# Data-Interpretation Methodologies for Practical Asset-Management

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

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

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

#### 2.1.1. Residual Minimization

#### 2.1.2. Traditional Bayesian Model Updating

#### 2.1.3. Error-Domain Model Falsification

#### 2.1.4. Modified Bayesian Model Updating

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

## 3. Full-Scale Applications

#### 3.1. Ponneri Bridge

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

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

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

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

#### 3.1.5. Interpretation of Identification Results

#### 3.1.6. Comparison of Computational Cost

#### 3.2. Crêt de l’Anneau Bridge, Switzerland

**Scenario 1**: Deflection measurements at five locations (S1–S5) were used and model uncertainty was ignored.**Scenario 2**: Deflection measurements at five locations (S1–S5) were used and model uncertainty was taken into account.**Scenario 3**: Deflection-measurements at 10 locations were used and model uncertainty was taken into account.

#### 3.2.1. Structural Identification (Scenario 1: Ignoring Model Bias)

^{6}parameter combinations were falsified as they produce residuals between measured and predicted deflections that do not comply with Equation (6) for all five measured locations. This indicates that either choice of model parameters and their ranges is erroneous or that uncertainty is under-estimated, as is the case here. In a similar way, modified BMU failed to find a starting point for 2000 randomly selected parameter combinations. Both methods led to the same conclusion that model predictions are incompatible with measurements, given the estimated combined uncertainties. However, modified BMU reached the conclusion with shorter simulation time.

