# Bridge Digital Twinning Using an Output-Only Bayesian Model Updating Method and Recorded Seismic Measurements

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## Abstract

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## 1. Introduction

- (i)
- Nonlinearity in bridge response behavior: Several approaches for linear FE model updating using modal properties, identified from ambient and/or operational data, exist in the literature (e.g., [6,7,8,9,10]). Nevertheless, these methods can update the bridge model only in the linear regime of response. Linear FE model updating methods provide limited insight into the nonlinear response behavior of the structure, which can occur during strong earthquake events.
- (ii)
- Measurement sparsity: Through the California Strong Motion Instrumentation Program (CSMIP) [11], the California Department of Conservation in collaboration with the California Department of Transportation (Caltrans) has instrumented several bridges and recorded their seismic responses during the past three decades. This valuable dataset has benefited the research community [12,13,14] and can provide the baseline for developing a DT platform for instrumented bridges across California. However, the collected data is often subjected to notable instrumentation sparsity. The sparsity in data poses important challenges to the uniqueness of the solution for the model updating technique and the accuracy of the resulting DT.
- (iii)
- Soil-Structure Interaction (SSI): The Foundation Input Motions (FIMs), which are the theoretical inputs to the soil-structure interactive system, are not explicitly measurable. FIMs can be different from the Free-Filed Motions (FFMs) due to the SSI effects [15]. The available knowledge on SSI effects is limited to analytical and numerical studies, and the in-situ and real-world effects of SSI on complex structures are not completely known [16,17,18,19,20]. Hence, developing DT for bridges using seismic measurements may require the estimation of the FIMs.

## 2. Sequential Bayesian Inference Method and Identifiability Analysis

#### 2.1. Sequential Bayesian Inference Method for Output-Only FE Model Updating

**φ**, the uncertainties of which are expressed with a Gaussian Probability Density Function (PDF). These uncertainties are propagated into the FE model $\widehat{y}=h\left(\phi \right)$, in which $h\left(\dots \right)$ is the nonlinear response function of the FE model. Next, a simulation (or prediction) error model $v\left(\phi \right)$is defined to correlate the FE-predicted response ($\widehat{y}$) with the measured response ($y$) collected by the sensors. Finally, the Bayes’ theorem is used to find the posterior PDF of the unknown parameters, which is then used as the prior PDF for the next sequence of measured responses.

#### 2.2. Formulation for the Identifiability Analysis

**,**is expressed as the difference between the a priori and a posteriori information entropy, which can be calculated as

**R**is the covariance matrix of the simulation error vector as defined earlier and $\widehat{\mathsf{\theta}}$ is a maximum a posteriori (MAP) estimate, which is approximated with the initial estimates based on the recommendations provided in [28]. To calculate the sensitivity terms (i.e., $\partial {\widehat{y}}_{t}/\partial \mathsf{\theta}$), a Finite Difference Method is employed. Then, the information gain of different model parameters is compared to sort out the model parameters with the highest information gain, which are relatively more likely to be identifiable unless they have strong dependence on other model parameters. The mutual entropy gain between ${\theta}^{i}$ and ${\theta}^{j}$, $\Delta M\left({\theta}^{i},{\theta}^{j}\right)$

**,**can be quantified through a mutual gain metric defined as

## 3. Verification Case Study Using the San Roque Canyon Bridge

#### 3.1. FE Model of the SRC Bridge

#### 3.2. Identifiability Analysis

#### 3.3. Verification Study Using Numerically Simulated Data

## 4. Case Study Using Real-World Earthquake Data

## 5. Digital Twin and Virtual Sensing Application

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic representation of the sequential estimation window approach. The estimation problem is completed at each estimation window iteratively and moves sequentially to the next estimation window.

**Figure 3.**Sequential Bayesian FE model updating flowchart based on the UT method for joint estimation of the unknown model parameters and the FIM time histories.

