Scholarly article on topic 'Parameter Uncertainty in Shallow Rainfall-triggered Landslide Modeling at Basin Scale: A Probabilistic Approach'

Parameter Uncertainty in Shallow Rainfall-triggered Landslide Modeling at Basin Scale: A Probabilistic Approach Academic research paper on "Earth and related environmental sciences"

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Abstract of research paper on Earth and related environmental sciences, author of scientific article — E. Arnone, Y.G. Dialynas, L.V. Noto, R.L. Bras

Abstract This study proposes a methodology to account for the uncertainty of hydrological and mechanical parameters in coupled distributed hydrological-stability models for shallow landslide assessment. A probabilistic approach was implemented in an existing eco-hydrological and landslide model by randomizing soil cohesion, friction angle and soil retention parameters. The model estimates the probability of failure through an assumed theoretical Factor of Safety (FS) distribution, conditioned on soil moisture content. The time-dependent and spatially distributed FS statistics are approximated by the First Order Second Moment (FOSM) method. The model was applied to the Rio Mameyes Basin, located in the Luquillo Experimental Forest in Puerto Rico.

Academic research paper on topic "Parameter Uncertainty in Shallow Rainfall-triggered Landslide Modeling at Basin Scale: A Probabilistic Approach"

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Procedía Earth and Planetary Science 9 (2014) 101 - 111

The Third Italian Workshop on Landslides

Parameter Uncertainty in Shallow Rainfall-Triggered Landslide Modeling at Basin Scale: a Probabilistic Approach

E. Arnonea,b*, Y.G. Dialynasb, L.V. Notoa, R.L. Brasb

aDipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali, Universita' degli Studi di Palermo, Viale delle Scienze ed. 8,

Palermo, 90128, Italy

bSchool of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive, Atlanta, GA, 30332, USA

Abstract

This study proposes a methodology to account for the uncertainty of hydrological and mechanical parameters in coupled distributed hydrological-stability models for shallow landslide assessment. A probabilistic approach was implemented in an existing eco-hydrological and landslide model by randomizing soil cohesion, friction angle and soil retention parameters. The model estimates the probability of failure through an assumed theoretical Factor of Safety (FS) distribution, conditioned on soil moisture content. The time-dependent and spatially distributed FS statistics are approximated by the First Order Second Moment (FOSM) method. The model was applied to the Rio Mameyes Basin, located in the Luquillo Experimental Forest in Puerto Rico.

© 2014 Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibilityofDipartimentodiIngegneriaCivile,Design,Edilizia e Ambiente,Seconda Universitdi Napoli.

Keywords: Rainfall-Triggered Landslides; Hydrological Modeling; Uncertainty; Probability.

1. Introduction

The use of coupled distributed hydrological-stability models for shallow landslide hazard assessment at catchment scale is common in the literature. The practice is to utilize the basin hydrological response, evaluated in terms of soil moisture and groundwater fields, to assess a spatially distributed Factor of Safety (FS) by using the

* Corresponding author. Tel.: +39-091-23896544; fax: +39-091-427121. E-mail address: elisa.arnone@unipa.it

1878-5220 © 2014 Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of Dipartimento di Ingegneria Civile, Design, Edilizia e Ambiente, Seconda Universit di Napoli. doi:10.1016/j.proeps.2014.06.003

infinite slope model1,2,3,4,5. Mechanical and hydrological soil properties play a crucial role in such an evaluation, and the importance of appropriately modeling soil water dynamics has been clearly demonstrated in some studies6,7.

