A Bayesian Network Approach to Coastal Storm Impact ModelingWiebke S. JägerPhD Student, Department of Hydraulic Engineering, Delft University of Technology, TheNetherlandsC. (Kees) den HeijerResearcher, Department of Hydraulic Engineering, Delft University of Technology, TheNetherlandsAnnelies BolleEngineer Advisor, International Marine & Dredging Consultants, Antwerp, BelgiumAnca M. HaneaResearch Fellow, Centre of Excellence for Biosecurity Risk Analysis, The University ofMelbourne, AustraliaABSTRACT: In this paper we develop a Bayesian network (BN) that relates offshore storm conditions totheir accompagnying flood characteristics and damages to residential buildings, following on the trend ofintegrated flood impact modeling. It is based on data from hydrodynamic storm simulations, informationon land use and a depth-damage curve. The approach can easily be applied to any site. We have chosenthe Belgian village Zeebrugge as a case study, although we use a simplified storm climate. The BNcan predict spatially varying inundation depths and building damages for specific storm scenarios anddiagnose under which storm conditions and where on the site the highest impacts occur.Coastal zones are very attractive to develop so-cial, industrial and recreational infrastructure. Theyhave rich natural resources, impressive landscapesand excellent navigation possibilities. In 2003 anestimated 23% of the world population lived in low-lying1 coastal areas (Small and Nicholls, 2003).The ongoing trend is a disproportionately rapidexpansion of economic activity, urban areas andtourist resorts. At the same time coasts are affectedby various hydro-meteorological phenomena, suchas wind, waves, tides and precipitation whichcan reach extraordinary magnitudes during stormsurges, hurricanes, typhoons or tsunamis. Result-ing floods threaten people, cause land loss, damageproperty, infrastructure and ecological habitats, anddestabilize economic activities.While coastal zone managers cannot influence1By low-lying coastal areas we mean areas both within100km of the shoreline and less than 100m above sea level.the occurrence of extreme events, they can applymeasures to reduce the accompanying risks in theshort, middle and long term.Researchers across many disciplines are dedi-cated to developing methodologies that identifyrisks and to helping decision makers design effec-tive risk reduction plans. They apply numericalhydrodynamic process models to assess the natu-ral coastal response and the extent of flooding dueto storms, e.g. XBeach (Roelvink et al., 2009),TELEMAC (Hervouet, 2000) or MIKE21 (Warrenand Bach, 1992), and use separate models to esti-mate economic, political, social, cultural, environ-mental and health-related impacts. Comprehensivereviews have been written on assessment methodsfor economic damage (Merz et al., 2010), on flood-related health impacts (Ahern et al., 2005; Hajatet al., 2005), and on estimation methods for lossof life (Jonkman et al., 2008b).112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Ongoing research is on the one hand directed to-wards improving the various consequence modelsand comparing them with each other (e.g. Schröteret al., 2014). On the other hand there is a trend to-wards integrating the separate modeling approachesinto a homogeneous framework. A GIS-based ap-proach to describe a spatially varying flood hazardand associated estimates of direct physical dam-ages to various objects, indirect economic damageand the loss of life has been proposed by (Jonkmanet al., 2008a).We continue on the trend of model integration.While Jonkman’s model presents the results of onetypical low probability-high impact flood scenariowith the help of maps, we attempt to compile im-pact estimates of many different storm scenarios ina discrete Bayesian network (BN). BNs are graphi-cal models that describe system relations in prob-abilistic terms. They can handle various sourcesand types of data enabling us to combine informa-tion on the topography and assets of the potentiallyaffected area with simulation data of flood scenar-ios and damage estimations from single disciplinemodels.More precisely, we relate flood impacts not onlyto flood characteristics, but also to offshore stormconditions, such as peak