# System Effects in Identifying Risk-Optimal Data Requirements for Digital
Twins of Structures

Domenic Di Francesco  Max Langtry  Andrew Duncan  Chris Dent  The Alan Turing
Institute for Artificial Intelligence and Data Science, The British Library,
2QR, John Dodson House, 96 Euston Rd, London NW1 2DB  Department of Civil
Engineering, Cambridge University, Trumpington Street, CB2 1PZ  Department of
Mathematics, Imperial College London, Huxley Building, South Kensington
Campus, London, SW7 2AZ  Department of Mathematics, University of Edinburgh,
James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD

###### Abstract

Structural Health Monitoring (SHM) technologies offer much promise to the risk
management of the built environment, and they are therefore an active area of
research. However, information regarding material properties, such as
toughness and strength is instead measured in destructive lab tests.
Similarly, the presence of geometrical anomalies is more commonly detected and
sized by inspection. Therefore, a risk-optimal combination should be sought,
acknowledging that different scenarios will be associated with different data
requirements.

Value of Information (VoI) analysis is an established statistical framework
for quantifying the expected benefit of a prospective data collection
activity. In this paper the expected value of various combinations of
inspection, SHM and testing are quantified, in the context of supporting risk
management of a location of stress concentration in a railway bridge. The
Julia code for this analysis (probabilistic models and influence diagrams) is
made available. The system-level results differ from a simple linear sum of
marginal VoI estimates, i.e. the expected value of collecting data from SHM
and inspection together is not equal to the expected value of SHM data plus
the expected value of inspection data. In summary, system-level decision
making, requires system-level models.

###### keywords:

Data-centric Engineering, Decision Analysis, Risk Management, Structural
Health Monitoring, Value of Information.

††journal: Journal of Reliability Engineering and System Safety Symbol | Meaning | Units
---|---|---
$a$ | risk mitigation action | $-$
$a^{*}$ | risk mitigation action associated with expected optimal utility | $-$
$A$ | set of available risk mitigation action(s) | $-$
$C_{fail}$ | cost of failure | $-$
$C_{rm}$ | cost of risk mitigation | $-$
$e$ | data collection activity | $-$
$e^{*}$ | data collection activity associated with expected optimal utility | $-$
$E$ | set of available data collection activities | $-$
$m$ | mean | $-$
$\Pr(fail)$ | probability of failure | $-$
$sd$ | standard deviation | $-$
$u$ | utility | $-$
$Var$ | variance | $-$
$z$ | measurement data | $-$
$f_{X}(x)$ | probability mass or density function for uncertain parameter, $x$ | $-$
$\alpha$ | shape parameter of Gamma distribution | $-$
$\beta_{R}$ | reliability index | $-$
$\gamma$ | scale parameter of Gamma distribution | $-$
$\delta$ | misalignment | $mm$
$\epsilon$ | measurement uncertainty parameter | $-$
$\theta$ | uncertain and unobserved model parameter(s) | $-$
$\rho$ | correlation coefficient | $-$
$\sigma_{L}$ | applied stress | $MPa$
$\sigma_{L-meas}$ | stress inferred from sensor data | $MPa$
$\sigma_{Y}$ | yield strength | $MPa$
$\Sigma$ | covariance matrix | $-$
$\Phi_{G}$ |  multivariate distribution defined by a Gaussian copula, associated covariance matrix and marginal distributions | $-$
Table 1: Nomenclature

##  1 Introduction

###  1.1 Risk Based Structural Integrity Management

Risk is defined as the expected consequences of uncertain outcomes [1], and
can therefore be used to rank decision alternatives consistently with
statistical decision theory [2]. For instance, it is expected to be worthwhile
investing in a maintenance activity if it is believed that the corresponding
net reduction in failure costs is greater than the costs required to complete
the work. By the same approach, the optimal activity (where multiple may be
available, including the option to take no action) is that for which this net
benefit is the greatest.

In industrial Risk Based Inspection (RBI) schemes, calculation complexity is
often cited as the reason that simplified heuristics are used, rather than the
above-mentioned principled statistical decision analysis [3, 4]. Due to the
emergence of novel data collection technologies and methods for scalable
probabilistic inference, it can be argued that these justifications for
avoiding quantitative statistical methods are now less valid [5]. The
principle of RBI is that resource allocation should be directed by risk, i.e.
as components in a system become increasingly likely to fail, or the
associated failures become increasingly costly (or both), their priority in
maintenance budgeting should also increase. Without an absolute scale of risk
(which can only be obtained from a fully quantitative risk analysis), it
cannot be determined for which components investment is expected to be
worthwhile, only their relative priorities can be identified, and not always
consistently. Simple rules will always have pragmatic and implementation
benefits, but can lead to sub-optimal resource allocation and excessive
(unquantified) risk. The barriers associated with the introduction of methods
from computational statistics will decrease as the cost of formulating and
performing calculations decreases and training in data-centric engineering
becomes widespread. The associated benefits of transparent, quantitative and
auditable decision-making are then expected to emerge.