The inability to fully characterize hydrological and geotechnical behavior of soil may have a significant impact on model results. Spatial variation of parameters is difficult to describe accurately, and measurement errors can also increase the natural variability of parameters. To account for this uncertainty, FS can be computed within a probabilistic framework, by considering soil parameters as random variables and thus, assigning them probability distributions instead of deterministic values. This practice has received considerable attention in the geotechnical engineering literature, which proposes different methodologies for modeling and analyzing the uncertainty related to the shear strength parameters (i.e. soil cohesion and friction angle) at hillslope scale8,9,10. Based on similar approaches, some studies have been conducted for basin scale applications within coupled hydrological-stability models5,11,12,13; in such applications, the probability of FS, conditioned to the soil moisture, is dynamically estimated across the basin, whereas the probability distributions of the shear strength parameters are time independent. However, the uncertainty of soil hydrological properties, which may be predominant in case of unsaturated conditions, is still neglected in most published literature. In particular, soil retention curve parameters are the most significant in determining the contribution of the soil matric suction to the equilibrium.

The probability distribution of FS can be derived numerically, analytically or through analytical approximations. The Monte Carlo simulation method uses independent sets of soil properties, generated through a priori assigned probability distributions8,10 at fixed topographic (i.e. slope) and hydrological (i.e. soil moisture) conditions to obtain a solution. However, such an approach may have significant computational cost for basin scale applications, since the above mentioned conditions change in time and space. The FS probability distribution can be analytically derived in the case where solely geotechnical parameters (i.e., cohesion and friction angle) are considered as random variables (e.g., for saturated conditions) and the infinite slope model is used for the slope stability analysis8,10. When the soil retention curve parameters are also assumed to be random (e.g. for unsaturated conditions), analytical derivation of FS distribution is not tractable8,10. In this case, the First Order Second Moment (FOSM) method14 is commonly used to estimate analytical approximations of the spatio-temporal FS statistics (i.e. mean and variance)15,16,17,18, to finally fit a theoretical probability distribution for FS and estimate the spatiotemporal dynamics of probability of failure.

In order to systematically account for the parameter uncertainty, we propose a probabilistic approach for coupled distributed hydrological-stability models based on the FOSM method, which was implemented in the tRIBS-VEGGIE (Triangulated Irregular Network (TIN)-based Real-time Integrated Basin Simulator - VEGetation Generator for Interactive Evolution) - Landslide module7. The proposed methodology was applied to the Rio Mameyes Basin, located in the Luquillo Experimental Forest in Puerto Rico, where previous landslide analyses have been carried out. The main purpose of the application is to demonstrate the model capabilities and highlight further possible improvements.

2. Methodology

2.1. Model Overview

The tRIBS-VEGGIE-landslide module7 is built upon the eco-hydrological model tRIBS-VEGGIE, which consists of a spatially distributed physically based hydrological model coupled to a model of plant physiology and spatial dynamics19. Basin hydrological response is simulated on an irregular spatial mesh which allows for the use of variable computational elements to describe the basin topography, by increasing the accuracy only in the most critical areas of the basin20. The model explicitly considers the spatial variability in land-surface descriptors and the corresponding moisture dynamics, stressing the role of topography in lateral redistribution. The infiltration module is responsible of the moisture fields computation and is based on a numerical approximation of the one dimensional Richards' equation21. The dynamics of each computational element are simulated separately, but spatial dependencies are introduced by considering the surface and subsurface moisture transfers among the elements along the direction of steepest descent, based upon the unsaturated hydraulic conductivity of the receiving cell. The soil retention and the unsaturated hydraulic conductivity are related to soil-moisture content through the Brooks and Corey22 (BC) parameterization scheme23, as a function of saturated hydraulic conductivity in the normal to the soil

surface direction, air entry bubbling pressure, and pore-size distribution index. Vegetation affects the soil moisture dynamics mainly by extracting soil water for the purposes of transpiration, considered as a soil moisture sink.

