A Risk-Cost Analysis for an Increasingly Widespread Monitoring of Railway Lines

Abstract

Structural Health Monitoring represents an essential tool for detecting timely failures that may cause potential damage to the railway infrastructure, such as extreme weather conditions, natural accidental phenomena, and heavy loads affecting tracks, bridges, and other structures over time. The more thorough the monitoring, the more exact the information can be derived. In this paper, we propose an optimal approach to ensure the maximum railway infrastructure reliability through increasingly widespread and effective monitoring, subject to a budget constraint. More in detail, considered a pre-existing network of zones, each of which is monitored by a set of fixed diagnostic sensors, our goal is to identify new additional areas in which to place the same set of sensors in order to evaluate the geometric and structural quality of the track simultaneously. A kriging technique is used to identify the riskiness of some unsampled locations in order to select the new areas to be monitored. Moreover, two different decision criteria have been introduced, both depending on the risk level of the occurrence of extreme phenomena under investigation and involving the analysis of monitoring and non-monitoring costs. A descriptive analysis of the procedure, which may be used to identify the additional zones to be monitored, is provided in the paper by illustrating the resolution algorithm of the problem. The methodology has been implemented in environment R by using simulated data.

Appendix

1.1 Universal Kriging Interpolation Technique

The Kriging interpolation method allows for estimating the value of a variable at an unsampled location by comparing it to the values of the variable at nearby locations that have been sampled. For each structural parameter ${\theta }_j$, $j=1, \dots , n_{\boldsymbol{\varTheta }}$, the interpolation technique is used to estimate the values of ${\theta }_j$ in each not monitored point identified by $P_j^* \in \mathcal {A}$.

According to the Kriging technique, the estimate of the parameter ${\theta }_j$ in $P_j^*$ can be computed as a weighted average, as follows,

$$\begin{aligned} \hat{{\theta }}_j (P_j^*)= \sum _{i=1}^{m} \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} \, {\theta }_j(P^i_{k,j}) , \end{aligned}$$

(1)

where $\lambda _{i,k}$ are unknown weights that, under the stationarity assumption, are subject to

$$\begin{aligned} \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k}=1 . \end{aligned}$$

(2)

Since Kriging is designed to provide the most accurate estimate of the parameter ${\theta }_j$, the weights $\lambda _{i,k}$ are chosen in order to minimise the estimate variance $\hat{\sigma }_j^2(P_j^*)$, that is,

$$\begin{aligned} \begin{aligned} \hat{\sigma }^2_j(P_j^*) =\text {E} [({{\theta }}_j (P_j^*)- \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} {\theta }_j(P^i_{k,j}))^2] . \end{aligned} \end{aligned}$$

(3)

Let a point $\tilde{P}_j \in \mathcal {A}$ and a fixed distance $\vec {h}$ be considered. For a given structural parameter ${\theta }_j$, the variogram function $\delta _j(\vec {h})$ is given by the following semivariance,

$$\begin{aligned} \delta _j(\vec {h})=\frac{1}{2} \hat{\sigma }^2 \left( {\theta }_j(\tilde{P}_j+\vec {h})-{\theta }_j(\tilde{P}_j)\right) \,, \end{aligned}$$

depending on merely the distance $\vec {h}$. As proven in [5], the estimate variance $\hat{\sigma }^2_j(P_j^*)$ can be rewritten as a function of the variogram $\delta _j(\widehat{{h}})$, where $\widehat{h}$ is the estimated distance between two points, that is,

$$\begin{aligned} \hat{\sigma }^2_j(P_j^*) =2\sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \delta _j(\widehat{h}_{*,ik})- \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \sum _{\tilde{i}=1}^m \sum _{\tilde{k}=1}^{n_{{\theta }_j}} \lambda _{i,k} \, \lambda _{\tilde{i},\tilde{k}} \, \delta _j(\widehat{h}_{ik,\tilde{i}\tilde{k}}), \end{aligned}$$

(4)

where $\delta _j(\widehat{h}_{*,ik})$ is the variogram depending on the distance $\widehat{h}_{*,ik}$ between the points $P_j^*$ and $P^i_{k,j}$, and similarly $\delta _j(\widehat{h}_{ik,\tilde{i}\tilde{k}})$ is the variogram depending on the distance $\widehat{h}_{ik,\tilde{i}\tilde{k}}$ between the points $P^i_{k,j}$ and $P^{\tilde{i}}_{\tilde{k}, j}$.

