**Contents**

## **About the Editors**

**Eleni Chatzi** is currently an Associate Professor and the Chair of Structural Mechanics at the Institute of Structural Engineering of the Department of Civil, Environmental and Geomatic Engineering (DBAUG) of ETH Zurich. She obtained her diploma (2004) and MSc (2006) in Civil ¨ Engineering, with honors, from the Department of Civil Engineering at the National Technical University of Athens (NTUA). In June 2010, she obtained her Ph.D. with distinction from the Department of Civil Engineering & Engineering Mechanics at Columbia University. In 2010, she was hired as the youngest Assistant Professor at ETH, and she was promoted to an Associate Professor in 2017.

Prof. Chatzi's research couples novel simulation tools with state-of-the-art monitoring methodologies for intelligent and data-driven assessment and diagnostics of engineered systems, with the goal of providing actionable tools able to guide operators and engineers in the management of their assets. A key aspect of her research lies in the extraction of quantifiable metrics that are indicative of structural performance across the component, system, and network levels. Her expertise lies in the area of structural health monitoring, with a strong focus on problems lying beyond the commonly adopted assumption of linear time-invariant systems. Her research spans a broad range of topics, including applications in emerging sensor technologies and structural control, methods for curbing uncertainties in structural diagnostics and life-cycle assessment, and advanced schemes for nonlinear/nonstationary dynamics simulations.

Prof. Chatzi serves as editor for numerous peer-reviewed international journals with particular focus on system identification methods and topics relating to SHM. Since 2016, she has coordinated the joint ETH Zurich & University of Zurich Ph.D. Programme in Computational Science. She is ¨ currently leading the ERC Starting Grant WINDMIL on the topic of "Smart Monitoring, Inspection and Life-Cycle Assessment of Wind Turbines", awarded by the European Research Council. Her work in the domain of self-aware infrastructure was recognized with the 2020 Walter L. Huber Research Prize, awarded by the American Society of Civil Engineers (ASCE). She is also a recipient of the 2020 EASD Junior Research Prize in the area of Computational Structural Dynamics, awarded by the European Association of Structural Dynamics (EASD).

**Vasilis K. Dertimanis**, Senior Assistant & Lecturer: Vasilis was born in Greece. He received his diploma in Mechanical Engineering from the University of Patras, Greece, and his Ph.D. from the National Technical University of Athens (NTUA), Greece, in the area of modeling and identification of faults in mechanical and structural systems. His research interests lie in the areas of structural identification and health monitoring, linear and nonlinear state estimation, active and passive structural control, and hybrid testing and optimization.

Vasilis has served as a senior researcher in the NTUA Vehicles Laboratory, Machine Design Laboratory, and Laboratory for Earthquake Engineering. He has also participated as a Marie Curie experienced researcher in the EU-funded SmartEN ITN project. For more than a decade, he has been in parallel self-employed as a freelance engineer and inspector, as an instructor in training seminars on transportation of dangerous goods by road/rail, and as a measurement engineer and structural vibration analyst.

Since January 2014, Vasilis has been a member of the Chair of the Structural Mechanics at ETH Zurich, and since May 2017, he has been Senior Assistant (Oberassistent) actively supporting the Chair in research and teaching.

## *Editorial* **Sensor Networks in Structural Health Monitoring: From Theory to Practice**

**Vasilis Dertimanis \* and Eleni Chatzi**

Department of Civil, Environmental and Geomatic Engineering, ETH Zürich, Stefano-Franscini-Platz 5, 8093 Zürich, Switzerland<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>
Received: 5 October 2020; Accepted: 12 October 2020; Published: 13 October 2020

The growing attention that structural health monitoring (SHM) has enjoyed in recent years can be attributed, amongst other factors, to the advent of low-cost and easily deployable sensors. The enabling technology has brought forth a new era of structural diagnostic means and is continuously redefining the tools for information processing, data reduction/compression, feature extraction, and smart assessment.

Within the SHM community, novel data-driven or hybrid methods are being developed, demonstrated on field deployments on large-scale systems, including transport, wind energy, and building infrastructure, promoting the actionability of SHM as an essential resource for life-cycle and resilience management. Nonetheless, in optimizing these deployments, a number of open issues remain with respect to the sensing side. These are associated with the type, configuration, and eventual processing of the information acquired from these sensors in order to deliver continuous behavioral signatures of the monitored structures.

Within this context, the aim of this Special Issue is to discuss the latest advances in the field of sensor networks for SHM. The focus lies in both active research on the theoretical foundations of optimally deploying and operating sensor networks, as well as in those technological developments that might designate the next generation of sensing solutions targeted for SHM.

Initiating form the data interpretation side, Argyris et al. [1] introduce a Bayesian framework for finite element model updating using experimental modal data. The novelty lies in utilization of a new likelihood formulation, which enables inclusion of the modal shapes, based on probabilistic treatment of the modal amplitude coherence (MAC) value between the model-predicted and measured mode shapes. The actual case study of the Metsovo bridge is utilized for validation. The modal characteristics of the bridge are identified based on operational modal analysis methods by means of a network of reference and roving sensors. The transitional Markov chain Monte Carlo algorithm is employed for model updating. The method delivers robust uncertainty bounds, outperforming conventional Bayesian formulations, which result in unrealistically thin uncertainty bounds for the model parameters, when a large (redundant) number of sensors is employed.

Moving away from the Bayesian paradigm, Pai et al. [2] describe a framework for practical asset management. In their work, the authors try to relax strong assumptions on uncertainty placed by typical data-interpretation methods, such as Bayesian model updating and residual minimization to better address practical engineering challenges. Error-domain model falsification is compared against these two more established approaches in terms of the ability to provide robust, accurate, engineer-friendly, and computationally inexpensive results for engineering model updating. Via demonstration on two full-scale case studies, the authors argue that error-domain model falsification is able to incorporate, iteratively and transparently, the incremental information gain, stemming from the sensing networks to facilitate updating of structural models at low additional computational cost.

Moving to utilization of sensing for triggering control actions, Azimi and Molaei Yeznabad [3] exploit a wireless-sensor network (WSN) as a main enabler for seismic vibration mitigation. Inspired by

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swarm intelligence, they propose a new control method, termed swarm-based parallel control (SPC), to improve seismic performance and minimize pounding hazards. This is achieved via theWSN-enabled exchange of information among adjacent buildings at corresponding floor levels. Three-dimensional time histories are simulated under earthquake excitation, with the response mitigated via use of semi-actively controlled magnetorheological (MR) dampers. The proposed control algorithm is assessed against fuzzy logic control (FLC), as well as passive on/off methods. Results indicate that in case of failure in the control system, as well as structural damage, the proposed WSN-driven SPC method can sense damage and accordingly update the control forces in the adjacent buildings to alleviate pounding.

A key factor in successful modal identification, model updating, and control is the chosen configuration of sensors. In this special issue, Barthorpe and Worden [4] o ffer a review of advances in sensor placement optimization (SPO) strategies for structural health monitoring (SHM). In recent years, this has been primarily examined in the context of damage identification. This work formally defines the SPO for SHM problem, o ffers an overview of the current state of the art, and proceeds to identify promising emergent trends. Some of the major (still) open questions pertain to robustness to modelling uncertainty, benign measurement influences, and sensor failures. A lack of su fficient studies on practical validation of the developed sensor designs is further remarked. It is finally argued that, since such a design is carried out at an o ffline design stage, it is not required to devote more e ffort in accelerating the proposed optimization schemes.

The issue of sensing extends beyond what is deemed as traditional civil infrastructure. In their work, Al-Obaidi et al. [5] study the transport processes of solids due to geophysical flows via use of a novel sensing mechanism; a miniaturized instrumented particle that can provide a direct, non-intrusive, low-cost, and accessible method for the assessment of coarse sediment particle entrainment. The presented instrumented particle is fitted with a triaxial MEMs accelerometer, a magnetometer and angular displacement sensors, which enable monitoring of the particle's 3D motion at a configurable rate of 200–1000 Hz. An appropriate filter is used for inertial sensor fusion, uncertainty mitigation, and noise reduction. The instrumented particle is tested under controlled lab conditions, reproducing initiation of the destabilization of a bed surface in an open channel flow. The work showcases the potential of custom designed and appropriately calibrated instrumented particles in the context of monitoring the possible instigation of water infrastructure hazards.

The final contribution underlines the importance of delivering methods able to tackle the challenge of uncertainty in the environmental and operational conditions (EOCs) that a monitored system is exposed to, and more importantly, do so in absence of availability of physics-based models. In a conventional setting, varying dynamic response can be captured via use of parameter-varying extensions of the original autoregressive with exogenous input (ARX) model. Yet, EOCs tend to vary in time-scales (days, weeks, months) that are su fficiently longer to the system dynamics. Tatsis et al. [6] propose a distributed framework for structural identification, damage detection, and localization by extending the conventional ARX structure. Here, the long time-scale dependence on EOCs is captured via a Gaussian process regression, while a distributed residual generation algorithm is set up for damage detection. Finally, damage localization is demonstrated via computation of the ARX-inferred mode shape curvatures. The diagnostic capabilities of the proposed scheme under varying EOCS are verified on two simulated case studies of increasing complexity.

This special issue has proposed contributions across the complete SHM information chain; from sensor design to configuration (placement), data interpretation, and triggering of reactive action (control). The featured papers o ffer an overview of the state-of-the-art and further proceed to introduce novel methods and tools. Several of these particularly target the treatment of uncertainty, which inherently describes the sensed information and the behavior of monitored systems. It is hoped that the readers will appreciate this Special Issue enhancing their knowledge in the area and helping them advance their ongoing research and innovation activities in sensing for SHM.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

©2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Bayesian Model-Updating Using Features of Modal Data: Application to the Metsovo Bridge**

**Costas Argyris 1,†, Costas Papadimitriou 1,\*, Panagiotis Panetsos 2 and Panos Tsopelas 3**

Received: 31 January 2020; Accepted: 29 May 2020; Published: 3 June 2020

**Abstract:** A Bayesian framework is presented for finite element model-updating using experimental modal data. A novel likelihood formulation is proposed regarding the inclusion of the mode shapes, based on a probabilistic treatment of the MAC value between the model predicted and experimental mode shapes. The framework is demonstrated by performing model-updating for the Metsovo bridge using a reduced high-fidelity finite element model. Experimental modal identification methods are used in order to extract the modal characteristics of the bridge from ambient acceleration time histories obtained from field measurements exploiting a network of reference and roving sensors. The Transitional Markov Chain Monte Carlo algorithm is used to perform the model updating by drawing samples from the posterior distribution of the model parameters. The proposed framework yields reasonable uncertainty bounds for the model parameters, insensitive to the redundant information contained in the measured data due to closely spaced sensors. In contrast, conventional Bayesian formulations which use probabilistic models to characterize the components of the discrepancy vector between the measured and model-predicted mode shapes result in unrealistically thin uncertainty bounds for the model parameters for a large number of sensors.

**Keywords:** Bayesian inference; model updating; modal identification; structural dynamics; bridges

## **1. Introduction**

The evaluation of the actual dynamic characteristics of structures, such as modal frequencies, modal damping ratios and mode shapes, through vibration measurements, as well as the development of high-fidelity finite element (FE) models, has been attracting an increasing research effort worldwide. Measured response data of structures mainly under ambient vibrations offer an opportunity to study quantitatively and qualitatively their dynamic behavior. These vibration measurements can be used for estimating the modal properties of structures, as well as for updating the corresponding FE models used to simulate their behavior [1,2]. The information for the calibrated FE models and their associated uncertainties is useful for checking design assumptions, for validating the assumptions used in model development, for improving modeling and exploring the adequacy of the different classes of FE models, and for carrying out more accurate robust predictions of structural response. These models are representative of the initial structural condition of the structure and can be further used for structural health-monitoring purposes [3–7].

Bayesian methods for ambient (operational) modal identification [8–18] and structural model updating [19–31] are used to develop high fidelity FE models of structures using modal properties

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identified from ambient vibration measurements. Due to the large size of civil infrastructure, the mode shapes are assembled from a number of sensor configurations that include optimally-placed reference sensors as well as moving sensors [32]. The modal properties are then integrated within Bayesian model-updating formulations to calibrate the parameters of large-scale FE models, as well as their associated uncertainty. The goal is to develop accurate and reliable models of the actual structures that are proven to closely simulate their behavior.

As far as the computational part is concerned, for complex posterior distributions, stochastic simulation algorithms such as Transitional Markov Chain Monte Carlo (TMCMC) [33] can be conveniently used to sample from the posterior distribution for parameter estimation, model selection and uncertainty propagation purposes. These methods require a large number of forward model runs which can increase the computational effort to excessive levels if one simulation for a high-fidelity large-order FE model requires several minutes or even hours to complete. For that purpose, fast and accurate component mode synthesis (CMS) techniques, consistent with the FE model parameterization [34,35], are used to achieve drastic reductions in computational effort. Further computational savings are achieved by adopting a parallelized version of the TMCMC algorithm to efficiently distribute the computations in available multi-core CPUs [36,37].

