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Probabilistic Approach to Physically Based Rainfall-Induced Shallow Landslide Modeling
Massimiliano Alvioli
CNR IRPI (via della Madonna Alta 126, 06128, Perugia, Italy)

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Physically based landslide modeling
DEFINITION
A landslide is the movement of a mass of rock, debris, or earth down a slope, under the influence of gravity. (Varnes 1978, Cruden & Varnes 1996)
We discuss shallow landslides and their occurrence in relation to rainfall events. Deep-seated landslides are not included.
Shallow landslides are caused primarily by rainfall and can be studied using empirical and/or physically based approaches.
A landslide is the movement of a mass of rock, debris, or earth down a slope, under the influence of gravity. We will discuss shallow landslide and their occurrence in relation to rainfall events. Landslides are shallow when they only interest the superficial layer of soil and are caused primarily by rainfall. They can be studied using empirical and/or physically based approaches.
References:
Varnes, D.J., Slope movements: types and processes. In: Schuster, R.L., Krizek, R.J. (Eds.), Landslide Analysis and Control, National Academy of Sciences, Special Re- port 176. Transportation Research Board, Washington D.C., pp. 11–33.
Cruden, D.M., Varnes, D.J., Landslide types and processes. In: Turner, A.K., Schuster, R.L. (Eds.), Landslides, Investigation and Mitigation, Special Report 247. Transportation Research Board, Washington D.C., pp. 36–75. ISSN: X, ISBN: X.
M. Alvioli

3. LANDSLIDE FORECASTING3
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LANDSLIDE FORECASTING
Landslide forecasting can be performed by means of rainfall thresholds or physically based models
(Guzzetti et al., 2007).
Existing warning systems are mainly based on rainfall thresholds and precipitation forecasts (Rossi et al., 2012). Integration with physically based models is under way.
Landslide forecasting can be performed by means of rainfall thresholds (Guzzetti et al., 2007) or physically based models, in conjunction with precipitation forecasts. Such kind of system is currently used over the whole Italy as a warning system. Daily runs of statistical models are preformed to prepare a bullettin to be provided to national institutions. Integration in the system of physically based predictions is desirable and it is under way.
References:
Guzzetti F., Peruccacci S., Rossi M., Stark C.P., Rainfall thresholds for the initiation of landslides in Central and Southern Europe. Meteorology and Atmospheric Physics, 98,
Rossi M., et al., SANF: a national warning system for rainfall-induced landslides in Italy. Proceedings 11th International & 2nd North American Symposium on Landslides, June 2-8, 2012, Banff, Alberta, Canada.
M. Alvioli

4. PHYSICALLY BASED MODELS
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PHYSICALLY BASED MODELS
Rainfall infiltration and pore pressure are a main trigger of slope failures.
We adopt a probabilistic extension of the Transient Rainfall Infiltration and GRid-based Slope stability model – TRIGRS (Baum et al., 2002; 2008)
In the model, shallow landslides are described within the infinite-slope approximation, for saturated and unsaturated conditions, and their stability is assessed thorough a factor of safety.
Physically based, distributed models use local soil characteristic as an input of a system of equations. Their solution provide a means of assessing the stability of the area under specifc rainfall forcing. Other key inputs of the models are soil initial conditions, rainfall infiltration rates and downhill routing of the excess water. The stability (or instability) of a unit of slope, e.g. a grid cell, is determined by calculating a factor of safety, a positive number which is <1 for unstable conditions.
References:
Baum, R., Savage, W., Godt, J.W., TRIGRS - a Fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis., U.S. Geological Survey Open-file Report, 424, 61.
Baum, R., Savage, W., Godt, J.W., TRIGRS - a Fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis, version 2.0., U.S. Geological Survey Open-file Report, 1159, 75.
M. Alvioli

5. INFINITE-SLOPE APPROXIMATION5
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INFINITE-SLOPE APPROXIMATION
All forces not resolvable on planes parallel to the ground surface are neglected.
Need to specify a maximum plausible failure depth Z, as the landslide thickness.
An infinite slope geometry is a rigorous, lowest order approximation of a multi-dimensional landslide geometry if H<<L, where H is the possible slip surface depth and L is the possible landslide length or witdh (Iverson 2000). A maximum plausible faiulre depth H can be defined. An upper bound for H can be identified on the basis of geological setting, in which strong rock underlies a weaker layer of soil. In the model, lateral stresses and inter-cell forces are neglected. This is also a reasonable assumption for shallow landslides, for which the depth is small as compared to the landslide width.
References:
Iverson, R.M, Water Resources Research, 36(7), 1897.
Iverson (2000)
M. Alvioli

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FACTOR OF SAFETY
The stability of a grid cell on a slope is governed by the balance of vertical component of gravity (Fc) against the resisting stress due to basal Coulomb friction (Ff), plus pore pressure (Fw) (Richards, 1931). Failure occurs at depth Z, measured vertically from the surface, if at that depth:
The balance of gravitational and resisting forces acting on a soil unit is used to asses the the local stability of the slope. Pressure gradient is an additional force exerted in water due to rainfall infiltration in the porous soil. The stability of a grid cell is governed by FS, the factor of safety.
In the following, for the calculation of the factor of safety, we will refer to the model impelmented in the TRIGRS software, originally developed by U.S. Geological Survey, and its extension TRIGRS-P, modified by Italian CNR-IRPI to account for a Monte-Carlo based determination of the input parameters.
References:
Richards L.A., Capillary conduction of liquids through porous mediums. Journal of Applied Physics, 1, 318–333.
M. Alvioli