The problem of minimising the estimation variance $\hat{\sigma }^2_j(P_j^*)$ in (3) is subjected to two constraints. The first one is related to the condition in (2). A further constraint is required to take into account the non-stationary trend of the parameter ${\theta }_j$. In this case, we can assume that the mean of the parameter ${\theta }_j$ evaluated at point $P_j^*$, $M(P_j^*)$, is described by a polynomial function of the following type,

$$\begin{aligned} M(P_j^*)=\sum _{w=1}^W a_w g_w(P_j^*), \end{aligned}$$

(5)

where $g_w(P_j^*)$, $w=1, \dots , W$, are polynomial functions of a certain order. Since Kriging is an exact estimator, by using the (1) property, expression (5) can be written as a combination of the same coefficients $\lambda _{i,k}$, that is,

$$\begin{aligned} \begin{aligned} M(P_j^*)=\sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} \text {E}[{\theta }_j(P^i_{k,j})]=\sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} M(P^i_{k,j}) . \end{aligned} \end{aligned}$$

(6)

Combining the two Eqs. (5) and (6), the following statement holds,

$$\begin{aligned} \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} \left( \sum _{w=1}^W a_w g_w(P^i_{k,j}) \right) = \sum _{w=1}^W a_w \left( \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} g_w(P^i_{k,j}) \right) , \end{aligned}$$

(7)

from which the second constraint of the optimization problem results

$$\begin{aligned} \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} \, g_w(P^i_{k,j})=g_w(P_j^*) . \end{aligned}$$

(8)

Therefore, for each parameter ${\theta }_j$, the optimization problem to be solved can be formulated in the following way,

$$\begin{aligned} \begin{aligned} \min _{\lambda _{i,k}} \hat{\sigma }^2_j(P_j^*)=\min _{\lambda _{i,k}} \left( 2\sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \delta _j(\widehat{h}_{*,ik})- \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \sum _{\tilde{i}=1}^m \sum _{\tilde{k}=1}^{n_{{\theta }_j}} \lambda _{i,k} \, \lambda _{\tilde{i},\tilde{k}} \, \delta _j(\widehat{h}_{ik,\tilde{i}\tilde{k}}) \right) \\ \\ \text {s. t.} {\left\{ \begin{array}{ll} \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k}=1 \\ \\ \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} \, g_w(P^i_{k,j})=g_w(P_j^*) \quad \quad \forall \, w=1, \dots , W . \end{array}\right. } \end{aligned} \end{aligned}$$

(9)

The optimization problem in (9) requires the Lagrangian approach. Assuming that $(\boldsymbol{\lambda }, \boldsymbol{\mu })=(\lambda _{1,1}, \dots , \lambda _{m,{n_{{\theta }_j}}}, \mu _0, \mu _1, \dots , \mu _W)$, where $\boldsymbol{\mu }$ is the lagrangian multipliers vector, the problem is solved by deriving the first order conditions of the following Lagrangian function,

$$\begin{aligned} \mathcal {L}(\boldsymbol{\lambda }, \boldsymbol{\mu }) = \hat{\sigma }^2_j(P_j^*) - \mu _0 \left( 1- \sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} \right) - \sum _{w=1}^W \mu _w \left( g_w(P_j^*)-\sum _{i=1}^m \sum _{k=1}^{n_{{\theta }_j}} \lambda _{i,k} \, g_w(P^i_{k,j}) \right) \,. \end{aligned}$$

(10)

Fig. 7.

Interpolation surface obtained via the Universal Kriging technique.

An example of a graphical representation of the metamodel built via the Kriging interpolation is shown in Fig. 7.