# Dimensional Upgrade Approach for Spatial-Temporal Fusion of Trend Series in Subsidence Evaluation

## Abstract

**:**

## 1. Introduction

## 2. Study Background

#### 2.1. Study Area

^{3}/year [8,21]. The maximum subsidence rate is 6.1 cm/year and the continuous subsidence area (subsidence rate larger than 3.0 cm/year) was about 309.1 km

^{2}in 2014 [42]. Taiwan High Speed Rail passes across the subsidence areas, which is a safety concern [43]. Subsidence in Jhuoshuei River Alluvial Fan is thus an important issue for the Taiwanese government.

#### 2.2. Subsidence Monitoring

## 3. Methodology

#### 3.1. Nonlinear Poroelastic Model

^{−1}= λ + 2µ, where λ and µ are Lame’s constants (µ = E/2(1 + v), and λ = Ev/(1 + v)(1 − 2v), where E is Young’s modulus, and v is Poisson’s ratio); a is the dimensionless coefficient of the effective stress; Q

^{−1}is the compressibility introduced by Biot; and κ is the Darcy conductivity, which is related to hydraulic conductivity K by K = λwκ, where λw is the unit weight of water (= 9810 N/m

^{3}).

#### 3.2. Grey System Model

^{(1)}is a canonical series written as:

^{(0)}is the original data series (i.e., leveling data series in this study) and N is the data number. Substituting the evaluated a and b into Equation (5) and letting the initial condition be x

^{(1)}(1) = x

^{(0)}(1) yields:

^{(1)}at the n + 1 point. Using back differential for ${x}^{(1)}(1)={x}^{(0)}(1)$ yields:

#### 3.3. Model Settings

#### 3.4. Fusion Method

_{1}to NPM

_{1,}NPM

_{2,}and NPM

_{3}are d

_{11,}d

_{12,}and d

_{13,}respectively (i.e., di

_{j}is the distance from GSM

_{i}to NPM

_{j}). The slopes of the subsidence trend for GSM

_{i}and NPM

_{j}between time t and t −1 can be written, respectively, as:

_{i}and NPM

_{j}are the evaluated subsidence values for the GSM at position i and the NPM at position j, respectively.

_{i}and NPM

_{j}, the mean slope of the subsidence trend for NPM

_{j}with respect to GSM

_{i}can be written as:

_{j}with respect to GSM

_{i}in the time interval dt and m is the number of monitoring points for the NPM (i.e., the number of MCMWs). A distance weighting between the points for the GSM and the NPM is considered in calculating the mean slope of the NPM. The influence decreases with increasing distance between the GSM and the NPM.

_{i}in the time interval dt is larger than the mean slope of NPM

_{j}in the same time interval, the slope of GSM

_{i}is substituted by the mean slope of NPM

_{j}. The slope criterion can be written as:

## 4. Results and Discussion

#### 4.1. Results for Nonlinear Poroelastic Model

^{2}values are good for the MCMWs except for the Jiaxing well. The reason for the poor fitting has been mentioned previously and is shown in Figure 5d. From the fitting results, the quantity of discharge is 0 to 2.58 × 10

^{−9}m/s and that of loading is 0 to 1.07 × 10

^{4}N/m

^{2}. Two and 15 MCMWs show zero discharge and zero loading, respectively. The main driving force for the land subsidence in Jhuoshuei River Alluvial Fan is groundwater pumping, which is expressed as discharge (pumping rate in unit area) in Table 3. The influence of top loading is relatively small, occurring only at 11 MCMWs. The quantities of loading are also small, lower than atmospheric pressure (= 1.013 × 10

^{5}N/m

^{2}).

#### 4.2. Results for Grey System Model

#### 4.3. Subsidence Fusion Results

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Study area of Jhuoshuei River Alluvial Fan, Taiwan, and distribution of leveling points and multi-level compaction monitoring wells. A–A’ is the line representing the hydrogeological profile shown in Figure 2.

**Figure 2.**One hydrogeological profile of Jhuoshuei River Alluvial Fan, Taiwan. The profile trace is shown in Figure 1.

**Figure 3.**Monitoring data from MCMWs at (

**a**) Xinghua and (

**b**) Yuanchang wells in spatial domain at various periods.

**Figure 4.**Schematic map of distribution of monitoring points. Grey system models (GSM)

_{i}and NPM

_{j}indicate monitoring points for grey system and poroelastic models, respectively. d

_{11,}d

_{12,}and d

_{13}are distances from GSM

_{1}to NPM

_{1,}NPM

_{2,}and NPM

_{3,}respectively.

