Bayesian Networks in Levee System Reliability
Kathryn Roscoe
PhD Candidate, (1) Dept. of Hydraulic Engineering, Delft University of Technology,
Delft, Netherlands, and (2) Dept. of Risk Analysis Water, Deltares, Delft, Netherlands
Anca Hanea
Research fellow, Centre of Excellence for Biosecurity Risk Analysis, The University of
Melbourne, Australia
ABSTRACT: We applied a Bayesian network to a system of levees for which the results of traditional reliability analysis showed high failure probabilities, which conflicted with the intuition and experience of those managing the levees. We made use of forty proven strength observations - high water levels with no evidence of failure - to refine the probability distributions of the random variables relevant for failure, and to improve the failure probability estimate of the system. We found that the use of these observations in the Bayesian network resulted in a decrease in the estimated faliure probability of over two orders of magnitude.
1. INTRODUCTION
Estimates of levee system reliability often conflict with experience and intuition. For example, we may compute a failure probability that is very high while no evidence of failure has been observed, or a very low failure probability when signs of failure have been detected. This conflict results in skepti-cism about the computed failure probabilities and an (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 observations to improve our reliability estimates. In this paper we describe the methodology to apply a Bayesian network at the spatial scale of a levee system, and show the details of an application to a levee system with proven strength, which means it has shown no evidence 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, we must omit certain details about the methodology and the application. Future publications will ad-dress these in their entirety.
2. BAYESIAN NETWORKS
Bayesian networks are a form of graphical model. Figure 1 presents a simple example; variables are represented by circular nodes, and arcs (arrows) be-tween nodes represent dependence. The idea of a Bayesian network is to simplify a joint probability distribution by coding dependence via the graphical structure, and letting each variable be represented by a conditional probability distribution. For exam-ple, in Figure 1 variables X1 and X2 are referred to
as the parents of X3, and variable X3is referred to as
the child of X1 and X2. Equation 1 shows the joint
distribution for this example network.
P(X1, X2, X3) = P (X1) · P (X2) · P (X3|X1, X2) (1)
1
X X2
3 X
Figure 1: Example three-variable Bayesian Network The Bayesian network described above suffers some shortcomings when it comes to reliability
analysis: (i) Efficient inference algorithms are al-most exclusively available for discrete distribu-tions, while in reliability analysis we typically have continuous 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 limit state functions (as they often are); (iii) all de-pendent (i.e. child) nodes must be represented by conditional distributions; however, we typically have marginal distributions, which we can obtain from data. Hybrid Bayesian networks address the first of these, allowing nodes to be described by both 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 Bayesian network that does not require discretization of con-tinuous distributions, and further that resolves all three shortcomings mentioned above. This network is described in the following section.
2.1. Non-parametric hybrid Bayesian network The non-parametric hybrid Bayesian network per-mits both continuous and discrete representation of variables, supports the inclusion of functional nodes, and only requires marginal distributions of each random variable in the network. Dependence between variables is captured via one-parameter conditional copulae (Joe (1997)), which are pa-rameterized by constant (conditional) rank correla-tions associated with each arc in the graph. Note that when Pearson product-moment correlations are available (which is commonly the case), we can use algorithms described in Kurowicka and Cooke (2006) to derive the conditional rank corre-lations. Hanea et al. (2006) showed that the condi-tional copulae, together with the one-dimensional marginal 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, initially
developed 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 using the assumption of the joint normal copula. Func-tional relationships can be represented with the use of functional nodes. Arcs connecting parent nodes to a functional child represent mathematical equa-tions rather than (conditional) copulae. For details about how sampling and inference are performed in the software, the reader is referred to Hanea et al. (2006).
3. MODELING LEVEE RELIABILITY
WITH A BAYESIAN NETWORK
We 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 in data-rich situations, and when failure of the levee is described 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 levee systemrefers to a large stretch of levees (typically tens of kilometers or more), within which are nu-merous levee segments (typically in the order of 1 kilometer) that are considered homogeneous. This means that while the random variables (e.g. soil permeability) fluctuate within the segment, the pa-rameters of their probability distribution are con-stant over the segment. The smallest spatial scale we consider is a levee cross section. This is a slice of the levee over which the values of the random variables are assumed to be constant.