A novel likelihood function formulation is introduced in this work, which treats mode shapes not as full vectors, but as scalars using features between the measured and model-predicted mode shapes such as the MAC value. Instead of following the conventional Bayesian approach of assigning a multivariable Gaussian distribution to the error vector quantifying the discrepancy between the measured and model predicted mode shapes, a truncated Gaussian distribution is proposed for the probabilistic modeling of the scalar MAC value between the model predicted and experimental mode shapes. This effectively reduces the number of data points in the likelihood and leads to different uncertainty quantification results compared to the classic vector-based likelihood formulation. It is demonstrated that the proposed formulation has certain desired properties which can not be obtained under the vector-based formulation for the likelihood.

The capabilities of the proposed modal-based Bayesian model-updating methodology are demonstrated by calibrating the parameters of a high-fidelity FE model developed for the Metsovo bridge, using modal properties experimentally identified from ambient vibration data. The FE model is parametrized with respect to the stiffnesses of the deck, piers and soil components of the bridge. Ambient acceleration time histories from multiple points along the bridge deck are used to extract the modal properties of the bridge experimentally, and the identified modal properties are used as data in the Bayesian model updating methodologies in order to perform inference about the model parameters. In order to explore the effect of soil–structure interaction, two classes of models are examined and compared using Bayesian model selection [26,38]. Comparisons between the vector-based and the proposed MAC-based likelihood formulations demonstrate the advantages of the MAC-based likelihood formulation.

This work is structured as follows. Section 2 presents the Bayesian inference framework for FE model parameter estimation using modal properties. Section 2.1.1 reviews existing likelihood formulations, while Section 2.1.2 present the new formulation for building the likelihood based on features between experimental data and model predictions. The use of model reduction techniques to alleviate the computational burden encountered with sampling techniques is summarized in Section 2.2. Section 2.3 briefly outlines the whole procedure of parameter estimation and uncertainty propagation using the TMCMC sampler. The field structure is introduced in Section 3, along with the unreduced and reduced FE models of the structure, and the experimental modal identification procedure. Section 4 presents the results of model updating based on the experimentally-identified modes and demonstrates the advantages of the proposed MAC-based likelihood formulation. Conclusions are summarized in Section 5.

#### **2. Bayesian Parameter Estimation Using Modal Data**

To apply the Bayesian formulation for parameter estimation of linear FE models, we consider that the data *D* consists of the squares of the modal frequencies, *ω*ˆ 2*r* , and the mode shapes *φ*ˆ*r* ∈ *RN*0,*r*, *r* = 1, . . . , *m*, experimentally estimated using vibration measurements, where *m* is the number of identified modes and *N*0,*r* is the number of measured mode shape components for mode *r*. Consider also a parameterized linear FE model class *M* ˜ of a structure and let *θ* ∈ *RNθ* be a vector of free structural model parameters to be estimated using the set of modal properties identified from vibration measurements.

Let *ωr*(*θ*) and *φr*(*θ*) ∈ *RN*0,*r* be the *r*-th modal frequency and mode shape at *N*0,*r* measured DOFs, respectively, predicted by the model for a given value *θ* of the model parameters. The squares of the modal frequencies *ω*2*r* (*θ*) and the mode shape components *φr*(*θ*) = *Lrϕr*(*θ*) ∈ R*N*0,*r* are computed from the full mode shapes *ϕr*(*θ*) ∈ R*n* that satisfy the eigenvalue problem:

$$[\mathcal{K}(\underline{\theta}) - \omega\_r^2(\underline{\theta})\mathcal{M}(\underline{\theta})] \underline{\mathcal{P}}\_r(\underline{\theta}) = \underline{\mathcal{Q}} \tag{1}$$

where *K*(*θ*) ∈ R*n*×*n* and *M*(*θ*) ∈ R*n*×*n* are the global stiffness and mass matrices respectively of the FE model of the structure, *n* is the number of model DOFs, and *Lr* ∈ *RN*0,*r*×*n* is an observation matrix, usually comprised of zeros and ones, that maps the *n* model DOFs to the *N*0,*r* observed DOFs for mode *r*. For a model with large number of DOFs, *N*0,*r n*.

The likelihood *p*(*D*|*θ*, *M*˜ ) is the probability of observing the measured data *D* under the model *M* ˜ for parameters equal to *θ*. It is used in Bayes rule to update the posterior distribution *p*(*θ*|*D*, *M*˜ ) of the model parameters *θ* as follows:

$$p(\underline{\theta}|D,\tilde{M}) = \frac{p(D|\underline{\theta},\tilde{M})\ p(\underline{\theta}|\tilde{M})}{p(D|\tilde{M})},\tag{2}$$

where *p*(*θ*|*M*˜ ) is the prior distribution of the model parameters and *p*(*D*|*M*˜ ) is the evidence of the model class, selected so that *p*(*θ*|*D*, *M*˜ ) integrates to one.

### *2.1. Likelihood Formulation*

The likelihood formulation is of critical importance in Bayesian inference. To build the likelihood, one needs to assume a probabilistic relation between the model predictions and experimental data in order to account for unavoidable model error as well as experimental or measurement error. There is not just one way to do that, and different likelihood formulations can lead to different results. Therefore, Bayesian inference is subjective in the sense that different likelihood models can be tried using the same data, and the inference results might differ significantly. Prediction error equations, which relate the model predictions with the experimental data probabilistically, are used to formulate the likelihood. Depending on the nature of the data, different prediction error equations can be used for different subsets of the entire data set.

For the modal frequencies, the most common choice is the uncorrelated Gaussian error assumption for each modal frequency (e.g., [39,40]). Specifically, the prediction error equation for the *r*-th modal frequency is taken as:

$$
\omega\_r^2 = \omega\_r^2(\underline{\theta}) + \varepsilon\_{\omega\_r \text{ \textquotedblleft}} \tag{3}
$$

where *εωr* is the prediction error for the *r*-th modal frequency taken to be Gaussian with zero mean and standard deviation *σωrω*ˆ 2*r* . The unknown parameter *σωr* is included in the parameter set *θ* to be estimated from the data. This formulation for the modal frequencies assumes that each modal frequency is uncorrelated with the rest. Then, the likelihood term for the *r*-th modal frequency is the

probability of observing the measured frequency given specific values of the model parameters *θ*, derived from Equation (3) in the form:

$$p(\hat{\omega}\_r^2|\underline{\theta}) = N(\hat{\omega}\_r^2; \omega\_r^2(\underline{\theta}), \sigma\_{\omega\_r}^2 \hat{\omega}\_r^4),\tag{4}$$

where *N*(*x*; *μ*, *σ*2) denotes the univariate Gaussian PDF evaluated at point *x* with mean *μ* and variance *σ*2.

However, as far as the mode shapes are concerned, the prediction error formulation can be more complex due to the fact that they are vectors with multiple components. Again we make the assumption that all mode shapes are uncorrelated with each other and therefore we can treat each mode shape individually, just like the modal frequencies. Two formulations are presented next. The first one is a review of existing formulations, while the second one is a novel formulation based on features between model predicted and experimentally identified mode shapes.

#### 2.1.1. Formulation Using Probabilistic Models for Mode Shape Vectors

An often-used formulation for the prediction error is to assume that the discrepancy vector between the measured mode shape vector and the model predicted mode shape vector follows a zero-mean multivariable Gaussian distribution with a specified covariance matrix. The prediction error equation for the *r*-th mode shape is then

$$
\underline{\Phi}\_r = \beta\_r(\underline{\theta}) \underline{\Phi}\_r(\underline{\theta}) + \underline{\varepsilon}\_{\underline{\Phi}\_r} \tag{5}
$$

where *εφr* is the prediction error vector for the *r*-th mode shape taken to be Gaussian with zero mean and covariance matrix *σ*2 *φr* Σ*φr*, where the matrix Σ*φr* specifies the possible correlation structure between the components of the prediction error vector of the *r*-th mode shape, the unknown scalar *σ*2 *φr* is included in the parameter set to be estimated, and

$$\left|\beta\_r(\underline{\theta}) = \underline{\oint}\_r \underline{\Phi}\_r(\underline{\theta}) \right\rangle \left| \left| \underline{\Phi}\_r(\underline{\theta}) \right| \right|^2 \tag{6}$$

is a normalization constant such that the measured mode shape *φ*ˆ *r* at the *N*0,*r* measured DOFs is closest to the model mode shape *βr*(*θ*)*φr* (*θ*) predicted by the particular value of *θ*, and ||*z*||2 = *z*T*z* is the usual Euclidean norm. The scalar *βr*(*θ*) is introduced in Equation (6) to account for the fact that the measured modeshape *φ*ˆ *r* is normalized to have Euclidean norm equal to one, while the model predicted modeshape *φr* (*θ*) is mass normalized. The scalar *βr*(*θ*) is derived by minimizing the distance ||*φ*ˆ *r* − *βr*(*θ*)*φr* (*θ*)|| between the measured mode shape and the scaled version of the model predicted mode shape.

It is important to note in this approach that the number of data points used for each mode shape is equal to the number of measured DOFs *N*0,*r* for that particular mode. For a spatially uncorrelated model for the prediction error *εφr* (diagonal Σ*φr* matrix) each mode shape component counts as a new independent data point in the likelihood. From the Bayesian Central Limit Theorem, the posterior uncertainty is expected to reduce without bounds as the number of mode shape components is increased. However, as the number of measured DOFs increases, the sensors become very close to one another, providing almost the same information content that should not further reduce the posterior uncertainty of the model parameters. The closeness of the sensors depends on the wavelength of the considered measured mode shape. Two sensors are close and are expected to provide redundant information if their distance is a fraction of the wave length of the corresponding mode shape. Therefore, a spatially uncorrelated model for the prediction error vector *εφr* of the mode shape would not yield the expected behavior regarding posterior uncertainty as the number of mode shape components increases.

A remedy to this is to introduce a correlation model between the components of the prediction error vector of the mode shape, leading to a non-diagonal covariance matrix Σ*φr*. However, a correlation function should be postulated to describe the spatial correlation between two mode shape components (sensors) as a function of their distance, where the closer two sensors get the more they are correlated. Several correlation functions exist in the literature [41]. The problem is that one cannot know beforehand which correlation function is the proper one for the particular application at hand. This decision of the correlation function might turn out to be extremely difficult to make in practice, because in practical situations one normally has slight to none available information regarding the correlation nature of the prediction error vector. Selecting the proper correlation function might be challenging and failure to do so could easily lead to erroneous results as was demonstrated in [41]. Finding the proper correlation function is not the goal of this work. More on that issue can be found in [41–43]. Herein two cases of correlation models are examined: uncorrelated and exponentially correlated models.

For the simplest case of uncorrelated mode shape prediction error vectors the covariance matrix simplifies to a diagonal matrix:

$$\Sigma\_{\underline{\hat{\Phi}}\_{r}} = ||\underline{\hat{\Phi}}\_{r}||^{2} / N\_{0,r} \, I\_{\prime} \tag{7}$$

with *I* being the *N*0,*r* × *N*0,*r* identity matrix, while for the exponentially correlated model the identity matrix *I* is replaced by the correlation matrix *Rr* whose (*i*, *j*)-th element is given by the exponential correlation function:

$$R\_r(i,j) = \exp(-\mathbf{x}\_r(i,j)/\lambda\_r),\tag{8}$$

where *xr*(*i*, *j*) is the Euclidean distance between the *i*-th and *j*-th mode shape components (sensors) for the *r*-th mode, and *λr* is the correlation length for the *r*-th mode which is a parameter to be identified.

Using Equation (5), the likelihood term for the *r*-th mode shape is the probability of observing the measured mode shape for given model parameters *θ*, given by

$$p(\underline{\hat{\phi}}\_r|\underline{\theta}) = N(\underline{\hat{\phi}}\_r; \beta\_r(\underline{\theta})\underline{\phi}\_r(\underline{\theta}), \sigma\_{\underline{\hat{\phi}}\_r}^2 \Sigma\_{\underline{\hat{\phi}}\_r}(\underline{\theta})), \tag{9}$$

where *N*(*x*; *μ*, Σ) denotes the multivariate Gaussian PDF evaluated at point *x* with mean vector *μ* and covariance matrix Σ. Following the work of Papadimitriou et al. [44] which was based on the same prediction error Equation (5), the likelihood function in Equation (9) can be expressed in terms of the MAC values between the measured and model predicted mode shapes.

Slightly different prediction error equations for the mode shapes have been proposed in the literature (e.g., [39,40]), including versions that do not require the use of the mode correspondence [4,6]. In all these alternatives, the likelihood formulation for the mode shapes is based on a probabilistic description of individual components of a vector and thus they fall into the category discussed in this subsection.