###  1.2 System Effects

Recent technological developments in data collection for engineering
structures have largely focused on Structural Health Monitoring (SHM). This
can broadly be described as the process of fitting sensors to structural
components (either during construction of new builds, or as retrofits to
existing structures) so that local conditions can be measured in operation
[6]. A typical application is that of installing strain gauges to better
understand the applied loads at particular locations. Alongside developments
in data analysis, across many industries, this may have a transformative
impact on the risk management of much of the built environment.

The expected value of this data, which can be quantified in the context of a
decision analysis (see Section 2), may be especially high when there is little
other information available regarding the local environmental conditions.
However, in instances where these are considered to be adequately
characterised in prior models, it may not be worthwhile installing SHM
systems. Rather, one of the other features of a structural integrity
assessment may benefit from measurement.

It is therefore important to recognise that the value of SHM data may be
influenced by the present understanding of the wider structural system, which
can be modelled by integrating multiple sources of information. A simple
example is shown in Figure 1, commonly referred to as the Engineering Critical
Assessment (ECA) of fracture mechanics triangle [7], which describes the key
sources of information in an assessment of structural condition. As an
example, consider the diagram in Figure 2 of two plates (base metal) joined by
a weld consisting of weld metal and a Heat Affected Zone (HAZ), which contains
anomalies. Here, data from SHM may provide an indirect and imperfect
measurement of the applied stress, $\sigma_{L}$. Inspection, often referred to
as Non-Destructive Testing/Evaluation (NDT/NDE), can provide similarly
imperfect detection and sizing of the presence/extent of damage111Note that
differing technologies are better suited to different types of damage (see
Annex T in [8]). Finally, material testing can provide a better understanding
of the properties of the weld, HAZ, and base metal. All of these remain
important inputs in empirical fracture mechanics models [9]. As SHM
technologies develop, it may be possible to reliably infer material and
geometrical properties from the data they provide. The methods and
calculations discussed in this paper are, in principle, agnostic to the source
of the data. Rather, they are concerned with a mathematical characterisation
of the quality of data required by a model, in the context of the underlying
decision problem for which the model is required, see Section 2.

Figure 1: Features of an assessment of structural condition Figure 2:
Geometric features of welded steel connection, including plate misalignment,
$\delta$

By considering the inter-dependencies between multiple sources of information,
engineers can better assess the overall health of a structure and make more
principled risk management decisions.

###  1.3 Probabilistic Digital Twins

There are various, and often contradictory definitions of the term _digital
twin_. A UK government report [10] proposes the following definition, which is
considered suitably broad:

> Digital twins are realistic digital representations of physical things. They
> unlock value by enabling improved insights that support better decisions,
> leading to better outcomes in the physical world.

The calculations and discussions in this paper are regarding how mathematical
representations of decision problems can be used to obtain transparent and
replicable (auditable) risk management strategies for structures. Such
analysis is conditional on an underlying model, which may be considered to be
a digital twin of the structure. Note that differing requirements have been
specified for what constitutes a _digital twin_ , such as the requirement for
a 3-dimensional model, or for at least one source of data to be in the form of
a (near) real-time stream.

In practice, the form and constituent components of a digital twin (or any
engineering model) should be based on the requirements of the underlying
decision problem that the model is intended to support. The decision to invest
in a sensing system should be based on the extent to which the data is
expected to facilitate improved risk management (see Section 2), and not so
that the model can be categorised as a digital twin222Unless there are
additional benefits to this, in which case they should be made explicit in the
model and accounted for.. Engineers have a legal [11] and ethical [12]
responsibility to develop and maintain safe systems, and this link between
underlying risk management decisions and structural calculations/analysis
should always be clearly defined, as it will be scrutinised following a
failure [13]. Digital twins should be developed to effectively support
decisions subject to uncertainty, such as what quality, and quantity of data
is required for a structure.

##  2 Decision Analysis

###  2.1 Conceptual Introduction

The need to support high consequence decisions is what dictates requirements
for various engineering analysis. For instance, material selection during a
design may be informed by stress analysis, or maintenance investments can be
informed by fracture mechanics assessments. However, unless the purpose is
made explicit, the results of such calculations may require some intermediary
interpretation before they can be used to support a decision. This means that
there is an opportunity to introduce unquantified and undocumented bias into
the analysis.

Influence diagrams [14] offer a concise graphical representation of the
components of a decision problem, namely uncertainties, costs and possible
actions. Domain knowledge is needed to design influence diagrams, as the
causal model used effects the results of any calculations performed, with the
addition or removal of arrows potentially leading to entirely different
outcomes [15].