**Figure 5.**Four MCMWs for demonstrating fitting results. (

**a**) Xinghua; (

**b**)Xizhou; (

**c**) Yunchang; and (

**d**) Jiaxing wells. White triangles show data used for model verification.

**Figure 6.**Subsidence distribution in Jhuoshuei River Alluvial Fan in 2039 obtained using GSM. Triangle symbols numbered #1 and #2 are locations of demonstration points used for fusion results in Figure 8.

**Figure 7.**Subsidence distribution with fusion in Jhuoshuei River Alluvial Fan in 2039 obtained using NPM and GSM. Triangle symbols numbered #1 and #2 are locations of demonstration points used for fusion results in Figure 8.

**Figure 8.**Comparison of results obtained with and without fusion for two leveling points marked in Figure 7.

**Table 1.**Basic information for multi-level compaction monitoring wells (MCMWs) adopted in this study.

County | MCMWName | Monitoring Depth (m) | Install Time (month-year) | E (TWD97) | N (TWD97) |
---|---|---|---|---|---|

Changhua | Xinjie | 300 | May-98 | 179967 | 2644391 |

Xigang | 300 | May-97 | 177633 | 2639733 | |

Xinghua | 300 | October-03 | 188363 | 2643201 | |

Xinsheng | 300 | April-08 | 188341 | 2648279 | |

Hunan | 300 | September-05 | 196984 | 2649404 | |

Xizhou | 300 | October-07 | 198873 | 2638772 | |

Zhutang | 300 | October-07 | 191773 | 2639649 | |

Yunlin | Feng’an | 300 | July-96 | 171858 | 2631893 |

Haifeng | 200 | December-94 | 171149 | 2629139 | |

Xinxing | 300 | July-96 | 170720 | 2626355 | |

Lunfeng | 200 | December-94 | 169441 | 2624157 | |

Jianyang | 200 | December-94 | 163505 | 2614756 | |

Dongguang* | 300 | August-09 | 175782 | 2616754 | |

Jinhu | 200 | December-94 | 163597 | 2608018 | |

Yiwu* | 300 | August-09 | 167840 | 2604973 | |

Chanlin | 300 | April-08 | 173087 | 2608156 | |

Erlun | 300 | April-08 | 190428 | 2629865 | |

Fengrong | 300 | April-08 | 179784 | 2632015 | |

Yuanchang | 300 | January-03 | 179484 | 2616803 | |

Kecuo | 300 | October-03 | 182074 | 2613831 | |

Neiliao | 300 | October-07 | 184141 | 2611722 | |

Tuku | 300 | September-03 | 187771 | 2620610 | |

Xiutan | 300 | August-06 | 183651 | 2617396 | |

Huwei | 300 | March-06 | 192040 | 2623605 | |

Guangfu | 300 | October-07 | 189083 | 2626507 | |

Zhengmin* | 330 | August-09 | 189570 | 2622974 | |

Longyan | 300 | March-06 | 179249 | 2624490 | |

Zhennan | 300 | October-07 | 202938 | 2621719 | |

Jiaxing | 300 | April-08 | 194874 | 2616145 |

^{*}Stations not adopted in this study due to short monitoring period.

MCMW Name | Hydraulic Conductivity (m/s)a | Porositya | Young’s Modulus (N/m^{2})b | Poisson’s Ratio b |
---|---|---|---|---|

Xinjie | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Xigang | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Xinghua | 5.39 × 10^{−8} | 0.421 | 1.00 × 10^{7} | 0.3 |

Xinsheng | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Hunan | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Xizhou | 8.54 × 10^{−7} | 0.469 | 1.00 × 10^{6} | 0.3 |

Zhutang | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Feng’an | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Haifeng | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Xinxing | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Lunfeng | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Jianyang | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Dongguang | 5.34 × 10^{−8} | 0.452 | 1.00 × 10^{7} | 0.3 |

Jinhu | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Yiwu | 4.05 × 10^{−9} | 0.474 | 1.00 × 10^{7} | 0.3 |

Chanlin | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Erlun | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Fengrong | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Yuanchang | 1.66 × 10^{−8} | 0.431 | 1.00 × 10^{7} | 0.3 |