water level and maximumsignificant wave height. This has two advantages.First, the BN can make spatially varying conse-quence predictions for an impending storm in real-time and it can thus support emergency managersin urgent decision making. In contrast a new sim-ulation with a hydrodynamic process model wouldbe computationally expensive and time consuming.Second, the BN can facilitate round table discus-sions of e.g. planners. It enables them to instantlycompare the effect of risk reduction measures for avariety of storm scenarios, as long as these measurehave been included in the model set up.In this article we develop and describe a proto-type of this BN and apply it to a case study site.We use the implementation of the software Netica(Norsys, 2014).Our study site is the old town of Zeebrugge, lo-cated on the North Sea coast of Belgium, which ismainly residential. The storm scenarios, however,are synthetic due to data limitations. While the net-work structure can be applied to any site, the quan-titative component is site specific. It implicitly con-tains site topology or other unique features, such asflood defenses, which determine if flooding occursand, if so, the spatial extent of the flooding.As a first step, we focus on the prediction ofphysical damage to residential buildings that havebeen in direct contact with floodwater. We plan toadd other damages to the network later on in thesame manner.1. BASIC CONCEPTS OF DISCRETE BAYESIANNETWORKSBNs have been applied numerous times as toolsfor decision-making under uncertainty. Henrik-sen et al. (2007) conclude that they are very valu-able for negotiations and discussions between man-agers, experts, stakeholders and representatives ofthe general public, among others, because they aretransparent and flexible models. In the context offloods, Garrote et al. (2007) combine BNs and de-terministic rain run-off models to forecast flooding,and Vogel et al. (2012, 2013) use BNs to estimatedamages resulting from river floods. In coastalenvironments they have been applied to predicterosion and shoreline retreat (Den Heijer, 2013;Gutierrez et al., 2011; Hapke and Plant, 2010). Atthe moment of writing we are not aware of applica-tions to coastal flooding.Discrete BNs are probabilistic graphical mod-els that represent a high-dimensional probabilitydistribution over a finite set of discrete variablesX1,X2, ...,Xn (Pearl, 1988; Jensen, 1996). The coreof the representation is a directed acyclic graph(DAG) whose nodes represent random variablesand whose arcs indicate a direct influence from"parent node" to "child node". Because the graphstructure stipulates that each variable is condition-ally independent of all predecessors given its par-ents, the joint distribution can be economically fac-torized using the chain rule:P(X1,X2, ...,Xn) =n∏i=1P(Xi | Pa(Xi)), (1)where Pa(Xi) denotes the set of parent nodes of Xiin the graph. Together, the DAG and a specification212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 1: BN structureof P(Xi | Pa(Xi)), for i = 1, ..,n, or P(Xi) in caseof no parents, uniquely specify a joint distributionover X1,X2, ...,Xn.A main use of BNs is updating: Once new evi-dence on one or more variables is obtained, the ef-fect can be propagated through the network usingBayes’ theorem. Evidence can be propagated bothforward and backward, which enables predictive aswell as diagnostic reasoning.2. DESIGN OF THE COASTAL STORM IMPACTMODELThis section motivates and describes the design ofthe BN, i.e. the definition of random variablesand the structure as shown in Figure 1. The par-ent nodes of the network characterize the hydro-dynamic forcing, i.e. peak water level and maxi-mum significant wave height, and the location ofbuildings in terms of areas. They influence spa-tially varying inundation depths, which in turn aretranslated into relative and absolute building dam-age with a simple depth-damage-curve and by as-suming an average building value.2.1. Storm ScenariosExtreme hydraulic conditions are commonly char-acterized in terms of peak water level, maximumsignificant wave height and period, predominantwave angle, and storm duration. Naturally, dataon these hydraulic variables is rare. Since recently,copulas are being used to represent their