# Influence of Rainfall Pattern on Wetness Index for Infinite Slope Stability Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Slope Stability

_{w}, D and D

_{w}are the unit weights of soils and water, and the depth of soils and infiltrated water, respectively. As presented in Equation (6), the FS is significantly affected by the infiltrated water; thus, the depth of the infiltrated water should be considered to determine the slope stability.

#### 2.2. Temporal Distribution of Design Rainfall

#### 2.2.1. The Yen and Chow Model

#### 2.2.2. The Mononobe Model

_{t}is the cumulative rainfall depth (mm) up to time t, R

_{T}is the design rainfall depth (mm) and T is the design rainfall duration (h). In general, the constant n is assumed to be 2/3 in Asian regions [58,59]. After locating the rainfall peak, the next largest rainfall intensity is located alternately around the rainfall peak by turn. The Mononobe model can be classified into three types, advanced, delayed and centered, depending on the location of the rainfall peak. When the rainfall peak is located before the center of the storm duration, we call it “advanced type”, but if the case is the opposite, it is indicated as the “delayed type”. The “centered type” means the rainfall peak is at the mid-point of the total duration. The centered type Mononobe model has been applied to distribute the design rainfall depth by analyzing the rainfall data mainly in Asia, including Korea and Indonesia [60,61,62].

#### 2.2.3. The Alternating Block Method

#### 2.2.4. The Huff Model

^{2}) and extracted a normalized model of temporal rainfall distribution. That is, the observed rainfall events are transformed into dimensionless ones by normalizing the time by the total rainfall duration, and the cumulative rainfall depth is also normalized by the total rainfall amount. The normalized rainfall distributions are separated into four types (i.e., first, second, third and fourth quartiles), dividing the normalized duration with respect to the frequent occurrence of the rainfall peak over the rainfall duration. Then, only those rainfall events locating their peak times within the same quartile are collected. For each type, the cumulative probability distributions are derived by applying the smoothing scheme, which indicates that the normalized cumulative rainfall at each normalized time is expressed by its exceedance probability [66]. The cumulative probabilities of exceedance occurrence for 10% to 90% with increments of 10% are derived. As a result, the second quartile 50% Huff model represents a cumulative rainfall pattern that should be exceeded in about half of the storms with which their rainfall peaks locate the second quartile of the normalized durations.

## 3. Results and Analyses

#### 3.1. Landslides in July 2011 in Seoul, Korea

^{3}and internal friction angle of 20° were adopted for the estimation of the FS as expressed in Equation (10). The wetness index indicated by black line varied similar to the rainfall intensity up to 9 am (see Figure 2), but decreased after the highest peak rainfall intensity. Note that the water infiltrated into the soil continuously flowed along the slope; thus, the wetness index might have decreased when the rainfall intensity was insufficient. Meanwhile, the FS varied contrary to the wetness index, and it began to decrease under the value of 1.0 around 8 a.m. In this study, the slope stability is determined by the FS value of 1.0, whether the state is stable or unstable, and the beginning of the unstable time is well-matched with the slope failure moment as shown in Figure 2. Thus, the pre-determined factors from previous investigations are reliable to be used in further model analysis.