The landslide module is based on the assessment of the Factor of Safety (FS) by applying the limit equilibrium analysis of the infinite slope model. The model dynamically computes FS across the study area through the following equation7:

FS =■

yszn sin« tana yszn \0S -0,

tan^ T„¥b | x tan^ (i)

where c' is the effective soil cohesion, ysis the total unit weight of soil (varying with soil moisture), ywis the water unit weight, zn is the soil depth measured along the normal direction to the slope; a is the slope angle, </> is the soil friction angle, y/b is the air entry bubbling pressure, X is the pore-size distribution index, 9w is the volumetric water content, and 0r and 8s are the residual and saturated soil moisture contents, respectively. X, 0r and 8s are the BC22 parameters used to model the soil retention curve. As a result of the multi-layer representation of soil moisture within tRIBS-VEGGIE, the final product of the module is a spatially distributed vertical dynamic FS profile that takes into account the local moisture and soil conditions within each computational element.

2.2. FS distribution

The implemented probabilistic framework for FS includes the analytical approximation of FS statistical properties (i.e., mean and variance), variant in time and space, by using the FOSM method, and fitting a theoretical probability distribution function for FS (see section 3.3).

The FOSM method applies the first order Taylor expansion, from which first and second order moments are derived. More precisely, if we consider a function of several random variables, X1,.,Xn: Y=y(X1,...,Xn), taking the Taylor series expansion about the mean pX1, ..., pX„, and retaining terms up to first order leads to the following expressions of first and second moments24:

E[Y] = y(MXi,..;Mxn)

Var[Y ] = E

Var (X,.) + 2£Z

i=i j+i

dy "SY:

Cov(X, Xj (3)

where E(Y) and Var(Y) are the mean and the variance, respectively, of the dependent variable Y, n is the number of random variables, Var(X) is the variance of random variables Xi, and Cov (Xi, X) is the covariance between Xi and Xj, which is known given the corresponding correlation coefficient, pXixj. Analytical expressions ofy derivatives with respect to the random variables require that y is continuously differentiable.

In this analysis, the soil parameters c', tanfi, 0r, 8s,yb,and X of the FS formulation (Eq.1) are assumed as random variables. Eq. 1 is thus continuously differentiable with respect to each random variable. Given the marginal statistics of the above mentioned parameters it is possible to estimate the temporally and spatially variant mean and variance of FS, i.e., E[FS] and Var(FS), respectively. Then, an assumed theoretical distribution function is used to estimate the probability of FS, limited to a two-parameter distribution. At this point, Monte Carlo experiments are required to identify the best fit theoretical distribution of FS (see Section 3.3). Under saturated conditions, it is possible to obtain the analytical solution of the FS probability distribution and the use of the FOSM method is not necessary. In fact, the soil retention curve parameters do not directly affect the FS formulation, which results in a linear function of random variables c' and tanfi. In this case, the distribution of FS can be analytically derived, assuming that cohesion and friction angle are uncorrelated normal random variables5,8,10. Consequently, FS follows a normal distribution, and the associated statistics (i.e., mean and variance of FS) are estimated analytically.

The procedure has been implemented into the tRIBS-VEGGIE-Landslide model to dynamically evaluate the

probability of landslide occurrence. For each timestep and at each layer (i) of the soil column cell, the model computes the probability that FS is equal or lower than 1 (implying failure, PrF) assuming independent events among different layers. The probability that a landslide occurs somewhere in the soil column cell is evaluated as the probability of the union of events for the entire column (PrFC). The most probable depth of failure or, equivalently, the probability that the plane of failure at the given column is located at a certain depth, is given by the joint probability of FS being equal or lower than 1 at a certain depth, while there is no failure above (PrPFi). As a result, the model is capable of providing the spatial distribution of PrFC, and the maximum value of PrPFi along with the corresponding depth, which constitutes the most probable landslide depth. Moreover, the model allows for the visualization, at any selected cell, of the temporal and vertical profile of PrFi and PrPF.

3. Case Study

3.1. Basin Description

The Rio Mameyes basin, located in the Luquillo Experimental forest (LEF) in the island of Puerto Rico, was selected for testing the model. The LEF is part of both the Long-Term Ecological Research (LTER) and the Critical Zone Observatory (CZO) networks and has been a focal point for studies of landslide impacts on ecology, geomorphology, biology, disturbance and recovery of vegetation25,26,27,28.The basin has been modeled for dynamic landslide analysis previously by Lepore et al.7 using the tRIBS-VEGGIE-Landside model.

