*3.1. Intensity-Duration Thresholds*

A total of 225 landslide events were recorded during the study period (2010–18), which were triggered by rainfall. The hourly intensities of all the rainfall events associated with the occurrence of landslides were calculated and plotted against the duration of events in hours in a logarithmic scale. The distribution of the events is fitted with the power-law distribution using an equation in the form

$$I = aD^{\beta} \tag{1}$$

$$\text{i.e., } \log(\text{I}) = \log(\alpha) + \beta \log(\text{D}) \tag{2}$$

where

I is Intensity of rainfall in mmh<sup>−</sup>1, D is Duration of rainfall event in hours, and α and β are empirical parameters, which is in the form of a straight line y = mx + c.

Use of this power-law equation has two fundamental assumptions. The first one is that with increase in the rainfall intensities, there is a nonlinear increase in the probability of occurrence of landslides. Below the threshold value, the likelihood of initiation of landslide is low, and above the threshold, the probability of occurrence of landslides increases nonlinearly. The second assumption is that the initiation of slides decreases as the duration of rainfall increases [2]. The term 'β' in Equation (1) defines this rate at which the critical intensity declines with the rise in duration. The frequentist approach of defining intensity-duration thresholds is used in this study. Empirical rainfall conditions which triggered landslides were first log-transformed and fitted using Equation (2), which is equivalent to the power-law in Equation (1). Using the Frequentist method, a best fit line for the distribution was obtained as I = 2.54D−0.16 (Figure 7) with a coefficient of determination (R2) of 0.04. The scattering of data results in a lower value of R2 and hence the uncertainty associated with the fitted line is evaluated with a confidence interval of 95%. Considering the uncertainties, Equation (1) gets modified to

$$\mathbf{I} = (\boldsymbol{\alpha} \pm \boldsymbol{\Lambda} \boldsymbol{\alpha}) \, \mathbf{D}^{(\boldsymbol{\beta} \pm \boldsymbol{\Lambda} \boldsymbol{\beta})} \tag{3}$$

The equation of the best fit line was obtained as I = 2.54D−0.16, with a confidence interval of <sup>I</sup> <sup>=</sup> (2.54 <sup>±</sup> 0.65)D(−0.16 <sup>±</sup> 0.05).

The approach is based on least square regression and the data is fitted using a power-law. The difference between the value on the best fit line log (If) and logarithm of event intensity log (I) for each event is calculated. This difference is termed as 'δI'. Kernel density estimation is used to determine the probability density function of the distribution of 'δI' and the result was fitted using a Gaussian function of the following form [40,41]:

$$f(x) = ae^{-\frac{(x-b)^2}{2c^2}}\tag{4}$$

where a and b are real constants and c is nonzero.

a,b,c, , R, and thresholds corresponding to various exceedance probabilities can be defined for the region. For a normally distributed random variable, a = <sup>1</sup> σ √ 2π, b <sup>=</sup> <sup>μ</sup> and c2 <sup>=</sup> <sup>σ</sup><sup>2</sup> where <sup>μ</sup> and <sup>σ</sup> are the mean and standard deviation of the distribution, respectively. Hence the equation becomes

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{\left(x-\mu\right)^2}{2\sigma^2}}\tag{5}$$

This equation is used to fit the distribution of 'δI', to determine the rainfall threshold as shown in Figure 8.

**Figure 7.** Rainfall Intensity vs. Duration (ID) plot on logarithmic scale for the Idukki district fitted using power-law.

**Figure 8.** Probability density function of the distribution of δI, fitted using a Gaussian distribution.

The data follows a distribution similar to the standard Gaussian distribution. Hence based on standard Gaussian distribution, a T5 line was plotted as in Figure 8, with an exceedance probability of 5%. The distance 'δ5' indicates the deviation of threshold line from the best fit line. This deviation was used to establish the intercept of the threshold line (Figure 9).

**Figure 9.** Intensity-duration threshold for the Idukki district on logarithmic scale.

From the threshold line, it can be inferred that for the minimum duration (24 hours), a continuous rainfall of 0.54 mmh−<sup>1</sup> can trigger landslides. The maximum duration of a rainfall event observed during the study period was 31 days. The obtained results predict that an intensity of 0.3 mmh−<sup>1</sup> over a period of 31 days can trigger landslides in the region. The confidence interval was obtained as I = (0.9 <sup>±</sup> 0.1)D(−0.16 <sup>±</sup> 0.05). The maximum number of events occurred at a duration of 7 days for which the minimum intensity to initiate a landslide event was found to be 0.4 mmh−1. The lesser value of thresholds for short duration events emphasizes the need for considering antecedent rainfall conditions for defining thresholds. Hence thresholds based on antecedent rainfall conditions are also defined for the area.
