# Accounting for Modeling Errors and Inherent Structural Variability through a Hierarchical Bayesian Model Updating Approach: An Overview

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## Abstract

**:**

## 1. Introduction

## 2. Uncertainty Quantification and Propagation through Hierarchical Bayesian Modeling

#### 2.1. Sources of Uncertainty

#### 2.1.1. Measurement Noise

#### 2.1.2. Changing Ambient and Environmental Conditions

#### 2.1.3. Modeling Errors

#### 2.2. Hierarchical Bayesian Model Updating

#### 2.2.1. Hyperparameters

#### 2.2.2. Error Function

**y**refers to time history measurements,

**x**is model-predicted counterparts and k is the time index.

#### 2.2.3. Model Updating Process

#### 2.3. Probabilistic Response Prediction

#### 2.4. Numerical Methods for Estimating Posterior Distributions

#### 2.4.1. Sampling Approach

#### 2.4.2. Simplified Approach for Estimating Map Values

## 3. Applications to Three Full-Scale Civil Structures

#### 3.1. Application 1: Footbridge at Tufts University Campus

#### 3.1.1. Test Structure and Measured Data

#### 3.1.2. Hierarchical Bayesian Modeling at Different Information Levels

#### 3.1.3. Model Updating Results and Response Predictions

#### 3.2. Application 2: Ten-Story RC Building in Utica, NY

#### 3.2.1. Test Structure and Measured Data

#### 3.2.2. Hierarchical Bayesian Model Updating with Zero-Mean and Non-Zero-Mean Error Function

#### 3.2.3. Model Predictions for Future Building Conditions

#### 3.3. Application 3: Two-Story RC Building in El Centro, California

#### 3.3.1. Test Structure and Measured Data

#### 3.3.2. Hierarchical Bayesian Modeling of Stiffness-Amplitude Relationship

**a**and

**b**are new higher-level hyperparameters which represent the intercept and slope of the linear relationship between stiffness mean ${\mathsf{\mu}}_{\theta}$ and vibration level ${\epsilon}_{t}$. The vibration level is defined as the average RMS of acceleration measurements. Based on a sensitivity analysis, the structure is divided into five substructures with five stiffness parameters, i.e., west (${\theta}_{1}$), east (${\theta}_{2}$), north (${\theta}_{3}$) and south (${\theta}_{4}$) walls of the 2nd story, and walls of the 1st story and all columns in the building (${\theta}_{5}$). Therefore,

**a**and

**b**are 5 × 1 vectors. The stiffness variance ${\Sigma}_{\theta}$ is assumed to be a constant diagonal matrix, as the variance of identified natural frequencies does not appear to be affected by different vibration levels (Figure 14). The error function is assumed to follow a zero-mean Gaussian distribution with a diagonal covariance matrix ${\Sigma}_{e}$. The hierarchical Bayesian method is applied to estimate the hyperparameters (

**a**,

**b**, and ${\Sigma}_{\theta}$) and the error covariance matrix (${\Sigma}_{e}$). In this application, the MAPs of updating parameters are computed using the simplified approach reviewed in Section 2.4.2, and then, MH within Gibbs sampling method was employed to sample the joint posterior PDF using the MAPs as starting point, to save computational efforts. This two-step sampling approach can provide the MAP values and the parameters estimation uncertainties. The MAP values of hyperparameters and error covariance using the simplified approach are reported in Table 3. It can be seen that all values of $\widehat{b}$ which represents the slope in Figure 14 have negative values, as expected, while the values of $\widehat{a}$ represent the mean of structural stiffness at zero vibration level.