65°4T\V 65°46'W 65'45'W

Fig. 1. Major characteristics of the Rio Mameyes Basin: (a) Digital Elevation Model, drainage system and meteorological station. The red polygon delineates the area where model validation was conducted based on soil moisture data (Lepore et al.7); (b) soil type distribution and Clay-Loam element selected for the analysis; (c) slope distribution. Modified from Lepore et al.7.

The basin has an area of 16.7 km2 and it is characterized by a strong gradient in elevation, which ranges from 104.2 m to 1046 m across a horizontal distance of 3 km (see Digital Elevation Model, DEM, in Fig.1a). Because of the strong gradient in elevation, meteorological variables (e.g. rainfall and temperature) vary consistently throughout the basin29, 30. The mean annual precipitation (MAP) varies from approximately 3000 mm, measured at an elevation of 352 m (Bisley Tower), to 5000 mm at higher elevations31 resulting in one of the wettest basins on the island. The analysis of the slope distribution (see Fig. 1d) reveals 10% of the basin area being characterized by slopes greater than 30°, and 30% of the basin area with slopes greater than 25°. In terms of vegetation, the Luquillo forest is characterized by a combination of lower montane wet tropical, wet subtropical and rain forest32. The predominant vegetation type of the basin is the Tabonuco forest (Dacryodes excelsa), present in the wet subtropical

and subtropical rain forest life zones, typically within the 150-600 m elevation range. Further details on description of the area and available data are provided by Lepore et al.7.

3.2. Data and Model Parameters

The model inputs used in this study (i.e., meteorological forcing, soil properties, and model parameters) are the same as defined in Lepore et al.7. In particular, the hydrological soil properties have been set and tuned through a validation/confirmation procedure based on soil moisture data. Final values of the main soil parameters are given in Table 1. Moreover, the case of anisotropy ratio (i.e., the ratio of the saturated hydraulic conductivities in the directions parallel to the slope and normal to the slope) equal to 100 is analyzed. Meteorological data (wind speed and direction, air temperature, cloud cover, relative humidity, rainfall, incoming shortwave radiation) are obtained from the Bisley meteorological tower (lat. 18.31, long. 65.74, 352 m) (see Fig. 1a), which has been working continuously since 2002, with the exception of brief interruptions. In particular, the model was forced using a continuous series for the year 2008, during which a significant rainfall event was recorded between the 27th and 28th April, 2008 with a peak rainfall intensity of 100 mm/hr. The initial conditions of the basin correspond to a stable equilibrium condition obtained via 1.5 years of spin-up simulation. The model converges to this solution, even starting with different initial conditions, after about six months of simulation. This type of spin-up is discussed in7,

Table l.Hydrological and mechanical soil properties and their statistics.