#### 3.2.2. Structural Identification (Scenario 2: Five Measurement Locations)

#### 3.2.3. Structural Identification (Scenario 3: 10 Measurements Locations)

#### 3.2.4. Practical Aspects of Data Interpretation

## 4. Discussion of Results

#### 4.1. Ponneri Bridge Case Study

#### 4.2. Crêt de l’Anneau Bridge Case-Study

## 5. Conclusions

- EDMF incorporates new information such as changes to uncertainty definitions and additional measurements iteratively. Bayesian model-updating methodologies and residual minimization must be restarted each time.
- EDMF is computationally more efficient than Bayesian model updating methodologies in an iterative data-interpretation framework, especially when grid sampling is used in combination with parallel computing.
- EDMF and modified BMU provide updated bounds on parameter values, which is more interpretable for practicing engineers than posterior parameter distributions that are obtained using traditional BMU.
- Residual minimization provides single optimal parameter values and, while this is compellingly useful in practice, it is not accurate in the presence of the biased uncertainties that are common in engineering modeling.
- EDMF involves a procedure that is more compatible with typical engineering procedures. For example, it is customary to define target reliability levels at the beginning. EDMF follows this procedure, whereas Bayesian approaches leave this to the end.
- Accuracy is assessed using leave-one-out cross-validation, which is computationally inexpensive when EDMF with grid sampling is used and computationally expensive when traditional BMU methodology is used.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- World Economic Forum; The Boston Consulting Group. Shaping the Future of Construction: A Breakthrough in Mindset and Technology; World Economic Forum: Cologny, Switzerland, 2016. [Google Scholar]
- World Economic Forum. Strategic Infrastructure, Steps to Operate and Maintain Infrastructure Efficiently and Effectively; World Economic Forum: Cologny, Switzerland, 2014. [Google Scholar]
- Brühwiler, E. Extending the service life of Swiss bridges of cultural value. Proc. Inst. Civ. Eng. Eng. Hist. Herit.
**2012**, 165, 235–240. [Google Scholar] [CrossRef][Green Version] - Smith, I.F.C. Studies of Sensor-Data Interpretation for Asset Management of the Built Environment. Front. Built Environ.
**2016**, 2, 8. [Google Scholar] [CrossRef] - World Economic Forum; The Boston Consulting Group. Shaping the Future of Construction: Inspiring Innovators Redefine the Industry; World Economic Forum: Cologny, Switzerland, 2017. [Google Scholar]
- Lynch, J.P.; Loh, K.J. A summary review of wireless sensors and sensor networks for structural health monitoring. Shock Vib. Dig.
**2006**, 38, 91–130. [Google Scholar] [CrossRef] - Taylor, S.G.; Raby, E.Y.; Farinholt, K.M.; Park, G.; Todd, M.D. Active-sensing platform for structural health monitoring: Development and deployment. Struct. Health Monit.
**2016**, 15, 413–422. [Google Scholar] [CrossRef][Green Version] - Frangopol, D.M.; Soliman, M. Life-cycle of structural systems: Recent achievements and future directions. Struct. Infrastruct. Eng.
**2016**, 12, 1–20. [Google Scholar] [CrossRef] - Der Kiureghian, A. Analysis of structural reliability under parameter uncertainties. Probab. Eng. Mech.
**2008**, 23, 351–358. [Google Scholar] [CrossRef] - Jiang, X.; Mahadevan, S. Bayesian validation assessment of multivariate computational models. J. Appl. Stat.
**2008**, 35, 49–65. [Google Scholar] [CrossRef] - Mottershead, J.E.; Friswell, M. Model updating in structural dynamics: a survey. J. Sound Vib.
**1993**, 167, 347–375. [Google Scholar] [CrossRef] - Soize, C. Stochastic models of uncertainties in computational structural dynamics and structural acoustics. In Nondeterministic Mechanics; Springer: Berlin, Germany, 2012; pp. 61–113. [Google Scholar]
- Soize, C. Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions. Int. J. Numer. Methods Eng.
**2010**, 81, 939–970. [Google Scholar] [CrossRef] - Görl, E.; Link, M. Damage identification using changes of eigenfrequencies and mode shapes. Mech. Syst. Signal Process.
**2003**, 17, 103–110. [Google Scholar] [CrossRef] - Beck, J.L. Bayesian system identification based on probability logic. Struct. Control Health Monit.
**2010**, 17, 825–847. [Google Scholar] [CrossRef] - Cross, E.J.; Worden, K.; Farrar, C.R. Structural health monitoring for civil infrastructure. In Health Assessment of Engineered Structures: Bridges, Buildings and Other Infrastructures; World Scientific: Hackensack, NJ, USA, 2013; pp. 1–28. [Google Scholar]
- Moon, F.; Catbas, N. Structural Identification of Constructed Systems. In Structural Identification of Constructed Systems; American Society of Civil Engineers: Reston, VA, USA, 2013; pp. 1–17. [Google Scholar]
