# A Landslide Probability Model Based on a Long-Term Landslide Inventory and Rainfall Factors

^{*}

## Abstract

**:**

## 1. Introduction

^{2}[21]. For instance, a research area of 4660 km

^{2}was subdivided into 12 analytical units with an average area of 388.3 km

^{2}[39], but excessively large analytical units make it impossible to identify the precise possible locations of landslides. In addition, the subdivision approaches employed in some studies run into the problem of incomplete coverage. For instance, although a 25 km

^{2}research area was subdivided into eight analytical units, the landslide prediction results only represented the paths of roads in the subdivisions and not the entire subdivisions because most landslides (94%) in the study occurred on roadside slopes [26]. Althuwaynee et al. [28] divided the research area into six circular analytical units with their centers at rain gauges, but the analytical units did not cover the entire research area and also overlapped. Although these studies subdivided their research areas into different analytical units, the units could not provide a landslide probability distribution with a finer spatial resolution because they were excessively large, or experienced problems such as incomplete coverage and overlap. If the method of subdividing a research area into analytical units is improved so that the units are smaller in area, the spatial resolution of the landslide probability estimation results could be improved. There are seven types of analytical units subdivided in research areas: grid cell, terrain unit, unique condition unit, slope unit, geo-hydrological unit, topographical unit, and administrative unit [46,47]. The slope units are suitable for landslide probability analysis because they express topographic features and slope characteristics. In this study, we consequently selected slope units as our analytical unit.

## 2. Research Area and Materials

#### 2.1. Environmental Setting of Taipei Water Source Domain

#### 2.2. Rainfall Data

#### 2.3. Landslide Inventory

^{2}and the average area was 2474 m

^{2}. The resulting distribution of landslides caused by the eight typhoon events was shown in Figure 4, which revealed that landslide sites were concentrated in the southwestern portion of the research area.

#### 2.4. Analytical Units and Rain Gauge Control Areas

## 3. Methods

#### 3.1. Analysis of Discrete Rainfall Groups

_{t}). After selecting rain gauges near the research area with rainfall data for recent years, we obtained daily rainfall data for the 1987–2016 period from the Water Resources Agency and Central Weather Bureau. This study calculated the effective accumulated rainfall based on rainfall for that day and rainfall during the previous 7 days using the method proposed by Jan [52]; this calculation was performed using Equation (1):

_{0}is the rainfall amount on that day, R

_{1}is the rainfall amount on the day before that day, and so on, and the weighting coefficient α = 0.7 proposed by Jan [52].

_{t}), we obtained a group of daily rainfall and effective accumulated rainfall (I, R

_{t}) for each day. The daily rainfall and effective accumulated rainfall were continuous variables and would not facilitate subsequent calculation of a joint cumulative distribution function, therefore we rounded off the daily rainfall and effective accumulated rainfall values to the 10th place and made them discrete variables. The group of daily rainfall and effective accumulated rainfall (I, R

_{t}) for each day was termed as “discrete rainfall group” in this study.

#### 3.2. Joint Cumulative Distribution Function

_{I},

_{Rt}(I

_{i}, R

_{tj})) of each discrete rainfall group (I

_{i}, R

_{tj}) was defined [54] as shown in Equation (2):

_{t}are multiples of 10 and the probability values in other places are 0.

_{i}, R

_{tj}) from the origin, the greater its probability value. The probability of a discrete rainfall group on the curved surface expressed the cumulative probability of all discrete rainfall groups, which were nearer to the origin than this discrete rainfall group (I

_{i}, R

_{tj}).

#### 3.3. Selection of a Rainfall Probability Threshold

#### 3.4. Poisson Probability Model

#### 3.5. Conditional Probability

## 4. Results and Discussion

#### 4.1. Joint Cumulative Distribution Functions of the Rain Gauges

_{t}) by employing Equation (1), which yielded rainfall and effective accumulated rainfall for each day. We then rounded off the daily rainfall and effective accumulated rainfall values to the 10th place, which yielded discrete rainfall groups including both daily rainfall and effective accumulated rainfall. The next step was establishing frequency tables for different discrete rainfall groups, which we used to show the frequency of the discrete rainfall groups. Figure 5 shows the frequency of discrete rainfall groups at the Bihu rain gauge with daily rainfall and effective accumulated rainfall (R

_{t}) ranging from 0 to 100 mm. The depth axis represents daily rainfall, the horizontal axis represents the effective accumulated rainfall (R

_{t}), and the vertical axis represents the frequency of a discrete rainfall group. We then calculated the cumulative frequency of each discrete rainfall group on this basis, and this represented the frequency of all discrete rainfall groups with values lower than that of any designated discrete rainfall group. The cumulative frequency was then divided by the total frequency of all discrete rainfall groups, which yielded the cumulative probability of each discrete rainfall group.

_{t}) are shown within a 0–300 mm range. The joint cumulative distribution functions have areas with gentler slopes indicating fewer and more dispersed discrete rainfall groups within a certain interval, and areas with steeper slopes indicating more and more concentrated discrete rainfall groups within a certain interval.

#### 4.2. Selection of Rainfall Probability Thresholds of the Rain Gauges

#### 4.3. Landslide Probability Analysis Employing a Rainfall Probability Threshold and a Long-Term Landslide Inventory

#### 4.4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Average daily rainfall within the Taipei Water Source Domain, 2000–2015. The dots represent the eight typhoon events causing the corresponding landslide inventory.

**Figure 3.**Average daily rainfall and standard deviation of the eight typhoon events in each control area of the rain gauge divided by the modified Thiessen polygon method.

