# Analyzing GNSS Measurements to Detect and Predict Bridge Movements Using the Kalman Filter (KF) and Neural Network (NN) Techniques

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Tabiat Bridge and Structural Health Monitoring System Description

## 3. Methodology

#### 3.1. Time-Series Denoising Using the Kalman Filter (KF) Approach

#### 3.2. Extracting the Movement Components

#### 3.2.1. Semi-Static Component

#### 3.2.2. Static and Short-Period Components

#### 3.2.3. Dynamic Component

_{2}-norm of a vector $y$ and the optimum filter can be defined by the minimum RMSE and MSE, and maximum NRMSE.

#### 3.3. Neural Network (NN) for Prediction

#### 3.4. Least Squares Harmonic Estimation (LS-HE) Method

## 4. Results and Discussions

#### 4.1. Data collection and Preparation

#### 4.2. Data Pre-Processing

#### 4.3. Bridge Movement Evaluation

#### 4.4. Frequency Domain Evaluation

#### 4.5. Evaluation of the Bridge Movement Prediction Model

^{−7}, 2.945 × 10

^{−7}, and 3.108 × 10

^{−7}m in the North, East, and Up directions, while these values for Station 2 are found to be 7.623 × 10

^{−7}, 2.651 × 10

^{−7}, and 6.425 × 10

^{−6}m. For the dynamic component, the RMSEs calculated between the model and the actual dynamic component, in Station 1, are found to be 2.478 × 10

^{−6}m in the North direction, 1.983 × 10

^{−6}m in the East, and 4.2 × 10

^{−6}m in the Up direction. For Station 2, the dynamic model is fitted with 2.335 × 10

^{−6}, 2.19 × 10

^{−6}, and 3.477 × 10

^{−6}m errors in the North, East, and Up direction, respectively. The abilities of the NN model in the prediction of semi-static and dynamic components are estimated with the RMSE values, which are summarized in Table 6. We also simulate the semi-static and dynamic components of the M022 station using this NN. The results of fitting in the prediction mode demonstrate the RMSEs of 3.894 × 10

^{−5}, 6.98 × 10

^{−5}, and 2.727 × 10

^{−4}m in the North, East, and Up directions, respectively.

## 5. Conclusions

- The Kalman filter technique can be considered as a precise technique to de-noise the coordinates time-series. This method improves the uncertainty of data approximately from 0.024 to 0.0013 m.
- The least squares harmonic estimation method was found to be efficient for extracting dominant frequencies of the dynamic component of the bridge movement, especially the step-wise statistical test avoids extracting non-meaningful dominant frequencies. The numerical results obtained from this method indicate that the bridge performance is natural under the load effect during the monitoring time.
- Using the neural network method can be considered as an appropriate technique to forecast the dynamic and semi-static components of the bridge for 15 min. Here, the prediction model obtained an accuracy of about 6 × 10
^{−5}and 4 × 10^{−5}m in the dynamic and semi-static components.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Tabiat Bridge plan and position (in Tehran, Iran, latitude: 35°41′47.75″ and longitude: 51°11′45.55″); (

**b**) GNSS-based monitoring system along the bridge; (

**c**) permanent station location; (

**d**) distance between the monitoring and permanent stations.

**Figure 5.**The correlation coefficient of the semi-static movement for the stations along the bridge. (The circles’ radius and color are associated with the numerical value of correlation)

**.**

**Figure 6.**The static components of Stations 1 and 2: (

**a**) Horizontal displacement of the East versus North; (

**b**) time-series in the Up direction.

**Figure 7.**Evaluation criteria of the dynamic component for Station 1: (

**a**) North; (

**b**) East; and (

**c**) Up direction.

**Figure 8.**The dynamic component of the bridge’s movements: (

**a**) North; (

**b**) East; and (

**c**) Up directions.

**Figure 9.**Short-period component time-series and periodogram diagram for the: (

**a**) East of Station 1; (

**b**) east of Station 2; and (

**c**) east of Station M022.

${\widehat{X}}_{K|K-1}=\mathsf{\Phi}{\widehat{X}}_{K-1|K-1}$ |

${Q}_{{\widehat{X}}_{K|K-1}}=\mathsf{\Phi}{Q}_{{\widehat{X}}_{K-1|K-1}}{\mathsf{\Phi}}^{T}+{Q}_{P}$ |