#### 3.2. Impact of Rainfall Pattern on Slope Stability

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Oh, S.R.; Lee, G.H. Slope stability analysis at catchment scale using spatially-distributed wetness index. J. Korean Geogr.
**2014**, 3, 111–126. [Google Scholar] - Rahardjo, H.; Lim, T.T.; Chang, M.F.; Fredlund, D.G. Shear-strength characteristics of a residual soil. Can. Geotech. J.
**1995**, 32, 60–77. [Google Scholar] [CrossRef] - Fourie, A.B. Predicting rainfall-induced slope instability. Proc. Inst. Civ. Eng. Geotech. Eng.
**1996**, 119, 211–218. [Google Scholar] [CrossRef] - Borga, M.; Dalla Fontana, G.; Da Ros, D.; Marchi, L. Shallow landslide hazard assessment using a physically based model and digital elevation data. Environ. Geol.
**1998**, 35, 81–88. [Google Scholar] [CrossRef] - Ng, C.W.W.; Shi, Q. Influence of rainfall intensity and duration on slope stability in unsaturated soils. Q. J. Eng. Geol. Hydrogeol.
**1998**, 31, 105–113. [Google Scholar] - Kim, S.Y.; Park, J.; Cha, W.; Lee, J.S.; Carlos Santamarina, J. Soil response during globally drained and undrained freeze–thaw cycles under deviatoric loading. J. Geotech. Geoenviron.l Eng.
**2021**, 147, 06020030. [Google Scholar] [CrossRef] - Chae, B.G.; Lee, J.H.; Park, H.J.; Choi, J. A method for predicting the factor of safety of an infinite slope based on the depth ratio of the wetting front induced by rainfall infiltration. Nat. Hazards Earth Syst. Sci.
**2015**, 15, 1835–1849. [Google Scholar] [CrossRef] [Green Version] - Beven, K.J.; Kirkby, M.J. A physically based, variable contributing area model of basin hydrology/Un modèle à base physique de zone d’appel variable de l’hydrologie du bassin versant. Hydrol. Sci. J.
**1979**, 24, 43–69. [Google Scholar] [CrossRef] [Green Version] - Barling, R.D.; Moore, I.D.; Grayson, R.B. A quasi-dynamic wetness index for characterizing the spatial distribution of zones of surface saturation and soil water content. Water Resour. Res.
**1994**, 30, 1029–1044. [Google Scholar] [CrossRef] - Zhao, B.; Dai, Q.; Han, D.; Zhang, J.; Zhuo, L.; Berti, M. Application of hydrological model simulations in landslide predictions. Landslides
**2020**, 17, 877–891. [Google Scholar] [CrossRef] - Fredlund, D.G.; Rahardjo, H. Soil Mechanics for Unsaturated Soils; John Wiley & Sons: Hoboken, NJ, USA, 1993. [Google Scholar]
- Muntohar, A.S.; Liao, H.J. Rainfall infiltration: Infinite slope model for landslides triggering by rainstorm. Nat. Hazards
**2010**, 54, 967–984. [Google Scholar] [CrossRef] - Montgomery, D.R.; Dietrich, W.E. A physically based model for the topographic control on shallow landsliding. Water Resour. Res.
**1994**, 30, 1153–1171. [Google Scholar] [CrossRef] - Cardinali, M.; Galli, M.; Guzzetti, F.; Ardizzone, F.; Reichenbach, P.; Bartoccini, P. Rainfall induced landslides in December 2004 in south-western Umbria, central Italy: Types, extent, damage and risk assessment. Nat. Hazards Earth Syst. Sci.
**2006**, 6, 237–260. [Google Scholar] [CrossRef] - Guzzetti, F.; Peruccacci, S.; Rossi, M.; Stark, C.P. Rainfall thresholds for the initiation of landslides in central and southern Europe. Meteorol. Atmos. Phys.
**2007**, 98, 239–267. [Google Scholar] [CrossRef] - Peruccacci, S.; Brunetti, M.T.; Gariano, S.L.; Melillo, M.; Rossi, M.; Guzzetti, F. Rainfall thresholds for possible landslide occurrence in Italy. Geomorphology
**2017**, 290, 39–57. [Google Scholar] [CrossRef] - Gariano, S.L.; Brunetti, M.T.; Iovine, G.; Melillo, M.; Peruccacci, S.; Terranova, O.; Guzzetti, F. Calibration and validation of rainfall thresholds for shallow landslide forecasting in Sicily, southern Italy. Geomorphology
**2015**, 228, 653–665. [Google Scholar] [CrossRef] - Deganutti, A.M.; Marchi, L.; Arattano, M. Rainfall and debris-flow occurrence in the Moscardo basin (Italian Alps). In Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment; Millpress Science Publishers: Rotterdam, The Netherlands, 2000; pp. 67–72. [Google Scholar]
- Zhou, W.; Tang, C.; Van Asch, T.W.; Zhou, C. Rainfall-triggering response patterns of post-seismic debris flows in the Wenchuan earthquake area. Nat. Hazards