As an example, consider the case of an inspection for geometric anomalies on a
structure. If predicted to be associated with sufficiently large stress
concentration effects (characterised by a Stress Concentration Factor, SCF),
the relevant components were replaced and tensile testing specimens machined
from them. When compared with pre-commissioning test data, the yield strength,
$\sigma_{Y}$ was found to be relatively high at these locations of high SCF.
This association can be described in various ways. In Figure 3, diagram (a)
proposes that a high strength causes a high SCF, and diagram (b) proposes that
a high SCF causes a high strength. A more suitable causal model identifies
that the proof load that the structure was exposed to is a confounding
variable333A simple explanation for this would be that, knowing a component
has been exposed to a certain load provides evidence for: the stresses and the
extent of damage present at the time. A large load at the presence of a high
SCF would require a relatively high strength to not fail., as shown in Figure
4. The reason that this representation matters, is because it will impact risk
management decisions. If a modelling team erroneously considered diagram (b)
in Figure 3, then any interventions that impact the SCF would also be assumed
to change the material strength. Domain expertise is required to design the
structure of influence diagrams, and prior models. However, they can then be
used to find associated expected optimal decision strategies.

Implications of such features in DAGs are discussed in more detail in [16]
from a causal inference perspective, and in [17] with reference to engineering
applications.

Figure 3: DAGS (a) and (b) conveying alternative causal explanations for an
association between $\sigma_{Y}$ and SCF Figure 4: Influence diagram proposing
non-causal association between $\sigma_{Y}$ and SCF

As suggested in Section 1.3, deciding whether or not to invest resources in
collecting and analysing data is an on-going challenge in engineering.
Fundamentally, this requires an evaluation of the question:

_how_ and _to what extent_ is the proposed data expected to facilitate
improved risk management?

Value of Information (VoI) analysis provides a replicable and quantitative
framework for addressing this question, as demonstrated by the example
calculations in this paper, and in recent scientific literature [18, 19, 20,
5]. It achieves this by considering the engineering models and the underlying
decision problem jointly. This helps target analysis in various ways,
including identifying when and where purchasing data of a specified quality is
expected to be worthwhile.

Applying VoI to structural integrity management raises the question, ”what is
it about data that provides value?” The mechanism is that data reduces
statistical (epistemic) uncertainty, for example, as more tensile tests are
completed, engineers can better understand the strength of a material.

As the uncertainty reduces, decision problems become simpler444Here, _become
simpler_ means that there is less uncertainty associated with an estimate of
an expected optimal decision. In the extreme (where there is no uncertainty
remaining) it is possible to simply compare known outcomes. and decision
makers may identify new expected optimal strategies, or simply expose
themselves to less risk as a consequence of more precisely quantifying
characteristics of their system.

A key challenge of VoI analysis is that the value that data will eventually
provide to a decision maker, will depend on the result of the measurement(s)
obtained. However, this needs to be evaluated before the measurement is taken.
This challenge is common to Bayesian experimental design [21], which is also
concerned with optimising data collection. The solution is to use a prior
model that describes the uncertainty in the quantity of interest. The
calculation compares the risk if a decision maker were to act based on this
prior model, with the expected risk when some plausible data is incorporated.
This comparison answers the initial question of how and to what extent data is
expected to improve risk management decision making. When this difference is
quantified on a monetary scale (as is typically the case in VoI analysis), it
can be interpreted as how much an engineering team should be willing to pay
for the data.

Some important features of VoI analysis are summarised below:

  * 1.

Challenge in generalising results. The expected value of data in the context
of solving a decision problem is case-specific. Changing a single element of
the problem may or may not change the results, and this outcome may not be
intuitive.

  * 2.

Quantity vs. Quality trade-off. For instance, for a given bridge, it may be
more worthwhile to collect small amounts of load data than large amounts of
high-resolution displacement data. Similarly the value of load measurements
may vary with location over the span.

  * 3.

Data is imperfect. Considering perfect information provides a useful upper
bound and simplifies calculations. Accounting for the imperfect features
(precision, bias, reliability, missingness) of data in the likelihood function
produces more realistic results at greater computational expense.

###  2.2 Formal Definitions

Consider the influence diagram in Figure 5. Here, the structural reliability
of a system $\beta_{R}$ is defined based on some set of uncertain parameters,
$\theta$. It determines the expected consequence of failure (along with a cost
of failure). A set of risk mitigation options, $A$, which include the option
to take no action, are available to the operator. The selection of this action
is required to find the cost of risk mitigation. The optimal action, $a^{*}$,
is defined as that which maximises the expected utility (or, equivalently,
minimises the expected cost), as defined in Equation 1 [22, 1].

$a^{*}=\operatorname*{arg\,max}_{a\in\mathcal{A}}\mathop{\mathbb{E}}_{\theta\sim
f_{\Theta}(\theta)}\big{[}u(a,\theta)\big{]}$ (1)

Figure 5: Generic influence diagram for mitigating risk of structural failure
Figure 6: Generic influence diagram for quantifying the expected value of
structural data

VoI analysis, originally proposed in [23], considers the opportunity to
collect data and investigates how this would impact the decision analysis. The
basic problem in Figure 5 is extended in Figure 6 to introduce data collection
opportunities, $\mathcal{E}$, including the option to proceed with no
additional data. Since this data, $z$, will generally be some indirect
measurement of $\theta$, it will influence the estimate of $\beta_{R}$. The
two decision variables (risk mitigation and data collection) can be jointly
optimised, as shown in Equation 2. The expected value of a specific data
collection activity, $e_{i}$, is defined as the difference between the
expected utility with and without the data. This result can be interpreted as
how much the operator should be willing to pay for the data, meaning that it
can directly inform budgeting decisions regarding data acquisition.