- Sanayei, M.; Imbaro, G.R.; McClain, J.A.; Brown, L.C. Structural model updating using experimental static measurements. J. Struct. Eng.
**1997**, 123, 792–798. [Google Scholar] [CrossRef] - Beven, K.J. Uniqueness of place and process representations in hydrological modelling. Hydrol. Earth Syst. Sci. Discuss.
**2000**, 4, 203–213. [Google Scholar] [CrossRef][Green Version] - Mottershead, J.E.; Link, M.; Friswell, M.I. The sensitivity method in finite element model updating: A tutorial. Mech. Syst. Signal Process.
**2011**, 25, 2275–2296. [Google Scholar] [CrossRef] - McFarland, J.; Mahadevan, S. Error and variability characterization in structural dynamics modeling. Comput. Methods Appl. Mech. Eng.
**2008**, 197, 2621–2631. [Google Scholar] [CrossRef] - McFarland, J.; Mahadevan, S. Multivariate significance testing and model calibration under uncertainty. Comput. Methods Appl. Mech. Eng.
**2008**, 197, 2467–2479. [Google Scholar] [CrossRef] - Rebba, R.; Mahadevan, S. Validation of models with multivariate output. Reliab. Eng. Syst. Saf.
**2006**, 91, 861–871. [Google Scholar] [CrossRef] - Beck, J.L.; Katafygiotis, L.S. Updating models and their uncertainties. I: Bayesian statistical framework. J. Eng. Mech.
**1998**, 124, 455–461. [Google Scholar] [CrossRef] - Kennedy, M.C.; O’Hagan, A. Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B
**2001**, 63, 425–464. [Google Scholar] [CrossRef] - Brynjarsdóttir, J.; O’Hagan, A. Learning about physical parameters: The importance of model discrepancy. Inverse Probl.
**2014**, 30, 114007. [Google Scholar] [CrossRef] - Li, Y.; Xiao, F. Bayesian Update with Information Quality under the Framework of Evidence Theory. Entropy
**2019**, 21, 5. [Google Scholar] [CrossRef] - Simoen, E.; Papadimitriou, C.; Lombaert, G. On prediction error correlation in Bayesian model updating. J. Sound Vib.
**2013**, 332, 4136–4152. [Google Scholar] [CrossRef] - Goulet, J.A.; Smith, I.F.C. Structural identification with systematic errors and unknown uncertainty dependencies. Comput. Struct.
**2013**, 128, 251–258. [Google Scholar] [CrossRef][Green Version] - Pasquier, R.; Smith, I.F. Robust system identification and model predictions in the presence of systematic uncertainty. Adv. Eng. Inform.
**2015**, 29, 1096–1109. [Google Scholar] [CrossRef][Green Version] - Pai, S.G.; Nussbaumer, A.; Smith, I.F. Comparing structural identification methodologies for fatigue life prediction of a highway bridge. Front. Built Environ.
**2018**, 3, 73. [Google Scholar] [CrossRef] - Reuland, Y.; Lestuzzi, P.; Smith, I.F. Data-interpretation methodologies for non-linear earthquake response predictions of damaged structures. Front. Built Environ.
**2017**, 3, 43. [Google Scholar] [CrossRef] - Pasquier, R.; Smith, I.F.C. Iterative structural identification framework for evaluation of existing structures. Eng. Struct.
**2016**, 106, 179–194. [Google Scholar] [CrossRef][Green Version] - Reuland, Y.; Lestuzzi, P.; Smith, I.F. A model-based data-interpretation framework for post-earthquake building assessment with scarce measurement data. Soil Dyn. Earthq. Eng.
**2019**, 116, 253–263. [Google Scholar] [CrossRef] - Zhang, Y.; O’Connor, S.M.; van der Linden, G.W.; Prakash, A.; Lynch, J.P. SenStore: A scalable cyberinfrastructure platform for implementation of data-to-decision frameworks for infrastructure health management. J. Comput. Civ. Eng.
**2016**, 30, 04016012. [Google Scholar] [CrossRef] - Worden, K.; Farrar, C.R.; Manson, G.; Park, G. The fundamental axioms of structural health monitoring. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences; The Royal Society: London, UK, 2007; Volume 463, pp. 1639–1664. [Google Scholar]
- Pavlovskis, M.; Antucheviciene, J.; Migilinskas, D. Application of MCDM and BIM for evaluation of asset redevelopment solutions. Stud. Inform. Control