#### 2.1.2. Formulation Using Probabilistic Models for MAC Values

The previous formulation uses the mode shapes as full vectors in the likelihood function. Herein we propose a novel formulation for including the mode shapes in the likelihood function which is based on the MAC value between the experimental and model predicted mode shape. The MAC value, defined as *MAC*(*u*, *v*) = *uTv*/(||*u*|| ||*v*||) between two vectors *u* and *v*, is the most common way to measure the similarity between two mode shape vectors. It is a scalar measure which varies from 0 to 1 with a value of 1 indicating a perfect match. The scaling of the mode shapes is not important for the MAC value which means that no normalization is needed for either the experimental or model predicted mode shape.

In the new formulation the experimental mode shape is not compared with the model predicted mode shape in an element-wise fashion, but rather based on its MAC value. This reduces the number of data points used in the likelihood for each mode shape from *N*0,*r* to just 1. Therefore, instead of calculating the probability of observing the experimental mode shape vector given the model predicted mode shape vector (for some given model parameter values), we calculate the probability of their MAC value taking a value of 1, implying that they match perfectly.

In contrast to the previous vector formulation of the likelihood, the MAC value is a univariate quantity and therefore requires a univariate distribution to model it. Taking into account the fact that the MAC value is strictly bounded in the interval [0, 1], a Truncated Gaussian distribution is used, although there are many other choices of candidate distributions. The Gaussian distribution is preferred because of its known properties. This leads to the following prediction error equation for the MAC value of the *r*-th mode shape:

$$
\hat{MAC}\_r = \hat{MAC}\_r(\underline{\theta}) + \varepsilon\_{\hat{MAC}\_r\prime} \tag{10}
$$

where *MACr*(*θ*) = *MAC*(*φ*ˆ*r*, *φr*(*θ*)) is the model-predicted MAC value, defined as the MAC value between the experimental *r*-th mode shape and the model predicted *r*-th mode shape for the given values of the model parameters *θ*. The term *εMACr* is the error in the *r*-th MAC value (analogous to the error in the *r*-th frequency), assumed to follow a univariate zero-mean Gaussian distribution with standard deviation equal to *σMACr*. The standard deviation *σMACr* is a measure of "how far" the observed MAC value *MAC* ˆ *r* can be from the model-predicted MAC value *MACr*(*θ*) due to model and experimental errors. This can be thought of as completely analogous to the error term for the modal frequencies in (3). The resulting Gaussian with mean *MACr*(*θ*) and standard deviation *σMACr* is truncated in 0 and 1 which results in the Truncated Gaussian distribution.

An important issue that should be addressed when using MAC values is the fact that although the MAC is a scalar value, it depends on the number of mode shape components used. This needs to be taken into account in the formulation in order to avoid erroneous results. For example, if only two components of a mode shape are used, there is a chance that the MAC value turns out to be very close to 1 (provided that those two components match well between the two mode shapes). However, if a large number of components is used, due to small errors in each component there is the chance that the MAC value is significantly lower than 1, which would mean that the case with two components would yield a larger MAC value. However, the case of large number of components components is expected to be much more informative than the case of two components since the more components we have the better we know the actual geometry of the mode shape. This naturally leads to the conclusion that the number of mode shape components must be taken into account, assigning higher preference to MAC values calculated with more components than MAC values calculated with less components.

One way to account for this in a Bayesian framework is through manipulation of the MAC value standard deviation parameter *σMACr*. We seek a formula through which to define *σMACr* that depends on the number of mode shape components *N*0,*r*. Although there is not only one way to achieve this, the following formula is used:

$$
\sigma\_{\rm MAC\_7} = \sigma\_{\rm MAC\_7}^{'} \left( 1 + \frac{1}{\sqrt{N\_{0,r}}} \right),
\tag{11}
$$

where *σMACr* is the parameter to be inferred from data. The first term in Equation (11) describes the uncertainty present in the MAC value that exists independently of the number of sensors. This uncertainty exists even for a large number of sensors and is due to model and experimental errors in the individual components and can not be reduced further. The second term in Equation (11) depends on the number of sensors and gets smaller as the number of sensors is increased, which reduces the standard deviation of the MAC value. This way more weight (less uncertainty) is given to MAC values calculated with more sensors. These are modeling choices within the Bayesian framework, much like the choice of Gaussian PDFs for the likelihood, independent data, etc. Alternative formulations could also be postulated. In particular, the two terms in Equation (11) can be weighted differently but this falls outside the scope of the present work.

*J. Sens. Actuator Netw.* **2020**, *9*, 27

Then the likelihood term for the MAC value of the *r*-th mode shape is the probability of observing a MAC value of 1 for given values of the model parameters *θ* (indicating a perfect match between the experimental and model predicted mode shapes), given by the Truncated Gaussian PDF:

$$p(\hat{MAC}\_r = 1 | \underline{\theta}) = TN(1; \hat{MAC}\_r(\underline{\theta}), \sigma\_{\hat{MAC}\_r}^2, 0, 1), \tag{12}$$

where *TN*(*x*; *μ*, *σ*2, *a*, *b*) denotes the Truncated Gaussian PDF evaluated at point *x* with mean *μ*, variance *σ*2 and truncation limits *a* and *b*.

#### 2.1.3. Likelihood Formulation Combining Modal Frequencies and Mode Shapes

The parameter set *θ* of the structural model class *M*˜ is augmented to include the parameters *σωr* and *σφr* or *σ MACr* related to the prediction error models. For simplicity, in order to avoid having too many parameters, the three prediction error parameters are assumed to be the same for all modes and therefore their dependence on *r* is dropped.

The total likelihood function is easily calculated as the product of the individual likelihoods for the frequencies and mode shapes, given their independence. For the vector formulation of the mode shapes the likelihood is:

$$p(D|\underline{\theta}, \tilde{M}) = \prod\_{r=1}^{m} p(\hat{\omega}\_r^2 | \underline{\theta}) \prod\_{r=1}^{m} p(\hat{\underline{\theta}}\_r | \underline{\theta}) \tag{13}$$

where *p*(*ω*ˆ 2 *r* |*θ*) and *p*(*φ*ˆ *r* |*θ*) are given by (4) and (9) respectively. For the MAC formulation of the mode shapes the likelihood is:

$$p(D|\mathcal{\boldsymbol{\theta}}, \tilde{\boldsymbol{M}}) = \prod\_{r=1}^{m} p(\hat{\omega}\_r^2 | \boldsymbol{\xi}) \prod\_{r=1}^{m} p(\boldsymbol{M} \hat{\boldsymbol{\Lambda}} \mathcal{C}\_r = 1 | \boldsymbol{\xi}) \tag{14}$$

where *p*(*ω*ˆ 2 *r* |*θ*) and *p*(*MAC*ˆ *r* = 1|*θ*) are given by (4) and (12), respectively.

## *2.2. Computational Tools*

The transitional Markov chain Monte Carlo algorithm (TMCMC) [33] is used for estimating the parameters of FE models by drawing samples from the posterior probability density function of the model parameters. Markov chain Monte Carlo algorithms, including TMCMC used in this work, require a moderate to very large number of repeated system analyses to be performed over the space of uncertain parameters. Consequently, the computational demands depend highly on the number of system analyses and the time required for performing a system analysis. For FE models with large number of DOFs, this can increase substantially the computational effort to excessive levels. Computational savings are achieved by adopting parallel computing algorithms to efficiently distribute the computations in available multi-core CPUs [36,37,45].

In addition, fast and accurate CMS techniques [46], consistent with the finite element model parameterization, are integrated with Bayesian techniques to reduce efficiently and drastically the FE model and thus reduce the computational effort [34,35]. CMS techniques are widely used to analyze structures in a reduced space of generalized coordinates. CMS involves dividing the structure into a number of substructures (components), obtaining reduced-order models of the substructures keeping a fraction of the substructure modes, and then assembling a reduced order model for the entire structure using the kept substructure modes and interface degrees of freedom between substructures. Additional substantial reductions can be achieved by reducing the number of interface DOF using characteristic interface modes through a Ritz coordinate transformation [34]. However, for methods involving re-analyses due to variations in the values of the uncertain model parameters the reduction for computing the system modes has to be repeated for each re-analysis. This gives rise to a substantial computational overhead that arises from the model reduction at component level, and from assembling the component mass and stiffness matrices to form the reduced global system mass and stiffness

matrices. The main objective in methods involving re-analyses of models with varying properties is to completely avoid the re-analysis at the component level as well as the re-assembling of the reduced global matrices at the system level.

It has been shown that when the partition of the structure into substructures is guided by certain parameterization schemes, the reduced global mass and stiffness matrices derived using CMS techniques can be represented exactly by an expansion of these matrices in terms of scalar functions of the model parameters, with coefficient matrices computed and assembled once from a single CMS analysis of a reference structure [34,47,48]. This representation allows one to re-compute the reduced global stiffness and mass matrices for different values of the model parameters from these expansions, avoiding expensive re-analyses involved in CMS procedure. Dramatic reduction in computational effort has been reported without compromising the accuracy in the modal properties predicted by the reduced model.

The reduction achieved by applying the CMS technique in the FE model of the Metsovo bridge is described in Section 3.3.

#### *2.3. Outline of Procedure*

Given the parameterized FE model of a structure, a parameterized reduced FE model is first obtained using CMS. This amounts to forming the reduced global stiffness and mass matrices as a function of the model parameters *θ*. The TMCMC sampler was used to sample from the posterior PDF in Equation (2), where the likelihood function is given either by Equations (4), (9) and (13) for the vector-based formulation or by Equations (4), (12) and (14) for the MAC-based formulation. The modal properties involved in the likelihood function are computed for each TMCMC sample in the model parameter space using the reduced FE model. Specifically, for each one of the two likelihood formulations presented in Sections 2.1.1 and 2.1.2, the reduced stiffness and mass matrices are used in Equation (1) to predict the modal properties *ωr*(*θ*) and *φr* (*θ*) for different values of the model parameter set *θ*. The sample points *θ*(*j*) , *j* = 1, ... , *N*, obtained from the TMCMC sampler populate the posterior PDF of the model parameters. These samples are subsequently used to depict the uncertainties in the model parameters and propagate uncertainties in output Quantity of Interest (QoI) by providing estimates of the modal frequencies *ωr*(*θ*) and MAC values *MAC*(*φ*ˆ *r* , *φr* (*θ*)), *j* = 1, ... , *N*, using Equation (1) for the reduced FE model. Results of uncertainty quantification are expressed in terms of marginal distributions for the model parameters, as well as useful simplified measures of uncertainty, such as mean and credible intervals of the output QoI.

#### **3. Application to Metsovo Bridge**

#### *3.1. Description of Bridge*

The ravine bridge of Metsovo (Anthohori–Anilio tunnel) of Egnatia Motorway is crossing the deep ravine of Metsovitikos river, 150 m over the riverbed. A picture of the bridge is shown in Figure 1. This is the highest bridge of the Egnatia Motorway, with the height of the tallest pier equal to 110 m. The total length of the bridge is 537 m. The bridge has 4 spans of length 44.78 m, 117.87 m, 235 m, 140 m and 3 piers of which the closest to the left side of the bridge (Figure 1), with height of 45 m, supports the boxbeam superstructure through pot bearings (movable in both horizontal directions), while the other two piers, with heights 110 m and 35 m, connect monolithically to the structure.

#### *3.2. Finite Element Model of Bridge*

The detailed geometry of the bridge is complicated because the deck and the piers have variable cross-sections and the deck is also inclined. A high fidelity FE model of the bridge is created using three-dimensional tetrahedral quadratic Lagrange finite elements. The model takes into account the potential soil-structure interaction by modeling the soil with large blocks of material and embedding the piers and abutments into these blocks. The nominal values of the moduli of elasticity of the deck and piers are selected to be the values used in design: 37 GPa for the deck and 34 GPa for the piers. The nominal value of the soil is taken to be 1 GPa. The largest size of the elements in the mesh is of the order of the thickness of the hollow deck cross-section. The size of the FE mesh is chosen to predict the first 20 modal frequencies and mode shapes of the bridge with sufficient accuracy. Several mesh sizes were tried, and an accuracy analysis was performed in order to find a reasonable trade-off between the number of degrees of freedom (DOF) of the model and the accuracy in the predicted modal frequencies. A mesh of 830,115 DOFs was kept for the bridge-soil model. This mesh was found to cause errors of the order of 0.1–0.5% in the first 20 modal frequencies, compared to the smallest possible mesh sizes which had approximately 3 million DOFs.

**Figure 1.** Picture of the Metsovo bridge.

The intent is to build a high fidelity model that could, in future studies, be extended locally to incorporate nonlinear mechanisms activated during strong motion or deterioration phenomena. In this study the focus is to update a baseline linear model using low-intensity vibration measurements. In future studies, the availability of higher-intensity vibration measurements will provide data for improving modeling and updating parameters of nonlinear models introduced to represent localized nonlinear phenomena activated due to large vibrations or deterioration due to various damage mechanisms. Simplified beam models, although adequate for design purposes, are inadequate to use for setting up digital twins of structures so that are reliable under various operating conditions. Simplified modeling, for example with beam elements, does not offer an adequate representation of the system dynamics over the dynamic range activated by various operational conditions. Such simplified models are often inadequate for monitoring purposes and involve large model errors even for operational conditions under which the structure may be assumed to behave linearly.