$e^{*},a^{*}=\operatorname*{arg\,max}_{\begin{subarray}{c}a\in\mathcal{A}\\\
e\in\mathcal{E}\end{subarray}}\mathop{\mathbb{E}}_{\begin{subarray}{c}\theta\sim
f_{\Theta}(\theta)\\\ z\sim
f_{Z}(z\mid\theta)\end{subarray}}\big{[}u(a,e,f_{\Theta}(\theta\mid z)\big{]}$
(2)

$VoI(e_{i})=\mathop{E}_{\begin{subarray}{c}\theta\sim f_{\Theta}(\theta),\\\
z\sim
f_{Z}(z\mid\theta)\end{subarray}}\big{[}u(e_{i},a^{*},f_{\Theta}(\theta\mid
z)\big{]}-\mathop{E}_{\theta\sim
f_{\Theta}(\theta)}\big{[}u(a^{*},f_{\Theta}(\theta))\big{]}$ (3)

The basic calculation procedure is outlined in Algorithm 1. The term
_preposterior_ in this algorithm is used in engineering to note that the prior
model is being updated using hypothesised data, rather than actual
measurements (which are not available at the time of evaluating whether
obtaining these measurements is risk-optimal) [24]. The steps involving
identifying the optimal action, and associated expected optimal utility
require solving an influence diagram, i.e. finding the strategy (combination
of available actions) that maximises the expected utility. In this paper, this
optimisation has been solved as a mixed-integer linear program using an
extension [25] to the Julia mathematical optimisation library [26, 27]. This
approach includes the constraint that all (decision) variables are discrete,
$a\in\mathbb{Z}^{n}$, representing the binary option to implement an action or
not. Each combination of actions $a\in A$ can therefore be assigned a discrete
index, and the maximum expected utility over this set will be associated with
$a^{*}$ [28].

Algorithm 1 Quantitative estimate of expected value of data collection

1\. Identify optimal action $a^{*}$, with respect to the expected utility,
$\mathop{E}[u]$, given prior distribution(s) $f_{\Theta}(\theta)$

2\. Find associated expected prior utility $E[u(a^{*})]$

for hypothesised measurement, $z_{e_{i}}$, in $\theta\sim f_{\Theta}(\theta)$
do

3a. Define likelihood function $f_{Z}(z_{e_{i}}\mid\theta)$ describing
information content of data, from measurement activity $e_{i}$

3b. Update prior model to obtain _preposterior_ distribution,
$f_{\Theta}(\theta\mid z_{e_{i}})$

3c. Identify $a^{*}$ given preposterior distribution

3d. Find associated expected prior utility $E[u(a^{*},z_{e_{i}})]$

end for

4\. Calculate expected value of proposed measurement:

$VoI=\dfrac{1}{Nsamples}\times\sum_{n=1}^{Nsamples}E[u(a^{*},z_{e_{i}}(n))]-E[u(a^{*})]$

###  2.3 Practical Considerations: Qualitative Example

A desirable feature of VoI analysis is that it is a quantitative and
replicable method for identifying risk-optimal data collection activities, to
the extent that the associated models provide a valid representation of the
true system. For instance, some pipelines in the USA are believed to be over
$50$ years old, and still in operation despite missing design documentation
[29]. There are instances where the material (grade of linepipe steel) is
unknown. Although it may be of interest to retrofit SHM sensors to such a
structure, a VoI analysis is likely to identify that it is expected to be more
worthwhile to understand the material properties. Depending on the intended
future operation, it may also be worthwhile to collect other types of data
too, but as argued in this paper, there should be a transparent, risk-based
justification for any measurements or interventions.

As a more detailed example, consider the Morandi cable stayed bridge, which
collapsed in Genoa (Italy) in 2018, resulting in death of 43 people. As
outlined in [30, 31], previous attempts to investigate the condition of the
concrete had failed to obtain the required information.

Figure 7: Schematic diagram of failed section of Morandi bridge

The means by which data collection from the bridge could have prevented the
catastrophic failure are considered below.

  * 1.

Strain gauges: measured strains may have triggered some intervention if cables
failed sequentially, with sufficient time between failures to detect the
redistribution of load.

  * 2.

Inspection of cables: provided that the inspection was able to obtain
measurements of the damage present in the cables (an earlier attempt had not,
[31]), this would have identified an important factor in the reduced
reliability of the bridge.

  * 3.

Material testing of cables and concrete: in the failure investigation,
concrete was measured to be approximately three times weaker than the expected
strength. This was attributed to illegal activities during the construction of
the bridge in the 1960’s. Testing of samples prior to the collapse could have
identified this.