3.1. Reliability of a levee cross section
We begin by considering the reliability of a cross section. We build our Bayesian network based on the 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 positive when the levee is reliable and negative when the levee fails. We include a failure node in the
net-work, F, which is 0 when Z ≥ 0 and 1 when Z < 0. As an example, assume that the limit state function depends 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 X3are shown as clear circular
nodes, representing random variables, and Z and F are shown as a circular nodes with black edges, rep-resenting functional nodes. Note that in this exam-ple, the random variables are independent of each other (no arcs between them), but this does not have to be the case.
1
X X2 X3
Z
F
Figure 2: Example of a Bayesian network for cross sectional levee failure probability
The Bayesian network is sampled taking into ac-count any defined correlations between variables (see section 2.1 for details). The failure probability can then be estimated in a standard way for Monte Carlo sampling; assume we have N samples, then we can estimate the failure probability according to Equation 2, where fjis the value of the failure node
F (1 or 0) for the jth sample.
ˆ Pf = 1 N N
∑
j=1 fj (2)3.2. Reliability of a levee segment
Homogeneous levee segments can be long, on the order of a few kilometers. The failure probability of a cross section is therefore a poor representa-tion of the failure probability of the entire segment. So instead of representing the failure probability by a single cross section, we represent it by multiple cross sections, and take care to honor the spatial autocorrelation of the variables between cross sec-tions. We show in Figure 2 an example of how the Bayesian network would look for a levee segment represented by three cross sections.
In the example in Figure 3, superscripts indicate the cross section. So for example, X12 indicates
1 1 X 2 1 X 2 2 X X23 3 1 X 1 2 X 1 3 X 2 3 X 1 Z Z2 Z3 1 F 2 F F3 S F 3 3 X
Figure 3: Bayesian network for a levee segment, in this example represented by three cross sections
variable X1 in the second cross section. Similarly,
F1, F2, and F3 represent the failure nodes for the first, 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 = 0
1, if ∃i s.t. Fi = 1 (3)
The ideal number of cross sections will depend on the characteristics of the levee and the failure mechanism being considered. In our method, we initially investigate each segment separately; we al-low the number of cross sections to increase itera-tively, each time computing the failure probability of the segment. When subsequent increases in the number of cross sections have no more effect on the failure probability, we know we have enough cross sections 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 autocorrelation of our variables can be modeled by one of a num-ber of valid autocorrelation formulas (Wenum-ber and Talkner (1993)).
3.3. Reliability of a levee system
Once we have determined the ideal number of cross sections to represent each of the levee segments in our system, we are ready to build the Bayesian net-work of our entire levee system. This essentially
consists of connecting the Bayesian networks of the segments. There are two important considerations when 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 potentially reduced water levels in the river or canal – on the failure probability of the downstream segments. We will address each of these two considerations in the following paragraphs.
It is important to determine which variables are expected to be correlated between levee segments. In general, levee segments are typically delineated by considering the length over which variables can be described by a single probability distribution. This often comes down to notable physical at-tributes, for example a change in stratigraphy. In such cases, it is reasonable to consider resistance variables between segments to be independent. On the other hand, load variables, like the water level in a river, are typically highly correlated between neighboring segments.
We can take system behavior into account by making the failure node of a downstream segment a function of the failure node for an upstream seg-ment. There are different ways to describe such a function, the most simple being that we only allow failure of a segment when upstream segments have not already failed (this is reasonable for small water systems). It is also possible to assign a probability with which this is the case.
4. CASE STUDY
We 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 setting of the levees, the failure mechanism we considered, and the details of the application.
4.1. Case description
We considered a system of canal levees in the Netherlands, about 45 km north of Amsterdam, and looked specifically at the failure mechanism pip-ing (also known as under-seepage). A map of the
system, including the levee segment numbers, is presented in Figure 4. The system protects a low-lying area known as Heerhugowaard, and contains the city of the same name. It is split into 18 levee segments. We considered our levee system to be composed of just three of these segments (9, 11, and 12). The reason for this is that only the south-ern and westsouth-ern levee segments (segment 5 through 12 in Figure 4) cause significant damage if they fail, and of these segments, 9, 11, and 12 were the real ’weak links’ in the sense that they had substan-tially higher computed failure probabilities than the others. The water board responsible for the lev-ees is highly skeptical about these failure probabili-ties, because they have never observed any evidence of piping. This made it an ideal case to apply a Bayesian network and make use of its capabilities to incorporate the ’proven strength’ of the levees.