#### *3.3. Model Reduction Using CMS*

The time required for a complete run of the FE model is approximately 2 min on a 8-core 3.20 GHz computer. Due to the thousands of forward model runs for different values of the model parameters that are required by the Bayesian computational tools, it is necessary to reduce the time required for a single model run. Model reduction is used to reduce the model size and thus the computational effort to manageable levels. Specifically, the parameterization-consistent CMS technique [34,35] based on the Graig-Bampton method [46] is applied to the bridge-soil FE model.

For this, the bridge is divided into 16 physical components with 15 interfaces between the components. Specifically, the deck is divided into six components or substructures of length 120 m, 120 m, 60 m, 50 m, 117 m and 70 m each. One component is assigned to each one of the three piers. Two components are introduced for the left and right abutments of the bridge. Five more components are introduced for the large solid blocks representing the flexibility of the soil at the

connections with the three piers and the two abutments. This partition into component is one of the many alternative ones, introduced herein to demonstrate the capabilities of CMS technique for model reduction. Usually the partition of the structure into components is guided by the purpose of the analysis or the structural health monitoring goals. For example, components may be introduced to monitor and select models of nonlinearities activated by various operational conditions in isolated (localized) parts of a structure. The partition of a structure into components facilitates monitoring of the structural health, allowing the identification of the location and severity of sparse damage within a small subset of substructures.

For each component, it is selected to retain all modes that have frequency less than *ω*max = *ρωc*, where *ωc* = 3.52 Hz is the cut-off frequency selected to be equal to the 20th modal frequency of the nominal FE model. The *ρ* values affect the computational efficiency and accuracy of the CMS technique. For *ρ* = 5 selected for most components, a total of 170 internal DOFs out of the 814,080 are retained for all 16 components. The total number of DOFs of the reduced model is 16,205 which also includes 16,035 interface DOFs. It is clear that more than an order of magnitude reduction in the number of DOFs is achieved using CMS. The largest fractional error between the modal frequencies computed using the complete FE model and the ones computed using the CMS technique for *ρ* = 5 falls below 0.2%. Thus a very good accuracy is achieved.

The large number of the interface DOFs can be reduced by retaining only a fraction of the constrained interface modes [34,49]. For each interface, only the modes that have frequency less than *ω*max = *νωc* are retained, where *ν* is user and problem dependent. For *ν* = 200 selected for most interfaces, the largest fractional error for the lowest 20 modes of the structure falls below 0.43%. In particular, for *ν* = 200 and *ρ* = 5 the reduced system has 1891 DOFs from which 170 generalized coordinates are fixed-interface modes for all components and the rest 1721 generalized coordinates are constrained interface modes [34]. A trade-off was made between reducing the model as much as possible (fewer kept DOFs) and keeping the accuracy of the predicted modal frequencies as close as possible to those of the unreduced model. It should be noted that further reductions are possible using an enhanced substructuring technique where the dynamics contribution of several kept modes is replaced by their static contribution [47].

Thus, using CMS a drastic reduction in the number of DOFs is obtained which can exceed two orders of magnitude, without sacrificing the accuracy with which the lowest 20 modal frequencies are computed. The time to solution for one run of the reduced model is of the order of a few seconds which should be compared to approximately 2 min required for solving the unreduced FE model.

Moreover, for nonlinear models of structures, especially models where local nonlinearities are mainly activated, the model reduction techniques can also be applied to reduce the models of components of the structure that behave linearly under various operational conditions [35,48].

#### *3.4. Experimental Modal Identification*

The testing system consist of a movable array of servo-accelerometers that are usually being installed on the bridge deck (sidewalks or pavement surface) or inside the box beam internal voids to measure the vibrations (accelerations) of the bridge under ambient excitations. The available measurement system consisted of five triaxial and three uniaxial accelerometers paired with a 24-bit data recording system, a GPS module for synchronization between sensors, and a battery pack. The system is wireless and can be easily moved from one location in the structure to another. The recorder can connect with a laptop through wired (Ethernet) or wireless (Wi-Fi) connection to be set up in the desired way (sampling rate, recording duration, repeater recordings etc) or view the measurements while they are being recorded for quality checking. Given the limited number of sensors and the large length of the deck, the entire length of the deck was covered in 13 sensor configurations, shown in Figure 2. For each configuration the recording lasted 20 min at a sampling rate of 100 Hz. Each triaxial sensor was positioned on the bridge sidewalks such that it measures along the transverse, vertical and longitudinal directions of the bridge deck. One triaxial and three uniaxial

sensors (one vertical and two horizontal transverse) remained in the same position throughout the measurements, in order to provide common measurement points amongst different configurations such as to enable the assembling of the total mode shape from partial mode shape components measured from the different configurations [30,32]. The use of more than one reference sensors per direction guarantees the redundancy of the measuring scheme in case one sensor is placed at the node of the modeshape. The wireless feature of the measurement system allows the execution of all recordings over the 13 sensor configurations in a single day. The recorded responses are mainly due to road traffic, which ranged from light vehicles to heavy trucks, and environmental excitation such as wind loading, which classifies this case as ambient (operational) modal identification.

**Figure 2.** Measured locations in the bridge. Solid grey circles represent the locations of the triaxial sensors measuring along the longitudinal, transverse and vertical direction of the bridge deck. Solid blue circle shows the location of the reference triaxial sensor. Solid blue rectangle denotes the locations of the two uniaxial sensors measuring along the transverse direction. Solid blue triangle denotes the location of uniaxial sensor measuring along the vertical direction.

The Bayesian operational modal analysis methodology [9,10] is used to estimate the modal frequencies, mode shapes and damping ratios for each sensor configuration. The mode shapes are assembled from the local mode shapes of each configuration using the methodology proposed by Au [32]. The full mode shapes are produced at all 159 sensor locations covered by the 13 sensor configurations. The components along the longitudinal direction of the bridge deck are ignored. Only the components along the transverse and vertical direction of the bridge deck are processed. The output-only vibration measurement for some of the 13 sensor configurations were not reliable enough to estimate the mode shape components at higher modes. As a result, it was not possible to assemble the mode shapes for more than 12 modes. Thus these mode shapes were excluded from the analysis. Specifically, the first 20 modal frequencies and modal damping ratios of the bridge were identified, along with 11 mode shapes. The mode shapes of all the modes up to the 12th were identified, except the 10th mode which was very poorly identified and also excluded from the date set. Table 1 presents the mean and standard deviation of the experimentally identified modal frequencies for all 20 identified modes of the Metsovo bridge. It also compares the identified frequencies and mode shapes with those predicted by the nominal FE model. In particular, the experimental and nominal model predicted mode shapes are compared using their MAC value which is a scalar measure of correlation between two mode shapes ranging from 0 to 1, with a value of 1 indicating perfect correlation. The identified mode shapes are shown in Figures A1–A4 of Appendix A and compared with the corresponding mode shapes predicted by the nominal FE model of the bridge. From both the MAC values of Table 1 and mode shapes of Figures A1–A4 it can be clearly seen that the mode shapes predicted by the nominal FE model match very accurately the corresponding experimentally identified

mode shapes with MAC values higher than 0.95 for the 11 identified mode shapes (except mode 9 which has a MAC value of 0.87). However, there appears to be a significant mismatch between the experimental and nominal FE model modal frequencies which indicates that a finite element model updating should be performed in order to achieve a closer fit between the model predicted and the experimentally identified modal frequencies.

**Table 1.** First 20 experimentally identified modal frequencies of the the Metsovo bridge compared with the modal frequencies predicted by the nominal FE model of the bridge. MAC values between identified and model predicted mode shapes are also shown.

#### **4. Model Updating Results**

The FE model of the bridge–soil system is parameterized using three parameters associated with the modulus of elasticity of the deck (*θ*1), piers (*θ*2) and soil (*θ*3). The model parameters multiply the nominal values of the corresponding moduli of elasticity for the deck (37 GPa ), the piers (34 GPa) and the soil (1 GPa). The nominal values for the deck and piers are reasonable estimates since they are the moduli of elasticity of the concrete used in design and therefore their updated values of *θ*1 and *θ*2 are expected to lie close to 1. However, as far as the soil is concerned, its nominal value is only a rough estimate, based on soil property measurements conducted at the site of the bridge. Therefore, its nominal value should be dealt with a large uncertainty in the model updating procedure. These modeling considerations regarding the initial parameter uncertainties are taken into account in the Bayesian framework through the prior PDF. It should be noted that a simplified uniform parameterization involving a small number of parameters is considered in order to avoid possible unidentifiability issues and enable the comparison between the two different likelihood formulations.

#### *4.1. Model Updating Using Modal Frequencies Only*

First, the FE model of the bridge-soil system is updated using only a subset of the experimentally identified modal frequencies. This approach allows one to use the rest of the frequencies in order to validate the updated model by checking its predictive capabilities with data that was not used in the updating. Specifically, the first 15 identified modal frequencies are used to estimate the model parameters and their uncertainty, while the other five modal frequencies are used in order to validate the updated model. For 11 out of the 20 modes we use mode correspondence through the MAC values

to associate the experimentally identified and model predicted modal properties. It was found that the *i*-th experimentally identified mode corresponds to the *i*-th mode predicted by the model. For modes higher than 12 for which there no mode shape identified from experimental data, we match the modal frequencies based only on the number of mode identified or predicted by the FE model, with modal frequencies arranged in an ascending order.

## 4.1.1. Flexible-Soil Model

The prior distribution for the parameters are assumed to be uniform with bounds in the domain [0.5, 1.5] × [0.5, 1.5] × [0.1, 1000] for the deck, pier and soil parameters respectively, and in the domain [0.001, 1] for the prediction error parameter *σω*. The domain for the soil parameter was deliberately chosen much larger in order to account for the large uncertainty in the values of the soil stiffness and be able to explore the full effect of the soil stiffness on the model behavior.

Model updating results are obtained using the parallelized TMCMC algorithm [33,36] for the bridge–soil FE model. The TMCMC is used to generate samples from the posterior PDF of the structural model and prediction error parameters. These samples represent the posterior PDF and therefore our updated state of knowledge about the parameters given the experimental data. After the posterior samples are drawn the parameter uncertainty is propagated to the predictions of the first 20 modal frequencies of the bridge. This is done in order to check the fit of the updated model with the experimental frequencies that were used to perform the model updating, but also with the next five modal frequencies that were not included in the data set. In all TMCMC runs, the following selection is made: *TolCov* = 1.0, *β* = 0.2 [33]. The number of samples used per TMCMC stage are 1000, resulting in a total runtime of approximately 10 minutes using the reduced 1.891 DOF model in a 8-core 3.20 GHz computer.

The TMCMC samples which represent the posterior PDF are visualized through their marginal distributions and two-dimensional (2D) projections in Figure 3. The sample statistics are shown in Table 2. The posterior parameter uncertainty is propagated through the model using the samples to yield the robust model predictions of the lowest 20 modal frequencies. The fit is shown in Figure 4. The predicted modal frequencies are normalized with respect to the experimentally-identified frequencies for comparison convenience. Therefore, values close to 1 are close to the experimental frequencies. The improvement achieved by the updated model compared to the nominal model is evident. For most modes the experimental frequency lies within the predicted 5–95% interval or very close to it, and in all cases the error is of the order of 3–4% which should be compared to the error of the nominal model which is of the order of 10% to 20% for some modes. This is a strong indication of the need for model updating in order to improve the accuracy and predictive capability of the updated model.

Regarding the parameters, it can be seen that the updated values of the deck and pier parameters lie close to 1 as expected, and slightly below it. The mean values for the deck and pier stiffness parameters are estimated to be approximately 0.95 and 0.98 times their nominal values with uncertainties of the order of 5% and 12% respectively. From the (*θ*1 − *θ*2) 2D projection of Figure 3 it is evident that a negative correlation exists between the deck and pier stiffnesses. This is reasonable since an increase in the stiffness of the deck can be counterbalanced by a decrease in the stiffness of the piers such that the modal frequency values are maintained, and vice versa.

As far as the updated soil stiffness is concerned, the only (but important) new information that is acquired by the model updating is that its value can be arbitrarily large, as long as it exceeds a threshold. The threshold value appears to be approximately 70 which is the minimum value that the updated soil parameter can attain, as seen from its posterior marginal distribution in Figure 3. A value of 70 implies a soil modulus of elasticity of 70 GPa which is more than double of the updated (and nominal) value of the pier modulus of elasticity (34 GPa). The soil parameter can increase substantially above this value without affecting the fit with the experimental data, that is, without causing any variation in the predicted modal frequencies of the model. Considering that the uniform prior bound for the soil stiffness was set to [0.1, 1000] it is obvious that lower values which would attribute to the soil some

flexibility similar to that of the piers are not preferred. In addition, the large posterior uncertainty in soil property indicates that the modal frequencies are insensitive to the values of the soil modulus of elasticity for these high values of the soil property. This insensitivity is due to the low vibration levels recording from ambient operational conditions of the bridge.