Parameter Description Units Clay - Loam Sandy - Loam Silty - Clay Clay

Ks Saturated hydraulic conductivity [mm/hr] 50.0 50.0 50.0 10.0

M«S Mean of saturated soil moisture, 0s [mm3/mm3] 0.56 0.55 0.55 0.53

M«R Mean of residual soil moisture, 0r [mm3/mm3] 0.075 0.041 0.051 0.09

M-x Mean of pore-size distribution index, ^ [-] 0.200 0.322 0.127 0.130

Mvb Mean of air entry bubbling pressure, yb [mm] -250 -150 -340 -370

M« Mean of soil effective cohesion, c' [N/m2] 3000 3000 3000 3000

^ Mean of soil friction angle, ^ [°] 25 25 25 25

tfc Standard deviation of c' [N/m2] 1200 1200 1200 1200

Standard deviation of ^ [°] 2.5 2.5 2.5 2.5

Standard deviation of ^b [mm] 290 210 390 600

^es Standard deviation of 0 s [mm3/mm3] 0.054 0.076 0.064 0.048

tfeR Standard deviation of 0r [mm3/mm3] 0.007 0.004 0.022 0.011

ax Standard deviation of X [-] 0.113 0.145 0.094 0.098

Pvb-es Coefficient of correlation yb-0s [-] 0 0 0 -0.216

p vb-eR Coefficient of correlation yb-0R [-] 0.203 0 0 0.154

p yb-X Coefficient of correlation [-] 0.151 0.274 0 0.128

Pes-eR Coefficient of correlation 0s-0r [-] 0.307 0 0 0

Pes-x Coefficient of correlation 0s-^ [-] 0.168 0 0 0

PeR-x Coefficient of correlation 0r-^ [-] 0.429 0.518 0.476 0.442

Several studies have provided a comprehensive description of parameter distributions and statistical properties for different soil types belonging to the USDA soil classification system. Lumb34 studied four typical soil formations and concluded that the Gaussian or a closely related distribution can adequately describe natural soils in terms of cohesion and friction angle, which agrees with Wu and Kraft35, and Langejan36. In general, the normality assumption for c' and tan$ has been used in several studies5,8,10,11,12,37,38. Statistical properties of geotechnical parameters are given in literature by Lumb39, Fredlund and Dahlman40, Schultze41, among others. Moreover, Matsuo and Kuroda42 and Lumb39 showed that cohesion and friction angle display negligible correlation, and hence statistical independence between c and tanp has been assumed in several studies8,10,43,44,45. Therefore, independence between geotechnical parameters was assumed in the present study.

Statistical properties of the BC22 soil retention parameters are given in literature as well. Brakensiek et al.46 analysed 1,085 sets of soil retention data from reports by Rawls47 and Holtan48, and suggested transformations of the BC parameters that allow their characterization as normal distributions and give their statistical properties (i.e.,

means, variances and cross-correlation coefficients) for different soil types in the USDA soil classification. McCuen et al.49 analysed the same data-set and provided BC parameters statistics and demonstrated the parameter variation across different soil textural classes. Rawls et al.50 analyzed a total of 5,350 soil samples from a 1978 soil survey, and provided BC parameters marginal statistics. Carsel and Parrish51 analysed approximately 5,700 samples from a soil database compiled by Carsel et al.52 and provide statistics of van Genuchten53 soil retention parameter statistics and corresponding transformations to normality. Their results were used by Meyer et al.54, who applied parameter equivalencies to numerically derive BC parameter statistics and marginal distributions. Schaap and Leij55 three data bases56,57,58 (to a total size of 2,130 samples) provided van Genuchten's (VG) soil retention parameter statistics. Flores et al.59 used the Schaap and Leij49 results and the same data sets to fit marginal distributions to BC parameters by transforming VG to BC parameters.

In this study, the Brakensiek et al.46 results are used, since they provided marginal and joint statistics of the BC parameters, and suggested parameter transformations to normality (and thus, supporting the use of joint normal distribution for BC parameters) without using any VG - BC parameter equivalencies. The values of standard deviation of the residual soil moisture are, those given by McCuen et al.49. According to the Brakensiek et al.46 statistical description of BC parameters, X can reach values very close to zero. However, as ^^0 ^FS , and dFS/dA^K>, which can be unacceptable since the implemented FOSM method requires cFS/dX<m and in general, may not be efficient in the presence of highly non-linear relationships. Therefore, the distribution of X was truncated at a lower limit for X (Amin), and the corresponding statistics were computed. The standard deviations of cohesion and friction angle are assigned as the 40% and 10% of the mean, according to Fredlund and Dahlman40 and Lumb34 respectively. Values of soil parameters and statistical properties are reported in Table 1.