**2014**, 70, 1417–1435. [Google Scholar] [CrossRef] - Baum, R.L.; Godt, J.W. Early warning of rainfall-induced shallow landslides and debris flows in the USA. Landslides
**2010**, 7, 259–272. [Google Scholar] [CrossRef] - Segoni, S.; Leoni, L.; Benedetti, A.I.; Catani, F.; Righini, G.; Falorni, G.; Rebora, N. Towards a definition of a real-time forecasting network for rainfall induced shallow landslides. Nat. Hazards Earth Syst. Sci.
**2009**, 9, 2119–2133. [Google Scholar] [CrossRef] [Green Version] - Aleotti, P.; Chowdhury, R. Landslide hazard assessment: Summary review and new perspectives. Bull. Eng. Geol. Environ.
**1999**, 58, 21–44. [Google Scholar] [CrossRef] - Goetz, J.N.; Brenning, A.; Petschko, H.; Leopold, P. Evaluating machine learning and statistical prediction techniques for landslide susceptibility modeling. Comput. Geosci.
**2015**, 81, 1–11. [Google Scholar] [CrossRef] - Park, H.-J.; Jang, J.-Y.; Lee, J.-H. Physically based susceptibility assessment of rainfall-induced shallow landslides using a fuzzy point estimate method. Remote Sens.
**2017**, 9, 487. [Google Scholar] [CrossRef] [Green Version] - Zêzere, J.L.; Pereira, S.; Melo, R.; Oliveira, S.C.; Garcia, R.A. Mapping landslide susceptibility using data-driven methods. Sci. Total Environ.
**2017**, 589, 250–267. [Google Scholar] [CrossRef] [PubMed] - Medina, V.; Hürlimann, M.; Guo, Z.; Lloret, A.; Vaunat, J. Fast physically-based model for rainfall-induced landslide susceptibility assessment at regional scale. Catena
**2021**, 201, 105213. [Google Scholar] [CrossRef] - Van Westen, C.J. The modelling of landslide hazards using GIS. Surv. Geophys.
**2000**, 21, 241–255. [Google Scholar] [CrossRef] - Huabin, W.; Gangjun, L.; Weiya, X.; Gonghui, W. GIS-based landslide hazard assessment: An overview. Prog. Phys. Geogr.
**2005**, 29, 548–567. [Google Scholar] [CrossRef] - Fell, R.; Corominas, J.; Bonnard, C.; Cascini, L.; Leroi, E.; Savage, W.Z. Guidelines for landslide susceptibility, hazard and risk zoning for land use planning. Eng. Geol.
**2008**, 102, 85–98. [Google Scholar] [CrossRef] [Green Version] - Park, H.J.; Lee, J.H.; Woo, I.K. Assessment of rainfall-induced shallow landslide susceptibility using a GIS-based probabilistic approach. Eng. Geol.
**2013**, 161, 1–15. [Google Scholar] [CrossRef] - Burton, A.; Bathurst, J.C. Physically based modelling of shallow landslide sediment yield at a catchment scale. Environ. Geol.
**1998**, 35, 89–99. [Google Scholar] [CrossRef] - Dai, F.C.; Lee, C.F. Landslide characteristics and slope instability modeling using GIS, Lantau Island, Hong Kong. Geomorphology
**2002**, 42, 213–228. [Google Scholar] [CrossRef] - Yilmaz, I.; Keskin, I. GIS based statistical and physical approaches to landslide susceptibility mapping (Sebinkarahisar, Turkey). Bull. Eng. Geol. Environ.
**2009**, 68, 459–471. [Google Scholar] - Gutierrez-Martin, A. A GIS-physically-based emergency methodology for predicting rainfall-induced shallow landslide zonation. Geomorphology
**2020**, 359, 107121. [Google Scholar] [CrossRef] - Ji, J.; Cui, H.; Zhang, T.; Song, J.; Gao, Y. A GIS-based tool for probabilistic physical modelling and prediction of landslides: GIS-FORM landslide susceptibility analysis in seismic areas. Landslides
**2022**, 19, 2213–2231. [Google Scholar] [CrossRef] - Guzzetti, F.; Peruccacci, S.; Rossi, M.; Stark, C.P. The rainfall intensity–duration control of shallow landslides and debris flows: An update. Landslides
**2008**, 5, 3–17. [Google Scholar] [CrossRef] - Saito, H.; Nakayama, D.; Matsuyama, H. Relationship between the initiation of a shallow landslide and rainfall intensity—Duration thresholds in Japan. Geomorphology
**2010**, 118, 167–175. [Google Scholar] [CrossRef] - Bogaard, T.; Greco, R. Invited perspectives: Hydrological perspectives on precipitation intensity-duration thresholds for landslide initiation: Proposing hydro-meteorological thresholds. Nat. Hazards Earth Syst. Sci.