**Figure 3.** Posterior marginal distributions and 2D sample projections of model parameters. *θ*1: deck, *θ*2: piers, *θ*3: soil, *σ*: prediction error, log-evidence = 2.527.

**Figure 4.** Uncertainty propagation to the first 20 modal frequencies compared with the experimental frequencies and nominal model predictions.

**Table 2.** Quantiles of posterior samples for all parameters. The x% quantile for a parameter is defined as the level that corresponds to x% of the values of the parameter over all TMCMC samples, arranged in increasing order, to fall below this level.

#### 4.1.2. Two-Parameter Stiff-Soil Model

The results obtained from the flexible-soil model suggest that the bridge appears to be fixed to the ground and the modal properties predicted by the model are insensitive to the soil modulus of elasticity. This leads to introducing a second model, which corresponds to eliminating the soil parameter by fixing its value to a large value as suggested by its posterior marginal distribution of Figure 3, simulating the very stiff soil conditions which were found from the first model. Therefore, the new two-parameter model has as parameters the modulus of elasticity of the deck (*θ*1) and piers (*θ*2), while the soil parameter is fixed to 100.

The posterior samples for the two-parameter model are visualized using their marginal distributions and 2D projections in Figure 5. The sample statistics are shown in Table 3. The posterior parameter uncertainty is propagated through the model using the samples to yield the robust model predictions of the lowest 20 modal frequencies. The fit is shown in Figure 6. Note that in Figure 6 the predictions of the nominal model are closer to the experimental due to the increase of the soil parameter to the fixed value of 100 in order to simulate the stiff-soil conditions, which led to an increase of the modal frequencies of the nominal model. It can be seen that, as expected, the model updating results both in terms of the updated values of the parameters and in terms of the fit with the data are almost identical to the results obtained from the three-parameter model in Figure 4.

**Figure 5.** Posterior marginal distributions and 2D sample projections of model parameters. *θ*1: deck, *θ*2: piers, *σ*: prediction error, log-evidence = 2.550.

**Table 3.** Quantiles of posterior samples for all parameters.

This is also confirmed using the Bayesian model selection framework [38] to compute the evidence *p*(*D*|*Mi*) for the two models, taking into account both the complexity of the models in the form of the number of its parameters and the fit they achieve with the data in order to obtain a trade-off between the two. The TMCMC algorithm provides the values of the evidence of each model as a by-product of the algorithm. Therefore, by performing model updating on both models they can be easily compared using their evidence values. The log-evidence for the three-parameter flexible-soil model was found to be 2.52, which is slightly less than the evidence value 2.55 of the two-parameter stiff-soil model. Bayesian model selection slightly rewards the stiff-soil model for having one less parameter than the flexible-soil model.

**Figure 6.** Uncertainty propagation to the first 20 modal frequencies compared with the experimental data and nominal model.

#### *4.2. Model Updating Using Modal Frequencies and Mode Shapes*

Next we also include the mode shapes into the dataset used for model updating. Both the vector-based (Section 2.1.1) and the MAC-based (Section 2.1.2) formulations of the likelihood are used to update the deck and pier model parameters of the two-parameter FE model. Regarding the vector-based likelihood formulation, two cases of mode shape component correlation are examined, namely the uncorrelated and exponentially correlated cases. For the exponentially correlated case, two correlation lengths are examined: *λr* = 100 m and *λr* = 500 m for all *r* values.

A crucial aspect of the analysis is to examine the effect of the number of mode shape components (sensors) used in the likelihood function on the model parameter uncertainty and uncertainty in model predictions. In order to study this effect, five different sensor configurations are considered with 8, 14, 26, 52, 105 measured DOF. For each configuration the sensors are selected to be uniformly spread along the bridge deck. In addition, the configuration with a larger number of measured DOF includes the measured DOF contained in configurations with smaller number of DOF. In this way, the information contained in the data of a configuration with a given number of measured DOF, includes the information contained in the data of a configuration with smaller number of DOF. In each sensor configuration case, half of the DOF are transverse (sensors measuring in the transverse direction) and half are vertical. The longitudinal DOF were not included due to their negligible contribution in the identified mode shapes compared to the transverse and vertical components. The transverse and vertical DOF were selected to be in the same point, that is, eight DOF correspond to four different pairs of transverse and vertical DOF in the same point. The case of 52 DOF corresponds to the complete set of measured DOF in the one side of the bridge, while the case of 105 DOFs corresponds to measured DOFs on both sides of the bridge. Due to the type of the vertical and transverse mode shapes, the mode shape components at one side of the bridge provide exactly the same information as the mode shape components at the opposite side of the bridge. So the case of 105 measured DOFs should not be expected to provide additional information as compared to the case of 52 sensors.

Figures 7 and 8 show the posterior parameter uncertainty for the deck and pier parameters of the model as a function of the number of sensors, for each case of likelihood formulation. The posterior uncertainty for each parameter is shown in terms of the 5%, 50% and 95% quantiles of the marginal posterior samples obtained from the TMCMC algorithm for the corresponding parameter.

It should be noted that the vector-based likelihood formulations for the mode shapes (uncorrelated and the two exponentially correlated models with spatial correlation lengths of 100 and 500) result in a steady reduction in the posterior uncertainty of both model parameters (deck and pier parameters), as the number of mode shape components used in the likelihood is increased. This is in agreement with the Bayesian theory of parameter estimation, which suggests that as the number of data points used in the likelihood is increased the posterior uncertainty is reduced. Indeed, in the vector-based formulations, the mode shapes are treated as vectors of size equal to the number of their used components. Therefore, the total number of data points used in the likelihood is increased as we use more of the identified mode shape components.

However, as more mode shape components are used, the locations of the sensors become increasingly closer to each other. The shorter characteristic length corresponding to the lowest 10 identified mode shapes is approximately 130 m as one can observe from Figures A1–A4 of Appendix A. As the number of sensors increase to 25 or higher, the shortest distance between sensors becomes a fraction of the characteristic length of the identified mode shapes and so there is redundant information contained in the measured mode shape data. In fact no new information is expected from sensors placed at a distance that is sufficiently smaller than the characteristic length of a mode shape. Especially in the case of 52 and 105 DOFs (which correspond to the entire set of identified mode shape components in one side and both sides of the bridge) we do not expect the inclusion sensor information from the second side to further reduce the posterior uncertainty. This is because the identified mode shape components at the two sides of the bridge are almost identical, and therefore including the second side does not provide any new information about the transverse and vertical mode shapes. The same holds true to a lesser degree for the other cases of DOFs because the sensors are getting closer as we use more of them and contain very similar information. So we would expect the posterior uncertainty to initially reduce as we increase the number of sensors, but only up to a certain point, and then remain practically constant as we include more sensors due to the redundant information provided from the closely spaced sensors or sensors placed at opposite sides of the bridge.

This expected behavior is opposite to what is observed using the vector-based formulations of the likelihood for the mode shapes. Even adding the sensors at the second side of the bridge (which provide identical information with the sensors in one side) seems to further reduce the posterior parameter uncertainty for both the deck and pier parameters. Correlated prediction error models have been suggested to alleviate this situation, and have been successful in some cases, but these correlated prediction error models are very difficult to postulate correctly in practice and could otherwise lead to erroneous results [41].

**Figure 7.** Deck parameter posterior uncertainty versus number of sensors. The MAC case corresponds to the MAC-based likelihood formulation, while "uncorrelated", "correlated 100" and "correlated 200" cases correspond to vector-based likelihood formulations with exponential spatially correlated prediction error given by Equation (8) with *λr* = 0, 100 and 200 respectively.

**Figure 8.** Pier parameter posterior uncertainty versus number of sensors. Description of curves as in Figure 7.

A totally different behavior is observed under the MAC-based likelihood formulation for the mode shapes. The posterior uncertainty does decrease at first, but then it stabilizes and is practically unaffected by the inclusion of more sensors after some point. Specifically, we see that when the number of sensors increase from 8 to 14 and 26, there is a reduction in the uncertainty, but after that point the uncertainty gets stabilized and is not affected by doubling the number of sensors to 52 and eventually to 105. This

happens because in the MAC-based formulation the mode shapes are not utilized as vectors, but as scalar MAC values, reducing the effective number of independent data points for each mode shape to one, instead of as many as the number of mode shape components. It is also important to note that the overall parameter uncertainty is much larger compared to the vector-based formulations indicating that no significant information gain occurs by further increasing the number of sensors.

A quantitative assessment is given in Table 4 which shows the 5–95% credible interval for the posterior PDF of the deck and pier model parameters for different number of sensors under the vector-based (uncorrelated) and MAC-based likelihood formulations. It can be seen that the vector-based (uncorrelated) formulation keeps reducing the posterior uncertainty of the model parameters as the number of sensors increase, whereas the uncertainty is maintained for the MAC-based likelihood formulation for 26, 52 and 105 sensors.

> **NDOF Uncorrelated MAC Deck Pier Deck Pier** 8 0.070 0.087 0.104 0.241 14 0.056 0.083 0.100 0.216 26 0.041 0.064 0.091 0.190 52 0.032 0.045 0.088 0.191

**Table 4.** Deck and pier posterior uncertainty (90% credible interval) versus the number of sensors for the vector-based (uncorrelated) and MAC-based formulations of likelihood.

Uncertainty Propagation to Modal Frequencies and MAC Values

105

 0.021 0.033 0.087 0.186

The uncertainty in the model parameter values is propagated to modal frequencies in Figure 9 and MAC values in Figure 10. Results are presented for the vector-based (uncorrelated) and MAC-based formulations using 105 DOFs for the mode shapes (sensors on both sides of the bridge). The modal frequency predictions in Figure 9 are normalized with respect to the experimental modal frequencies.

The larger posterior parameter uncertainties obtained with the MAC-based likelihood formulation result in larger uncertainties in the predicted modal frequencies compared to the uncertainties predicted by the vector-based (uncorrelated) likelihood formulation. The experimental frequencies are included within the 5–95% credible intervals predicted by the MAC-based likelihood formulation for 10 out of the 15 modes (the black horizontal line crosses the 5–95% interval except for 5 modes), while the modal frequency predictions obtained from the uncorrelated model do not include the experimental frequencies within the 5–95% credible intervals, except from only three modes (4-th, 6-th and 15-th modes). Therefore, the predictions obtained from the vector-based likelihood formulation have more error associated with them compared with those obtained from the MAC formulation, when checking against the experimentally identified modal frequencies. Thus, the MAC-based likelihood formulation has better predictive capabilities than the vector-based likelihood formulation, in the sense that the predicted uncertainty bounds either fully contain or are closer to the experimental modal frequencies.

The predicted MAC values presented in Figure 10 have also larger uncertainties as expected under the MAC formulation, but the difference is not as obvious as in the modal frequencies (except for mode shapes 11 and 12 which have large uncertainties in their MAC value). The MAC values are well above 0.95 (with the exception of mode shape 9 which has a MAC value of 0.88), indicating a very close match between the experimental and model predicted mode shapes. Note that the MAC values obtained from the nominal model (green circles in Figure 10) are contained within the credible intervals of the vector-based likelihood formulation. This indicates that the mode shapes are highly insensitive to changes in the values of the two model parameters. However, it was demonstrated that inclusion of the mode shapes in the likelihood function does play an important role in the resulting posterior uncertainty of the model parameters, thereby affecting the uncertainty in the predicted modal frequencies.

**Figure 9.** Posterior robust predictions for the modal frequencies normalized by the experimental modal frequencies under the vector-based (uncorrelated) and MAC-based formulations for the likelihood. Vertical lines (with three circles) for each mode number show the 5%, 50% and 95% quantiles. The isolated green circles show predictions from the nominal FE model.

**Figure 10.** Posterior robust predictions for the mode shapes compared with the experimental mode shapes using the MAC value, under the vector-based (uncorrelated) and MAC-based formulations for the likelihood. Vertical lines (with three circles) for each mode number show the 5%, 50% and 95% quantiles. The isolated green circles show predictions from the nominal FE model.

Based on the uncertainty results for the model parameters presented in Figures 7 and 8, it is expected that the uncertainty in the modal frequencies and mode shapes presented in Figures 9 and 10 will be unaffected for the MAC-based likelihood formulation when the number of sensors is reduced from 102 to 54 or 27. However, for the vector-based likelihood formulation the uncertainty in the modal frequencies and MAC values is expected to increase due to the increase in the parameter uncertainties in Figures 7 and 8 when the number of sensors is reduced from 102 to 54 or 27.

