Probabilistic modeling of shallow landslide initiation using regional scale random fields

Abstract

Regional mapping of landslide susceptibility aims to identify zones of potential instability across geological settings. Given their predictive capabilities, physically based, deterministic models are useful tools for landslide triggering studies at regional scale. However, they rely on detailed input parameters that are rarely available for large areas. To address these limitations, this work proposes a computational framework to incorporate the spatial uncertainty of input data into physically based, landslide hazard zonation models through the use of regional scale random fields (RSRF). For this purpose, input parameters are treated as spatially correlated random variables with assigned statistical attributes, while a vectorization strategy is used to reduce the computational cost of large-scale stochastic analyses. Deterministic simulations based on a hydro-mechanical model are then performed for multiple Monte Carlo realizations to compute maps of failure probability (p_f). The methodology was applied to a well-documented series of rainfall-induced shallow landslides in a volcanic site for which field measurements were available to constrain the statistical variability of the hydraulic conductivity and treat this parameter as an RSRF. To analyze the results, four classes of landslide susceptibility characterized by different p_f thresholds were used. Such classes were mapped over the study zone and throughout the storm event, allowing a direct comparison with the spatio-temporal evidence of landslide triggering. The results indicate that (i) uncertainty analyses neglecting the role of spatial correlation may lead to non-conservative estimates of landslide susceptibility and (ii) there is an interval of spatial correlation distance that optimizes the performance of the model, thus providing an indirect estimate of the heterogeneity of the site. Such results highlight the benefits of accounting for the uncertainty of the soil properties in regional-scale models and offer a new predictive stochastic framework to assess the implications of future rainfall scenarios over large areas.

Appendix

This section provides a description of the field equations and their implementation into the hydro-mechanical model previously proposed by Lizarraga and Buscarnera (2018). Although the model is here described in uncoupled form, it can be readily extended to coupled scenarios by accounting for the influence of soil volume change on pore pressure generation (Wu et al., 2016)

For unsaturated slopes, transient infiltration can be simulated by enforcing the water mass balance (Richards, 1931):

$$ {nC}_w(h)\frac{\partial h}{\partial t}=\nabla .\left[K(h)\nabla \left(h+z\right)\right] $$

(2)

where n is the porosity, h is the pressure head induced by capillarity, t is time, z is the vertical coordinate, K(h) is a hydraulic conductivity function (HCF), and C_w(h) is the unsaturated storage coefficient (i.e., the rate of change of degree of saturation S_r(h) with respect to h). The above partial differential equation (PDE), supplemented with constitutive relations, initial and boundary conditions, constitute the initial-boundary-value problem to solve for each slope unit within the landscape. A Galerkin spatial discretization and an explicit forward temporal scheme were used to discretize the above PDE and solve it numerically (Chen et al., 2006; Zienkiewicz et al., 1999).

The PDE (2) requires the definition of a water retention curve (WRC) and a hydraulic conductivity function (HCF). An exponential decay function is used (Gardner, 1958) for the WRC:

$$ {nS}_r={\theta}_r+\left({\theta}_s-{\theta}_r\right)\exp \left({\alpha}_{\mathrm{WRC}}h\right) $$

(3)

and similarly, for the HCF:

$$ K={K}_s\exp \left({\alpha}_{\mathrm{HCF}}h\right) $$

(4)

where θ_s and θ_r are the saturated and residual volumetric water content, respectively. The material constants α_WRCand α_HCF control the suction sensitivity of S_r and K, respectively. The hydrologic constitutive equations above have been chosen for their simplicity and widespread use in analytical solutions for regional analyses (Srivastava and Yeh, 1991; Baum et al., 2010). However, the proposed model does not involve limitations on the mathematical form of WRC and HCF, as it allows to employ more sophisticated nonlinear, hysteretic expressions, depending on the specific application.

Stability analyses require the definition of a factor of safety (FS) which can be derived in light of material stability considerations (Buscarnera and di Prisco, 2011, 2013). Although several forms of FS have been proposed in the literature (Duncan et al., 2014; Lu and Godt, 2008), here an expression previously calibrated for the soils of the study area selected for this note will be used (Lizárraga et al., 2017), as follows:

$$ \mathrm{FS}=\frac{\tan {\phi}^{\hbox{'}}}{\tan \alpha}\left(1+\frac{ks}{\sigma^{net}}\right) $$

(5)

where ϕ^′ and α are the friction angle of the layer and its slope inclination, respectively, σ^net is the normal net stress, and k is a parameter that quantifies the effect of suction s, on the shearing resistance (Fredlund et al., 1978). A summary of the input parameters is shown in Table 1.

Table 1 Summary of material parameters and input values