**2018**, 18, 31–39. [Google Scholar] [CrossRef] [Green Version] - Ma, T.; Li, C.; Lu, Z.; Wang, B. An effective antecedent precipitation model derived from the power-law relationship between landslide occurrence and rainfall level. Geomorphology
**2014**, 216, 187–192. [Google Scholar] [CrossRef] - Pennington, C.; Dijkstra, T.; Lark, M.; Dashwood, C.; Harrison, A.; Freeborough, K. Antecedent precipitation as a potential proxy for landslide incidence in South West United Kingdom. In Landslide Science for a Safer Geoenvironment: Vol. 1: The International Programme on Landslides (IPL); Springer International Publishing: New York, NY, USA, 2014; pp. 253–259. [Google Scholar]
- Zhao, B.; Dai, Q.; Han, D.; Dai, H.; Mao, J.; Zhuo, L.; Rong, G. Estimation of soil moisture using modified antecedent precipitation index with application in landslide predictions. Landslides
**2019**, 16, 2381–2393. [Google Scholar] [CrossRef] - Hong, M.; Kim, J.; Jeong, S. Rainfall intensity-duration thresholds for landslide prediction in South Korea by considering the effects of antecedent rainfall. Landslides
**2018**, 15, 523–534. [Google Scholar] [CrossRef] - De Luca, D.L.; Versace, P. A comprehensive framework for empirical modeling of landslides induced by rainfall: The Generalized FLaIR Model (GFM). Landslides
**2017**, 14, 1009–1030. [Google Scholar] [CrossRef] - Sirangelo, B.; Versace, P. A real time forecasting model for landslides triggered by rainfall. Meccanica
**1996**, 31, 73–85. [Google Scholar] [CrossRef] - Capparelli, G.; Biondi, D.; De Luca, D.L.; Versace, P. Hydrological and complete models for forecasting landslides triggered by rainfalls. Proc. First Ital. Workshop Landslides
**2009**, 1, 162–173. [Google Scholar] - Capparelli, G.; Versace, P. FLaIR and SUSHI: Two mathematical models for early warning of landslides induced by rainfall. Landslides
**2011**, 8, 67–79. [Google Scholar] [CrossRef] - Na, W.; Yoo, C. Evaluation of rainfall temporal distribution models with annual maximum rainfall events in Seoul, Korea. Water
**2018**, 10, 1468. [Google Scholar] [CrossRef] [Green Version] - Alfieri, L.; Laio, F.; Claps, P. A simulation experiment for optimal design hyetograph selection. Hydrol. Process. Int. J.
**2008**, 22, 813–820. [Google Scholar] [CrossRef] - D’Odorico, P.; Fagherazzi, S.; Rigon, R. Potential for landsliding: Dependence on hyetograph characteristics. J. Geophys. Res. Earth Surf.
**2005**, 110, F01007. [Google Scholar] [CrossRef] [Green Version] - Ibsen, M.-L.; Casagli, N. Rainfall patterns and related landslide incidence in the Porretta-Vergato region, Italy. Landslides
**2004**, 1, 143–150. [Google Scholar] [CrossRef] - Tsai, T.-L. The influence of rainstorm pattern on shallow landslide. Environ. Geol.
**2008**, 53, 1563–1569. [Google Scholar] [CrossRef] - Tsai, T.-L.; Wang, J.-K. Examination of influences of rainfall patterns on shallow landslides due to dissipation of matric suction. Environ. Earth Sci.
**2011**, 63, 65–75. [Google Scholar] [CrossRef] - Ran, Q.; Hong, Y.; Li, W.; Gao, J. A modelling study of rainfall-induced shallow landslide mechanisms under different rainfall characteristics. J. Hydrol.
**2018**, 563, 790–801. [Google Scholar] [CrossRef] - Fan, L.; Lehmann, P.; Zheng, C.; Or, D. Rainfall intensity temporal patterns affect shallow landslide triggering and hazard evolution. Geophys. Res. Lett.
**2020**, 47, e2019GL085994. [Google Scholar] [CrossRef] - Acharya, G.; De Smedt, F.; Long, N.T. Assessing landslide hazard in GIS: A case study from Rasuwa, Nepal. Bull. Eng. Geol. Environ.
**2006**, 65, 99–107. [Google Scholar] [CrossRef] - Yen, B.C.; Chow, V.T. Design hyetographs for small drainage structures. J. Hydraul. Div.
**1980**, 106, 1055–1076. [Google Scholar] [CrossRef] - Mononobe, N. Hydraulics; Iwanami Shoten: Chiyoda, Japan, 1993; pp. 351–356. [Google Scholar]
- Jeong, J.H.; Yoon, Y.N. Design Practices for Water Resources; Goomi Press: Seoul, Republic of Korea, 2007. (In Korean) [Google Scholar]
