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

Bayesian networks in levee system reliability Roscoe, Kathryn; Hanea, Anca Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Bayesian Networks in Levee System ReliabilityKathryn RoscoePhD Candidate, (1) Dept. of Hydraulic Engineering, Delft University of Technology,Delft, Netherlands, and (2) Dept. of Risk Analysis Water, Deltares, Delft, NetherlandsAnca HaneaResearch fellow, Centre of Excellence for Biosecurity Risk Analysis, The University ofMelbourne, AustraliaABSTRACT: We applied a Bayesian network to a system of levees for which the results of traditionalreliability analysis showed high failure probabilities, which conflicted with the intuition and experienceof those managing the levees. We made use of forty proven strength observations - high water levelswith no evidence of failure - to refine the probability distributions of the random variables relevant forfailure, and to improve the failure probability estimate of the system. We found that the use of theseobservations in the Bayesian network resulted in a decrease in the estimated faliure probability of overtwo orders of magnitude.1. INTRODUCTIONEstimates of levee system reliability often conflictwith experience and intuition. For example, wemay compute a failure probability that is very highwhile no evidence of failure has been observed, ora very low failure probability when signs of failurehave been detected. This conflict results in skepti-cism about the computed failure probabilities andan (understandable) unwillingness to make impor-tant management decisions based upon them.Bayesian networks are ideal in these circum-stances because they allow us to use observationsto improve our reliability estimates. In this paperwe describe the methodology to apply a Bayesiannetwork at the spatial scale of a levee system, andshow the details of an application to a levee systemwith proven strength, which means it has shown noevidence of failure during high water levels. We be-gin with background about Bayesian networks, in-cluding brief specifics about our choice of network.Because of the limited length of this paper, wemust omit certain details about the methodologyand the application. Future publications will ad-dress these in their entirety.2. BAYESIAN NETWORKSBayesian networks are a form of graphical model.Figure 1 presents a simple example; variables arerepresented by circular nodes, and arcs (arrows) be-tween nodes represent dependence. The idea of aBayesian network is to simplify a joint probabilitydistribution by coding dependence via the graphicalstructure, and letting each variable be representedby a conditional probability distribution. For exam-ple, in Figure 1 variables X1 and X2 are referred toas the parents of X3, and variable X3 is referred to asthe child of X1 and X2. Equation 1 shows the jointdistribution for this example network.P(X1,X2,X3) = P(X1) ·P(X2) ·P(X3|X1,X2) (1)1X 2X3XFigure 1: Example three-variable Bayesian NetworkThe Bayesian network described above sufferssome shortcomings when it comes to reliability112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015analysis: (i) Efficient inference algorithms are al-most exclusively available for discrete distribu-tions, while in reliability analysis we typically havecontinuous distributions, and are particularly in-terested in the tails; (ii) it cannot perform infer-ence when a functional node has been observed,which would exclude observations of ’failure’ or’no failure’ when these are represented by limitstate functions (as they often are); (iii) all de-pendent (i.e. child) nodes must be representedby conditional distributions; however, we typicallyhave marginal distributions, which we can obtainfrom data. Hybrid Bayesian networks address thefirst of these, allowing nodes to be described byboth discrete and continuous distributions. A num-ber of these have been developed in recent years(Langseth et al. (2009), Straub and Der Kiureghian(2010), Neil et al. (2007)), and often involve dis-cretization, which has drawbacks (Langseth et al.