Chloride-induced corrosion was known to be a threat, given the marine
environment that the bridge was exposed to. Propagating uncertainty in the
condition of the concrete (and perhaps also the strength of the concrete)
through a decision analysis could have shown that investments in data
collection and subsequent risk mitigation may have been worthwhile. Without
formalising this procedure, maintenance decisions lose transparency, as
different interpretations of reports, calculations and confounding variables
can be used to justify any outcome.

In 2018 steel supports were added to the span of the bridge that failed, see
Figure 7. Strengthening repairs are a valid form of risk mitigation, but will
not necessarily be the best option. Such interventions should be justified
using a decision analysis, with a transparent path from the intervention to
the specific source of risk that is being targeted. In this case,
strengthening the concrete towers did not improve the reliability of the
corroding cables. Alternative options may include limiting the traffic
(operational cyclic loads) on the bridge, or replacing parts with new
components with more precisely known properties.

There are organisational challenges in integrating some novel methods of data
collection and analysis into the business processes of the asset owner, in
this case Autostrade per l’Italia. The introduction of data-driven management
may improve safety by benefitting from recent research, but any existing
workflows that are superceded should be carefully considered, as subject
matter expertise (for instance, in defining influence diagrams and prior
probabilistic models), is not presently straightforward to automate,
particularly for specific assets. In this case, a generic review may have
identified the threat of Chloride-induced corrosion, but a knowledge of the
design and operational history would be able to confirm the extent to which
this had been mitigated, or measured. The absence of useful data should have
been evident in an increased uncertainty in the cable condition, which may
then have identified some available risk mitigation as a worthwhile
investment. These concepts are demonstrated quantitatively, in Section 3.

##  3 Quantitative Example: Risk Management for a Railway Bridge

###  3.1 Introduction

A bridge in Stafforshire, UK, was designed to carry passenger trains and was
constructed with an SHM system [32]. Specifically, Fibre Bragg Grating (FBG)
strain gauges were installed at multiple locations, a diagram of their
arrangement is shown in Figure 8. Here, these sensors were placed on the main
I-beams ($20$ each), across the span of the bridge, and on two selected
transverse beams ($7$ each). Tensile strains of approximately $25\mu$ strain
were recorded during a train passing [33].

Figure 8: Schematic diagram of strain gauge placement on elements in
Staffordshire railway bridge

This bridge is used as an example to demonstrate how maintenance decisions
subject to uncertainty can benefit from additional data collection. The values
(e.g. material properties, loads, and costs) used in the calculation do not
precisely match those for the Staffordshire rail bridge, but in Section 3.2
are argued to be representative of such a problem.

The maintenance decision problem is represented by the influence diagram in
Figure 9, which has been solved as part of this example. Here, each of the
elements of the ECA triangle in Figure 1 is considered (as shown by the dashed
lines), with options to collect data, or reduce the risk at each of the three
elements. These elements then jointly inform (an uncertain estimate of) the
reliability index, $\beta_{R}$. The load, and geometry in particular, may vary
along the span of the bridge. The influence diagram in Figure 9, and
associated probabilistic models in Section 3.2 define the conditions at a
single joint. Misaligment at flange connections, such as those in the
Staffordshire railway bridge, or at welded connections, such as in Figure 2,
both concentrate stresses.

Figure 9: Extension of ”ECA triangle” (see dashed lines) to influence diagram
representation of structural integrity management decision problem

Specifically, an upcoming annual maintenance window for the bridge is
considered. The failure mechanisms considered are listed below:

  * 1.

Over-stress: This limit state is defined as the combined effect of the applied
load and the stress concentration exceeding the yield strength.

  * 2.

Fatigue: The repeated application of loads, which may individually be
insufficient to cause failure, exceeding a permissible limit, as defined by an
SN curve555This model proposes a linear relationship (on a logarithmic scale)
between the amplitude of repeated stress cycles, $S$, and the number of cycles
at $S$ before fatigue failure, $N$.. Test data has been simulated from a curve
for a class D joint 666This category is considered representative for the hot-
spot stress (at locations where cracks are more likely to initiate) of a range
of welded joints and flange connections [34], and a probabilistic model has
been fit to account for the variability. See the calculation for complete
details [35].

The probability of failure, $Pr(fail)$, is defined as the probability of
exceeding of at least one of these limit states, see Equation 4. This is
calculated using probabilistic models of yield strength, applied stress, and
stress concentration. The available actions considered in the decision problem
are presented in Table 2. Note that this analysis considers that _any
combination_ of these actions can be selected, including the option to take no
action, and to collect all sources of data and implement all available risk
mitigation.