**2016**, 25, 293–302. [Google Scholar] [CrossRef] - Vinogradova, I.; Podvezko, V.; Zavadskas, E. The recalculation of the weights of criteria in MCDM methods using the bayes approach. Symmetry
**2018**, 10, 205. [Google Scholar] [CrossRef] - Kaganova, O.; Telgarsky, J. Management of capital assets by local governments: An assessment and benchmarking survey. Int. J. Strateg. Prop. Manag.
**2018**, 22, 143–156. [Google Scholar] [CrossRef] - Re Cecconi, F.M.N.; Dejaco, M.C. Measuring the performance of assets: a review of the Facility Condition Index. Int. J. Strateg. Prop. Manag.
**2018**, 23, 187–196. [Google Scholar] [CrossRef] - Ljung, L. Perspectives on system identification. Annu. Rev. Control
**2010**, 34, 1–12. [Google Scholar] [CrossRef][Green Version] - Chang, C.C.; Chang, T.; Xu, Y. Adaptive neural networks for model updating of structures. Smart Mater. Struct.
**2000**, 9, 59. [Google Scholar] [CrossRef] - Kuok, S.C.; Yuen, K.V. Investigation of modal identification and modal identifiability of a cable-stayed bridge with Bayesian framework. Smart Struct. Syst.
**2016**, 17, 445–470. [Google Scholar] [CrossRef] - Behmanesh, I.; Moaveni, B. Accounting for environmental variability, modeling errors, and parameter estimation uncertainties in structural identification. J. Sound Vib.
**2016**, 374. [Google Scholar] [CrossRef] - Friswell, M.; Penny, J.; Garvey, S. A combined genetic and eigensensitivity algorithm for the location of damage in structures. Comput. Struct.
**1998**, 69, 547–556. [Google Scholar] [CrossRef] - Ding, Z.; Huang, M.; Lu, Z. Structural damage detection using artificial bee colony algorithm with hybrid search strategy. Swarm Evolut. Comput.
**2016**, 28, 1–13. [Google Scholar] [CrossRef] - Gökdağ, H. Comparison of ABC, CPSO, DE and GA Algorithms in FRF Based Structural Damage Identification. Mater. Test.
**2013**, 55, 796–802. [Google Scholar] [CrossRef] - Gökdağ, H.; Yildiz, A.R. Structural damage detection using modal parameters and particle swarm optimization. Mater. Test.
**2012**, 54, 416–420. [Google Scholar] [CrossRef] - Majumdar, A.; Maiti, D.K.; Maity, D. Damage assessment of truss structures from changes in natural frequencies using ant colony optimization. Appl. Math. Comput.
**2012**, 218, 9759–9772. [Google Scholar] [CrossRef] - Beck, J.L.; Au, S.K. Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation. J. Eng. Mech.
**2002**, 128, 380–391. [Google Scholar] [CrossRef] - Ching, J.; Chen, Y.C. Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. J. Eng. Mech.
**2007**, 133, 816–832. [Google Scholar] [CrossRef] - Boulkaibet, I.; Mthembu, L.; Marwala, T.; Friswell, M.I.; Adhikari, S. Finite element model updating using the shadow hybrid Monte Carlo technique. Mech. Syst. Signal Process.
**2015**, 52, 115–132. [Google Scholar] [CrossRef] - Dubbs, N.; Moon, F. Comparison and implementation of multiple model structural identification methods. J. Struct. Eng.
**2015**, 141, 04015042. [Google Scholar] [CrossRef] - Proverbio, M.; Costa, A.; Smith, I.F.C. Adaptive Sampling Methodology for Structural Identification Using Radial-Basis Functions. J. Comput. Civ. Eng.
**2018**, 32, 1–17. [Google Scholar] [CrossRef] - Robert-Nicoud, Y.; Raphael, B.; Smith, I. System Identification through Model Composition and Stochastic Search. J. Comput. Civ. Eng.
**2005**, 19, 239–247. [Google Scholar] [CrossRef] - Schwer, L.E.; Mair, H.U.; Crane, R.L. Guide for verification and validation in computational solid mechanics. Am. Soc. Mech. Eng.
**2006**, 10, 2006. [Google Scholar] - Alvin, K. Finite element model update via Bayesian estimation and minimization of dynamic residuals. AIAA J.
**1997**, 35, 879–886. [Google Scholar] [CrossRef] - Katafygiotis, L.S.; Beck, J.L. Updating models and their uncertainties. II: Model identifiability. J. Eng. Mech.
**1998**, 124, 463–467. [Google Scholar] [CrossRef] - Katafygiotis, L.S.; Papadimitriou, C.; Lam, H.F. A probabilistic approach to structural model updating. Soil Dyn. Earthq. Eng.
**1998**, 17, 495–507. [Google Scholar] [CrossRef][Green Version] - Ching, J.; Beck, J.L. New Bayesian model updating algorithm applied to a structural health monitoring benchmark. Struct. Health Monit.
**2004**, 3, 313–332. [Google Scholar] [CrossRef] - Yuen, K.V.; Beck, J.L.; Katafygiotis, L.S. Efficient model updating and health monitoring methodology using incomplete modal data without mode matching. Struct. Control Health Monit.
**2006**, 13, 91–107. [Google Scholar] [CrossRef] - Muto, M.; Beck, J.L. Bayesian updating and model class selection for hysteretic structural models using stochastic simulation. J. Vib. Control
**2008**, 14, 7–34. [Google Scholar] [CrossRef] - Ntotsios, E.; Papadimitriou, C.; Panetsos, P.; Karaiskos, G.; Perros, K.; Perdikaris, P.C. Bridge health monitoring system based on vibration measurements. Bull. Earthq. Eng.
**2009**, 7, 469. [Google Scholar] [CrossRef] - Goller, B.; Schueller, G. Investigation of model uncertainties in Bayesian structural model updating. J. Sound Vib.
**2011**, 330, 6122–6136. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sohn, H.; Law, K.H. Bayesian probabilistic damage detection of a reinforced-concrete bridge column. Earthq. Eng. Struct. Dyn.
**2000**, 29, 1131–1152. [Google Scholar] [CrossRef] - Beck, J.L.; Au, S.K.; Vanik, M.W. Monitoring structural health using a probabilistic measure. Comput.-Aided Civ. Infrastruct. Eng.
**2001**, 16, 1–11. [Google Scholar] [CrossRef] - Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation; Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, USA, 2005. [Google Scholar]
- Popper, K. The Logic of Scientific Discovery; Routledge: Abingdon-on-Thames, UK, 1959. [Google Scholar]