**Figure 4.**Distribution of slope units and landslide sites caused by the eight typhoon events in the Taipei Water Source Domain.

**Figure 8.**The exceedance probability that at least one rainfall event will exceed the threshold of discrete rainfall group within any one year in each rain gauge control area.

**Figure 9.**The landslide probability of slope units in the Taipei Water Source Domain when rainfall exceeds the threshold of discrete rainfall group.

**Figure 10.**The probability that at least one rainfall event exceeds the threshold of discrete rainfall group and one landslide will also occur during the future one-year period within the Taipei Water Source Domain.

Typhoon Event | Date (MM/DD/YYYY) | Average Rainfall at the Date (mm) | Number of New Landslide Sites | Smallest Landslide Area (m^{2}) | Largest Landslide Area (m^{2}) | Total Area of Landslides (m^{2}) | Average Area of Landslides (m^{2}) |
---|---|---|---|---|---|---|---|

Xangsane | 11/01/2000 | 326.67 | 42 | 326 | 19,619 | 131,148 | 3123 |

Nari | 09/16/2001 | 538.05 | 92 | 107 | 68,032 | 261,650 | 2844 |

Aere | 08/24/2004 | 465.57 | 97 | 140 | 27,270 | 239,856 | 2473 |

Sinlaku | 09/13/2008 | 348.18 | 32 | 475 | 21,101 | 71,111 | 2222 |

Morakot | 08/07/2009 | 219.83 | 173 | 16 | 118,108 | 1,016,448 | 5875 |

Parma | 10/05/2009 | 221.79 | 302 | 47 | 49,369 | 484,785 | 1605 |

Megi | 10/21/2010 | 262.35 | 47 | 407 | 27,318 | 118,874 | 2529 |

Soudelor | 08/08/2015 | 478.99 | 589 | 257 | 48,041 | 1,075,263 | 1826 |

**Table 2.**True positive rate (TPR), true negative rate (TNR), positive predictive value (PPV), and Youden’s index for Bihu rain gauge at different rainfall probability thresholds.

Rainfall Probability Threshold | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 |

Number of landslide events predicted correctly | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

Number of rainfall events triggering landslides actually | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

TPR | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 100% |

Number of no landslide events predicted correctly | 0 | 0 | 92 | 92 | 173 | 237 | 237 | 292 | 332 | 361 |

Number of rainfall events triggering no landslides actually | 608 | 608 | 608 | 608 | 608 | 608 | 608 | 608 | 608 | 608 |

TNR | 0% | 0% | 15% | 15% | 28% | 39% | 39% | 48% | 55% | 59% |

PPV | 1.3% | 1.3% | 1.5% | 1.5% | 1.8% | 2.1% | 2.1% | 2.5% | 2.8% | 3.1% |

Youden’s index | 0% | 0% | 15% | 15% | 28% | 39% | 39% | 48% | 55% | 59% |

Rainfall Probability Threshold | 0.55 | 0.60 | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | 0.95 | 1.00 |

Number of landslide events predicted correctly | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 7 | 0 |

Number of rainfall events triggering landslides actually | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

TPR | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 100% | 88% | 0% |

Number of no landslide events predicted correctly | 407 | 431 | 469 | 494 | 526 | 549 | 575 | 588 | 602 | 608 |

Number of rainfall events triggering no landslides actually | 608 | 608 | 608 | 608 | 608 | 608 | 608 | 608 | 608 | 608 |

TNR | 67% | 71% | 77% | 81% | 87% | 90% | 95% | 97% | 99% | 100% |

PPV | 3.8% | 4.3% | 5.4% | 6.6% | 8.9% | 11.9% | 19.5% | 28.6% | 53.8% | - |

Youden’s index | 67% | 71% | 77% | 81% | 87% | 90% | 95% | 97% | 87% | 0% |

**Table 3.**TPR, TNR, PPV, and Youden’s index for all rain gauges at a rainfall probability threshold of 0.95.

Rainfall Gauge | Bihu | Fushan (3) | Tatungshan | Pinglin (4) | Sihdu | Taiping | Quchi |
---|---|---|---|---|---|---|---|

Number of landslide events predicted correctly | 7 | 4 | 8 | 5 | 8 | 7 | 7 |

Number of rainfall events triggering landslides actually | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

TPR | 88% | 50% | 100% | 63% | 100% | 88% | 88% |

Number of no landslide events predicted correctly | 602 | 586 | 652 | 598 | 563 | 549 | 610 |

Number of rainfall events triggering no landslides actually | 608 | 599 | 662 | 605 | 583 | 565 | 629 |

TNR | 99% | 98% | 98% | 99% | 97% | 97% | 97% |

PPV | 53.8% | 23.5% | 44.4% | 41.7% | 28.6% | 30.4% | 26.9% |

Youden’s index | 87% | 48% | 98% | 61% | 97% | 85% | 84% |

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**MDPI and ACS Style**

Wu, C.-Y.; Yeh, Y.-C.
A Landslide Probability Model Based on a Long-Term Landslide Inventory and Rainfall Factors. *Water* **2020**, *12*, 937.
https://doi.org/10.3390/w12040937

**AMA Style**

Wu C-Y, Yeh Y-C.
A Landslide Probability Model Based on a Long-Term Landslide Inventory and Rainfall Factors. *Water*. 2020; 12(4):937.
https://doi.org/10.3390/w12040937

**Chicago/Turabian Style**

Wu, Chun-Yi, and Yen-Chu Yeh.
2020. "A Landslide Probability Model Based on a Long-Term Landslide Inventory and Rainfall Factors" *Water* 12, no. 4: 937.
https://doi.org/10.3390/w12040937