- Danasla, M.A.; Kusuma, G.J.; Tuheteru, E.J.; Gautama, R.S. Hydrology model establishment of pit Lake: Extreme event rainfall data analysis. In Proceedings of the IOP Conference Series: Earth and Environmental Science, Bandung, Indonesia, 23–24 June 2021; Volume 883, p. 012048. [Google Scholar]
- Yoo, C.; Jun, C.; Park, C. Effect of rainfall temporal distribution on the conversion factor to convert the fixed-interval into true-interval rainfall. J. Hydrol. Eng.
**2015**, 20, 04015018. [Google Scholar] [CrossRef] - Prayuda, D.D. Temporal and spatial analysis of extreme rainfall on the slope area of Mt. Merapi. J. Civ. Eng. Forum
**2015**, 21, 1285–1290. [Google Scholar] - Priambodo, S.; Montarcih, L.; Suhartanto, E. Hourly rainfall distribution patterns in Java island. MATEC Web Conf.
**2019**, 276, 04012. [Google Scholar] [CrossRef] [Green Version] - Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology, International ed.; MacGraw-Hill, Inc.: Singapore, 1988; Volume 149. [Google Scholar]
- Viessman, W.; Lewis, G.L.; Knapp, J.W.; Harbaugh, T.E. Introduction to Hydrology; Prentice Hall, Pearson Education, Inc.: Hoboken, NJ, USA, 1989. [Google Scholar]
- Huff, F.A. Time distribution of rainfall in heavy storms. Water Resour. Res.
**1967**, 3, 1007–1019. [Google Scholar] [CrossRef] - Azli, M.; Rao, A.R. Development of Huff curves for peninsular Malaysia. J. Hydrol.
**2010**, 388, 77–84. [Google Scholar] [CrossRef] - Kim, J.; Jeong, S.; Regueiro, R.A. Instability of partially saturated soil slopes due to alteration of rainfall pattern. Eng. Geol.
**2012**, 147, 28–36. [Google Scholar] [CrossRef] - KGS (Korean Geotechnical Society). Final Report on the Cause of Landslides in Umyeonsan(Mt.) Area and the Establishment of Restoration Measures; Report No. KGS11-250; Korean Geotechnical Society: Seoul, Republic of Korea, 2011. (In Korean) [Google Scholar]
- SI (The Seoul Institute). Final Report on the Cause of Landslides in Umyeonsan(Mt.)—Complementary Investigation; Report No. 51-6110000-000649-01; The Seoul Institute: Seoul, Republic of Korea, 2014. (In Korean)
- Ministry of Construction and Transportation (MOCT). 1999 Report on the Development of Water Resources Management Techniques: Design Rainfall Temporal Distribution; Ministry of Construction and Transportation: Seoul, Republic of Korea, 2000. (In Korean)
- Ministry of Land, Transport and Maritime Affairs (MLTM). Research on the Improvement of Probability Rainfall; Ministry of Land, Transport and Maritime Affair: Seoul, Republic of Korea, 2019. (In Korean)
- Yoo, C.; Na, W. Analysis of a Conventional Huff Model at Seoul Station and Proposal of an Improvisation Method. J. Korean Soc. Hazard. Mitig.
**2019**, 19, 43–55. (In Korean) [Google Scholar] [CrossRef] - Bogaard, T.A.; Greco, R. Landslide hydrology: From hydrology to pore pressure. Wiley Interdiscip. Rev. Water
**2016**, 3, 439–459. [Google Scholar] [CrossRef] - Tsai, T.L.; Yang, J.C. Modeling of rainfall-triggered shallow landslide. Environ. Geol.
**2006**, 50, 525–534. [Google Scholar] [CrossRef] - Yang, W.Y.; Li, D.; Sun, T.; Ni, G.H. Saturation-excess and infiltration-excess runoff on green roofs. Ecol. Eng.
**2015**, 74, 327–336. [Google Scholar] [CrossRef] - Wicki, A.; Lehmann, P.; Hauck, C.; Seneviratne, S.I.; Waldner, P.; Stähli, M. Assessing the potential of soil moisture measurements for regional landslide early warning. Landslides
**2020**, 17, 1881–1896. [Google Scholar] [CrossRef] [Green Version] - Jungerius, P.D.; Ten Harkel, M.J. The effect of rainfall intensity on surface runoff and sediment yield in the grey dunes along the Dutch coast under conditions of limited rainfall acceptance. Catena
**1994**, 23, 269–279. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**10-min hyetograph with cumulative rainfall for 27 July 2011 at Seocho station in Seoul, Korea.