- Šidák, Z. Rectangular confidence regions for the means of multivariate normal distributions. J. Am. Stat. Assoc.
**1967**, 62, 626–633. [Google Scholar] [CrossRef] - Goulet, J.A.; Michel, C.; Smith, I.F.C. Hybrid probabilities and error-domain structural identification using ambient vibration monitoring. Mech. Syst. Signal Process.
**2013**, 37, 199–212. [Google Scholar] [CrossRef][Green Version] - Goulet, J.A.; Coutu, S.; Smith, I.F.C. Model falsification diagnosis and sensor placement for leak detection in pressurized pipe networks. Adv. Eng. Inform.
**2013**, 27, 261–269. [Google Scholar] [CrossRef][Green Version] - Moser, G.; Paal, S.G.; Smith, I.F. Performance comparison of reduced models for leak detection in water distribution networks. Adv. Eng. Inform.
**2015**, 29, 714–726. [Google Scholar] [CrossRef][Green Version] - Vernay, D.G.; Raphael, B.; Smith, I.F.C. A model-based data-interpretation framework for improving wind predictions around buildings. J. Wind Eng. Ind. Aerodyn.
**2015**, 145, 219–228. [Google Scholar] [CrossRef][Green Version] - Pasquier, R.; Goulet, J.A.; Acevedo, C.; Smith, I.F.C. Improving Fatigue Evaluations of Structures Using In-Service Behavior Measurement Data. J. Bridge Eng.
**2014**, 19, 4014045. [Google Scholar] [CrossRef][Green Version] - Pasquier, R.; Angelo, L.D.; Goulet, J.A.; Acevedo, C.; Nussbaumer, A.; Smith, I.F.C. Measurement, Data Interpretation, and Uncertainty Propagation for Fatigue Assessments of Structures. J. Bridge Eng.
**2016**, 21. [Google Scholar] [CrossRef] - Pai, S.G.S.; Smith, I.F.C. Comparing Three Methodologies for System Identification and Prediction. In Proceedings of the 14th International Probabilistic Workshop, Ghent, Belgium, 22 December 2017; Caspeele, R., Taerwe, L., Proske, D., Eds.; Springer International Publishing: Berlin, Germany, 2017; pp. 81–95. [Google Scholar] [CrossRef]
- Goulet, J.A.; Smith, I.F.C. Performance-driven measurement system design for structural identification. J. Comput. Civ. Eng.
**2012**, 27, 427–436. [Google Scholar] [CrossRef] - Goulet, J.A.; Smith, I.F.C. Predicting the usefulness of monitoring for identifying the behavior of structures. J. Struct. Eng.
**2012**, 139, 1716–1727. [Google Scholar] [CrossRef] - Papadopoulou, M.; Raphael, B.; Smith, I.F.C.; Sekhar, C. Optimal sensor placement for time-dependent systems: Application to wind studies around buildings. J. Comput. Civ. Eng.
**2015**, 30, 4015024. [Google Scholar] [CrossRef] - Papadopoulou, M.; Raphael, B.; Smith, I.F.; Sekhar, C. Evaluating predictive performance of sensor configurations in wind studies around buildings. Adv. Eng. Inform.
**2016**, 30, 127–142. [Google Scholar] [CrossRef] - Reuland, Y.; Lestuzzi, P.; Smith, I.F. Measurement-based support for post-earthquake assessment of buildings. Struct. Infrastruct. Eng.
**2019**, 5, 1–16. [Google Scholar] [CrossRef] - Sychterz, A.C.; Smith, I.F. Using dynamic measurements to detect and locate ruptured cables on a tensegrity structure. Eng. Struct.
**2018**, 173, 631–642. [Google Scholar] [CrossRef][Green Version] - Reuland, Y.; Pai, S.G.; Drira, S.; Smith, I.F. Vibration-based occupant detection using a multiple-model approach. In Proceedings of the IMAC XXXV—Structural Dynamics Challenges in Next Generation Aerospace Systems, Garden Grove, CA, USA, 30 January–2 February 2017; Society for Experimental Mechanics (SEM): Bethel, CT, USA, 2017. [Google Scholar]
- Kohavi, R. A study of cross-validation and bootstrap for accuracy estimation and model selection. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), Montreal, QC, Canada, 20–25 August 1995; Volume 14, pp. 1137–1145. [Google Scholar]
- APDL. Mechanical Applications Theory Reference, 13th ed.; ANSYS Release 13.0; ANSYS Inc.: Canonsburg, PA, USA, 2010. [Google Scholar]
- Vrouwenvelder, T. The JCSS probabilistic model code. Struct. Saf.
**1997**, 19, 245–251. [Google Scholar] [CrossRef]