## **5. Conclusions**

A Bayesian framework was presented for FE model updating of structures using experimentally identified modal frequencies and mode shapes. A novel way for including the mode shapes into the likelihood formulation was proposed by assigning a probability model to the MAC values between the experimentally identified and model predicted mode shapes, summarizing the information in the mode shapes in scalar features instead of vectors as it is conventionally done in existing formulations. The MAC-based likelihood formulation provides uncertainty bounds of the model parameters which are consistent with expectations as the number of sensors increases, while the vector-based likelihood formulation fail to properly account for the redundant information contained in the mode shape components, especially for relatively closely spaced sensors. The merits of the new likelihood formulation in relation to existing formulations were explored by updating the FE model of the Metsovo bridge. A high fidelity FE element model of hundreds of thousand of DOF was developed to accurately model the dynamic behavior of the bridge. TMCMC was used to perform the model updating, while model reduction techniques were effectively employed to drastically reduce the computational effort to manageable levels.

It was demonstrated that the model-updating results obtained from the MAC-based likelihood formulation differ significantly from the ones obtained by classical vector-based likelihood formulations. Specifically, the posterior parameter uncertainty was found to be stabilized as the number of sensors in the mode shapes are increased or the distance between sensors is relatively less than the characteristic lengths of the identified mode shapes, or as extra mode shape components (at the opposite side of the bridge), containing redundant information, are added. In contrast, the uncertainty in the model parameters for the classical vector-based likelihood formulation is decreasing as the number of sensors increases, which is counter-intuitive since it does not take into account the redundant information contained in measurements. This decrease in uncertainty is observed for spatially uncorrelated and exponentially correlated prediction error models considered in this study. Propagating the uncertainty in modal frequencies and MAC values, it is demonstrated that the MAC-based likelihood formulation provides wider uncertainty bounds that contain the experimental data. In contrast, the uncertainty bound predicted by the vector-based likelihood formulation fail to fit the experimental data since there is a significant distance of the experimental data from the predicted uncertainty bounds.

**Author Contributions:** Conceptualization, C.A. and C.P.; methodology, C.A., P.T., P.P. and C.P.; software, C.A.; validation, C.A.; formal Analysis, C.A.; investigation, C.A. and P.P.; resources, C.A. and P.P.; writing—original draft preparation, C.A.; writing—review and editing, C.P., P.T. and P.P.; visualization, C.A.; supervision, C.P. All authors have read and agreed to the published version of the manuscript.

**Acknowledgments:** We thank Ivan Siu Kui Au for providing a copy of the Bayesian Operational Modal Analysis software used in this study to estimate the modal properties of the bridge.

**Conflicts of Interest:** The authors declare no conflict of interest. *J. Sens. Actuator Netw.* **2020**, *9*, 27

#### **Appendix A. Supplementary Data**

**Figure A1.** Comparison between the experimentally identified (left column) and nominal FE model predicted (right column) mode shapes of the Metsovo bridge. Modes 1–3.

**Figure A2.** Comparison between the experimentally identified (left column) and nominal FE model predicted (right column) mode shapes of the Metsovo bridge. Modes 4–6.

**Figure A3.** Comparison between the experimentally identified (left column) and nominal FE model predicted (right column) mode shapes of the Metsovo bridge. Modes 7–9.

**Figure A4.** Comparison between the experimentally identified (left column) and nominal FE model predicted (right column) mode shapes of the Metsovo bridge. Modes 10–12.

## **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Data-Interpretation Methodologies for Practical Asset-Management**

#### **Sai G. S. Pai 1,2,\*, Yves Reuland 1 and Ian F. C. Smith 1,2**

Received: 23 April 2019; Accepted: 11 June 2019; Published: 22 June 2019

**Abstract:** Monitoring and interpreting structural response using structural-identification methodologies improves understanding of civil-infrastructure behavior. New sensing devices and inexpensive computation has made model-based data interpretation feasible in engineering practice. Many data-interpretation methodologies, such as Bayesian model updating and residual minimization, involve strong assumptions regarding uncertainty conditions. While much research has been conducted on the scientific development of these methodologies and some research has evaluated the applicability of underlying assumptions, little research is available on the suitability of these methodologies to satisfy practical engineering challenges. For use in practice, data-interpretation methodologies need to be able, for example, to respond to changes in a transparent manner and provide accurate model updating at minimal additional cost. This facilitates incremental and iterative increases in understanding of structural behavior as more information becomes available. In this paper, three data-interpretation methodologies, Bayesian model updating, residual minimization and error-domain model falsification, are compared based on their ability to provide robust, accurate, engineer-friendly and computationally inexpensive model updating. Comparisons are made using two full-scale case studies for which multiple scenarios are considered, including incremental acquisition of information through measurements. Evaluation of these scenarios suggests that, compared with other data-interpretation methodologies, error-domain model falsification is able to incorporate, iteratively and transparently, incremental information gain to provide accurate model updating at low additional computational cost.

**Keywords:** probabilistic data-interpretation; Bayesian model updating; error-domain model falsification; iterative asset-management; practical applicability; computation time

## **1. Introduction**

Improving living conditions and a global trend in migration of population from rural to urban centers result in increasing demand for civil infrastructure [1]. However, most present-day infrastructure has been built in the second half of the twentieth century and is close to the end of its designed service life. The deficit between demand and supply has been estimated to be USD 1 trillion in 2014 [2] and is increasing. Replacement of all aging infrastructure is unsustainable. However, civil infrastructure are generally designed using conservative models and, thus, they may possess reserve capacity beyond code requirements [3,4]. For quantification of such reserve capacity, better understanding of structural behavior is needed. Thus, decision making regarding asset-management actions such as repair, retrofit and replacement, is enhanced [5].

Measurements of structural response can be interpreted using physics-based models in order to enhance understanding of structural behavior. Increased availability and reduced cost of sensing techniques [6,7] and computational tools [8] have made model-based data interpretation feasible. However, all models are idealizations of reality [9]. Conservative modeling assumptions lead to large uncertainty, with systematic and correlated errors at measurement locations [10]. Many researchers have studied uncertainties that affect interpretation of civil-infrastructure response [11–13]. Improvement in quantification of uncertainties can help improve accuracy of data interpretation.

Interpretation of measurements using a physics-based model is referred to as structural identification. Due to the presence of uncertainties, structural identification, which is an abductive task, is an ill-posed problem. Methodologies for solving such inverse problems have been studied by many researchers [14–17]. In practical applications, residual minimization (also called model calibration) is the most commonly used model-based measurement-interpretation method. For residual minimization, optimal values of parameters governing model behavior are estimated by minimizing the residual between model predictions and measurements [18]. Although popular among practicing engineers due to its simplicity, residual minimization may provide inaccurate results [19]. An assumption made by residual-minimization methods is that the difference between model predictions and measurements is governed only by the choice of parameters [20]. This implies that systematic bias between the approximate model and measurements is not taken into account during parameter estimation. In other words, the difference between model predictions and measurements is assumed to be distributed as zero-mean uncertainty forms [10,21–23].

Another methodology that has gathered much interest from the data-interpretation community is Bayesian model updating (BMU). Traditionally, BMU employs an independent zero-mean Gaussian likelihood function [24]. Model parameters, considered as probabilistic distributions are updated using this likelihood function. Model-parameter combinations that provide predictions whose error with measurements are low are attributed higher likelihood. Many developments over the traditional implementation of BMU have been made to account for the presence of model bias [25–27]. However, mis-evaluation of systematic bias and correlations leads to inaccurate estimation of model parameters [28–32].

Goulet and Smith [29] presented a multi-model data-interpretation methodology called error-domain model falsification (EDMF). In EDMF, model-parameter instances are falsified when their predictions are not compatible with measurements. Compatibility is determined based on falsification thresholds that are computed based on the uncertainties affecting identification of parameter values. Estimation of uncertainties involves information available from tests, guidelines and engineering heuristics. EDMF has been shown to provide more accurate identification and predictions compared with traditional BMU and residual minimization [29–32].

Data interpretation for asset management is an iterative task, with re-evaluations required as new information becomes available. New information can be: new measurements; change in uncertainty conditions; and new diagnostic information. Pasquier and Smith [33] presented an iterative sequence-free data-interpretation framework using EDMF and highlighted the iterative nature of data interpretation. A similar framework for post-earthquake assessment using EDMF was presented by Reuland et al. [34]. Zhang et al. [35] developed a data-interpretation tool that employs BMU with the goal of assisting asset managers. Except from these studies, most present day research in structural identification has focused on damage detection within a sequential framework [36]. In addition, no research is available that evaluates the usefulness of data-interpretation methodologies such as BMU and residual minimization within iterative frameworks to assist asset managers faced with real-world challenges. Model-based data-interpretation has the potential to enhance asset management strategies. These strategies have been developed by many researchers using, for example, multi-criteria decision making [37,38] and asset-performance metrics [39,40].

Application of model-based data-interpretation methodologies for full-scale structures presents challenges such as unidentifiability [41] and, above all, constraints on computational cost [42]. Structural identification of full-scale structures, unlike laboratory experiments, are affected by environmental conditions such as wind [43] and temperature [44]. Moreover, numerical models of full-scale structures are computationally expensive. Many researchers have suggested using efficient sampling methods to alleviate constraints on computational cost. Residual minimization has been implemented with optimization algorithms such as genetic algorithms [45], artificial bee colony optimization [46,47], particle swarm optimization [48] and ant colony optimization [49] to reduce computational cost. BMU has been implemented using Markov-Chain Monte Carlo (MCMC) sampling [50], transitional MCMC sampling [51], and evolutionary MCMC sampling [52]. EDMF has traditionally been implemented using grid sampling, which is computationally expensive [53]. Adaptive-sampling strategies such as radial-basis functions [54] and probabilistic global search optimization [55] have been implemented to improve sampling efficiency for EDMF. While use of these search methods decreases computational cost, their efficiency for use in an iterative framework for data-interpretation has not been studied.

In this paper, several methodologies are compared based on their ability to efficiently incorporate new information and changing uncertainty definitions to provide accurate structural identification. Comparisons have been made using two full-scale bridge case studies to evaluate applicability of these data-interpretation methodologies outside of well-controlled laboratory environments.

#### **2. Model-Based Data-Interpretation for Asset Management**

Model-based data interpretation of civil infrastructure is difficult due to many scientific and practical challenges. To allow asset managers to exploit potential reserve capacity safely, accurate interpretation of measurement data is necessary. In addition, civil infrastructure (such as bridges and tunnels) form critical components in transportation networks. Their failure can cause loss of life and cascading disruptions to economies due to loss of connectivity. Thus, in addition to accuracy, ease of interpretation of data-interpretation results is imperative to asset managers.

In Figure 1, typical steps involved in using model-based data-interpretation is presented. The sequence shown in the flowchart is general and a sequence-free framework for more specific asset management tasks was presented by Pasquier and Smith [33]. In this paper, the steps involving model-based data-interpretation and validation are discussed.

**Figure 1.** Flowchart detailing typical steps involved in use of model-based data-interpretation methodologies for asset management.

In Figure 1, site investigations help collect data useful for modeling and understanding sources of uncertainty. The task of sensing involves collecting data pertaining to structural response during either a load test or in-service conditions.

The task of model-based data-interpretation in Figure 1 consists of quantification of uncertainties, determination of a model class for identification and use of measurement data to update the values of parameters defined by the model class. Interpretation of measurement data may be carried out using various methodologies and some of these are described in Section 2.1.

A key task that is generally absent in most applications of model-based data interpretation is validation before making predictions. The validation task, subsequent to data interpretation (see Figure 1), has been recommended to be carried out using a leave-one-out cross-validation strategy in this paper. This is explained in Section 2.2.

Data interpretation, as shown in Figure 1, is iterative, especially as new information becomes available over the service life of the structure. New information takes the form of new measurements, improved understanding of uncertainties, improved understanding leading to a new model class, etc. Lack of validation of data-interpretation results may also necessitate iterations. Therefore, methodologies for data interpretation should be amenable to new information and flexible to changes.

In Section 2.1, data-interpretation methodologies that are available in the literature are described. These methodologies make implicit assumptions regarding estimation and quantification of uncertainties affecting structural identification. Methodologies also differ in the sampling strategies used to obtain appropriate solution(s). Accuracy of these methodologies in interpreting measurement data is dependant on the validity of assumptions. A leave-one-out cross-validation method to assess accuracy and precision of data-interpretation is explained in Section 2.2. In this paper, a comparison of data-interpretation methodologies based on iterative applications is presented. The objective of these comparisons and validation checks are to help engineers select a suitable methodology to interpret measurements using physics-based models.

#### *2.1. Background of Data-Interpretation Methods*

In this paper, four data-interpretation methodologies are compared with respect to their ability to provide accurate identification and incorporate new information in an iterative manner. In the following, the four methodologies are briefly introduced and their inherent assumptions are discussed.