**Figure 4.**SRC bridge and its instrumentation layout [29]: (

**a**) deck cross-section, (

**b**) pier cross-section, (

**c**) elevation view, and (

**d**) plan view and installed accelerometers.

**Figure 5.**Earthquake data used for model updating: (

**a**) recorded acceleation responses at the SRC bridge for earthquakes listed in Table 1, (

**b**) pseudo spectral acceleration of FIMs projected in the longitudinal direction, and (

**c**) pseudo spectral acceleration of FIMs projected in the transverse direction. The period of the first longitudinal and transverse modes of the SRC (calculated based on the prior model) are shown in the figure.

**Figure 6.**Schematic representation of the SRC bridge FE model. The parameter numbers in the parentheses are listed in Table 2.

**Figure 7.**Relative information gain of model parameters. Parameters are defined in Table 2.

**Figure 8.**Relative mutual entropy gain between model parameter pairs: (

**a**) without scaling, (

**b**) with scaling.

**Figure 9.**Estimation error time histories for the unknown model parameters through the model updating technique using numerically simulated data.

**Figure 10.**Comparison between true and estimated FIMs in (

**a**) longitudinal and (

**b**) transverse directions, and (

**c**) RRMSE of the estimated FIMs.

**Figure 11.**Comparison between measured and estimated responses obtained from the updated FE model at: (

**a**) channel 4, (

**b**) channel 5, (

**c**) channel 6, (

**d**) channel 8, (

**e**) channel 9. Part (

**f**) is the corresponding RRMSEs.

**Figure 12.**Estimation time histories of (

**a**) ${E}_{ave}$, (

**b**) ${k}_{L}^{b}$, (

**c**) ${k}_{T}^{b}$, (

**d**) ${a}_{0}$, and (

**e**) ${a}_{1}$ for different earthquakes. The estimates are normalized to the initial estimates.

**Figure 13.**The variation in final estimates of (

**a**) ${E}_{ave}$, (

**b**)${k}_{L}^{b}$, (

**c**)${k}_{T}^{b}$, (

**d**) ${a}_{0}$, and (

**e**) ${a}_{1}$ for different earthquakes. The red dashed lines show the nominal values.

**Figure 14.**The final estimates of the Rayleigh damping ratio as a function of freqeuncy for different earthquakes.

**Figure 15.**Comparison between the recorded FFMs and estimated FIMs in the longitudinal direction for (

**a**) 2003 San Simeon, (

**b**) 2004 IslaVista, (

**c**) 2013 IslaVista, (

**d**) 2017 Montecito, and (

**e**) 2018 Santa Cruz, earthquake events. Part (

**f**) is the RRMSE between the estimated FIMs and recorded FFMs.

**Figure 16.**Comparison between the recorded FFMs and estimated FIMs in the transverse direction for (

**a**) 2003 San Simeon, (

**b**) 2004 IslaVista, (

**c**) 2013 IslaVista, (

**d**) 2017 Montecito, and (

**e**) 2018 Santa Cruz, earthquake events. Part (

**f**) is the RRMSE between the estimated FIMs and recorded FFMs.

**Figure 17.**Comparison between measured and estimated (from updated model) responses at different measurement channels for 2003 San Simeon earthquake: (

**a**) channel 4, (

**b**) channel 5, (

**c**) channel 6, (

**d**) channel 8, (

**e**) channel 9. Part (

**f**) is the corresponding RRMSEs.

**Figure 18.**Comparison between measured and estimated (from updated model) responses at different measurement channels for 2004 IslaVista earthquake: (

**a**) channel 4, (

**b**) channel 5, (

**c**) channel 6, (

**d**) channel 8, (

**e**) channel 9. Part (

**f**) is the corresponding RRMSEs.

**Figure 19.**Comparison between measured and estimated (from updated model) responses at different measurement channels for 2013 IslaVista earthquake: (

**a**) channel 4, (

**b**) channel 5, (

**c**) channel 6, (

**d**) channel 8, (

**e**) channel 9. Part (

**f**) is the corresponding RRMSEs.

**Figure 20.**Comparison between measured and estimated (from updated model) responses at different measurement channels for 2017 Montecito earthquake: (

