Functioning and precipitation-displacement modelling of rainfall-induced deep-seated landslides subject to creep deformation

Abstract

We propose an approach to study the hydro-mechanical behaviour and evolution of rainfall-induced deep-seated landslides subjected to creep deformation by combining signal processing and modelling. The method is applied to the Séchilienne landslide in the French Alps, where precipitation and displacement have been monitored for 20 years. Wavelet analysis is first applied on precipitation and recharge as inputs and then on displacement time-series decomposed into trend and detrended signals as outputs. Results show that the detrended displacement is better linked to the recharge signal than to the total precipitation signal. The infra-annual detrended displacement is generated by high precipitation events, whereas annual and multi-annual variations are rather linked to recharge variations and thus to groundwater processes. This leads to conceptualise the system into a two-layer aquifer constituted of a perched aquifer (reactive aquifer responsible of high-frequency displacements) and a deep aquifer (inertial aquifer responsible of low-frequency displacements). In a second step, a new lumped model (GLIDE) coupling groundwater and a creep deformation model is applied to simulate displacement on three extensometer stations. The application of the GLIDE model gives good performance, validating most of the preliminary functioning hypotheses. Our results show that groundwater fluctuations can explain the displacement periodic variations as well as the long-term creep exponential trend. In the case of deep-seated landslides, this displacement trend is interpreted as the consequence of the weakening of the rock mechanical properties due to repeated actions of the groundwater pressure.

Appendix—wavelet analysis

Continuous wavelet transforms

The continuous wavelet transform (CWT) W _x (τ, a) of a time-series x(t) is given as follows:

$$ {W}_x\left(\tau,\ a\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}x(t){\varPsi}_{\tau, a}^{*}(t)\mathrm{d}t $$

(13)

where

$$ {\varPsi}_{\tau,\ a}=\frac{1}{\sqrt{a}}\varPsi \left(\frac{t-\tau }{a}\right) $$

(14)

represents a group of wavelet functions, Ψ _{τ, a} , based on a mother wavelet Ψ which can be scaled and translated, modifying the scale parameter a and the translation parameter τ, respectively. (*) corresponds to the complex conjugate. Wavelet functions have multi-scale properties, dilating or contracting a (a > 1; a < 1). When a increases, the wavelet covers a higher signal window. It allows the large-scale behaviour of x to be extracted. Conversely, when a decreases, the analysed signal window decreases, allowing local variations of x to be studied. Wavelet transform is thus characterised on the space scale by a window decreasing in width when we focus on local-scale structures (high frequency) and widening when we focus on large-scale structures (low frequency).

As in the Fourier analysis, a wavelet power spectrum (WPS, also called a scalogram) P _x (τ, a) can be defined as the wavelet transform of W _x (τ, a):

$$ {P}_x\left(\tau,\ a\right)=\left({W}_x\left(\tau,\ a\right)\ {W}_x^{*}\left(\tau,\ a\right)\right)={\left|{W}_x\left(\tau,\ a\right)\right|}^2 $$

(15)

The choice of the appropriate analysis wavelet depends on the nature of the signal and on the type of information to be extracted from the time-series (De Moortel et al. 2004). Statistical significance level was estimated against a red noise model (Torrence and Compo 1998; Grinsted et al. 2004). As CWTs are applied to time-series of finite length, edge effects may appear on the scalogram, leading to the definition of a cone of influence (COI) as the region where such effects are relevant (Torrence and Compo 1998). The COI is marked as a shadow in the scalogram.

The covariance of two time-series x and y is estimated using a cross wavelet spectrum (XWT, also called a cross-scalogram) W _xy (τ, a), which is defined as:

$$ {W}_{xy}\left(\tau,\ a\right)=\left({W}_x\left(\tau,\ a\right)\ {W}_y^{*}\left(\tau,\ a\right)\right) $$

(16)

XWT reveals an area with a high common power value, but Maraun and Kurths (2004) reported that it appears unsuitable for significance testing of the interrelation between two series. These authors recommend the use of wavelet coherence (WTC) which is a measure of the intensity of covariance of the two series in the time-scale space. Beginning with the approach of Torrence and Webster (1999), the WTC of two time-series x and y is defined as:

$$ {C}_{xy}^2\left(\tau,\ a\right)=\frac{{\left|S\left({a}^{-1}{W}_{xy}\left(\tau,\ a\right)\right)\right|}^2}{S\left({a}^{-1}{\left|{W}_x\left(\tau,\ a\right)\right|}^2\right)\cdot S\left({a}^{-1}{\left|{W}_y\left(\tau,\ a\right)\right|}^2\right)} $$

(17)

where S is a smoothing operator in both time and scale (see Torrence and Webster (1999) and Jevrejeva et al. (2003) for detailed mathematical expressions). The 5 % significance level of WTC against red AR1 noise is estimated using Monte Carlo methods (Grinsted et al. 2004).

Multi-resolution analysis

In order to implement the wavelet transform on sampled signals, the discrete wavelet transform (DWT) can be used to discretise the scale and location parameters j and k, respectively. The discrete form of the wavelet transform of a time-series x(t) is given according to Eq. (18):

$$ {W}_x\left({\tau}_0,\ {a}_0\right)={\displaystyle \sum_{-\infty}^{+\infty }}x(t){\varPsi}_{\tau_0,\ {a}_0}^{*}(t)\mathrm{d}t $$

(18)

where

$$ {\varPsi}_{\tau_0,\ {a}_0}=\frac{1}{\sqrt{a_0^j}}\varPsi \left(\frac{t-k{a}_0^j{\tau}_0}{a_0^j}\right) $$

(19)

with a ^j₀ being the scale parameter, τ ₀ the translation parameter and k and j integers. ${\varPsi}_{\tau_0,\ {a}_0}^{*}$ corresponds to the complex conjugate of ${\varPsi}_{\tau_0,\ {a}_0}$.

Multi-resolution analysis (MRA) is able to study signals represented at different resolutions. It can be used to decompose a signal into a progression of successive approximations and details in increasing order of resolution. Choosing particular values of a ₀ and τ ₀, in Eq. (8), namely a ₀ = 2 and τ ₀ = 1, corresponds to the dyadic case used in MRA. The aim is to reduce/increase the resolution by a factor of 2 between two scales. Therefore, the approximation of a signal x(t) at a resolution j, denoted by A _x ^j and the detail of the same function at a resolution j, denoted by D _x ^j, are defined by:

$$ {A}_x^j(t)={\displaystyle \sum_{k=-\infty}^{+\infty }}{C}_{j,k}{\Psi}_{j,k}(t) $$

(20)

$$ {D}_x^j(t)={\displaystyle \sum_{k=-\infty}^{+\infty }}{D}_{j,k}{\Phi}_{j,k}(t) $$

(21)

where Φ _j,k (t) is a scaled and translated basis function called the scaling function, which is determined with Ψ _j,k (t) when a wavelet is selected. C _j,k is the scaling coefficient given the discrete sampled values of x(t) at resolution j and location k. It is calculated from Φ _j,k (t) in a similar way for the wavelet coefficient D _j,k from Ψ _j,k (t) (see Kumar and Foufoula-Georgiou (1997) for detailed mathematical expressions).

The signal x(t) can be reconstructed from the approximation and detail components as:

$$ x(t)={A}_x^j(t)+{\displaystyle \sum_{j=1}^J}{D}_x^j(t) $$

(22)

where J is the highest resolution level considered. Since MRA ensures that variance is well captured in a limited number of resolution levels, analysis of energy distribution in the sampling time-series across scales gives a good idea of the energy distribution across frequencies.