8 2 14 9 15 17 3 4 10 1 18 6 16 5 11 7 13 12
Figure 4: Location of the levee system
4.2. Failure mechanism
In our current case study, we focus on the piping failure mechanism. Figure 5 provides an illustra-tion that supports the following descripillustra-tion. When the pressure difference between the outside water
level (h in Figure 5) and the landside water level (hls) is great enough, it can increase the soil pore water pressure in the aquifer (sand layer) to the point 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 can begin to transport from the aquifer onto the landside of the levee. What follows is an eroded pipe within the aquifer, allowing water from the landside of the levee to start filling in the pipe, as sand continues to erode. If the pipe reaches the waterside, the levee will essentially be resting on a film of water, which is a very unstable situation, and is likely to lead to collapse of the levee.
hls h
Sand Clay
Figure 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 is described 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. In the interest of brevity, we omit the formulas from this paper; they are described in detail in Schweck-endiek (2014). The variables that are used in the formulas are described in Table 1.
4.3. Data
In 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 = deterministic
Variable Description Distribution
D0 Thickness of aquifer logn
D Thickness of blanket layer logn
L Distance, waterside levee toe to landside water logn
θ Bedding angle of sand norm
d70 70th-percentile of sand grain diameter logn
η Drag coefficient logn
γwc Volumetric weight of blanket layer logn
γk Volumentric weight of sand logn
mu Error in critical pressure difference, for uplift logn
mh Error in actual pressure difference, for uplift logn
ms Error in piping model (Sellmeijer) logn
k Permeability of aquifer logn
hls Water level on landside of levee norm
d70m Reference value for d70 det
g Gravitational constant det
γw Volumetric weight of water det
ν Viscosity of water det
Table 2 shows the distribution parameters for all of the 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; when needed, water is pumped into the canal from the lower-lying protected area, as long as the water level in the canal does not exceed the maximum tolerated level. In this canal, that level is exactly equal to the Dutch datum, known as Amsterdam Ordinance 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 any water level above the maximum tolerated level had an exceedance probability of zero. In this way, we account for the regulated aspect of the canal. Fig-ure 6 shows the exceedance probability curve for the water level, and Table 3 shows parameters of the GPD.
4.4. Building the Bayesian network
We began by building the cross sectional Bayesian network for each of our three levee segments. We connected probabilistic nodes with functional nodes according to the limit state functions (see
Figure 2 for an example). We assumed that all
of our variables (see Table 1) are independent of each other. This assumption mirrors the one used in the Dutch reliability model used for levee systems (Steenbergen et al. (2004)); the validity - or at least
Table 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 is
fully correlated over the segment.
Variables M(S9) M(S11) M(S12) SD dx
D0 15 0.1M 200
D 0.3 0.01 0.01 0.1M 200
L 37.5 17.75 39 0.1M 3000
θ 37 3 600
d70 3.15E-04 2.62E-04 3.15E-04 0.15M 180
η 0.25 0.05M ∞ γwc 16 0.05M 300 γk 18 0.05M 300 mu 1 0.1M ∞ ms 1 0.1M ∞ mh 1 0.08M ∞
k 1.74E-04 9.26E-05 1.74E-04 M 600
hls -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 level
Shape Scale Threshold # Peaks 0.0651 0.0318 -0.3984 100 100 101 102 103 104 105 −0.4 −0.2 0 0.2 0.4 Return period
Canal level (m + AOD)
GPD fit
Regulated maximum Corrected curve
Figure 6: Water level exceedance probabilities for the Schermer Canal
the sensitivity - of this assumption should be tested, but was not considered in the current research. We compared the cross-sectional failure probabilities computed by the Bayesian network (see Table 4) with those from a traditional Monte Carlo approach with 100,000 samples and found the results identi-cal to two significant digits.
We then began adding cross sections within each segment, taking care to autocorrelate all of our vari-ables. The autocorrelation function that was used
for the variables in our network is shown in Equa-tion 4. The subscripts i and j refer to the cross sections. The scaling parameter dx determines how
quickly the correlation decreases over distance, and was given for each variable in Table 2. The distance between neighboring cross sections, ∆x, depends on the length of the segment and the number of cross sections. ρi j = exp |i − j| · ∆x dx 2 (4) Equation 4 computes what is known as the Pear-son product moment correlation - a measure of lin-ear correlation. These values were subsequently converted to conditional rank correlation coeffi-cients, as needed by the Bayesian network (see sec-tion 2.1). The details of this process are beyond the scope of this paper.
For each segment we iteratively increased the number of cross sections until the failure proba-bility of the segment reached an asymptote. We chose 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 7 shows the results for Segment 9. The other seg-ments look similar; Table 4 summarizes the results, providing the cross section failure probability, the ideal number of cross sections, and the segment failure probability for each of the three segments.