3.3. Monte Carlo experiments

Monte Carlo experiments were conducted to numerically derive empirical distributions of FS, with given hydrological and topographical conditions, and identify the best fit of analytical distribution functions. The empirical FS distributions were obtained for different sets of soil depths, slope and soil moisture, and for the four soil types of the study area, i.e. clay, clay-loam, silty-clay, sandy-loam. The fit of several theoretical distributions was evaluated. The selection was based both on Kolmogorov-Smirnov goodness of fit statistics and visible comparison between empirical and theoretical distributions, focusing on mass around the value of FS=1, in order to focus on the reproduction of the probability of failure. Fig.2 shows the comparison of three theoretical distributions (i.e., Normal, Gamma and Lognormal) against the empirical FS distribution in terms of quantiles (Q-Q plots) numerically obtained for a set of values of slope, depth of failure surface and volumetric water content (equal to 40°, 1000 mm and 0.3 mm3/mm3 respectively), and for the statistical parameters reported in Table 1. Similar analyses were carried out for different set of hydrological and topographic conditions, not shown here for sake of brevity.

In all cases, the lognormal distribution (red dots) is the closest to the grey line (ideal fit), demonstrating that it best captures the probability of failure and the FS distribution and therefore constitutes a better fit compared to other two-parameter distributions. This is also supported by the results reported by Frattini et al.11, who demonstrated that the lognormal function properly describes the FS distribution for different soil formations.

Based on the Monte Carlo experiments results, the lognormal distribution was implemented to model the FS distribution within tRIBS-VEGGIE-Landslide and evaluate the probability of failure across the basin. However, it is worth pointing out that the best fit distribution may vary for different sets of soil parameter statistics (i.e., other than the ones used in this paper).

Fig. 2. Empirical FS quantiles (numerically derived) vs theoretical FS quantiles of Lognormal (red), Gamma (blue), and Normal (purple) distributions for (a) clay, (b) clay loam, (c) sandy loam, and (d) silty clay. The used values of slope, depth of failure surface and volumetric water content are equal to 40°, 1000 mm and 0.3 mm3/mm3 respectively, while the used values of statistical parameters are reported in Table 1.

3.4. Model results

In this section the results from the model application are described and discussed. As explained in section 2.2., the model output includes: the spatial distribution of the probability that failure occurs at any depth within a given element-soil column (PrFC), for specified maximum soil thickness; the spatial distribution of the most probable depth of failure; at selected elements and each layer, the probability that FS<1 (PrF,); and finally the probability that the plane of failure is located at the corresponding depth (PrPF).

Fig. 3 illustrates screenshots of PrFC at the time of the maximum rainfall intensity (t1, 103 mm/hr), and two hours later (t2, 24.4 mm/hr) (Fig.3a and b, respectively). The likelihood of failure occurrence is particularly high (Fig.3a, red regions) in the steepest part of the basin (see slope distribution in Fig. 1d), with 10% of the basin area resulting in a PrFC higher than 0.8, as depicted by the frequency distribution reported in Fig.4a. Values higher than 0.5 correspond to 19% of the basin (8.5% is between PrFC of 0.2 and 0.5, Fig.4a). Note that in the south-eastern region, the PrFC rarely reaches values less than 0.2. This part of the basin is characterized by clayey soil corresponding to higher variation coefficient of BC parameter y/b, which increases suction uncertainty. This leads to greater FS variance and higher uncertainty around stability, which in turn results in higher probability of failure. PrFC between 0.2 and 0.5 (yellow class in Fig.3a and b) characterizing 13% of total area (Fig.4a), is spread across the basin. At t2 (Fig.3b and Fig.4a), the percentage of basin with high PrFC (8.8%) does not decrease significantly. The percentage of the area corresponding to PrFC between 0.2 and 0.5 drops to 7.5%, with a decrease in PrFC especially in the central and the north-eastern part of the basin. No significant changes between t1 and t2 are observed for clayey soils, characterized by lower soil moisture dynamics. In general, at t2 lower values of PrFC are more frequent compared to t1, since soil moisture reduces across the basin.