#### 3.3.3. Time History Response Predictions

## 4. Summary and Conclusions

- (1)
- In the application of the Dowling Hall footbridge, different information levels are considered for stiffness hyperparameters formulation and compared for their performance. It is found that the stiffness variability is reduced when information about ambient temperatures and excitation amplitudes is considered, thus providing tighter confidence intervals for model-predictions. In this study, error function must be included in model predictions to provide realistic confidence bounds.
- (2)
- In the application of the 10-story building, the effects of error bias are studied, and it is found that more accurate and tighter prediction bounds are obtained when the error bias ${\mathsf{\mu}}_{e}$ of the fifth mode (which was observed to be biased) is considered. It is also found that the estimated error distribution may not be valid outside the calibration range. Therefore, special precautions should be taken when the calibrated model is used for extrapolation.
- (3)
- In the application of the two-story building, the stiffness mean vector is assumed to have a linear relationship with the vibration levels. Accurate predictions are observed for modal parameters and acceleration time histories using the calibrated model when the stiffness-amplitude dependency is explicitly considered, while inaccurate results are observed when this correlation is neglected.

- (a)
- The hierarchical framework is capable of quantifying structural inherent variability and modeling errors, through postulating probability distributions for structural parameters, and estimating the hyperparameters of these distributions. The estimated structural parameters uncertainty would converge to a constant variation level depending on MAP values of hyperparameters, while parameters uncertainty using classical Bayesian methods is reduced infinitely with more data.
- (b)
- More accurate and robust prediction bounds are achieved by hierarchical framework through propagating of parameters variability and error function. This is often more valuable than just obtaining an accurate prediction fit with measurements. Moreover, more reasonable prediction bounds are obtained compared to the classical Bayesian approach, even when only propagating parameters variability, which is especially useful for predictions of unobserved quantities where error function estimate is not available.
- (c)
- Different relationships and factors that contribute to structural parameters uncertainty can be embedded into the hyperparameters, e.g., ambient temperature and excitation amplitude, in the hierarchical framework, which would reduce the parameters variability and provide tighter prediction confidence bounds.
- (d)
- The hierarchical framework is capable of quantifying the residual prediction errors, by estimating the distribution parameters of error function. The inclusion and propagation of error function into response predictions is important and necessary in some cases when a significant amount of uncertainties is retained in the error function, e.g., Applications 1–3. Considering non-zero-mean error distribution in the presence of error bias reduces the error covariance matrix, thus providing tighter confidence bounds.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Comparison of deterministic, classical Bayesian and hierarchical Bayesian methods: In deterministic approach, optimal structural parameter ${\theta}^{opt}$ is estimated; In classical Bayesian methods, most probable structural parameter $\widehat{\mathsf{\theta}}$ is estimated with its estimation uncertainty; At hierarchical level 1, ${\mathsf{\mu}}_{\theta}$ and ${\Sigma}_{\theta}$ are estimated, and at hierarchical level 2, α, β, and λ are estimated

**Figure 3.**South-east view of Dowling Hall footbridge in (

**a**) above-freezing, and (

**b**) below-freezing conditions [54].

**Figure 4.**Effects of temperatures and excitation levels on identified natural frequencies of Dowling Hall footbridge.

**Figure 5.**Estimated stiffness mean (

**a**) and variance (shown as CoV) (

**b**) at information levels 1, 2 and 3 [54].

**Figure 6.**Loading scenarios I (left) and II (right) [54].

**Figure 7.**Comparison between model-predicted and identified natural frequencies in two loading scenarios (left column: loading scenario I; right column: loading scenario II), with/without accounting for error function (top row: without error function; bottom row: with error function).

**Figure 8.**(

**a**) South-west view of 10-story building; (

**b**) location of removed walls for moderate and severe damage states.

**Figure 9.**Model-predicted (gray) vs. identified (black) natural frequencies, using zero-mean error function (upper triangular subplots denote predictions without error function considered; lower triangular subplots denote predictions with propagation of error function).

**Figure 10.**Model-predicted (gray) vs. identified (black) natural frequencies considering non-zero-mean error function for 5th mode, with the inclusion of error function.

**Figure 11.**Model-predicted (with the propagation of error function) (gray) vs. identified (black) natural frequencies of the 10-story building at moderate damage state, considering non-zero-mean error function for 5th mode.

**Figure 12.**Model-predicted (gray) vs. identified (black) natural frequencies of the 10-story building at severe damage state, considering non-zero-mean error function for 5th mode (upper triangular subplots denote predictions without error function considered; lower triangular subplots denote predictions with propagation of error function).