$\Pr(fail)=\Pr(\textrm{over-stress}\cup\textrm{fatigue})$ (4)

Action | Action type | Child node | Description
---|---|---|---
testing | data collection | $\sigma_{Y}$ | reduces uncertainty in material properties
repair | risk mitigation | $\sigma_{Y}$ |  strengthening repair increases capacity to withstand load. Stress is multiplied by $0.75$
inspection | data collection | SCF | reduces uncertainty in local geometry
replace | risk mitigation | SCF |  replacing misaligned component reduces stress concentration effects (if correctly installed). SCF is multiplied by $0.0$
SHM | data collection | $\sigma_{L}$ | reduces uncertainty in applied load
reduce operation | risk mitigation | $\sigma_{L}$ |  limiting the frequency of train passage reduces the number of stress cycles experienced, increasing fatigue life. Cyclic loading frequency is mulitplied by $0.5$
Table 2: Summary of intervention options available to operator and qualitative
explanation of their effect on downstream nodes

###  3.2 Decision Problem and Utility Function

The decision problem described by the influence diagram in Figure 9 is defined
by probabilistic models of the uncertain parameters (including the effect of
the various interventions, which are summarised in Table 2), and the costs of
each outcome. Note that these models describe a specific element, at which
some misalignment, and therefore stress concentration, is believed to be
present.

The prior model for stress, $\sigma_{L}$, is shown in Equation 5. The mean
value of $50MPa$ is approximately equivalent to the (elastic) stress
associated with an SHM measurement of $25\mu$ strain [33] and an elastic
modulus of $210GPa$, as assumed in [32]. A joint prior model of stress
concentration factor, $SCF$, and yield strength, $\sigma_{Y}$, is presented in
Equation 8. Here some dependency is proposed to account for the effect of
previous proof load testing, which makes the combination of relatively low
material strength and relatively high stress concentration to be less likely
(see Figure 3).

$\sigma_{L}\sim\mathcal{N}(m_{\sigma_{L}},sd_{\sigma_{L}})\\\ $ (5)

$\mu_{\sigma_{L}}\sim\mathcal{N}(m=50,sd=5)\\\ $ (6)

$\sigma_{\sigma_{L}}\sim LogNormal(m=6,sd=3)\\\ $ (7)

$SCF,\sigma_{Y}\sim\Phi_{G}\Bigg{(}\begin{matrix}\Gamma(\alpha_{SCF},\gamma_{SCF})\\\
LogNormal(\mu_{\sigma_{Y}},\sigma_{\sigma_{Y}})\end{matrix},\Sigma(SCF,\sigma_{Y})\Bigg{)}$
(8)

$\alpha_{SCF}\sim\mathcal{N}\bigg{(}m=2,sd=\dfrac{1}{2}\bigg{)},\;\alpha_{SCF}\geq
0\\\ $ (9)

$\gamma_{SCF}\sim\mathcal{N}\bigg{(}m=\dfrac{1}{2},sd=\dfrac{1}{2}\bigg{)},\;\gamma_{SCF}\geq
0\\\ $ (10)

$\mu_{\sigma_{Y}}\sim\mathcal{N}(m=400,sd=20)\\\ $ (11)

$\sigma_{\sigma_{Y}}\sim LogNormal(m=10,sd=3)\\\ $ (12)

$\Sigma(SCF,\sigma_{Y})=\begin{bmatrix}Var[SCF]&\rho(SCF,\sigma_{Y})\cdot\sqrt{Var[SCF]}\\\
\rho(SCF,\sigma_{Y})\cdot\sqrt{Var[SCF]}&\sigma_{\sigma_{Y}}^{2}\end{bmatrix}$
(13)

$Var[SCF]=\alpha_{SCF}\times\gamma_{SCF}^{2}$ (14)

$\rho(SCF,\sigma_{Y})=\dfrac{2}{3}$ (15)

An implementation of this decision analysis using the above models has been
made available using the Julia programming language [36] (with supporting
optimisation libraries [25, 26]). Samples from these priors have been obtained
(using latin hypercube sampling [37]) for prior predictive checks, and solving
the risk management decision problems [35]. These samples are then used to
estimate the expected value of combinations of material testing, SHM and
inspection.

The utility function in this calculation is defined in Table 3, providing the
cost of implementing the various interventions is defined. Note that every
combination of the risk mitigation actions in Table 2 is considered in the
calculation. A site visit cost (which is only incurred as part of a risk
mitigation activity) is associated with each action. All costs in this example
are normalised, so that the consequence of failure is set to $1.0$. Risk
mitigation costs can therefore be considered as proportions of the failure
cost.

Intervention or outcome | Cost components | Total cost
---|---|---
repair |  repair $(0.025)$ $+$ site visit $(0.01)$ | $0.035$
replace |  replace $(0.075)$ $+$ site visit $(0.01)$ | $0.085$
reduce operation |  reduce operation $(0.05)$ $+$ site visit $0.01$ | $0.05$
repair and replace |  repair $(0.025)$ $+$ replace $(0.075)$ $+$ site visit $(0.01)$ | $0.11$
repair, replace, and reduce operation |  repair $(0.025)$ $+$ replace $(0.075)$ $+$ reduce operation $(0.05)$ $+$ site visit $(0.01)$ | $0.16$
failure | failure $(1.0)$ | $1.0$
Table 3: Evaluations of utility function for various combinations of
interventions

As an example, the expected utility associated with each risk mitigation
option is evaluated using Equation 16. This incorporates the cost of failure,
$C_{fail}$ and of implementing any risk mitigation, $C_{rm}$. Note that
maximising the expected utility (see Equations 1 and 2) is equivalent to
minimising the expected cost. As shown in Table 4, the optimal action,
$a^{*}$, conditional on the prior models of $\sigma_{Y}$, $SCF$, and
$\sigma_{L}$ is to not invest in any risk mitigation measures in the upcoming
window.