**Figure 3.**Temporal variation in wetness index (m) and factor of safety (FS) for 26–27 July 2011 at Seocho station in Seoul, Korea.

**Figure 4.**10-min hyetograph with cumulative rainfall for different types of rainfall pattern: (

**a**) uniform distribution; (

**b**) Yen and Chow model; (

**c**) Mononobe model; (

**d**) alternating block model; (

**e**) Huff model (second quartile).

**Figure 5.**Temporal variation in wetness index (m) and factor of safety (FS) for different types of rainfall pattern: (

**a**) uniform distribution; (

**b**) Yen and Chow model; (

**c**) Mononobe model; (

**d**) alternating block model; (

**e**) Huff model (second quartile).

**Table 1.**Slope stability determination using the factor of safety [56].

Factor of Safety | Slope Stability | Remarks |
---|---|---|

FS > 1.5 | Stable | Only major destabilizing factors lead to instability |

1.25 < FS < 1.5 | Moderately stable | Moderate destabilizing factors lead to instability |

1 < FS < 1.25 | Quasi-stable | Minor destabilizing factors can lead to instability |

FS < 1 | Unstable | Stabilizing factors are needed for stability |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Na, W.; Jun, C.; Kim, S.Y.
Influence of Rainfall Pattern on Wetness Index for Infinite Slope Stability Analysis. *Water* **2023**, *15*, 2535.
https://doi.org/10.3390/w15142535

**AMA Style**

Na W, Jun C, Kim SY.
Influence of Rainfall Pattern on Wetness Index for Infinite Slope Stability Analysis. *Water*. 2023; 15(14):2535.
https://doi.org/10.3390/w15142535

**Chicago/Turabian Style**

Na, Wooyoung, Changhyun Jun, and Sang Yeob Kim.
2023. "Influence of Rainfall Pattern on Wetness Index for Infinite Slope Stability Analysis" *Water* 15, no. 14: 2535.
https://doi.org/10.3390/w15142535