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

**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.

**Figure 3.**(

**a**) Plan view of the bridge; and (

**b**) section X-X showing details of the two steel girders.

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

**Figure 5.**Samples from updated parameter distributions obtained using: (

**a**) EDMF; (

**b**) modified BMU; and (

**c**) traditional BMU before retrofit of the Ponneri Bridge.

**Figure 6.**Samples from updated parameter distributions obtained using: (

**a**) EDMF; (

**b**) modified BMU; and (

**c**) traditional BMU after retrofit of the Ponneri Bridge.

**Figure 7.**Updated knowledge of parameter ${K}_{A,z}$ obtained through structural identification with measurements before retrofit actions (Case 2 in Table 4). EDMF and modified BMU provide updated bounds for parameter ${K}_{A,z}$, while traditional BMU provides an informed (inaccurate) marginal PDF of ${K}_{A,z}$.

**Figure 8.**A change in prior distribution size requires simulation of only additional model instances to evaluate their compatibility with measurements when using EDMF.

**Figure 9.**Comparison of computational cost. The simulations were carried out on a Intel(R) Xeon(R) CPU X5650 @2.67GHz processor with 24 cores.

**Figure 11.**(

**a**) Elevation of Crêt de l’Anneau Bridge; and (

**b**) cross-section of a typical span, showing the location of deflection sensors placed on Spans II and IV.

**Figure 12.**Leave-one-out cross validation for the five deflection measurements used in Scenario 2. mBMU and EDMF were accurate for all five measurements, while the measurements fell outside 95th-percentile bounds for deflection ${\delta}_{1}$ in the case of tBMU.

**Figure 13.**Comparison of computation times for the three probabilistic data-interpretation methodologies. Computation time is for parallel computing using up to 24 cores and is presented in relative time with respect to simulation time for EDMF in Scenario 1 (A).

**Figure 14.**Leave-one-out cross validation for sensors 7 (

**A**,

**C**) and 8 (

**B**,

**D**) before (

**A**,

**B**) and after (

**C**,

**D**) including measurements at the same span in structural identification for the Cret-de-l’Anneau bridge. Although inclusion of measurements from the same span increases precision, 95th-percentile bounds are not compatible with measurements in both cases. In addition, displacement is overestimated for sensor 7 (

**C**) and underestimated for sensor 8 (

**D**), which shows that results are not always conservative.

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

Parameter | Distribution |
---|---|

Young’s modulus of elasticity of steel, ${E}_{s}$ (GPa) | U(195, 215) |

Longitudinal stiffness of support at end A, ${k}_{A,z}$ (log N/mm) | U(3, 6) |

Condition | ${\mathit{E}}_{\mathit{s}}$ (GPa) | ${\mathit{k}}_{\mathit{A},\mathit{z}}$ (log N/mm) |
---|---|---|

Before retrofit | 210 | 4 |

After retrofit | 210 | 7 |

**Table 3.**Distribution of uncertainty sources affecting structural identification. Uncertainties were estimated relative (%) to design model predictions.

Scenario | Model Bias | Measurement Uncertainty |
---|---|---|

1 | U(−38, 2) | N(0, 2) |

2 | U(−40, 8) | N(0, 2) |

**Table 4.**Cases of structural identification considered for comparison of four data-interpretation methodologies.

Scenario | Condition | Description |
---|---|---|

1 | Before retrofit | Without model bias |

2 | Before retrofit | With model bias |

3 | After retrofit (replacement of bearing) | Without re-evaluating prior PDFs |

4 | After retrofit | After re-evaluating prior PDFs |

Description | Units | Distribution |
---|---|---|

Stiffness of supports (longitudinal) | log N/mm | U(3.5, 5.0) |

Stiffness of supports (vertical) | log N/mm | U(3.5, 5.5) |

Young’s modulus of steel | GPa | U(190, 220) |

Young’s modulus of concrete | GPa | U(30, 50) |

Gerber joint (longitudinal) | log N/mm | U(4.0, 6.0) |

Deck to girder connection (longitudinal) | log N/mm | U(4.0, 5.5) |

**Table 6.**Primary parameters with their parameter ranges (prior distributions) for the Crêt-de-l’Anneau bridge.

Parameter | Description | Units | Range |
---|---|---|---|

${\theta}_{1}$ | Deck-to-girder connection stiffness (longitudinal) | log (N/mm) | U(4.0,5.5) |