multivari-ate distributions at offshore locations (e.g. De Waaland van Gelder, 2005; Corbella and Stretch, 2013;Li et al., 2014). However, the hydrodynamic pro-cess model requires near-shore conditions as in-put. The transformation of the joint distributionof hydraulic variables from offshore to near-shoreis complex and has, to our knowledge, been rarelydescribed in the literature up to now (Bolle et al.,2014; Leyssen et al., 2013). Also for our case studysite this information is not yet available. Therefore,we assume a simplistic synthetic storm climate withthe intention to extend the model in the future.This storm climate consists of 25 realistic stormscenarios. They are combinations of five waterlevel time series with different peak water lev-els, z, varying between 6.35m and 7.9m and fivewave time series with different maximum signif-icant wave heights, h, varying between 5.2m and6.2m. This choice covers a range of storms withreturn periods from about 100 years to more than10.000 years. For each combination a 46 hoursstorm is simulated, which corresponds to three hightides.For simplicity we assume Z and H to be inde-pendent random variables (see the two left nodesin Figure 5) with discrete uniform probabilities ofoccurrence in 100 years, i.e. 20%, where the timeframe is chosen arbitrarily. Hence, each storm cli-mate scenario occurs within the next 100 years witha probability of 4%. This is a strong assumption anddoes not reflect the storm climate at Zeebrugge re-alistically. However, this assumption is unproblem-atic for applications in real-time decision making,because Z and H will be fixed to the (forecasted)values of the impending or occurring storm.2.2. Residential Buildings on the SiteThe case study site is divided into four areas, as il-lustrated by Figure 2. The parcels correspond to ad-ministrative districts, but other division criteria arepossible as well, e.g. based on topography. Howmany residential houses lie within each area canbe extracted from a cadastral map and is listed inTable 1. We introduce a node A to the networkto represent the location of an arbitrary residentialbuilding. If we randomly select a house, just likedrawing a ball from an urn, the probability that itis within area a is proportional to the number ofhouses in a. This defines the probability distribu-tion A. Note that it is independent of the storm sce-nario.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 2: Residential buildings and areas at case studysiteTable 1: Number of residential buildings per areaArea 1 2 3 4Number of buildings 283 759 383 2732.3. Maximum Inundation Depth and DamageWe obtain maximum inundation patterns throughnumeric simulation of storm scenarios. The simu-lations focus on overtopping North of the old townand do not take into account flooding from the basinin the West. Because NNW is the most critical wavedirection for this effect, it is used in all scenarios.The overtopping discharge time series is input fora TELEMAC 2D model, which calculates the dy-namic behavior of the flooding on land and fromwhich the maximum inundation depth can be in-ferred for each grip point, and by interpolation foreach house. An example is given in Figure 3.We introduce a node maximum inundation depth(of an arbitrary house under an arbitrary stormscenario, more details in section 3), I, to theBN which is Z, H and A’s child, and discretizeits distribution into four intervals {i1, i2, i3, i4} ={[0m] ,(0m,0.5m] ,(0.5m,1m] ,(1m,2m]}. Then theconditional probabilities can be specified asP(I = i j | A = a,Z = z,H = h) (2)=ni j,a,z,hnafor j = 1...4, where na is the number of houses inarea a and na,i j,z,h is the number of houses in areaa with maximum inundation depth i j under stormscenario {z,h}.Figure 3: Example of an Inundation map for Zeebruggeand surroundings. The North Sea is to the North.Figure 4: Depth-damage curve for Flanders, BelgiumThe relative damage per house (in terms of max-imum possible damage), d, is calculated with thedepth-damage curve for residential houses in Flan-ders, Belgium, by Vanneuville et al. (2006). Thiscurve is depicted in Figure 4 and provides a func-tional relationship between I and D. Assuming anaverage value per house, v, an indication can begiven for the absolute damage per