Fig.3 c, d report maps of the depth of the most probable plane of failure, obtained by excluding the portion of the basin with PrFC lower than 0.5, at t1 and t2, respectively. The corresponding relative frequency distribution is shown in Fig.4b. The most frequent depth for the former case (Fig.3c) corresponds to the range of 200-600 mm, mainly distributed in the upstream north-central part of the basin. This is followed by the range of 850-1250 mm, most of which occur in the clayey area (south-east). That is, because steeper slopes at high soil moisture conditions can be prone to failure at shallower depths. Only in a few cases (0.04 of total, see Fig.4b) the most probable plane of failure corresponds to deeper depths. At t2 (Fig.3d), the probability that failure is initiated at shallower depths decreases, and landslides tend to occur at deeper depths (i.e. 1250-1500 mm) with higher frequency. This can be

given to soil moisture reduction at t2, which decreases destabilizing forces. Thus, greater soil weight is required for landslide initiation, which sets the most probable plane of failure deeper, compared to tl.

0 600 1 000 2»

/ • V*

Most Prob. Plane of fail, [mm]

[ Pr|FS<=l]<0 5

■I 700 -600 600 950

IBI B50 1250 i 1250 ■ 1500

Fig. 3.Spatial distribution of PrFC (a) at the time of the maximum rainfall intensity (t1,103 mm/hr) and (b) two hours later (t2, 24.4 mm/hr). Spatial distribution of the corresponding depth of the most probable plane of failure (c) and (d).

Fig. 4. (a) Frequency distribution of the probability of failure of soil column across the basin (PrFC); (b) frequency distribution of depth of the

most probable plane of failure across the basin, for PrFC> 0.5.

The temporal evolution of landslide likelihood is analysed at element scale for a selected element within the Clay-Loam soil type (CL) with a slope of 52° (see Fig.1b). Fig.5 shows the profiles in time and depth, of soil moisture (Fig.5b), and the probability that the plane of failure at the given element is located at certain depth, PrPFi (Fig.5c) as a response to the rainfall forcing shown in Fig.5a. Around the rainfall peak (tl) higher values of soil moisture are reached at shallow depths. The green closed region in panel c depicts the peak of PrPFi, located between 350-450 mm, at the time of the heavy rainfall. Thus, green color (Fig.5c) indicates the area of most probable depth and time of landslide occurrence. The profile can also be interpreted as the temporally variant probability function of landslide depth, which has its maximum within the range of 300-500 mm, consistently with the characteristics of the shallow landslides.

Fig. 5.Time varying model output at selected cell: (a) rainfall forcing; (b) soil moisture profile; (c) probability of plane of failure profile (i.e. probability that plane of failure at the given element is located at certain depth. Clay-Loam, slope 52°.

4. Conclusions

Natural variability of soil parameters significantly affects the results of hydrological and geomorphological modeling in a basin but a full characterization of this variability cannot be commonly achieved in a distributed modelling context. Here we propose a probabilistic approach for rainfall-induced landslide modeling which takes into account geotechnical and hydrological parameter uncertainty. The methodology considers soil strength and retention curve parameters as random variables, and approximations of FS statistics are estimated by means of the First Order Second Moment (FOSM) method.

The procedure was implemented into the tRIBS-VEGGIE-Landslide model and applied to the Rio Mameyes Basin, Luquillo Experimental Forest in Puerto Rico, as a demonstration of the model capabilities and required improvements. The model output provides a dynamic hazard classification of the basin in terms of probability of failure initiation conditioned on soil moisture. The location and the timing of the most probable plane of failure are estimated by computing the joint probability of FS being less than 1 at each depth.

Future work may include investigations on soil characterization in terms of mechanical and statistical properties. A sensitivity analysis of the model to statistical parameters will improve our understanding of model behavior as soil characteristics change. Additional Monte Carlo experiments to evaluate how the FS distribution changes with varying parameter statistics are also needed. Future developments may also involve the analysis of the effect of the gradient in precipitation on landslide hazard evaluation, which strongly characterizes the basin under study.

Acknowledgements

This work was funded by a subcontract from the University of Pennsylvania's Luquillo Critical Zone Observatory effort, funded by the National Science Foundation. ARRA funds were utilized for the effort.

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