(2009)). We know of only one type of Bayesiannetwork that does not require discretization of con-tinuous distributions, and further that resolves allthree shortcomings mentioned above. This networkis described in the following section.2.1. Non-parametric hybrid Bayesian networkThe non-parametric hybrid Bayesian network per-mits both continuous and discrete representationof variables, supports the inclusion of functionalnodes, and only requires marginal distributions ofeach random variable in the network. Dependencebetween variables is captured via one-parameterconditional copulae (Joe (1997)), which are pa-rameterized by constant (conditional) rank correla-tions associated with each arc in the graph. Notethat when Pearson product-moment correlationsare available (which is commonly the case), wecan use algorithms described in Kurowicka andCooke (2006) to derive the conditional rank corre-lations. Hanea et al. (2006) showed that the condi-tional copulae, together with the one-dimensionalmarginal distributions and the conditional indepen-dence statements implied by the graph uniquely de-termine the joint distribution.The non-parametric Bayesian network is im-plemented in the UniNet software, initiallydeveloped at Delft University of Technology(Morales Nápoles et al. (2007)) by Lighttwist Soft-ware (http://www.lighttwist.net/). In this imple-mentation, both sampling and inference are usingthe assumption of the joint normal copula. Func-tional relationships can be represented with the useof functional nodes. Arcs connecting parent nodesto a functional child represent mathematical equa-tions rather than (conditional) copulae. For detailsabout how sampling and inference are performed inthe software, the reader is referred to Hanea et al.(2006).3. MODELING LEVEE RELIABILITYWITH A BAYESIAN NETWORKWe describe in this section how we model levee re-liability at different spatial scales. We keep this sec-tion general, and fill in the specifics when we de-scribe our case study. This method is best used indata-rich situations, and when failure of the levee isdescribed by a formula (which is the case for sev-eral important failure mechanisms). Bayesian net-works can be excellent tools in data-scarce situa-tions, as well as in cases where the failure mech-anism is not analytically formulated, but the ap-proach we describe here would need to be modified.Before diving into details, we would like to clar-ify some terminology about spatial scales. A leveesystem refers to a large stretch of levees (typicallytens of kilometers or more), within which are nu-merous levee segments (typically in the order of 1kilometer) that are considered homogeneous. Thismeans that while the random variables (e.g. soilpermeability) fluctuate within the segment, the pa-rameters of their probability distribution are con-stant over the segment. The smallest spatial scalewe consider is a levee cross section. This is a sliceof the levee over which the values of the randomvariables are assumed to be constant.3.1. Reliability of a levee cross sectionWe begin by considering the reliability of a crosssection. We build our Bayesian network based onthe formulaic representation of failure, which is of-ten postulated as a limit state function. Such a func-tion, typically denoted by the letter Z, is positivewhen the levee is reliable and negative when thelevee fails. We include a failure node in the net-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015work, F , which is 0 when Z ≥ 0 and 1 when Z < 0.As an example, assume that the limit state functiondepends on three variables: X1, X2, and X3. Fig-ure 2 shows what the Bayesian network for the fail-ure probability of the cross section might look like.Variables X1, X2, and X3 are shown as clear circularnodes, representing random variables, and Z and Fare shown as a circular nodes with black edges, rep-resenting functional nodes. Note that in this exam-ple, the random variables are independent of eachother (no arcs between them), but this does not haveto be the case.1X 2X 3XZFFigure 2: Example of a Bayesian network for crosssectional levee failure probabilityThe Bayesian network is sampled taking into ac-count any defined correlations between variables(see section 2.1 for details). The failure probabilitycan then be estimated in a standard way for MonteCarlo sampling; assume we have N samples, thenwe can estimate the failure probability according toEquation 2, where f j is the value of the failure nodeF (1 or 0) for the jth sample.Pˆf =1NN∑j=1f j (2)3.2. Reliability of a levee segmentHomogeneous levee segments can be long, on theorder of a few kilometers. The failure probabilityof a cross section is therefore a poor representa-tion of the failure probability of the entire segment.So instead of representing the failure probability bya single cross section, we represent it by multiplecross sections, and take care to honor the spatialautocorrelation of the variables between cross sec-tions. We show in Figure 2 an example of how theBayesian network would look for a levee segmentrepresented by three cross sections.In the example in Figure 3, superscripts indicatethe cross section. So for example, X21 indicates11X 