${\theta}_{2}$ | Vertical stiffness support A | log (N/mm) | U(3.5, 5.5) |

${\theta}_{3}$ | Vertical stiffness support B | log (N/mm) | U(3.5, 5.5) |

${\theta}_{4}$ | Vertical stiffness support C | log (N/mm) | U(3.5, 5.5) |

${\theta}_{5}$ | Vertical stiffness support D | log (N/mm) | U(3.5, 5.5) |

${\theta}_{6}$ | Gerber joint stiffness (longitudinal) | log (N/mm) | U(4.0, 6.0) |

Source | Distribution |
---|---|

Model bias (%) | U(−5, 15) |

Secondary parameters (%) | U(−1.5, 0.5) |

Surrogate model (mm) | U(−0.1, 0.1) |

Measurement (mm) | M(0, 0.15) |

**Table 8.**Updated parameter ranges (posterior distributions) for Scenario 1. mBMU failed to find a starting point, while EDMF falsified the entire initial model set. MAP refers to the maximum a-posteriori estimate.

Parameter | RM | tBMU | mBMU | EDMF |
---|---|---|---|---|

${\theta}_{1}$ | 5.5 | $\left[5.1,5.5\right]$, MAP = $5.5$ | - | - |

${\theta}_{2}$ | 5.5 | $\left[3.6,5.4\right]$, MAP = $4.8$ | - | - |

${\theta}_{3}$ | 5.5 | $\left[3.7,5.4\right]$, MAP = $4.8$ | - | - |

${\theta}_{4}$ | 5.5 | $\left[4.8,5.5\right]$, MAP = $5.4$ | - | - |

${\theta}_{5}$ | 5.5 | $\left[4.9,5.5\right]$, MAP = $5.4$ | - | - |

${\theta}_{6}$ | 5.8 | $\left[4.8,6.0\right]$, MAP = $5.7$ | - | - |

**Table 9.**Accuracy and precision established using a leave-one-out cross-validation approach. For mBMU and EMDF, the entire model class was rejected and, thus, no updated parameter values could be validated. For tBMU, absence of accuracy indicates that uncertainties were mis-evaluated.

Leave-One-Out Cross-Validation | RM | tBMU | mBMU | EDMF |
---|---|---|---|---|

Accuracy | No | No | - | - |

Precision | 1 | 0.96 | - | - |

Parameter | RM | tBMU | mBMU | EDMF |
---|---|---|---|---|

${\theta}_{1}$ | 5.5 | $\left[4.9,5.5\right]$, MAP = $5.4$ | $\left[4.8,5.5\right]$ | $\left[4.8,5.5\right]$ |

${\theta}_{2}$ | 5.5 | $\left[3.5,5.2\right]$, MAP = $4.0$ | $\left[3.5,5.5\right]$ | $\left[3.5,5.5\right]$ |

${\theta}_{3}$ | 5.5 | $\left[3.5,5.4\right]$, MAP = $4.6$ | $\left[3.5,5.5\right]$ | $\left[3.5,5.5\right]$ |

${\theta}_{4}$ | 5.5 | $\left[4.6,5.5\right]$, MAP = $5.4$ | $\left[4.4,5.5\right]$ | $\left[4.4,5.5\right]$ |

${\theta}_{5}$ | 5.5 | $\left[4.7,5.5\right]$, MAP = $5.4$ | $\left[4.6,5.5\right]$ | $\left[4.6,5.5\right]$ |

${\theta}_{6}$ | 5.8 | $\left[4.3,6.0\right]$, MAP = $5.8$ | $\left[4.3,6.0\right]$ | $\left[4.0,6.0\right]$ |

**Table 11.**Accuracy and precision established using a leave-one-out cross-validation approach for Scenario 2.

Leave-One-Out Cross Validation | RM | tBMU | mBMU | EDMF |
---|---|---|---|---|

Accuracy | No | No | Yes | Yes |

Precision | 1 | 0.84 | 0.74 | 0.74 |

**Table 12.**Updated parameter ranges (posterior distributions) for Scenario 3 involving deflection measurements at ten locations.