house in AC, dAC.As an example, the BN here has v = 100000 AC andis represented as a constant node in the figures.23. INTERPRETATION OF THE COASTAL STORMIMPACT MODELThe resulting BN is shown in Figure 5. At its heartis node I. This node can be interpreted in two ways,which are described in separate sections below. Thesame applies to the two damage nodes, which aremerely translations from the maximum inundation2Note that constant nodes do not have arcs in Netica412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 5: BN with prior distributionsFigure 6: Updated BN for Z = 7.1m and H = 5.7mdepth to units of impact and will therefore not bediscussed individually.3.1. In General: (Conditional) ProbabilitiesOne possibility, the conventional one, is to inter-pret node I as the maximum inundation depth ofa single house. The prior probability distributionof this node (Figure 5) represents the uncertaintyabout the true maximum inundation value for an ar-bitrary house whose location at the site is unknownas well as the storm scenario by which they are af-fected. We can reduce this uncertainty by condi-tioning, for example, on Z = 7.1m and H = 5.7m.Now the distribution represents the uncertainty inthe inundation for a house at an unknown locationdue to the storm with peak water level 7.1m andmaximum significant wave height 5.7m. This isshown in Figure 6. By conditioning on A = 2 (Fig-ure 7) we obtain the distribution for a house underthis storm that is located in area 2. It is importantto realize that the uncertainty does not stem fromthe physical modeling. It arises, because the vari-ous houses in area 2 experience different inundationdepths. In that sense it reflects the unknown exactlocation.Alternatively, we can reason backward, e.g. byconditioning on DAC = [47000AC,50000AC) (Figure8) to understand the conditions due to which top-most damage occurs. A house is most likely to suf-fer maximum damage if it is located in area 1 andthe more severe the storm climate is, foremost thepeak water level. Moreover, no house in area 4 willincur maximum damage and no house at all will in-cur maximum damage, if peak water level is 6.35m.3.2. In the Special Case of Forward Reasoning:(Conditional) ExpectationsUnless we reason backward, we can interpret nodeI in an alternative manner. Besides representing onerandom variable with four possible states, it repre-sents four random variables associated with (condi-tional) expectations.Looking back at equation (2), we notice that theright hand side is not just a conditional probabil-ity. It is also simply the fractions of houses in areaa with maximum inundation depth i j under storm512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 7: Updated BN for B= 2, Z = 7.1m and H = 5.7mFigure 8: Updated BN for DAC = [47000AC,50000AC)scenario {z,h}. If we define four new random vari-ables, the fractions of houses that are inundated byi j, Fj, j = 1...4, then{Fj | A = a,Z = z,H = h}=ni j,a,z,hna. (3)Figure 7 indicates that area 2 has 759 houses, ofwhich 18.7% are not flooded, 78.3% are inundatedup to 0.5m, and 3.03% are inundated between 0.5mand 1m.If we remove evidence for node A, as in Figure6, Netica uses the law of total probability and com-putes the distribution of I with its conditional prob-ability table and the marginal distribution of A:P(I = i j | Z = z,H = h) (4)=4∑a=1P(I = i j | A = a,Z = z,H = h) ·P(A = a) .This equals4∑a=1na,i j,z,hna·P(A = a) (5)and, using that (3) is a constant,4∑a=1E[Fj | A = a,Z = z,H = h]·P(A = a) . (6)This can be rewritten, using the law of total expec-tation, asE[Fj | Z = z,H = h]. (7)Hence, each bin j in Node I also represents the con-ditional expectation of the fraction of houses withmaximum inundation depth i j over all areas givenstorm scenario {z,h}. Note that because P(A = a)is proportional to the number of houses in area a,this coincides with{Fj | Z = z,H = h}= ni j,z,h, (8)where ni j,z,h is the total number of houses withmaximum inundation depth i j under storm scenario{z,h}. This reasoning with conditional expecta-tions can easily be extended to different condition-ing sets. For example, for the network in Figure 5we haveE[Fj]. (9)612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Thus, the bins j = 1...4 of node I in the BN