$E[u]=Pr(fail)\times-(C_{fail})-C_{rm}$ (16)

Intervention or outcome | $Pr(fail)$ | Risk mitigation cost(s) | Total cost
---|---|---|---
no action | $0.0357$ | $0$ |  $a^{*}$ $0.0357$
repair | $0.0101$ | $0.035$ | $0.0451$
reduce operation | $0.0271$ | $0.050$ | $0.0771$
replace | $0.000$ | $0.085$ | $0.085$
repair and reduce operation | $0.0069$ | $0.085$ | $0.0919$
repair and replace | $0.000$ | $0.110$ | $0.110$
repair, replace, and reduce operation | $0.000$ | $0.160$ | $0.160$
Table 4: Evaluations of utility function for prior decision analysis (with no
additional data collection)

The VoI analysis presented in Section 3.3 compares this outcome to the results
conditional on updated models of $\sigma_{Y}$, $SCF$, and $\sigma_{L}$, based
on simulated (hypothesised) data from testing, inspection, and SHM.

###  3.3 Results

####  3.3.1 Perfect Information

The expected value of perfect SHM data in the context of planning for an
upcoming annual maintenance window is presented in Figure 10. This plot is the
result of the decision analysis (expected optimal risk mitigation) associated
with each sample from the prior model of stress. As the colorbar indicates, in
instances where a relatively low stress concentration is inferred from the
inspection data, the expected optimal solution is not to invest in maintenance
activities. As the hypothesised misalignment measurement increases, so does
the expected cost, initially because the associated higher stress
concentration will expose the operator to more risk, but eventually then also
because the expected optimal decision transitions to a strategy of investing
in a strengthening repair to reduce the stress in the component. This
continues until the strategy transitions again, this time to replace the
component and remove any stress concentration effects due to poor
installation. The mean value of the expected costs from all of these
simulations is an estimate of the expected maintenance cost with perfect SHM
data, and is indicated by the dashed vertical line. This is compared to the
expected costs without this data, as indicated by the solid vertical line
(results presented in Table 4). The arrow pointing to the difference between
these lines is the expected value of the data.

Figure 10: Expected value of perfect inspection data

This analysis can be repeated for inspection and testing, as well as
combinations of these, i.e. assessing the influence of collecting multiple
sources of data. These results are presented in Figure 11.

The key finding from this calculation is that these estimates do not simply
sum linearly. For instance, the expected value of inspection, $VoPInsp$ was
estimated to be $0.0332$ and the expected value of SHM, $VoPSHM$, was found to
be $0.0167$. However, the expected value of collecting both types of data
together was was estimated to be $0.0334$, which is less than
$VoPInsp+VoPSHM$. Similarly, the expected value of testing alone was estimated
to be $0.0012$, but it is not expected to contribute any further value when
completed with either SHM or inspection (or both). This means that in
instances where data will be collected from an inspection or SHM system, the
maintenance strategy (and associated costs) is expected to be unchanged if
material testing is also completed.

When a data collection is being evaluated the decision problem changes and the
extent to which the new optimal action space now benefits from further
reductions in uncertainty (from other types of data) may change. These non-
linearities are introduced by the utility function, for example due to the
decision boundaries presented in Figure 10. This sometimes non-intuitive
transformation onto a utility scale makes it difficult to generalise results,
as the analysis is evidently dependent on the features of the specific
decision problem. It does, however, introduce the benefit of producing
interpretable, actionable results by providing an operator with a maximum
(generally) monetary value that they should be willing to spend, for a
specified type(s) of data. If a vendor quotes a higher price than the expected
value of the data, then the risk optimal solution is simply to proceed without
the data. What the results in Figure 11 demonstrate is that when there is the
opportunity to collect different types of data, these should be considered
jointly in a VoI analysis, or a sub-optimal data collection plan may be
identified.

Figure 11: Expected value of combinations of structural integrity data

####  3.3.2 Imperfect Information

Quantifying the expected value of imperfect data increases the complexity of
step 3b in Algorithm 1. Here, a likelihood function is required to describe
the imperfect features of the data, and the subsequent posterior distribution
is then propagated through the decision analysis. For example, in the case of
assessing the expected value of SHM, a measurement from the sensor no longer
removes all uncertainty from the estimate of stress (at the time and location
of measurement). Rather it is acknowledged that the the measurement may be an
under-estimate, or over-estimate of the true stress. This imprecision can be
easily incorporated into a Bayesian model, as shown in Figure 12 and Equation
17. Note that as the precision with which the stress can be estimated from the
SHM data increases, the expected value of this data asymptotically approaches
the expected value of perfect SHM data, as calculated in Figure 11.