21X 22X 32X31X12X 13X 23X1Z 2Z 3Z1F 2F 3FSF33XFigure 3: Bayesian network for a levee segment, in thisexample represented by three cross sectionsvariable X1 in the second cross section. Similarly,F1, F2, and F3 represent the failure nodes for thefirst, second, and third cross sections, respectively.These cross-sectional failure nodes are then con-nected to a failure node for the entire segment, FS,represented by the following function.FS ={0, if ∀i F i = 01, if ∃i s.t. F i = 1(3)The ideal number of cross sections will dependon the characteristics of the levee and the failuremechanism being considered. In our method, weinitially investigate each segment separately; we al-low the number of cross sections to increase itera-tively, each time computing the failure probabilityof the segment. When subsequent increases in thenumber of cross sections have no more effect on thefailure probability, we know we have enough crosssections to represent the spatial variability.We must describe the arcs between variables(see Figure 3) by a conditional rank correlation,which we can derive from Pearson product mo-ment correlations (see Section 2.1). When suffi-cient data are available, the spatial autocorrelationof our variables can be modeled by one of a num-ber of valid autocorrelation formulas (Weber andTalkner (1993)).3.3. Reliability of a levee systemOnce we have determined the ideal number of crosssections to represent each of the levee segments inour system, we are ready to build the Bayesian net-work of our entire levee system. This essentially312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015consists of connecting the Bayesian networks of thesegments. There are two important considerationswhen connecting them: (1) ensuring that the cor-relations between variables in the neighboring seg-ments are properly accounted for and (2) account-ing for any dependency between segment failures.This last point is often referred to as ’system be-havior’ and generally refers to the effect that an up-stream levee failure can have – due to potentiallyreduced water levels in the river or canal – on thefailure probability of the downstream segments. Wewill address each of these two considerations in thefollowing paragraphs.It is important to determine which variables areexpected to be correlated between levee segments.In general, levee segments are typically delineatedby considering the length over which variables canbe described by a single probability distribution.This often comes down to notable physical at-tributes, for example a change in stratigraphy. Insuch cases, it is reasonable to consider resistancevariables between segments to be independent. Onthe other hand, load variables, like the water levelin a river, are typically highly correlated betweenneighboring segments.We can take system behavior into account bymaking the failure node of a downstream segmenta function of the failure node for an upstream seg-ment. There are different ways to describe such afunction, the most simple being that we only allowfailure of a segment when upstream segments havenot already failed (this is reasonable for small watersystems). It is also possible to assign a probabilitywith which this is the case.4. CASE STUDYWe applied the methodology described in the previ-ous section to a system of levees in the Netherlands.In this section, we will describe the physical settingof the levees, the failure mechanism we considered,and the details of the application.4.1. Case descriptionWe considered a system of canal levees in theNetherlands, about 45 km north of Amsterdam, andlooked specifically at the failure mechanism pip-ing (also known as under-seepage). A map of thesystem, including the levee segment numbers, ispresented in Figure 4. The system protects a low-lying area known as Heerhugowaard, and containsthe city of the same name. It is split into 18 leveesegments. We considered our levee system to becomposed of just three of these segments (9, 11,and 12). The reason for this is that only the south-ern and western levee segments (segment 5 through12 in Figure 4) cause significant damage if theyfail, and of these segments, 9, 11, and 12 were thereal ’weak links’ in the sense that they had substan-tially higher computed failure probabilities than theothers. The water board responsible for the lev-ees is highly skeptical about these failure probabili-ties, because they have never observed any evidenceof piping. This made it an ideal case to apply aBayesian network and make use of its capabilitiesto incorporate the ’proven strength’ of the levees.821491517341011861651171312Figure 4: Location