# Dam Deformation Monitoring Data Analysis Using Space-Time Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Space-Time Kalman Filter Model

#### 2.1. Mathematical Model

#### 2.2. Spatial Fields $H$

#### 2.3. Parameters Estimation

- Use Kalman smoother to estimate the unknown state parameter $\mathit{\alpha}\left(t\right)$ with respect to the $\left(r\right)th$ iterated value ${\theta}_{r}$.
- E step: calculate the conditional expectation $\mathrm{G}\left({\theta}_{r}\right)=E(\mathrm{log}\left(l\right)|{\mathit{L}}_{1},{\mathit{L}}_{2},\mathrm{..},{\mathit{L}}_{m})$ of $\mathrm{log}\left(l\right)$ under the estimated $\mathit{\alpha}\left(t\right)$ distribution in step 1, where $E(\ast )$ is the expectation operator.
- M step: maximize $\mathrm{G}\left({\theta}_{r}\right)$, which yields the newly iterated value ${\theta}_{r+1}$.
- Replace ${\theta}_{r}$ with ${\theta}_{r+1}$, and repeat steps 1, 2, and 3 until the logarithm of joint likelihood function $\mathrm{log}\left(l\right)$ or the innovations form [23] stop increasing.

#### 2.4. Denoising, Space-Time Interpolation, and Prediction

## 3. Simulation Experiment

## 4. Application

#### 4.1. Description of Wuqiangxi Dam Tension Wire Alignment Data

#### 4.2. Filtering, Spatiotemporal Interpolation, and Prediction

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Flow diagram of the overall procedure of Space-Time Kalman Filter (STKF). EM algorithm, Expectation Maximization algorithm.

**Figure 3.**Location of the tension wire measuring points. The red points in the upper part show EX2 tension wire alignment measurement points; red points in the lower part show EX1 tension wire alignment measurement points; blue points are the inverted plumb measurement points.

**Figure 4.**Displacement sequences. (

**a**) Displacement sequence of the 20 points; (

**b**) Displacement sequence of EX2_21.

**Figure 6.**Filter and interpolated missing data results. (

**a**) Filter and interpolated missing data results of all points; (

**b**) Filter and interpolated missing data results of EX2_21.

**Figure 7.**Interpolated time series. (

**a**) Interpolation of all points; (

**b**) Filter and interpolation results of EX2_21.

**Figure 9.**Equally chosen 20 days of interpolated horizontal displacement for the whole dam in the spatio-temporal domain between 24 August 2005 and 17 March 2007.

**Table 1.**$RM{S}^{1}$ of Interpolation, Filter, and Prediction of each point (unit: mm). Interp, Interpolation; Pred, Prediction.

Site | Interp | Filter | Pred | Site | Interp | Filter | Pred | Site | Interp | Filter | Pred |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.019 | 0.022 | 0.030 | 9 | 0.039 | 0.043 | 0.059 | 17 | 0.014 | 0.019 | 0.038 |

2 | 0.024 | 0.025 | 0.056 | 10 | 0.037 | 0.037 | 0.073 | 18 | 0.009 | 0.017 | 0.029 |

3 | 0.026 | 0.029 | 0.051 | 11 | 0.036 | 0.036 | 0.077 | 19 | 0.004 | 0.012 | 0.003 |

4 | 0.030 | 0.032 | 0.055 | 12 | 0.034 | 0.034 | 0.068 | 20 | 0.008 | 0.013 | 0.008 |

5 | 0.033 | 0.035 | 0.061 | 13 | 0.031 | 0.032 | 0.064 | 21 | 0.015 | 0.017 | 0.033 |

6 | 0.035 | 0.035 | 0.081 | 14 | 0.028 | 0.029 | 0.055 | 22 | 0.023 | 0.025 | 0.040 |

7 | 0.037 | 0.037 | 0.091 | 15 | 0.024 | 0.027 | 0.042 | 23 | 0.032 | 0.034 | 0.057 |

8 | 0.037 | 0.037 | 0.072 | 16 | 0.019 | 0.022 | 0.038 |

Site | Position | Site | Position | Site | Position | Site | Position |
---|---|---|---|---|---|---|---|

EX2_1 | 0.5 | EX2_2 | 17.1 | EX2_3 | 41.6 | EX2_4 | 61.1 |

EX2_5 | 81.6 | EX2_6 | 97.1 | EX2_7 | 115.6 | EX2_8 | 134.1 |

EX2_10 | 168.6 | EX2_11 | 184.1 | EX2_12 | 205.6 | EX2_13 | 230.2 |

EX2_14 | 254.7 | EX2_15 | 279.2 | EX2_16 | 286.2 | EX2_17 | 303.7 |

EX2_18 | 329.2 | EX2_19 | 353.7 | EX2_20 | 378.2 | EX2_21 | 402.7 |

**Table 3.**$RM{S}^{2}$ Filter, Interpolation, and Prediction of each point (unit: mm). Interp, Interpolation; Pred, Prediction.

Site | Filter | Pred | Interp | Site | Filter | Pred | Interp |
---|---|---|---|---|---|---|---|

EX2_1 | 0.05 | 0.43 | 0.45 | EX2_12 | 0.16 | 1.07 | 0.52 |

EX2_2 | 0.05 | 0.09 | 0.15 | EX2_13 | 0.18 | 1.07 | 0.32 |

EX2_3 | 0.08 | 0.43 | 0.29 | EX2_14 | 0.18 | 1.14 | 0.45 |

EX2_4 | 0.12 | 0.38 | 0.41 | EX2_15 | 0.12 | 1.06 | 0.2 |

EX2_5 | 0.11 | 0.91 | 0.84 | EX2_16 | 0.12 | 1.05 | 0.19 |

EX2_6 | 0.11 | 0.31 | 1.12 | EX2_17 | 0.12 | 0.31 | 0.53 |

EX2_7 | 0.11 | 0.84 | 1.67 | EX2_18 | 0.2 | 0.70 | 0.74 |

EX2_8 | 0.09 | 0.44 | 0.76 | EX2_19 | 0.22 | 0.58 | 0.4 |

EX2_10 | 0.15 | 0.66 | 0.66 | EX2_20 | 0.23 | 0.41 | 0.42 |

EX2_11 | 0.12 | 0.93 | 0.28 | EX2_21 | 0.21 | 0.53 | 0.92 |

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**MDPI and ACS Style**

Dai, W.; Liu, N.; Santerre, R.; Pan, J.
Dam Deformation Monitoring Data Analysis Using Space-Time Kalman Filter. *ISPRS Int. J. Geo-Inf.* **2016**, *5*, 236.
https://doi.org/10.3390/ijgi5120236

**AMA Style**

Dai W, Liu N, Santerre R, Pan J.
Dam Deformation Monitoring Data Analysis Using Space-Time Kalman Filter. *ISPRS International Journal of Geo-Information*. 2016; 5(12):236.
https://doi.org/10.3390/ijgi5120236

**Chicago/Turabian Style**

Dai, Wujiao, Ning Liu, Rock Santerre, and Jiabao Pan.
2016. "Dam Deformation Monitoring Data Analysis Using Space-Time Kalman Filter" *ISPRS International Journal of Geo-Information* 5, no. 12: 236.
https://doi.org/10.3390/ijgi5120236