Figure 12: Simple DAG representation of measurement precision in SHM sensor
data

$\sigma_{L-meas}\sim N(\sigma_{L},\epsilon)$ (17)

The expected value of perfect data provides a useful upper bound, and may be
sufficient to support a decision (for instance if it is still below the cost
that is being quoted to obtain the data, then it can be concluded that
purchasing this data is not expected to be a risk optimal strategy). As data
becomes increasingly imprecise, the extent to which it will support risk
management decisions will decrease (or, at best, remain the same). This is
demonstrated by the sensitivity analysis results in Figure 13. Here, the value
on the x-axis is $\epsilon$ in Equation 17. Note that this pattern is true for
other features of imperfect information, such as bias, missingness,
reliability, and risk of obtaining the data. Examples of how to incorporate
these in a VoI analysis are provided in [38].

Figure 13: Sensitivity analysis of measurement uncertainty on expected value
of SHM data

####  3.3.3 Forecasting Multiple Maintenance Windows

Another way to extend this problem is to quantify the expected value of data
in the context of supporting multiple decision problems. Maintenance
strategies are not static and depend on planned data collection, or risk
mitigation interventions at other points in time. To address this, the
influence diagram in Figure 9 can be repeated sequentially to represent the
decisions that will need to be made in multiple future maintenance windows.
This representation is generally referred to as a dynamic Bayesian network (or
influence diagram) [39]. This can be considered a system effect in the
dependency between various elements in time, because they are jointly
optimised.

Solving the dynamic influence diagram provides the results in Table 5.
However, when simulating (indirect) measurements of stress from an SHM sensor,
the mean expected optimal cost again reduces and the expected VoI can be
quantified as shown in Figure 14. Note that in each case, a specific
combination of actions was identified, as was the case in the prior analysis
in Table 5. The frequency of optimal action pathways is visualised in Figure
15. Here, the width of each path is proportional to the number of samples for
which that action pathway minimises expected costs.

$a^{*}$ Year 1 |  $a^{*}$ Year 2 |  $a^{*}$ Year 3 | $E[u(a^{*})]$
---|---|---|---
strengthen | strengthen | no action | $0.155224$
Table 5: Optimal actions (w.r.t. expected utility) forecast for three
successive maintenance windows Figure 14: Prior samples of stress and expected
optimal preposterior utilities for the repeated decision problem using three
consecutive planning windows Figure 15: Sequences of optimal actions (w.r.t.
expected utility) forecasting over a three-year period

##  4 Conclusions

The key conclusions and propositions from this work are outlined below:

  * 1.

Influence diagrams (or similar formalisation of structural integrity
management decision problems) demonstrate the relationships/dependencies
between different elements of a risk mitigation system. This includes showing
which specific risks are mitigated by various interventions, and which
quantities are measured by different data collection activities. Subject
matter experts can agree this structure, which then lends itself to
probabilistic modelling (uncertainty quantification) and optimisation
(decision analysis) using modern software libraries. For instance, the
calculation associated with the quantitative example in this paper is
available in full in [35].

  * 2.

Value of information analysis can be used to formalise the process of managing
engineering data. This is contingent on a model, a mathematical description of
the underlying decision problem, and how the (prospective) data is related to
the quantities of interest. Ensuring (and transparently demonstrating) that
sufficient data is obtained to effectively manage the risk of engineering
systems aligns with the legal and ethical responsibilities of the profession.
Training in methods of data-centric engineering, and improvements to
industrial decision-support software are both expected to make VoI easier to
engage with and unlock its benefits for professional engineers.

  * 3.

The assessment procedure presented in this paper, namely

    1. (a)

Describing the relationship between the quantities of interest with each other
and prospective data collection & risk mitigation actions.

    2. (b)

identifying expected optimal maintenance actions, and;

    3. (c)

quantifying the expected benefit of collecting additional data, in the context
of further supporting this decision,

is generic. If SHM technologies allow for other quantities to be measured with
the same flexibility, then this approach will still be capable of finding the
expected optimal combination of data to purchase.

  * 4.

In this paper it has been demonstrated that when multiple measurement
opportunities are present, failing to solve this problem jointly (considering
inter-dependencies of collecting different combinations of various types of
data) can lead to sub-optimal data collection plans, i.e. either gathering
redundant information, or failing to collect information that becomes
beneficial only in combination with other measurements.

##  5 Acknowledgements

Domenic Di Francesco is supported by the Ecosystem Leadership Award under the
EPSRC Grant EP/X03870X/1, and The Alan Turing Institute, particularly the
Turing Research Fellowship scheme under that grant.

Max Langtry is supported by the EPSRC, through the CDT in Future
Infrastructure and Built Environment: Resilience in a Changing World Grant
EP/S02302X/1.

Chris Dent was supported by the Isaac Newton Institute for Mathematical
Sciences, in particular the Mathematical and Statistical Foundations of Data
Driven Engineering programme when work on this paper was undertaken. This work
was supported by the EPSRC Grant EP/R014604/1. He was also partially supported
by a grant from the Simons Foundation.

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