**a**) channel 4, (

**b**) channel 5, (

**c**) channel 6, (

**d**) channel 8, (

**e**) channel 9. Part (

**f**) is the corresponding RRMSEs.

**Figure 21.**Comparison between measured and estimated (from updated model) responses at different measurement channels for 2018 Santa Cruz earthquake: (

**a**) channel 4, (

**b**) channel 5, (

**c**) channel 6, (

**d**) channel 8, (

**e**) channel 9. Part (

**f**) is the corresponding RRMSEs.

**Figure 22.**Local level response monitored in SRC bridge using its digital twin during 2013 IslaVista earthquake event: (

**a**) moment-curvature response (about the transverse direction) at the lowest section of the east pier, and (

**b**) estimated stress-strain response in the extreme fiber at the lowest section of the west pier.

No. | Earthquake | Date | Distance (km) | PGA (g) | PSA in Transverse Direction (g) | PSA in Vertical Direction (g) | PSA in Longitudinal Direction (g) |
---|---|---|---|---|---|---|---|

1 | San Simeon | 22 December 2003 | 187.0 | 0.015 | 0.045 | 0.042 | 0.022 |

2 | IslaVista | 9 May 2004 | 27.2 | 0.016 | 0.026 | 0.047 | 0.013 |

3 | IslaVista | 29 May 2013 | 18.0 | 0.041 | 0.060 | 0.150 | 0.040 |

4 | Montecito | 23 April 2017 | 9.5 | 0.022 | 0.024 | 0.045 | 0.014 |

5 | Santa Cruz | 5 April 2018 | 67.9 | 0.016 | 0.021 | 0.058 | 0.019 |

**Table 2.**Candidate unknown model parameters. The model parameters are linked to their associated model components in Figure 6. The unknown parameters to be estimated using the model updating technique are highlighted in this table.

No. | Parameter | Description | Nominal Value |
---|---|---|---|

1 | ${E}_{d}$ | Elastic modulus of deck | 27.8 GPa |

2 | ${f}_{c,c}^{\u2019}$ | Compressive strength of column | 40.4 MPa |

3 | ${E}_{c}$ | Initial elastic modulus of column | 27.8 GPa |

4 | ${k}_{T}^{b}$ | Transverse elastomeric shear stiffness of bearing pad | $100\mathrm{MN}/\mathrm{m}$ |

5 | ${k}_{L}^{b}$ | Longitudinal elastomeric shear stiffness of bearing pad | $10\mathrm{MN}/\mathrm{m}$ |

6 | ${m}_{a}$ | Embankment mass for abutment | 53.0 kg |

7 | ${k}_{V}^{p}$ | Vertical soil-foundation stiffness under pier | $12.7\mathrm{G}\mathrm{N}/\mathrm{m}$ |

8 | ${c}_{V}^{p}$ | Vertical soil-foundation damping coefficient under pier | $240\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

9 | ${k}_{L}^{p}$ | Longitudinal soil-foundation stiffness under pier | $9.8\mathrm{GN}/\mathrm{m}$ |

10 | ${c}_{L}^{p}$ | Longitudinal soil-foundation damping coefficient under pier | $220\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

11 | ${k}_{T}^{p}$ | Transverse soil-foundation stiffness under pier | $9.8\mathrm{GN}/\mathrm{m}$ |

12 | ${c}_{T}^{p}$ | Transverse soil-foundation damping coefficient under pier | $190\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

13 | ${k}_{R,L}^{p}$ | Rotational soil-foundation stiffness under pier about the longitudinal axis | $170\mathrm{GN}.\mathrm{m}/\mathrm{rad}$ |

14 | ${c}_{R,L}^{p}$ | Rotational soil-foundation damping coefficient under pier about the longitudinal axis | $2.7\mathrm{GN}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ |

15 | ${k}_{R,T}^{p}$ | Rotational soil-foundation stiffness under pier about the transverse axis | $170\mathrm{GN}.\mathrm{m}/\mathrm{rad}$ |