withprior probabilities provide a summary of the distri-bution of each Fj in terms of the expected value.We can explore Fj’s distribution by conditioningon storm scenarios {z,h}: we find the values f j thatcorrespond to the probability P(Z = z,H = h) =P(Z = z) · P(H = h). Admittedly, the usefulnessof this information depends on how realistically thestorm climate is quantified. In our case it is com-pletely synthetic. Moreover, we can zoom in andout in space: we can obtain information per area orfor the entire case study site by conditioning nodeA, or not.4. CONCLUSIONIn this article we proposed a BN approach to coastalflood impact modeling. The BN links various off-shore storm conditions to flood depths and buildingdamages.To understand the implications of a specificstorm scenario it seems very useful to interpret thebins of nodes I, D and DAC as the (conditional)expectation of the fraction of houses that are in-undated by i j, have relative damage d j or abso-lute damage dAC j , respectively. Conditioned on ascenario the BN indicates corresponding spatiallyvarying inundation depths and building damages.Because we distinguish just four areas, the spa-tial detail is significantly less than the one of an in-undation or damage map: We can predict how manyhouses within an area have a specific flood depth,but we do not know which ones. If desired, the res-olution can be increased by adding bins to node A,the area in which a house is located.Nevertheless this BN approach has a couple ofadvantages over map-based approaches. The vari-ables of interest can be seen simultaneously, whileone map per variable is needed. They can eas-ily be compared across storms, by conditioning ondifferent water and wave heights, or across areas,by conditioning on areas. Admittedly, as yet, wehave treated only maximum inundation depth, rel-ative damage and absolute damage, but this qual-ity grows when more flood consequences are inte-grated. Additionally, the BN presents the exact per-centage of inundated and damaged houses, an infor-mation which is not apparent after a quick glance ona map.We can also interpret the BN results in the con-ventional way: I, D and DAC are the maximum inun-dation depth, relative damage and absolute damagefor a single house. Then we can diagnose underwhich conditions the highest flood depth and dam-age occur, which may help decision makers to de-sign risk reduction measures.Finally, we would like to point out that the BNcan be built gradually and improved continuously,according to data availabilities and simulation ca-pacities. Naturally its prediction and diagnosisvalue depends on the quality of underlying mod-els. Here it has to be noted that many consequencemodels, including damage curves, are "simple ap-proaches [...] to complex processes [...]" (Merzet al., 2010) and are associated with large, and oftenunkown, model uncertainties.In the future, we aim to use a realistic joint proba-bility distribution of hydraulic storm conditions, be-ing represented by continuous nodes3 and possiblyincluding storm duration and wave period or angleas additional variables. Then, the BN could givean indication of the flood risk to residential build-ings, as it links the damage extend to its probabilityof occurrence. And again, we have the ambitionto extend the approach to a wider range of damagecategories. Another step could be to take model un-certainties into account, for both the damage modelas well as the hydraulic model.ACKNOWLEDGMENTSThis work is funded by the RISC-KIT project (EUcontract 603458, Special thanksgo to Ap van Dongeren, Robert McCall and Os-waldo Morales Nápoles for the fruitful discussionsin preparation of this article.5. REFERENCESAhern, M., Kovats, R. S., Wilkinson, P., Few, R.,and Matthies, F. (2005). “Global health impacts offloods: epidemiologic evidence.” Epidemiologic re-views, 27(1), 36–46.Bolle, A., Blanckaert, J., and Leyssen, G. (2014).“Une méthode probabiliste pour la determination de3We mean continuous according to the definition of Net-ica.