1 4 8 12 16 20 24 0.35 0.4 0.45 0.5 0.55 0.6 0.65
Number of cross sections
Failure probability of segment
Figure 7: Finding the ideal number of cross sections to represent the spatial variability of the levee segment, shown here for levee segment 9
Table 4: Cross sectional failure probability (Pf,CS),
segment failure probability (Pf,Seg), number of cross
sections (# CS), and the length of the segment (Lseg)
Segment Pf,CS Pf,Seg # CS Lseg(m)
9 0.32 0.59 24 1000
11 0.85 0.99 12 2000
12 0.12 0.57 40 4000
Once we have the ideal number of cross sections for each of our segments, we are able to build the network for the levee system. To do this, we needed to 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 of
infinity (∞) were said to be fully correlated over the entire segment. However, only a few of these are also fully correlated over our entire system. These were the water level in the canal (h), and the model error in the uplift and piping models (mu and ms,
respectively). All other variables were taken to be uncorrelated between segments. In total, our sys-tem contained 943 variables.
To account for system behavior, we connected the segment failure nodes, and made each down-stream segment dependent on failure at the up-stream segment. Because this water system is so small, an upstream breach will drastically draw down the water level in the canal. We therefore as-sumed a downstream segment can only fail if an upstream segment remained reliable.
When we ran our Bayesian network, we found an annual 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). This probability is highly suspect, given that no evidence of piping has been observed in tens of years. 4.5. Incorporating proven strength
To update our network, we made use of forty cou-pled observations of high water levels and no piping evidence. For each observation, we sampled from the conditional network (conditional on the water level observation), and then selected the samples of each of our variables that led to ’no failure’ for all cross sections in the network. From these samples,
we estimated new means and standard deviations for all of our variables. We could then re-run the network to compute new failure probabilities based on 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 in Figure 8. The impact is substantial, a decrease in annual system failure probability from 0.999 to 0.009 after forty updates. We present on a log scale to highlight the minor differences after about 20 ob-servations. Essentially, the observed peaks are no longer the most extreme ones, so that subsequent observations are fairly similar, and the information gained is minimal. 0 10 20 30 40 10−3 10−2 10−1 100 Number of updates
System failure probability
Figure 8: Effect of updating on system failure probabil-ity
The variables with the most notable changes in posterior distribution were: η, hls, k, θ , ms, and L
(see Table 1 for descriptions). Figure 9 shows the prior and posterior distributions of these variables for Segment 11. The impact of the observations was most substantial for this segment, due to its extremely high prior failure probability (see Table 4), but posteriors in the remaining segments showed similar behavior.
5. DISCUSSION
Discussions with the water board following the completion of this research illuminated an interest-ing aspect missinterest-ing from the Bayesian network rep-resentation of the system. The piping mechanism depicted in Figure 5 assumes that a sand layer un-derlies the canal. In practice, there may be a clay
0.2 0.25 0.3 0 20 40 η −4 −2 0 1 2 h ls 0 1 2 3 4 5 x 10 −4 0 5 10 x 104 k 20 40 60 0 0.1 0.2 θ 0 1 2 0 2 4 6 m s 10 20 30 0 0.2 0.4 L
Figure 9: Prior (dashed line) and posterior (solid line) distributions for Segment 11
layer between the canal and the aquifer, essentially making the soil water pressure in the aquifer im-mune to the water level in the canal, and render-ing the piprender-ing 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 network with a node that represents the existence (or nonex-istence) of such a clay layer. The node will act like a switch; when the layer exists, the failure proba-bility of the segment will be zero, regardless of the value of the limit state functions for piping. When the layer does not exist, the limit state functions will determine whether failure occurs. We hope to use the proven strength observations to refine the prob-ability of the existence of the clay layer, in addition to refining the probability distributions of our input variables.
6. CONCLUSIONS
Our research shows that a Bayesian network is a 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) were used to improve the failure probability estimate of a three-segment levee system, reducing it by over two orders of magnitude, from 0.999 to 0.009.
7. ACKNOWLEDGMENTS
The authors would like to thank Kasper Lendering and Nelle van Veen for their support in describ-ing the physical system of Heerhugowaard, and for making data accessible. We are also grateful for the financial support of the Dutch Technology Founda-tion STW, which is part of the Netherlands Organ-isation for Scientific Research, and which is partly funded by the Ministry of Economic Affairs.
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