# Rainfall Threshold Estimation and Landslide Forecasting for Kalimpong, India Using SIGMA Model

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## Abstract

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## 1. Introduction

## 2. Study Area and Input Data

## 3. The SIGMA Model

_{k}, F(z

_{k}) defines the probability that the variable z takes a value less than z

_{k}, where k varies between 1 to n.

^{−1}(F(z)) → G

^{−1}(P

_{k}) = y

_{1–3}is the vector indicating the cumulated rainfall at time t and ${S}_{n}(\u2206)$ are the thresholds relative to number of days n and Δ [23]. In the case of slow movements, the algorithm ponders the effect of cumulative rainfall from 4 days up to 63 days [23]. The condition for crossing the threshold is given by:

## 4. Analysis

## 5. Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location details (

**a**) India; (

**b**) West Bengal; (

**c**) digital elevation model of Kalimpong (modified after [31]).

**Figure 2.**(

**a**) Yearly cumulative rainfall; (

**b**) Monthly distribution of landslide occurrence and average rainfall, (2010–2017) in mm.

**Figure 3.**Spatial distribution of rain gauges and landslide events during the study period. (modified after [15]).

**Figure 4.**(

**a**) Transformation of original cumulative distribution in the target distribution for Kalimpong town; (

**b**) An example of SIGMA (sistema integrato gestione monitoraggio allerta—integrated system for management, monitoring and alerting) curves (σ curves) for cumulative periods up to 100 days (2010–2015).

**Figure 6.**Visualization of calibration algorithm. The threshold value was raised till the cumulative rainfall curve of the event (F) is not crossing the threshold curve (standard threshold of 1.75 is optimized to 1.95).

Month | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
---|---|---|---|---|---|---|---|---|

June | 317 | 337 | 355 | 248 | 396 | 568 | 327 | 154 |

July | 666 | 678 | 433 | 424 | 371 | 534 | 870 | 812 |

August | 425 | 526 | 251 | 401 | 572 | 242 | 263 | 432 |

September | 268 | 384 | 467 | 113 | 265 | 331 | 367 | 288 |

Statistical Attributes | SIGMA Model |
---|---|

T1 = True positives | 13 |

F1 = False positives | 55 |

F2 = False negatives | 2 |

T2 = True negatives | 661 |

Negative predictive power = T2/(F2 + T2) | 1.00 |

Positive predictive power = T1/(T1 + F1) | 0.19 |

Misclassification rate = (F1 + F2)/(T1 + F1 + F2 + T2) | 0.08 |

Efficiency = (T1 + T2)/(T1 + F1 + F2 + T2) | 0.92 |

Odds ratio = (T1 + T2)/(F1 + F2) | 11.82 |

False negative rate = F2/(T1 + F2) | 0.13 |

False positive rate = F1/(F1 + T2) | 0.08 |

Specificity (Sp) = T2/(F1 + T2) | 0.92 |

Sensitivity (Sn) = T1/(T1 + F2) | 0.87 |

Likelihood ratio = Sn/(1 − Sp) | 11.28 |

Statistical Attributes | ID Threshold [10,35] | ED Threshold [15] | SIGMA (This Work) | ||
---|---|---|---|---|---|

Threshold | Lower Limit | Upper Limit | |||

T1 = True positives | 8 | 8 | 9 | 8 | 13 |

F1 = False positives | 98 | 93 | 117 | 75 | 55 |

F2 = False negatives | 7 | 7 | 6 | 7 | 2 |

T2 = True negatives | 618 | 623 | 599 | 641 | 661 |

Negative predictive power = T2/(F2 + T2) | 0.99 | 0.99 | 0.99 | 0.99 | 1.00 |

Positive predictive power = T1/(T1 + F1) | 0.08 | 0.08 | 0.07 | 0.10 | 0.19 |

Misclassification rate = (F1 + F2)/(T1 + F1 + F2 + T2) | 0.14 | 0.14 | 0.17 | 0.11 | 0.08 |

Efficiency = (T1 + T2)/(T1 + F1 + F2 + T2) | 0.86 | 0.86 | 0.83 | 0.89 | 0.92 |

Odds ratio = (T1 + T2)/(F1 + F2) | 5.96 | 6.31 | 4.94 | 7.91 | 11.82 |

False negative rate = F2/(T1 + F2) | 0.47 | 0.47 | 0.40 | 0.47 | 0.13 |

False positive rate = F1/(F1 + T2) | 0.14 | 0.13 | 0.16 | 0.10 | 0.08 |

Specificity (Sp) = T2/(F1 + T2) | 0.86 | 0.87 | 0.84 | 0.90 | 0.92 |

Sensitivity (Sn) = T1/(T1 + F2) | 0.53 | 0.53 | 0.60 | 0.53 | 0.87 |

Likelihood ratio = Sn/(1 − Sp) | 3.90 | 4.11 | 3.67 | 5.09 | 11.28 |

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**MDPI and ACS Style**

Abraham, M.T.; Satyam, N.; Kushal, S.; Rosi, A.; Pradhan, B.; Segoni, S.
Rainfall Threshold Estimation and Landslide Forecasting for Kalimpong, India Using SIGMA Model. *Water* **2020**, *12*, 1195.
https://doi.org/10.3390/w12041195

**AMA Style**

Abraham MT, Satyam N, Kushal S, Rosi A, Pradhan B, Segoni S.
Rainfall Threshold Estimation and Landslide Forecasting for Kalimpong, India Using SIGMA Model. *Water*. 2020; 12(4):1195.
https://doi.org/10.3390/w12041195

**Chicago/Turabian Style**

Abraham, Minu Treesa, Neelima Satyam, Sai Kushal, Ascanio Rosi, Biswajeet Pradhan, and Samuele Segoni.
2020. "Rainfall Threshold Estimation and Landslide Forecasting for Kalimpong, India Using SIGMA Model" *Water* 12, no. 4: 1195.
https://doi.org/10.3390/w12041195