${V}_{K}={y}_{K}-{A}_{K}{\widehat{X}}_{K|K-1}$ |

${K}_{K}={Q}_{{\widehat{X}}_{K|K-1}}{A}_{K}^{T}{\left({A}_{K}{Q}_{{\widehat{X}}_{K|K-1}}{A}_{K}^{T}+{R}_{K}\right)}^{-1}$ |

${\widehat{X}}_{K|K}={\widehat{X}}_{K|K-1}+{K}_{K}{V}_{K}$ |

${Q}_{{\widehat{X}}_{K|K}}=\left(I-{K}_{K}{A}_{K}\right){Q}_{{\widehat{X}}_{K|K-1}}$ |

**Table 3.**An overview of the observation, models and processing strategy used for the network of this study.

GNSS System(s) | GPS Only |
---|---|

Basic observable | carrier phase with an elevation angle cutoff of 7° and a sampling rate of 30 s. |

Modelled observable | Double differences of the ionosphere-free linear combination. |

Ground antenna phase center calibrations | IGS08 absolute phase-center variation model is applied. |

Tropospheric Model | A priori model is the GMF mapped with the DRY-GMF. |

Tropospheric Mapping Function | GMF |

Ionosphere | First-order effect eliminated by forming the ionosphere-free linear combination of L1 and L2. Second and third effect applied. |

Center orbit time | final |

Station coordinates | Coordinate constraints are applied at the Reference sites with standard deviation of 1 mm and 2 mm for horizontal and vertical components respectively. |

Ambiguity | Ambiguities are resolved in a baseline-by-baseline mode using the Code-Based strategy for 180–6000 km baselines, the Phase-Based L5/L3 strategy for 18–200 km baselines, the Quasi-Ionosphere-Free (QIF) strategy for 18–2000 km baselines and the Direct L1/L2 strategy for 0–20 km baselines. |

Terrestrial reference frame | IGS08 station around Iran (ankr, artu, drag, nico, polv, tehn, zeck) coordinates and velocities mapped to the mean epoch of observation. |

Statistical Parameter | Station 1 | Station 2 | ||||
---|---|---|---|---|---|---|

N | E | U | N | E | U | |

MAX | 0.1133 | 0.1465 | 0.2553 | 0.1145 | 0.1001 | 0.1934 |

STD | 0.03 | 0.05 | 0.057 | 0.035 | 0.045 | 0.058 |

Point | N | E | U |
---|---|---|---|

Station 1 | 0.00052 | 0.00028 | 0.00022 |

Station 2 | 0.00035 | 0.00027 | 0.00022 |

Station M022 | 0.00031 | 0.00026 | 0.00024 |

**Table 6.**The assessment of semi-static and dynamic components in the prediction mode using the neural network technique (NN).

Semi-Static | Dynamic | ||||||
---|---|---|---|---|---|---|---|

N | E | U | N | E | U | ||

RMSE | Station 1 | 4.736 × 10^{−5} | 2.676 × 10^{−5} | 0.0012 | 7.940 × 10^{−5} | 3.490 × 10^{−5} | 7.807 × 10^{−5} |

Station 2 | 9.772 × 10^{−5} | 4.371 × 10^{−5} | 0.0002 | 0.0001 | 6.616 × 10^{−5} | 0.0002 |

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**MDPI and ACS Style**

Forootan, E.; Farzaneh, S.; Naderi, K.; Cederholm, J.P. Analyzing GNSS Measurements to Detect and Predict Bridge Movements Using the Kalman Filter (KF) and Neural Network (NN) Techniques. *Geomatics* **2021**, *1*, 65-80.
https://doi.org/10.3390/geomatics1010006

**AMA Style**

Forootan E, Farzaneh S, Naderi K, Cederholm JP. Analyzing GNSS Measurements to Detect and Predict Bridge Movements Using the Kalman Filter (KF) and Neural Network (NN) Techniques. *Geomatics*. 2021; 1(1):65-80.
https://doi.org/10.3390/geomatics1010006

**Chicago/Turabian Style**

Forootan, Ehsan, Saeed Farzaneh, Kowsar Naderi, and Jens Peter Cederholm. 2021. "Analyzing GNSS Measurements to Detect and Predict Bridge Movements Using the Kalman Filter (KF) and Neural Network (NN) Techniques" *Geomatics* 1, no. 1: 65-80.
https://doi.org/10.3390/geomatics1010006