### 2.1.1. Residual Minimization

In residual minimization, a structural model is calibrated by determining model-parameter values that minimize the error between model predictions and measurements. A typical objective function for residual minimization is shown in Equation (1).

$$\hat{\theta} = \operatorname\*{argmin}\_{\theta} \sum\_{i=1}^{n\_y} \left( \frac{\mathcal{g}\_i(\theta) - \mathcal{g}\_i}{\mathcal{g}\_i} \right)^2 \tag{1}$$

In Equation (1), *θ* is the optimum model-parameter set obtained using measurements and *gi*(*θ*) − *y*ˆ*i* is the residual obtained between the model response, *gi*(*θ*), and measurement, *y*ˆ*i*, at measurement location *i*.

Predictions with models updated using residual minimization are limited to the domain of data used for calibration [56]. Therefore, calibrated model-parameter values may only be suitable for predictions that involve interpolation [56] and not for extrapolation (predictions outside the domain of data used for calibration) [19,20].

#### 2.1.2. Traditional Bayesian Model Updating

ˆ

Bayesian model updating (BMU) is a popular probabilistic data-interpretation methodology [24,57,58] based on Bayes' theorem. In BMU, prior information of model parameters, *P*(*θ*), is conditionally updated using a likelihood function *P*(*y* | *θ*) to obtain a posterior distribution of model parameters, *P*(*θ* | *y*), as shown in Equation (2).

$$P(\theta \mid y) = \frac{P(y \mid \theta) \cdot P(\theta)}{P(y)} \tag{2}$$

In Equation (2), *P*(*y*) is the normalization constant. *P*(*θ*) is the prior distribution of model parameters, which indicates prior available knowledge regarding parameter values. The likelihood function, *P*(*y* | *θ*), is the probability of observing the measurement data, *y*, for a specific set of model-parameter values, *θ*. The most commonly used likelihood function is a *L*2-norm-based Gaussian probability-distribution function (PDF), as shown in Equation (3).

$$P(\boldsymbol{y} \mid \boldsymbol{\theta}) \propto constant \cdot \exp\left[ -\frac{1}{2} \left( \boldsymbol{g} \left( \boldsymbol{\theta} \right) - \boldsymbol{y} \right)^{\mathrm{T}} \boldsymbol{\Sigma}^{-1} \left( \boldsymbol{g} \left( \boldsymbol{\theta} \right) - \boldsymbol{y} \right) \right] \tag{3}$$

In Equation (3), Σ is a covariance matrix that consists of variances and correlation coefficients of uncertainties related to each measured location. In most applications of BMU, uncertainties at measurement locations are assumed to be independent zero-mean Gaussian distributions [59–66]. In addition, the variance in uncertainty, *σ*2, is assumed to be the same for all measurement locations. This leads the covariance matrix to be a diagonal matrix, with all non-zero terms being equal. However, the assumption of a *L*2-norm-based Gaussian distribution for uncertainty [67] and uncorrelated error [28] is rarely satisfied and may lead to a biased updated probability distribution [29,30,32].

#### 2.1.3. Error-Domain Model Falsification

Error-domain model falsification (EDMF) is a data-interpretation methodology developed by Goulet and Smith [29]. EDMF is based on the assertion by Popper [68] that models cannot be validated by data; they can only be falsified. Model instances (instances of model-parameter values) are falsified based on information from measurements. Model instances that are not falsified form a candidate set, which is a subset of all possible parameter values based on the prior model parameter PDFs.

Generally, civil infrastructure are designed using conservative and simplified models. As a result, engineering models possess significant model bias from sources such as simplification of loading conditions, geometrical properties, material properties and boundary conditions. Extent of these uncertainties can only be estimated using engineering heuristics and usually takes the form of bounds.

Let  *mod*,*q* be the modeling uncertainty and  *meas*,*q* the measurement uncertainty, both at a measurement location *q*. Let the structure be represented by a physics-based model, *g*(*θ*). The true response of the structure at a measurement location is given by Equation (4).

$$y\_q + \epsilon\_{mars,q} = R\_q = \mathcal{g}\_q \left(\theta^"\right) + \epsilon\_{mod,q} \tag{4}$$

In Equation (4), *gq* (*θ*∗) is the model response at a measurement location *q* for the real values of the model parameters, *θ*∗. *yq* is the measured response of the structure at measurement location *q*. Rearranging the terms of Equation (4), a relationship among model response, *g*(*θ*), measurement, *yq*, and uncertainties,  *meas*,*q* and  *mod*,*q*, at location *q* is obtained, as shown by Equation (5).

$$
\varepsilon\_{\mathcal{J}} \left( \theta^\* \right) - y\_{\mathcal{J}} = \varepsilon\_{\text{meas},\eta} - \varepsilon\_{\text{mod},q} \tag{5}
$$

In Equation (5), the residual between model response, *g*(*θ*), and measurement, *yq*, at a sensor location, *q*, is equal to the combined model and measurement uncertainty. Engineers make design decisions based on predefined target reliability. Using the target reliability, *φ*, for identification, the criteria for falsification, thresholds *Thigh*,*q* and *Tlow*,*q*, are computed using Equation (6).

$$\boldsymbol{\Phi}^{1/m} = \int\_{T\_{\text{low},q}}^{T\_{\text{high},q}} f\_{\text{IL}\_{\text{c},q}} \left( \boldsymbol{\epsilon}\_{\text{c},q} \right) d \boldsymbol{\epsilon}\_{\text{c},q} \tag{6}$$

In Equation (6), *f Uc*,*q c*,*q* is the PDF of combined uncertainty at measurement location *q* and *φ* is the target reliability of identification. Thresholds, *Thigh*,*q* and *Tlow*,*q*, correspond to the shortest interval providing a probability equal to target reliability, *φ*. In Equation (6), the term 1/*m* is the Šidák correction [69], which accounts for *m* independent measurements used in identification of model parameters. In EDMF, compatibility of model predictions with measurements at each sensor location is considered as a hypothesis, which can have false positives and negatives. Inclusion of false positives as a candidate instance decreases precision of model updating, while falsely rejecting a model instance could potentially lead to falsification of true parameter values. Šidák correction controls the error rate such that the possibility of rejecting the true parameter values is lower than 1 − *φ*.

EDMF is traditionally carried out using grid sampling. In grid sampling, samples from prior distribution of model parameters, *θ*, are drawn. If *ns* samples are drawn from the prior distribution of each parameter, then these samples constitute a grid, which is called the initial model set (IMS). For *ns* samples drawn from *np* parameters, the total number of model instances in the IMS is *nnp s* .

Residual between model responses, *g*(*θ*), and measurements, *y*, are compared with the thresholds, *Tlow*,*q* and *Thigh*,*q*. If the residual between model response and measurements lies within the thresholds for all measurement locations, then the model instance is accepted. This criteria for falsification is shown in Equation (7).

$$T\_{low,q} \le g\_q\left(\theta\right) - y\_q \le T\_{high,q} \qquad \qquad q \in \{1...m\} \tag{7}$$

If predictions for a model instance, *θi*, does not satisfy Equation (7) for even one measurement location, then that model instance is falsified. All candidate model instances are considered equally likely and, thus, assigned a uniform probability density. Candidate models are used for making further predictions using the physics-based model with reduced parametric uncertainty [30]. The EDMF methodology has been applied to more than 20 full-scale systems since 1998 [4]. Recent applications include: model identification [70]; leak detection [71,72]; wind simulation [73]; fatigue life evaluation [74–76]; measurement-system design [77–80]; post-earthquake assessment [34,81]; damage localization in tensegrity structures [82]; and occupant localization [83].

EDMF when compared with BMU and residual minimization has been shown to provide accurate identification due to its robustness to correlation assumptions and explicit estimation of model bias based on engineering heuristics [29–32]. Although grid sampling carries some advantages with respect to practical applications and parallel computing, grid sampling remains computationally expensive [53].

#### 2.1.4. Modified Bayesian Model Updating

To alleviate shortcomings of traditional BMU (see Section 2.1.2), a box-car likelihood function is utilized for modified BMU, which is more robust to incomplete knowledge of uncertainties and correlations than standard *L*2-norm-based Gaussian likelihood functions. The box-car likelihood function is developed using an *L*∞-norm-based Gaussian likelihood function [67], which is defined as shown in Equation (8).

$$L\left(y \mid \theta\right) = \begin{cases} 1/2\sigma\_{\infty} & \text{for } \mu\_y - \sigma\_{\infty} \le y(\theta) - y \le \mu\_y + \sigma\_{\infty} \\ 0 & \text{otherwise.} \end{cases} \tag{8}$$

In Equation (8), parameters of the likelihood function, *μy* and *σ***∞**, are determined using Equations (9) and (10).

$$
\mu\_y = \frac{T\_{high} + T\_{low}}{2} \tag{9}
$$

$$
\sigma\_{\infty} = T\_{high} - \mu\_y \tag{10}
$$

In Equations (9) and (10), *Tlow* and *Thigh* are the thresholds computed for EDMF using Equation (6) for a target reliability of identification *φd*. Modified BMU using such a box-car likelihood distribution has been shown to provide results similar to those obtained using EDMF [31,32].

#### *2.2. Practical Challenges Associated with Model-Based Data Interpretation*

As stated before, data interpretation is an iterative task that requires exploring results and re-evaluating results in light of new information regarding uncertainties or new measurements. In addition, the task of data-interpretation may have to be repeated when identification results are found to be inaccurate due to a wrong model class. Assessing accuracy is a challenge as knowledge of true parameter values is unavailable. Accuracy can be approximated with cross-validation methods [84]. While enabling accuracy estimation, such validation strategies are limited to the domain of data used for identification. Moreover, when these methods indicate that structural identification is inaccurate, diagnostics are required to re-assess the assumptions made during identification.

Cross validation of structural-identification results can be conducted using several techniques, such as leave-one-out, hold-out and *k*-fold. Hold-out and *k*-fold cross-validation require large measurement datasets for identification and validation. In structural identification of civil infrastructure, measurements are typically scarce with few sensors that provide information about structural behavior. Thus, leave-one-out cross-validation is preferred over other strategies for validating model-updating results.

In leave-one-out cross validation, observation from one sensor is omitted (left-out) and structural identification is carried out using all remaining measurements. Updated model-parameter values are then used to predict the model response at the omitted sensor. If the omitted measurement is compatible with updated model predictions, then structural identification is concluded to be accurate for that sensor location. This procedure is repeated by omitting each sensor separately in order to assess accuracy at all measurement locations.

Consider that *nm* measurements are acquired during a load test. These measurements are used for updating a physics-based model of the structure, *g*(*θ*), which has parameters *θ* = [*θ*1, *θ*2, ..., *θnp* ], where *np* is the number of parameters. Prior to model updating, the initial prediction at a sensor location *j* is given by Equation (11).

$$\mathfrak{q}\_{j} = \mathfrak{g}\_{j}\left(\boldsymbol{\theta}\right) + \boldsymbol{\epsilon}\_{prod,j} \tag{11}$$

In Equation (11), *gj* (*θ*) is the model prediction at sensor *j* for model parameters *θ* and  *pred*,*j* is the model error from sources such as parameters not considered in the parameter vector *θ*, uncertainty in load and its position. The model-prediction distribution including model error before model updating is *qj*.

Let measurement from sensor *j* be excluded from model updating of parameters *θ* to perform leave-one-out cross-validation. The Sidâk correction for determining the threshold bounds, leaving one sensor out, is 1 *nm*−1 . Model parameters, *θ*, are updated to obtain candidate model parameters, *θ*. Model updating is performed using the four methodologies described in the previous sections. The following equations apply directly to EDMF and modified BMU. For traditional BMU, they can be calculated based on the 95th-percentile bound of the prediction distribution. Updated parameters, *θ*, are provided as input to the physics-based model, *g***(***θ--***)**, to predict the model response at the omitted sensor, *j*, as shown in Equation (12).

$$q\_{\dot{j}}^{\prime\prime} = \lg\left(\theta^{\prime\prime}\right) + \epsilon\_{pred,\dot{j}} \tag{12}$$

In Equation (12), *q j* is the distribution of updated model predictions at sensor location *j*. Depending on the uncertainties and relationships between model parameters and response, the prediction distributions obtained using Equations (11) and (12) are irregular. They are assumed to have uniform distributions based on the principle of maximum entropy. When bounds of the updated distribution of model predictions include the measured value, which has been left out for model updating, then identification is considered to be accurate.