of the levee system4.2. Failure mechanismIn our current case study, we focus on the pipingfailure mechanism. Figure 5 provides an illustra-tion that supports the following description. Whenthe pressure difference between the outside water412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015level (h in Figure 5) and the landside water level(hls) is great enough, it can increase the soil porewater pressure in the aquifer (sand layer) to thepoint that it causes the clay layer to uplift (i.e. rup-ture) on the landside of the levee. Once this occurs,if the pressure difference is great enough, sand canbegin to transport from the aquifer onto the landsideof the levee. What follows is an eroded pipe withinthe aquifer, allowing water from the landside of thelevee to start filling in the pipe, as sand continues toerode. If the pipe reaches the waterside, the leveewill essentially be resting on a film of water, whichis a very unstable situation, and is likely to lead tocollapse of the levee.hlshSandClayFigure 5: Progression of the piping mechanism, begin-ning with uplift (top) until the pipe is complete (bottom)In the model we use, the piping mechanism isdescribed by two limit state functions, one describ-ing uplift of the clay layer (uplift) and the other de-scribing the initiation of the pipe formation (pip-ing). Failure is considered to occur if both the up-lift and piping limit state function are negative. Inthe interest of brevity, we omit the formulas fromthis paper; they are described in detail in Schweck-endiek (2014). The variables that are used in theformulas are described in Table 1.4.3. DataIn this section we describe the prior probability dis-tributions of the variables in the Bayesian network.Table 1: Description and distribution types of the vari-ables used in the piping analysis; logn = lognormal,norm = normal, det = deterministicVariable Description DistributionD0 Thickness of aquifer lognD Thickness of blanket layer lognL Distance, waterside levee toe to landside water lognθ Bedding angle of sand normd70 70th-percentile of sand grain diameter lognη Drag coefficient lognγwc Volumetric weight of blanket layer lognγk Volumentric weight of sand lognmu Error in critical pressure difference, for uplift lognmh Error in actual pressure difference, for uplift lognms Error in piping model (Sellmeijer) lognk Permeability of aquifer lognhls Water level on landside of levee normd70m Reference value for d70 detg Gravitational constant detγw Volumetric weight of water detν Viscosity of water detTable 2 shows the distribution parameters for all ofthe variables relevant for the piping limit state func-tions (with the exception of the canal water level).The water level in the canal is regulated; whenneeded, water is pumped into the canal from thelower-lying protected area, as long as the waterlevel in the canal does not exceed the maximumtolerated level. In this canal, that level is exactlyequal to the Dutch datum, known as AmsterdamOrdinance Datum (AOD).We fit a generalized Pareto distribution to inde-pendent water level peaks above a selected thresh-old. We then modified the distribution so that anywater level above the maximum tolerated level hadan exceedance probability of zero. In this way, weaccount for the regulated aspect of the canal. Fig-ure 6 shows the exceedance probability curve forthe water level, and Table 3 shows parameters ofthe GPD.4.4. Building the Bayesian networkWe began by building the cross sectional Bayesiannetwork for each of our three levee segments.We connected probabilistic nodes with functionalnodes according to the limit state functions (seeFigure 2 for an example). We assumed that allof our variables (see Table 1) are independent ofeach other. This assumption mirrors the one used inthe Dutch reliability model used for levee systems(Steenbergen et al. (2004)); the validity - or at least512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 2: Input values for the three segments (S9, S11,and S12). Shown are the mean M of the distributions,the standard deviation SD, and the correlation length(in meters) dx. Note that for dx, ∞ means the variable isfully correlated over the segment.Variables M (S9) M(S11) M (S12) SD dxD0 15 0.1M 200D 0.3 0.01 0.01 0.1M 200L 37.5 17.75 39 0.1M 3000θ 37 3 600d70 3.15E-04 2.62E-04 3.15E-04 0.15M 180η 0.25 0.05M ∞γwc 16 0.05M 300γk 18 0.05M 300mu 1 0.1M ∞ms 1 0.1M ∞mh 1 0.08M ∞k 1.74E-04 9.26E-05 1.74E-04 M 600hls -3.6 -3.6 -2.85 0.1M ∞d70m 2.08E-04 – –g 9.81 – –γw 10 – –ν 1.00E-05 – –Table 3: GPD parameters, canal water levelShape Scale Threshold # Peaks0.0651 0.0318 -0.3984 100100101102103104105−0.4−0.200.20.4Return