*3.1. Statistical Analysis*

According to Caine [24], intensity–duration thresholds are defined as a power-law in the following form:

$$\mathbf{I} = \boldsymbol{\alpha} \mathbf{D}^{\beta} \tag{1}$$

where I is the mean intensity, expressed in mm h−1, D is the duration in hours and α and β are empirically derived parameters. α is the scaling constant, which defines the intercept, and β defines the slope of the power-law curve. In this study, two rainfall events are considered separate if there is no precipitation for a minimum period of 24 hours (one day) in between, and the frequentist method is adopted to establish a regional scale threshold for Wayanad district.

This approach uses the least square method to find the best fit line [34]. Taking into account the variation of intensity values, the values were first log-transformed to avoid problems in fitting the data. In a log vs. log plot, the data is fitted using a straight line with the equation

$$\text{Log I} = \text{Log}\infty + \beta \text{LogD} \tag{2}$$

which is equivalent to the power-law in Equation (1). From Equation (2), the values of α and β can be calculated, as Logα is the intercept and β is the slope of the straight line.

The difference in y coordinates of each event with the best fit line is then calculated and termed as δI, which is obtained by the following equation:

$$\text{\\$I} = \text{Log}[\text{I}\_{\text{f}}\text{D}] - \text{Log}[\text{I}(\text{D})] \tag{3}$$

where Log If(D) is the y co-ordinate on best-fit line, and Log I(D) is the mean intensity associated with each event. The distribution of δI is then fitted using a kernel density function of the form.

$$\text{\\$I = \text{Log}[I\_\text{f}(\text{D})] - \text{Log}[I(\text{D})]} \tag{4}$$

The data are found to follow a distribution similar to the standard Gaussian distribution. The Gaussian fit of the probability density function is shown in Figure 7 as the dotted line. The threshold lines of different exceedance probabilities were calculated using the fitted distribution of δI. The distance Δ between the best-fit line and the T line is used to calculate the intercept of threshold line in the log vs. log plot. An exceedance probability of 5% indicates that the probability of occurrence of landslides below this threshold is less than 5% [34]. The threshold of 5% exceedance is plotted according to the shift Δ.

**Figure 7.** The probability density function of distribution of δI, fitted using a kernel density function—example using merged dataset.
