A comprehensive framework for empirical modeling of landslides induced by rainfall: the Generalized FLaIR Model (GFM)

Abstract

In this paper, a new version of the hydrological model named FLaIR (Forecasting of Landslides Induced by Rainfall) is described, and it is indicated as GFM (Generalized FLaIR Model). Nonstationary rainfall thresholds, depending on antecedent precipitation, are introduced in this new release, which allows for a better prediction of landslide occurrences. Authors demonstrate that GFM is able to reproduce all the antecedent precipitation models (AP) proposed in technical literature as particular cases, besides intensity-duration schemes (ID) and more conceptual approaches, like Leaky Barrel, whose reconstruction with the first release of FlaIR model, which adopts only stationary thresholds, was already discussed in technical literature. Authors applied GFM for two case studies: 1) Gimigliano municipality, which is located in Calabria region (southern Italy) and where a consistent number of landslides occurred in the past years; in particular, during the period 2008–2010, this area (like the whole Calabria region) was affected by persistent rainfall events, which severely damaged infrastructures and buildings; 2) Barcelonnette Basin, which is located in the dry intra-Alpine zone (South French Alps). The high flexibility of GFM allows to obtain significant improvements in landslide prediction; in details, a substantial reduction of false alarms is obtained with respect to application of classical ID and AP schemes.

Appendix 1

In this Appendix, a numerical example is provided, in order to clarify the GFM use in real time for a user.

Let k ₁ = 1 day, k ₂ = 10 days, and ω = 0.7 be the parameter values for a MEF filter (Eq. 25). Table 9 reports, for each day, the values of filter (see also Fig. 23), cumulative Area (from which M* is equal to 35 days), the normalized cumulative Area (see also Fig. 24), and the ΔArea (useful for carrying out the convolution in numerical way, see also Fig. 25).

Table 9 Values of MEF for k ₁ = 1 day, k ₂ = 10 days, and ω = 0.7, cumulative Area (from which M* is equal to 35 days), normalized cumulative Area, and the ΔArea. The sum of all the ΔArea values is clearly equal to 1

Fig. 23

Values of MEF for k ₁ = 1 day, k ₂ = 10 days, and ω = 0.7

Fig. 24

Normalized cumulative area for the assumed MEF filter

Fig. 25

ΔArea for the assumed MEF filter

The sum of all the ΔArea values is clearly equal to 1.

From the analysis of Table 9, d is assumed equal to the first value for which the normalized cumulative Area is greater than (or equal to) ω, i.e., 3 days and consequently D = 32 days.

Table 10 contains 35 values of daily rainfall in chronological order (for simplicity, two constant values of daily rainfall are assumed: 5 mm for the first 32 days and 15 mm for the last 3 days, see also Fig. 26), ΔArea in inverse order with respect to Table 9 (aimed to carry out the convolution), and the products between rainfall and ΔArea for each day.

Table 10 Thirty-five values of daily rainfall in chronological order (Fig. 26), ΔArea in inverse order with respect to table 9 (aimed to carry out the convolution), and the products between rainfall and ΔArea for each day

Fig. 26

Values of daily rainfall

The sum of the first 32 products is equal to Y _D (t − d), i.e., about 1.25 mm/day, while the sum of the last three products is Y _d (t), i.e., about 11.25 mm/day; consequently, Y(t) is about 12.5 mm/day and, from Eqs. (3)–(4), ${R}_D^{*}\left(t-d\right)$ = 40.05 mm and ${R}_d^{*}(t)$ = 33.74 mm.

If a MEF–NT is assumed, then let α = 100 mm/mm^β and β = −0.4 be the parameter values for Eq. (12), from which ${R}_{d,cr}^{*}(t)$ is about 22.85 mm for ${R}_D^{*}\left(t-d\right)$ = 40.05, and Y _cr (t) is about 8.87 mm/day (Eq. 7). Thus, Y(t) > Y _cr (t) (or, in equivalent way, ${R}_d^{*}(t)$>${R}_{d,cr}^{*}(t)$), and then a landslide is predicted from GFM application.