periodCanal level (m + AOD)  GPD fitRegulated maximumCorrected curveFigure 6: Water level exceedance probabilities for theSchermer Canalthe sensitivity - of this assumption should be tested,but was not considered in the current research. Wecompared the cross-sectional failure probabilitiescomputed by the Bayesian network (see Table 4)with those from a traditional Monte Carlo approachwith 100,000 samples and found the results identi-cal to two significant digits.We then began adding cross sections within eachsegment, taking care to autocorrelate all of our vari-ables. The autocorrelation function that was usedfor the variables in our network is shown in Equa-tion 4. The subscripts i and j refer to the crosssections. The scaling parameter dx determines howquickly the correlation decreases over distance, andwas given for each variable in Table 2. The distancebetween neighboring cross sections, ∆x, depends onthe length of the segment and the number of crosssections.ρi j = exp(|i− j| ·∆xdx)2(4)Equation 4 computes what is known as the Pear-son product moment correlation - a measure of lin-ear correlation. These values were subsequentlyconverted to conditional rank correlation coeffi-cients, as needed by the Bayesian network (see sec-tion 2.1). The details of this process are beyond thescope of this paper.For each segment we iteratively increased thenumber of cross sections until the failure proba-bility of the segment reached an asymptote. Wechose a relatively pragmatic approach for deter-mining when the asymptote was reached, weigh-ing computational efficiency against the gain of in-creasing the number of cross sections. Figure 7shows the results for Segment 9. The other seg-ments look similar; Table 4 summarizes the results,providing the cross section failure probability, theideal number of cross sections, and the segmentfailure probability for each of the three segments.1 4 8 12 16 20 240.350.40.450.50.550.60.65Number of cross sectionsFailure probability of segmentFigure 7: Finding the ideal number of cross sections torepresent the spatial variability of the levee segment,shown here for levee segment 9612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 4: Cross sectional failure probability (Pf ,CS),segment failure probability (Pf ,Seg), number of crosssections (# CS), and the length of the segment (Lseg)Segment Pf ,CS Pf ,Seg # CS Lseg (m)9 0.32 0.59 24 100011 0.85 0.99 12 200012 0.12 0.57 40 4000Once we have the ideal number of cross sectionsfor each of our segments, we are able to build thenetwork for the levee system. To do this, we neededto determine which variables were correlated be-tween segments, and to account for system behav-ior. In Table 2, we provided the correlation lengths(dx) for each of the variables; those with a value ofinfinity (∞) were said to be fully correlated over theentire segment. However, only a few of these arealso fully correlated over our entire system. Thesewere the water level in the canal (h), and the modelerror in the uplift and piping models (mu and ms,respectively). All other variables were taken to beuncorrelated between segments. In total, our sys-tem contained 943 variables.To account for system behavior, we connectedthe segment failure nodes, and made each down-stream segment dependent on failure at the up-stream segment. Because this water system is sosmall, an upstream breach will drastically drawdown the water level in the canal. We therefore as-sumed a downstream segment can only fail if anupstream segment remained reliable.When we ran our Bayesian network, we found anannual failure probability of 0.999 for the levee sys-tem, which is unsurprising, given the failure prob-ability of 0.99 of Segment 11 (see Table 4). Thisprobability is highly suspect, given that no evidenceof piping has been observed in tens of years.4.5. Incorporating proven strengthTo update our network, we made use of forty cou-pled observations of high water levels and no pipingevidence. For each observation, we sampled fromthe conditional network (conditional on the waterlevel observation), and then selected the samples ofeach of our variables that led to ’no failure’ for allcross sections in the network. From these samples,we estimated new means and standard deviationsfor all of our variables. We could then re-run thenetwork to compute new failure probabilities basedon the posterior distributions of our variables.We performed the above procedure forty times,one for each of our coupled observations. The ef-fect on the system failure probability is shown inFigure 8. The impact is substantial, a decreasein annual system failure