**Figure 15.**Natural frequencies predictions with their identified counterparts [56].

**Figure 16.**Acceleration time history predictions and measured counterparts (light pink areas refer to 95% quantiles and black lines denote measured data) [56].

**Figure 17.**Acceleration time history predictions using ambient vibration level and measured counterparts (light pink areas refer to 95% quantiles and black lines denote measured data) [56].

Q | S | R | ϒ | $\mathit{\tau}$ | Y | $\mathit{\sigma}$ | $\mathbf{log}({\mathit{\sigma}}_{\mathit{e}})$ | |
---|---|---|---|---|---|---|---|---|

Mean | 1.013 | −0.005 | 0.198 | −1.101 | 3.147 | −0.013 | 0.040 | −4.254 |

Standard deviation | 0.0030 | 0.0001 | 0.0027 | 0.0513 | 0.0861 | 0.0004 | 0.0002 | 0.0012 |

**Table 2.**MAP values of hyperparameters and error covariance for zero-mean and non-zero-mean error function cases (estimated error mean/bias is shown in last row).

Zero-Mean Error Function | Non-Zero-Mean Error Function for 5th Mode | ||
---|---|---|---|

Hyperparameters | ${\widehat{\mu}}_{{\theta}_{1}}$ | 0.995 | 0.995 |

${\widehat{\mu}}_{{\theta}_{2}}$ | 1.132 | 1.129 | |

${\widehat{\sigma}}_{{\theta}_{1}}$ | 0.021 | 0.021 | |

${\widehat{\sigma}}_{{\theta}_{2}}$ | 0.015 | 0.014 | |

Error covariance (for eigenvalues) | ${\widehat{\sigma}}_{{\lambda}_{1}}$ | 0.010 | 0.011 |

${\widehat{\sigma}}_{{\lambda}_{2}}$ | 0.014 | 0.014 | |

${\widehat{\sigma}}_{{\lambda}_{3}}$ | 0.007 | 0.007 | |

${\widehat{\sigma}}_{{\lambda}_{4}}$ | 0.034 | 0.034 | |

${\widehat{\sigma}}_{{\lambda}_{5}}$ | 0.188 | 0.021 | |

Error mean | ${\widehat{\mu}}_{{\lambda}_{5}}$ | 0 | −0.185 |

$\widehat{\mathbf{a}}$ | $\widehat{\mathbf{b}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\theta}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{e}}\text{}(\mathit{\%})$ | ||||
---|---|---|---|---|---|---|---|

Hyperparameters | $({\theta}_{1})$ | 0.11 | −2.41 | 0.011 | Error covariance | $({\lambda}_{1})$ | 0.02 |

$({\theta}_{2})$ | 0.85 | −4.68 | 0.008 | $({\lambda}_{2})$ | 0.98 | ||

$({\theta}_{3})$ | 0.08 | −0.44 | 0.010 | $({\Phi}_{1})$ | 1.32 | ||

$({\theta}_{4})$ | 0.40 | −4.91 | 0.017 | $({\Phi}_{2})$ | 1.98 | ||

$({\theta}_{5})$ | 1.82 | −22.66 | 0.065 |

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**MDPI and ACS Style**

Song, M.; Behmanesh, I.; Moaveni, B.; Papadimitriou, C.
Accounting for Modeling Errors and Inherent Structural Variability through a Hierarchical Bayesian Model Updating Approach: An Overview. *Sensors* **2020**, *20*, 3874.
https://doi.org/10.3390/s20143874

**AMA Style**

Song M, Behmanesh I, Moaveni B, Papadimitriou C.
Accounting for Modeling Errors and Inherent Structural Variability through a Hierarchical Bayesian Model Updating Approach: An Overview. *Sensors*. 2020; 20(14):3874.
https://doi.org/10.3390/s20143874

**Chicago/Turabian Style**

Song, Mingming, Iman Behmanesh, Babak Moaveni, and Costas Papadimitriou.
2020. "Accounting for Modeling Errors and Inherent Structural Variability through a Hierarchical Bayesian Model Updating Approach: An Overview" *Sensors* 20, no. 14: 3874.
https://doi.org/10.3390/s20143874