Kecuo | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Neiliao | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Tuku | 4.48 × 10^{−6} | 0.447 | 1.00 × 10^{6} | 0.3 |

Xiutan | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

Huwei | 3.88 × 10^{−8} | 0.453 | 1.00 × 10^{7} | 0.3 |

Guangfu | 5.43 × 10^{−8} | 0.461 | 1.00 × 10^{7} | 0.3 |

Zhengmin | 5.27 × 10^{−8} | 0.467 | 1.00 × 10^{7} | 0.3 |

Longyan | 1.36 × 10^{−7} | 0.411 | 1.00 × 10^{7} | 0.3 |

Zhennan | 6.62 × 10^{−8} | 0.443 | 1.00 × 10^{7} | 0.3 |

Jiaxing | 8.17 × 10^{−8} | 0.449 | 1.00 × 10^{7} | 0.3 |

^{a}Hydraulic conductivity value of 8.17 × 10

^{−8}m/s and porosity value of 0.449 were calculated their means for other MCMWs.

^{b}Values of Young’s modulus and Poisson’s ratio were taken from Das [47].

County | Station | Loading (N/m^{2}) | Discharge (m/s) | R^{2} |
---|---|---|---|---|

Changhua | Xinjie | 1.07 × 10^{4} | 0 | 0.945 |

Xigang | 6.64 × 10^{3} | 0 | 0.710 | |

Xinghua | 5.02 × 10^{3} | 9.25 × 10^{−10} | 0.990 | |

Xinsheng | 0 | 1.44 × 10^{−9} | 0.991 | |

Hunan | 0 | 1.21 × 10^{−9} | 0.990 | |

Xizhou | 0 | 4.46 × 10^{−10} | 0.980 | |

Zhutang | 1.29 × 10^{3} | 1.04 × 10^{−9} | 0.993 | |

Yunlin | Feng’an | 0 | 4.43 × 10^{−9} | 0.917 |

Haifeng | 0 | 2.52 × 10^{−10} | 0.795 | |

Xinxing | 0 | 8.62 × 10^{−10} | 0.958 | |

Lunfeng | 0 | 4.37 × 10^{−10} | 0.898 | |

Jianyang | 7.77 × 10^{1} | 3.03 × 10^{−10} | 0.887 | |

Jinhu | 1.27 × 10^{3} | 3.26 × 10^{−10} | 0.813 | |

Chanlin | 0 | 4.45 × 10^{−10} | 0.505 | |

Erlun | 1.38 × 10^{3} | 3.14 × 10^{−10} | 0.962 | |

Fengrong | 0 | 7.84 × 10^{−10} | 0.893 | |

Yuanchang | 0 | 1.96 × 10^{−9} | 0.954 | |

Kecuo | 3.53 × 10^{1} | 1.37 × 10^{−9} | 0.972 | |

Neiliao | 4.21 × 10^{2} | 1.51 × 10^{−9} | 0.950 | |

Tuku | 0 | 2.58 × 10^{−9} | 0.867 | |

Xiutan | 0 | 1.29 × 10^{−9} | 0.948 | |

Huwei | 4.22 × 10^{3} | 7.87 × 10^{−10} | 0.985 | |

Guangfu | 0 | 9.46 × 10^{−10} | 0.978 | |

Longyan | 0 | 1.40 × 10^{−9} | 0.975 | |

Zhennan | 1.95 × 10^{3} | 7.78 × 10^{−11} | 0.639 | |

Jiaxing | 0 | 6.80 × 10^{−11} | 0.101 |

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**MDPI and ACS Style**

Wang, S.-J.
Dimensional Upgrade Approach for Spatial-Temporal Fusion of Trend Series in Subsidence Evaluation. *Entropy* **2015**, *17*, 3035-3052.
https://doi.org/10.3390/e17053035

**AMA Style**

Wang S-J.
Dimensional Upgrade Approach for Spatial-Temporal Fusion of Trend Series in Subsidence Evaluation. *Entropy*. 2015; 17(5):3035-3052.
https://doi.org/10.3390/e17053035

**Chicago/Turabian Style**

Wang, Shih-Jung.
2015. "Dimensional Upgrade Approach for Spatial-Temporal Fusion of Trend Series in Subsidence Evaluation" *Entropy* 17, no. 5: 3035-3052.
https://doi.org/10.3390/e17053035