Using leave-one-out cross validation, precision of structural identification can be quantified in addition to accuracy. Precision is a measure of variability either in updated model-parameter distributions or model predictions. Using leave-one-out cross-validation precision is estimated using the error between the updated model-prediction distributions and corresponding measurements. The model-prediction distribution at sensor *j* before and after model updating is given by Equations (11) and (12). The measurement at this sensor location is *yj*. The prediction error is the residual between model predictions and measurement. The prediction error distribution at sensor *j*, before and after model updating are uniform and their ranges are given by Equations (13) and (14).

$$\mathfrak{R}\_{\circ} = \frac{q\_{j,\max} - q\_{j,\min}}{y\_j} \tag{13}$$

$$\mathcal{R}^{\prime\prime}\_{\;j} = \frac{q\_{j,\max}^{\prime\prime} - q\_{j,\min}^{\prime\prime}}{y\_j} \tag{14}$$

In Equations (13) and (14), R*j* and R*j* are ranges of prediction-error distributions relative to the measurement at sensor *j* before and after model updating. For *nm* cases of leave-one-out cross-validation, *nm* prediction-error ranges before and after model updating are obtained. Considering prediction-error ranges for all cases of leave-one-out cross-validation before and after model updating, the precision index, *ϕ*, is defined as shown in Equation (15).

$$\varphi = \frac{\left(\mu\_{\mathfrak{H}} - \mu\_{\mathfrak{H}''}\right)}{\mu\_{\mathfrak{H}}} \tag{15}$$

In Equation (15), *μ*R and *μ*R are the mean of prediction-error ranges, before and after model updating, over all cases of sensors left out. Precision, *ϕ*, represents reduction in prediction error after model updating and ranges from 0 to 1. Precision, *ϕ*, equal to zero implies no gain in information from model updating. In such situations, *μ*R is equal to *μ*R , implying that on average over all cases of leave-one-out cross-validation, no reduction in prediction uncertainty is obtained. Precision index, *ϕ*, equal to one implies perfect model updating wherein updated parameter and consequently the prediction distributions have zero variability.

## **3. Full-Scale Applications**

In this section, with the help of two full-scale case studies, the application of four data-interpretation methodologies is compared. Comparisons are made with respect to their ability to provide accurate identification and transparently and iteratively incorporate new information.

## *3.1. Ponneri Bridge*

Ponneri Bridge, shown in Figure 2b, is a steel railway bridge located close to Chennai, India. It is part of a system of bridges (see Figure 2a) built in 1977 that comprises a railway crossing over the Arani river.

(a) System of bridges over the Arani river

(b) Instrumented bridge (Ponneri)

**Figure 2.** (**a**) System of railway-bridges over the Arani river in Chennai; and (**b**) instrumented bridge evaluated in this section, which is called Ponneri Bridge.

The behavior of each bridge in this system is independent of the others. One of the bridges in this system, referred to as Ponneri Bridge, is instrumented. This is the first bridge in the system for a train entering from Chennai and heading north. The bridge has a span of 18.3 m and is composed of two steel I-section girders, as shown in Figure 3. The two steel girders are connected through diagonal cross-bracing with a spacing of 1.6 m, which provides the bridge with stiffness in the transversal direction.

**Figure 3.** (**a**) Plan view of the bridge; and (**b**) section X-X showing details of the two steel girders.

A finite-element (FE) model of the Ponneri Bridge was developed in Ansys [85]. In the model, the steel girders were modeled using SHELL182 elements. Diagonal bracings (see Figure 3) connecting the steel girders are modeled using BEAM188 elements. The bridge was modeled as simply supported with perfect pin-conditioned support at end B (see Figure 3a) and a partially pin-conditioned support at end A. At end A, the support was modeled to have infinite vertical stiffness with stiffness in longitudinal direction (along the span) parameterized using zero-length spring elements, COMBIN14 (roller support). The bounds of this parameter, *KA*,*z*, was estimated using the FE model and is reported in Table 1. Other than stiffness of support at end A in longitudinal direction, the Young's modulus of steel, *Es*, was parameterized. The bounds for *Es*, reported in Table 1, were conservatively estimated based on the probabilistic model for modulus of elasticity of steel provided by Vrouwenvelder [86].

Using the FE model, strain measurements for the bridge were simulated at 16 locations, which are shown in Figure 4. Use of simulated measurements helped compare accuracy of model updating for various scenarios using four data-interpretation methodologies.

**Table 1.** Prior uncertainty distributions of parameters included in the FE model. The model parameters were assumed to have a uniform distribution (U).

**Figure 4.** Location of strain gauges for which measurements were simulated using a FE model of the Ponneri Bridge.

When simulating measurements using the FE model, model bias was introduced by assigning partial rotational rigidity to the boundary conditions at supports A and B. Other than the model bias introduced, the values of model parameters, *Es* and *kA*,*z*, assigned to simulate measurements for two conditions of the bridge are shown in Table 2.

**Table 2.** True values of model parameters *Es* and *kA*,*z* used to simulate measurements.

The before-retrofit scenario in Table 2 is the present condition of the bridge. The bridge was then assumed to be retrofitted by replacement of bearing at support A. The new bearing prevents translational movement of the bridge, similar to the bridge bearing at support B. This was represented by increased stiffness of the support A in the longitudinal direction in Table 2. For the two scenarios reported in Table 2, measurements were simulated for a train passing over the bridge. The train was positioned on the bridge such that it produces maximum strain at sensors close to mid-span, as shown in Figure 4.

Using the two scenarios of the Ponneri Bridge (see Table 2), the applicability of four data-interpretation methodologies to provide accurate structural identification was compared. As measurements were simulated, knowledge of the "true" parameter values helps assess identification accuracy in the presence of systematic bias.

Identification of parameters, *Es* and *KA*,*z*, had to be carried out in the presence of uncertainty from multiple sources, such as model bias (due to partial rotational stiffness of end conditions) and sensor noise (added to measurements). Estimations of these uncertainties are shown in Table 3.

The two sources of uncertainties shown in Table 3 were combined using Monte Carlo sampling to determine the updating criteria for traditional BMU, modified BMU and EDMF. For traditional BMU, a zero-mean *L*2-norm-based Gaussian likelihood function as defined by Equation (3) was developed. Falsification thresholds for EDMF were calculated using Equation (6). *L*∞-norm-based Gaussian likelihood function for modified BMU was developed using Equations (8)–(10). Using the Ponneri case study, the four data-interpretation methodologies were compared with respect to their ability to provide accurate structural identification as well as their amenability to incorporate changes in uncertainty definitions with respect to four identification scenarios, as shown in Table 4.

#### 3.1.1. Scenario 1: Structural Identification before Retrofit, Ignoring Model Bias

Structural identification of *Es* and *kA*,*z* was conducted with measurements recorded before retrofit of the Ponneri Bridge. The first iteration of data interpretation was carried out without taking into account model bias (Scenario 1 in Table 4).

For EDMF, an initial grid of model instances was generated based on the prior distribution of model parameters (see Table 1). A total of 273 model instances (13 *kA*,*z* instances multiplied by 21 *Es* instances drawn from the prior parameter distributions) were generated and provided as input to the FE model. Model instances whose responses were compatible with measurements at all sensor locations were accepted as part of the CMS. Falsification thresholds obtained by ignoring the model bias falsified all model instances. Thus, the model class is falsified. Falsification of the entire model class indicates mis-evaluation of uncertainties.

Modified BMU was conducted with a boxcar-shaped likelihood function (see Equation (8)). Samples from the joint posterior PDF of model parameters were drawn using MCMC sampling. The starting point was determined using MC sampling. Without taking the model bias into account in development of the likelihood function, after 1000 MC samples, no starting point was obtained, which suggests rejection of the model class by modified BMU, in a similar way to EDMF.

Traditional BMU was conducted with a zero-mean Gaussian likelihood function, without taking into account the model uncertainty. The posterior PDF thus obtained was precise and biased from the true parameter values used to simulate the measurements. Similar to traditional BMU, residual minimization returned parameter values, 215 GPa and 4.5 log N/mm, as optimal. These values were inaccurate and biased from the true parameter values (see Table 2).

#### 3.1.2. Scenario 2: Structural Identification before Retrofit, Considering Model Bias

Based on diagnostics obtained using EDMF and modified BMU, data-interpretation was repeated by including model bias (Scenario 2 in Table 4). EDMF, with re-calculated falsification thresholds and without requiring any new simulations, provided the CMS shown in Figure 5a.

Modified BMU was repeated by taking into account the model bias. Due to limitations on computational cost in using a FE model of a full-scale bridge, only 500 samples were drawn using MCMC sampling. A scatter plot of the samples drawn from the joint posterior PDF of model parameters, *Es* and *kA*,*z*, is shown in Figure 5b.

Traditional BMU was also repeated with a zero-mean Gaussian likelihood function using MCMC sampling (500 samples). A scatter plot of the samples drawn is shown in Figure 5c.

True values of model parameters

**Figure 5.** Samples from updated parameter distributions obtained using: (**a**) EDMF; (**b**) modified BMU; and (**c**) traditional BMU before retrofit of the Ponneri Bridge.

#### *J. Sens. Actuator Netw.* **2019**, *8*, 36

In Figure 5, the values of parameters used to simulate measurements are also shown for comparison. For EDMF and modified BMU, the "true" parameter values lie within the updated values of model parameters obtained, as shown in Figure 5a,b. However, for traditional BMU, the updated posterior distribution of model parameters does not include the "true" parameter values (see Figure 5c). Therefore, EDMF and modified BMU provide accurate structural identification for the Ponneri Bridge before retrofit actions, while traditional BMU does not. Since uncertainties were ignored in residual minimization, it was not repeated for changes in estimation of uncertainties.

#### 3.1.3. Scenario 3: Structural Identification after Retrofit, without Re-Evaluating Prior Parameter Distributions

Simulated measurements after retrofit were used for structural identification using the four data-interpretation methodologies (Scenario 3 in Table 4). Replacement of the bearing at support A increased the stiffness of the support in longitudinal direction. The parameter values used to simulate the bridge response (see Table 2) lie outside the bounds of the prior distribution of model parameter *kA*,*z* (see Table 1). Therefore, EDMF falsified the entire model class, which implies either uncertainty or other assumptions have to be re-evaluated. Similar to EDMF, modified BMU failed to find a starting point, which suggests that no parameter values from the prior distribution of model parameters have a non-zero likelihood.

Traditional BMU using MCMC sampling found a starting point and identified a posterior with low variability (high precision). As uncertainties affecting structural identification were not changed compared with Scenario 2, this precision was not due to decrease in uncertainty rather mis-evaluation of the prior distributions of model parameters, which have to be re-evaluated.

EDMF, in addition to falsifying the model class, provided diagnostics to re-assess the uncertainty definitions. A comparison of rejected model predictions at all sensor locations with falsification thresholds suggests that the prior distribution of model parameters have to be re-evaluated. Moreover, the error between model response and the thresholds is not the same at all sensor locations, suggesting a systematic source of uncertainty. The only systematic uncertainty source in the model class is *KA*,*z* and the prior distribution of this parameter was modified to be uniform with bounds [3, 8] log N/mm.

3.1.4. Scenario 4: Structural Identification after Retrofit, after Re-Evaluating Prior Parameter Distributions

Model parameters were identified with new prior distributions using the four data-interpretation methodologies. The updated parameter distributions obtained using the three probabilistic data-interpretation methodologies are shown in Figure 6 (Scenario 4 in Table 4).

In Figure 6, a scatter plot of samples from the updated parameter distributions is shown. Using post-retrofit measurements of the Ponneri Bridge, all three probabilistic data-interpretation methodologies provided accurate updated parameter distributions. Residual minimization provided 215 GPa and 7 log N/mm as optimal parameter values, which were biased from the "true" parameter values (see Table 2).

To summarize, EDMF and modified BMU provide accurate model updating for all scenarios, before and after retrofit (Cases 1–4 in Table 4). Traditional BMU (considering 95th percentile bounds) provided accurate identification only post-retrofit with appropriate estimation of priors (Case 4 in Table 4). Residual minimization did not provide accurate model updating for any scenario. The various data interpretation scenarios were typical iterations due to unsuccessful validation, as described in Figure 1.

*J. Sens. Actuator Netw.* **2019**, *8*, 36

True values of model parameters

**Figure 6.** Samples from updated parameter distributions obtained using: (**a**) EDMF; (**b**) modified BMU; and (**c**) traditional BMU after retrofit of the Ponneri Bridge.

#### 3.1.5. Interpretation of Identification Results

Residual minimization is the most commonly used data-interpretation methodology in practice, due to the simplicity of updating criteria (calibration) and ease of result interpretation (single optimal values). However, residual minimization is unable to provide accurate model updating in presence of large modeling uncertainties. Probabilistic data-interpretation methodologies account for the presence of modeling uncertainties, thereby improving accuracy, sometimes at the cost of transparency in understanding model-updating results.

EDMF provides a set of candidate models that are compatible with measurements. Modified BMU provides a bounded joint posterior PDF. Thus, posterior obtained using modified BMU can also be interpreted as updated bounds on model parameters, similar to those obtained using EDMF. Traditional BMU using uniform priors and a *L*2-norm-based Gaussian likelihood function provides an informed (not uniform) joint PDF as posterior. The normality of the posterior PDF is dependent on the information gained from measurements. In Figure 7, a comparison of results obtained from model updating using the three probabilistic methodologies for parameter *KA*,*z* is shown.