probability from 0.999 to0.009 after forty updates. We present on a log scaleto highlight the minor differences after about 20 ob-servations. Essentially, the observed peaks are nolonger the most extreme ones, so that subsequentobservations are fairly similar, and the informationgained is minimal.0 10 20 30 4010−310−210−1100Number of updatesSystem failure probabilityFigure 8: Effect of updating on system failure probabil-ityThe variables with the most notable changes inposterior distribution were: η , hls, k, θ , ms, and L(see Table 1 for descriptions). Figure 9 shows theprior and posterior distributions of these variablesfor Segment 11. The impact of the observationswas most substantial for this segment, due to itsextremely high prior failure probability (see Table4), but posteriors in the remaining segments showedsimilar behavior.5. DISCUSSIONDiscussions with the water board following thecompletion of this research illuminated an interest-ing aspect missing from the Bayesian network rep-resentation of the system. The piping mechanismdepicted in Figure 5 assumes that a sand layer un-derlies the canal. In practice, there may be a clay712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150.2 0.25 0.302040η−4 −2012hls0 1 2 3 4 5x 10−40510x 104 k20 40 6000.10.2θ0 1 20246ms10 20 3000.20.4LFigure 9: Prior (dashed line) and posterior (solid line)distributions for Segment 11layer between the canal and the aquifer, essentiallymaking the soil water pressure in the aquifer im-mune to the water level in the canal, and render-ing the piping mechanism impossible. Field mea-surements for the Heerhugowaard system have con-cluded that this clay layer exists at a number of lo-cations. We plan to extend the Bayesian networkwith a node that represents the existence (or nonex-istence) of such a clay layer. The node will act likea switch; when the layer exists, the failure proba-bility of the segment will be zero, regardless of thevalue of the limit state functions for piping. Whenthe layer does not exist, the limit state functions willdetermine whether failure occurs. We hope to usethe proven strength observations to refine the prob-ability of the existence of the clay layer, in additionto refining the probability distributions of our inputvariables.6. CONCLUSIONSOur research shows that a Bayesian network isa powerful and intuitive tool to approach relia-bility problems, when the results of more tradi-tional methods conflict with intuition and experi-ence. Proven strength observations (i.e. high wa-ter levels with no evidence of levee failure) wereused to improve the failure probability estimate ofa three-segment levee system, reducing it by overtwo orders of magnitude, from 0.999 to 0.009.7. ACKNOWLEDGMENTSThe authors would like to thank Kasper Lenderingand Nelle van Veen for their support in describ-ing the physical system of Heerhugowaard, and formaking data accessible. We are also grateful for thefinancial support of the Dutch Technology Founda-tion STW, which is part of the Netherlands Organ-isation for Scientific Research, and which is partlyfunded by the Ministry of Economic Affairs.8. REFERENCESHanea, A., Kurowicka, D., and Cooke, R. (2006). “Hy-brid Method for Quantifying and Analyzing BayesianBelief Nets.” Quality and Reliability Engineering In-ternational, 22(6), 613–729.Joe, H. (1997). Multivariate Models and DependenceConcepts. Chapman & Hall, London.Kurowicka, D. and Cooke, R. (2006). Uncertainty Anal-ysis with High Dimensional Dependence Modelling.Wiley.Langseth, H., Nielsen, T., Rumí, R., and Salmerón, A.(2009). “Inference in hybrid Bayesian networks.” Re-liability Engineering and System safety, 51, 485–498.Morales Nápoles, O., Kurowicka, D., Cooke, R., andAbabei, D. (2007). “Continuous-discrete distribu-tion free Bayesian belief nets in aviation safety withUNINET.” Technical Report TU Delft.Neil, M., Tailor, M., and D., M. (2007). “Inferencein Bayesian networks using dynamic discretisation.”Statistics and Computing, 17(3), 219–33.Schweckendiek, T. (2014). “On reducing piping uncer-tainties: A bayesian decision approach.” Ph.D. thesis,TU Delft, Delft, the Netherlands.Steenbergen, H., Lassing, B., Vrouwenvelder, A., andWaarts, P. (2004). “Reliability analysis of flood de-fence systems.” Heron, 49(1), 51–73.Straub, D. and Der Kiureghian, A. (2010). “BayesianNetwork Enhanced with Structural Reliability Meth-ods: Methodology.” Journal of Engineering Mechan-ics, 136(10), 1248–1258.Weber, R. O. and Talkner, P. (1993). “Some remarkson spatial correlation function models.” MonthlyWeather Review, 121(9), 2611–2617.8

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