# Value of Information of Structural Health Monitoring in Asset Management of Flood Defences

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## Abstract

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## 1. Introduction

## 2. Flood Defence Monitoring from an Asset Management Perspective

- The action to acquire information $i\in I$, where I is the set of all possible information acquiring actions.
- The outcome of the action to acquire information, $z\in Z$, where Z is the set of all possible outcomes. Note that z is used to update the belief of the decision maker about the state $\theta $ (see the last bullet).
- The action $a\in A$ following the obtained information, where A is the set of all possible actions. Here, it should be noted that this can be formulated by a decision rule which maps different outcomes $a\in A$ to outcomes $z\in Z$. This yields a decision rule $d\left(z\right)$ that assigns an a to each z. Hence, we use a set of decision rules $d\left(z\right)\in A$.
- The state of nature $\theta \in \Theta $ where $\Theta $ is the a priori set of all possible states of nature.

## 3. Methodology

#### 3.1. Time-Dependent Failure Probability Model

#### 3.2. Environment

#### 3.2.1. General Input

^{th}sample of a possible posterior state.

#### 3.2.2. Decision Model

#### 3.3. Decision-Making

#### 3.3.1. General Input

- Strategy a: no monitoring.
- Strategy b: monitoring is started if the failure probability ${P}_{\mathrm{f}},\mathrm{bel}>0.5\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{P}_{\mathrm{req}}$, where ${P}_{\mathrm{req}}$ is the reliability requirement. Monitoring is stopped after 25 years.
- Strategy c: continuous monitoring starting at $t=1$.

#### 3.3.2. Decision Model

#### 3.4. Evaluation

## 4. Case Study

## 5. Results

#### 5.1. Benefits of Monitoring for a Deterministic Future

#### 5.2. The Effect of Future Uncertainty on the Value of Information of Monitoring

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Kernel Density Estimates for Six Cases of Water Level Increase

**Figure A1.**Gaussian Kernel Density Estimates for six rates of water level increase after $t=50$. Bar plots indicate histograms of strategies b (in red) and c (in green).

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**Figure 1.**Decision tree for the choice whether to obtain information $i\in I$, modified from [10]. Levels I and A are choices by the decision maker, whereas levels Z and $\Theta $ are governed by chance. Level A consists of decision rules $d\left(z\right)$ which map actions to outcomes of z. The result is a utility over a combination of the levels $u(i,z,d(z),\theta )$.

**Figure 2.**Overview of the methodology. The top part shows in general how the input values are related to each other. The bottom part shows an influence diagram for the Bayesian decision model that is run for each sampled state. Dashed arrows indicate how general input is transferred to the model per sample. The dotted line in the middle indicates in which section (Section 3.2 or Section 3.3) the various parts are discussed. Parameters in blocks relate to Figure 1.

**Figure 3.**Two calculations of β in time for samples x

_{i}∈

**x**

_{n}for Case B. The left pane shows an unfavourable sample (β

_{state}> β

_{belief}), the right pane a favourable sample. Circled markers on the line for strategy b indicate presence of monitoring equipment. Dotted vertical lines indicate a reinforcement.

**Figure 4.**Value of Information (VoI) for cases A to D. Thick lines represent a Gaussian Kernel Density Estimation. Bars denote the histograms of the underlying samples. Dotted lines represent computed expected Value of Information for both strategies b and c, for which values are shown in the left top of each figure.

**Figure 6.**VOI for case B with default and lower threshold. Thick lines represent a Gaussian kernel Density Estimation and bars the histograms of underlying data. For B, it holds that ${P}_{\mathrm{thresh}}=1/10\phantom{\rule{3.33333pt}{0ex}}{\mathrm{year}}^{-1}$ for B low threshold ${P}_{\mathrm{thresh}}=1/2\phantom{\rule{3.33333pt}{0ex}}{\mathrm{year}}^{-1}$.

**Figure 7.**Cumulative water level increase $\Delta h\left(t\right)$ in centimeters per year for all considered scenarios.

**Figure 8.**Conditional and Relative VoI for different rates of water level increase after t = 50 for Case B. Relative VoI is the Conditional VoI normalized by the Total Cost.

**Table 1.**All input data for cases A, B, C and D. The top part shows all input distributions, the middle part shows the initial design point values for the influence of the strength uncertainty and epistemic part of the strength uncertainty, as well as the prior reliability index ${\beta}_{\mathrm{prior}}$. The bottom part shows the threshold ${P}_{\mathrm{thresh}}$ for the normal case and a lower threshold, safety standard ${P}_{\mathrm{norm}}$, planning period ${t}_{\mathrm{plan}}$ and discount rate r for each case.

Name | Unit | Distribution | Case | |||||
---|---|---|---|---|---|---|---|---|

Type | Values | A | B | C | D | |||

Input distributions | ${\mu}_{\mathrm{h}}\mathrm{c}$ | m +ref | Normal | $\mu $ | 7.56 | 6.12 | 5.76 | 5.88 |

$\sigma $ | 1.03 | 0.58 | 0.42 | 0.50 | ||||

${\sigma}_{\mathrm{irr}}$ | 0.2 | 0.2 | 0.2 | 0.2 | ||||

h | m +ref | Gumbel | a | 3.2 | ||||

b | 0.2 | |||||||

$\Delta h$ | mm/yr | Determ. | 8 | |||||

$\Delta {h}_{\mathrm{c}}$ | mm/yr | Determ. | 5 | |||||

Initial design point | ${\beta}_{\mathrm{prior}}$ | - | 3.96 | 4.00 | 4.01 | 4.03 | ||

${\alpha}_{\mathrm{h}}{\mathrm{c}}^{2}$ | - | 0.74 | 0.57 | 0.34 | 0.41 | |||

${\alpha}_{{\mu}_{\mathrm{h}}\mathrm{c}}^{2}$ | - | 0.69 | 0.51 | 0.28 | 0.39 | |||

Other input | ${P}_{\mathrm{thresh}}$ | -/yr | Default: 0.1 Lower: 0.5 | |||||

${P}_{\mathrm{norm}}$ | -/yr | 1/3000 | ||||||

${t}_{\mathrm{plan}}$ | yr | 50 | ||||||

r | %/yr | 3 |

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## Share and Cite

**MDPI and ACS Style**

Klerk, W.J.; Schweckendiek, T.; den Heijer, F.; Kok, M. Value of Information of Structural Health Monitoring in Asset Management of Flood Defences. *Infrastructures* **2019**, *4*, 56.
https://doi.org/10.3390/infrastructures4030056

**AMA Style**

Klerk WJ, Schweckendiek T, den Heijer F, Kok M. Value of Information of Structural Health Monitoring in Asset Management of Flood Defences. *Infrastructures*. 2019; 4(3):56.
https://doi.org/10.3390/infrastructures4030056

**Chicago/Turabian Style**

Klerk, Wouter Jan, Timo Schweckendiek, Frank den Heijer, and Matthijs Kok. 2019. "Value of Information of Structural Health Monitoring in Asset Management of Flood Defences" *Infrastructures* 4, no. 3: 56.
https://doi.org/10.3390/infrastructures4030056