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015l’événement de réference et ses consequences pourles ppl et ppri.” Congrès SHF: Modélisation opéra-tionnelle de grands transitoires hydrauliques.Corbella, S. and Stretch, D. D. (2013). “Simulating amultivariate sea storm using Archimedean copulas.”Coastal Engineering, 76, 68–78.De Waal, D. and van Gelder, P. (2005). “Modelling ofextreme wave heights and periods through copulas.”Extremes, 8(4), 345–356.Den Heijer, C. (2013). “The role of bathymetry, waveobliquity and coastal curvature in dune erosion pre-diction.” Doctoral dissertation, Delft University ofTechnology, Delft, The Netherlands.Garrote, L., Molina, M., and Mediero, L. (2007). “Prob-abilistic forecasts using bayesian networks calibratedwith deterministic rainfall-runoff models.” ExtremeHydrological Events: New Concepts for Security,Springer, 173–183.Gutierrez, B. T., Plant, N. G., and Thieler, E. R. (2011).“A Bayesian network to predict coastal vulnerabilityto sea level rise.” Journal of Geophysical Research,116(F2), F02009.Hajat, S., Ebi, K., Kovats, R., Menne, B., Edwards, S.,and Haines, A. (2005). “The human health conse-quences of flooding in europe: a review.” Extremeweather events and public health responses, Springer,185–196.Hapke, C. and Plant, N. (2010). “Predicting coastal clifferosion using a Bayesian probabilistic model.” Ma-rine Geology, 278(1-4), 140–149.Henriksen, H. J., Rasmussen, P., Brandt, G., von Bülow,D., and Jensen, F. V. (2007). “Public participationmodelling using Bayesian networks in managementof groundwater contamination.” Environmental Mod-elling & Software, 22(8), 1101–1113.Hervouet, J.-M. (2000). “TELEMAC modelling system:an overview.” Hydrological Processes, 14(13), 2209–2210.Jensen, F. (1996). An introduction to Bayesian networks.University College London Press, London.Jonkman, S. N., Bocˇkarjova, M., Kok, M., and Bernar-dini, P. (2008a). “Integrated hydrodynamic and eco-nomic modelling of flood damage in the Nether-lands.” Ecological Economics, 66(1), 77–90.Jonkman, S. N., Vrijling, J. K., and Vrouwenvelder, A.C. W. M. (2008b). “Methods for the estimation of lossof life due to floods: a literature review and a proposalfor a new method.” Natural Hazards, 46(3), 353–389.Leyssen, G., Mercelis, P., De Schoesitter, P., and Blanck-aert, J. (2013). “Generation of multivariate near shoreextreme wave conditions based on an extreme valuecopula for offshore boundary conditions..” EGU Gen-eral Assembly Conference Abstracts, Vol. 15, 10094.Li, F., van Gelder, P., Ranasinghe, R., Callaghan, D.,and Jongejan, R. (2014). “Probabilistic modelling ofextreme storms along the Dutch coast.” Coastal Engi-neering, 86, 1–13.Merz, B., Kreibich, H., Schwarze, R., and Thieken, A.(2010). “Review article "Assessment of economicflood damage".” Natural Hazards and Earth SystemScience, 10(8), 1697–1724.Norsys (1995-2014). Netica. <>.Pearl, J. (1988). Probabilistic reasoning in intelligentsystems: networks of plausible inference. MorganKaufmann.Roelvink, D., Reniers, A., van Dongeren, A., van Thielde Vries, J., McCall, R., and Lescinski, J. (2009).“Modelling storm impacts on beaches, dunes and bar-rier islands.” Coastal Engineering, 56(11-12), 1133–1152.Schröter, K., Kreibich, H., Vogel, K., Riggelsen, C.,Scherbaum, F., and Merz, B. (2014). “How use-ful are complex flood damage models?.” Water Re-sources Research, 50(4), 3378–3395.Small, C. and Nicholls, R. J. (2003). “A global analy-sis of human settlement in coastal zones.” Journal ofCoastal Research, 19(3), 584–599.Vanneuville, W., Maddens, R., Collard, C., Bogaert,P., de Maeyer, P., and Antrop, M. (2006). Impactop mens en economie t.g.v. overstromingen bekekenin het licht van wijzigende hydraulische condities,omgevingsfactoren en klimatologische omstandighe-den. Ghent University.Vogel, K., Riggelsen, C., Merz, B., Kreibich, H., andScherbaum, F. (2012). “Flood damage and influenc-ing factors: A Bayesian network perspective.” Pro-ceedings of the 6th European Workshop on Proba-bilistic Graphical Models, 347–354.Vogel, K., Riggelsen, C., Scherbaum, F., öter, K.,Kreibich, H., and Merz, B. (2013). “Challenges forbayesian network learning in a flood damage assess-ment application.” Proceedings of the 11th Interna-tional Conference on Structural Safety and Reliabil-ity, 3123–3130.Warren, I. and Bach, H. (1992). “MIKE 21: a modellingsystem for estuaries, coastal waters and seas.” Envi-ronmental Software, 7(4), 229–240.8


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items