16 | ${c}_{R,T}^{p}$ | Rotational soil-foundation damping coefficient under pier about the transverse axis | $2.7\mathrm{GN}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ |

17 | ${k}_{R,V}^{p}$ | Rotational soil-foundation stiffness under pier about the vertical axis | $290\mathrm{GN}.\mathrm{m}/\mathrm{rad}$ |

18 | ${c}_{R,V}^{p}$ | Rotational soil-foundation damping coefficient under pier about the vertical axis | $3.6\mathrm{GN}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ |

19 | ${k}_{L}^{a}$ | Longitudinal soil-foundation stiffness under abutment | $8.7\mathrm{GN}/\mathrm{m}$ |

20 | ${c}_{L}^{a}$ | Longitudinal soil-foundation damping coefficient under abutment | $170\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

21 | ${k}_{T}^{a}$ | Transverse soil-foundation stiffness under abutment | $37\mathrm{GN}/\mathrm{m}$ |

22 | ${c}_{T}^{a}$ | Transverse soil-foundation damping coefficient under abutment | $130\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

23 | ${k}_{V}^{a}$ | Vertical soil-foundation stiffness under abutment | $10\mathrm{GN}/\mathrm{m}$ |

24 | ${c}_{V}^{a}$ | Vertical soil-foundation damping coefficient under abutment | $150\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

25 | ${k}_{R,L}^{a}$ | Rotational soil-foundation stiffness under abutment about its longitudinal axis | $300\mathrm{GN}.\mathrm{m}/\mathrm{rad}$ |

26 | ${c}_{R,L}^{a}$ | Rotational soil-foundation damping coefficient under abutment about the longitudinal axis | $4.7\mathrm{GN}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ |

27 | ${k}_{R,V}^{a}$ | Rotational soil-foundation stiffness under abutment about the vertical axis | $240\mathrm{GN}.\mathrm{m}/\mathrm{rad}$ |

28 | ${c}_{R,V}^{a}$ | Rotational soil-foundation damping coefficient under abutment about the vertical axis | $390\mathrm{GN}.\mathrm{m}.\mathrm{s}/\mathrm{rad}$ |

29 | ${k}_{L}^{fs}$ | Far-field soil-embankment stiffness in longitudinal direction | $8.7\mathrm{GN}/\mathrm{m}$ |

30 | ${c}_{R}^{fs}$ | Far-field soil-embankment radiation damping coefficient in the longitudinal direction | $38\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

31 | ${c}_{L}^{fs,m}$ | Far-field soil-embankment material damping coefficient in the longitudinal direction | $140\mathrm{MN}.\mathrm{s}/\mathrm{m}$ |

32 | ${k}_{L}^{bs}$ | Soil-backwall initial stiffness in the longitudinal direction | ${10}^{5}\mathrm{GN}/\mathrm{m}$ |

33 | ${a}_{0}$ | Mass proportional Rayleigh damping coefficient | $0.6{\mathrm{s}}^{-1}$ |

34 | ${a}_{1}$ | Stiffness proportional Rayleigh damping coefficient | 0.003 s |

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**MDPI and ACS Style**

Ghahari, F.; Malekghaini, N.; Ebrahimian, H.; Taciroglu, E.
Bridge Digital Twinning Using an Output-Only Bayesian Model Updating Method and Recorded Seismic Measurements. *Sensors* **2022**, *22*, 1278.
https://doi.org/10.3390/s22031278

**AMA Style**

Ghahari F, Malekghaini N, Ebrahimian H, Taciroglu E.
Bridge Digital Twinning Using an Output-Only Bayesian Model Updating Method and Recorded Seismic Measurements. *Sensors*. 2022; 22(3):1278.
https://doi.org/10.3390/s22031278

**Chicago/Turabian Style**

Ghahari, Farid, Niloofar Malekghaini, Hamed Ebrahimian, and Ertugrul Taciroglu.
2022. "Bridge Digital Twinning Using an Output-Only Bayesian Model Updating Method and Recorded Seismic Measurements" *Sensors* 22, no. 3: 1278.
https://doi.org/10.3390/s22031278