Parameter | RM | tBMU | mBMU | EDMF |
---|---|---|---|---|

${\theta}_{1}$ | 5.5 | $\left[5.0,5.5\right]$, MAP = $5.4$ | $\left[5.0,5.5\right]$ | $\left[5.0,5.5\right]$ |

${\theta}_{2}$ | 5.5 | $\left[4.6,5.5\right]$, MAP = $5.4$ | $\left[4.6,5.5\right]$ | $\left[4.6,5.5\right]$ |

${\theta}_{3}$ | 5.5 | $\left[4.8,5.5\right]$, MAP = $5.3$ | $\left[4.8,5.5\right]$ | $\left[4.8,5.5\right]$ |

${\theta}_{4}$ | 5.5 | $\left[4.5,5.5\right]$, MAP = $5.3$ | $\left[4.4,5.5\right]$ | $\left[4.4,5.5\right]$ |

${\theta}_{5}$ | 5.5 | $\left[4.6,5.5\right]$, MAP = $5.3$ | $\left[4.5,5.5\right]$ | $\left[4.6,5.5\right]$ |

${\theta}_{6}$ | 5.8 | $\left[5.0,6.0\right]$, MAP = $5.9$ | $\left[4.8,6.0\right]$ | $\left[4.4,6.0\right]$ |

**Table 13.**Accuracy and precision established using a leave-one-out cross-validation approach for Scenario 3, involving ten deflection measurements. Precision is the mean value over the ten measurement locations, while accuracy needs to be validated over all measurement locations.

Leave-One-Out Cross Validation | RM | tBMU | mBMU | EDMF |
---|---|---|---|---|

Accuracy | No | No | Yes | Yes |

Precision | 1 | 0.85 | 0.82 | 0.81 |

**Table 14.**Summary of the accuracy of structural identification scenarios (see Table 4) evaluated for the Ponneri Bridge. mBMU is modified BMU and tBMU is traditional BMU. Checkmarks imply accurate identification and crosses imply inaccurate identification.

Case | Scenario | Description | RM | EDMF | mBMU | tBMU |
---|---|---|---|---|---|---|

1 | Before retrofit | Without model bias | ✗ | ✓ | ✓ | ✗ |

2 | With model bias | ✗ | ✓ | ✓ | ✗ | |

3 | After retrofit | Without re-evaluating prior PDFs | ✗ | ✓ | ✓ | ✗ |

4 | After re-evaluating prior PDFs | ✗ | ✓ | ✓ | ✓ |

**Table 15.**Summary of the accuracy of structural identification scenarios evaluated for the Crêt-de-l’Anneau Bridge. mBMU is modified BMU and tBMU is traditional BMU. Checkmarks imply accurate identification or model-class rejection and crosses imply inaccurate identification based on leave-one-out cross validation.

Scenario | Description | RM | EDMF | mBMU | tBMU |
---|---|---|---|---|---|

1 | Without model bias (deflection at 5 locations) | ✗ | ✓ | ✓ | ✗ |

2 | With model bias (deflection at 5 locations) | ✗ | ✓ | ✓ | ✗ |

3 | With model bias (deflection at 10 locations) | ✗ | ✓ | ✓ | ✗ |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Pai, S.G.S.; Reuland, Y.; Smith, I.F.C. Data-Interpretation Methodologies for Practical Asset-Management. *J. Sens. Actuator Netw.* **2019**, *8*, 36.
https://doi.org/10.3390/jsan8020036

**AMA Style**

Pai SGS, Reuland Y, Smith IFC. Data-Interpretation Methodologies for Practical Asset-Management. *Journal of Sensor and Actuator Networks*. 2019; 8(2):36.
https://doi.org/10.3390/jsan8020036

**Chicago/Turabian Style**

Pai, Sai G. S., Yves Reuland, and Ian F. C. Smith. 2019. "Data-Interpretation Methodologies for Practical Asset-Management" *Journal of Sensor and Actuator Networks* 8, no. 2: 36.
https://doi.org/10.3390/jsan8020036