7. FACTOR OF SAFETY Ψ(Z,t) is the pressure head7
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FACTOR OF SAFETY
When rainfall occurs, the factor of safety varies as a function of depth and time. It is convenient to split the factor of safety into a time-varying component and a steady background component. The steady component is usually known, once the slope and soil parameters are specified. The time-varying component is calculated as a function of rainfall intensity and duration, to account for transient rainfall effects. The model is suited for slopes under in the range 0o to 60o.
References:
Iverson, R.M, Water Resources Research, 36(7), 1897.
Ψ(Z,t) is the pressure head
solution at depth Z and time t
Iverson (2000)
M. Alvioli

8. FACTOR OF SAFETY Soil parameters: c = soil cohesion φ = friction angle8
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FACTOR OF SAFETY
Soil parameters:
c = soil cohesion
φ = friction angle
γS = soil unit weight
γW = unit weight of
groundwater
Ψ(Z,t) is the pressure head distribution at depth Z and time t.
The model does not describe the evolution after a given cell has failed, i.e. after the factor of safety in a given cell falls below unity. The pore pressure depends on depth Z and time t, and must be determined for each cell as a function of time, using well-defined equations, given a number of assumptions and boundary conditions.
References:
Iverson, R.M, Water Resources Research, 36(7), 1897.
Iverson (2000)
M. Alvioli

9. 2D GRID APPROACH Grid maps needed:9
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2D GRID APPROACH
Grid maps needed:
Hydrologically- consistent Digital Elevation Model
Map of flow directions
Map of slopes
Horizontal eterogeneity is considered by allowing material properties, rainfall, and other input values to vary from cell to cell. Input grids can be prepared using Geographic Information System (GIS) software, also used to visualize intermediate and final results. For example, GRASS (Neteler et al., 2012) GIS software can be used to build the flow directions and slope starting from a hydrologically consistent DEM. A homogenues and isotropic soil is assumed for each cell. Cells are independent. Only one-dimensional vertical infiltration is considered, and lateral flow is neglected.
References:
Raia, S., Alvioli, M., Rossi, M., Baum, R., Godt, J.W., Guzzetti, F., Geoscientific Model Developement (Discussion Paper).
Raia et al., (2013)
M. Alvioli

10. 2D GRID APPROACH Optional maps are: depth of the water table
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2D GRID APPROACH
Optional maps are:
depth of the water table
soil depth - we use the
model of DeRose (1996)
rainfall intensity
map of lithological zones
Geo-technical and hydrological parameters must be provided for each lithological zone;
Complex rainfall histories can be provided.
The study area can be partitioned into different zones. For each zone, the input parameters are specified independently. These include geo-technical and hydrological properties of the soil, water table depth and initial conditions. Different methods of solution (saturated, unsaturated) of the model equations are used each zone. In the model, the defintion of the various zones is provided by an additional grid map using a different integer value for each zone.
References:
DeRose, R., Relationships between slope morphology, regolith depth, and the incidence of shallow landslides in eastern Taranaki hill country, Zeitschrift fur Geomorphologie Supplementband, 105, 49–60.
M. Alvioli

11. RAINFALL INFILTRATION11
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RAINFALL INFILTRATION
Vertical infiltration of water in the soil is described using two different conditions:
water-saturated soil
(Baum et al., 2002)
water-unsaturated
soil (Baum et al., 2008)
Two altenative models are used for infiltration. Saturated conditions are described by Iverson (2000) equations, where there is a steady seepage component and a transient conponent which assumes a verical downward flow, peculiar of tension-saturated conditions. For unsaturated conditions , a two-layer system is adopted. The system consists of a saturated zone with a capillary fringe above the water table, and an unsaturated zone which extends to the ground surface.
References:
Baum, R., Savage, W., Godt, J.W., TRIGRS - a Fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis., U.S. Geological Survey Open-file Report, 424, 61; Open-file Report, 1159, 75.
Baum, R., Savage, W., Godt, J.W., TRIGRS - a Fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis, version 2.0., U.S. Geological Survey Open-file Report, 1159, 75.
M. Alvioli

12. RAINFALL INFILTRATION12
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RAINFALL INFILTRATION
Natural rainfall events are simulated using complex rainfall histories.
Different rainfall intensities are used for each time interval.
The model differential equations are solved
for each time step.
A realistic definition of the rainfall events is crucial for the model. Natural rainfall events are simulated using complex rainfall histories. Different rainfall intensities are used for each time interval. The model differential equations are solved for each time step. Various outputs can be saved at intermediate time steps, with increased computational cost if they are different from the input time steps.
The time scale for applicability of the model is such that lateral flow across neighboring cells is negligible. Since vertical infiltration is dominant in determining pore pressure changes early in a storm, lateral flow starts to be important during long storms.
M. Alvioli

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Physically based landslide modeling
GOVERNING EQUATIONS
The pressure head Ψ(Z,t) is the solution of the Richards equation (Richards, 1931):
TRIGRS adopts different techniques for the solution in the saturated and unsaturated scenarios.
The pressure head Ψ(Z,t) is the solution of the Richards equation. Kz is the vertical hydraulic conductivity, which depends on ψ; Z is the slope-normal vertical coordinate; Θ is the water content; t is time. TRIGRS adopts different techniques for the solution in the saturated and unsaturated scenarios.
References:
Richards L.A., Capillary conduction of liquids through porous mediums. Journal of Applied Physics, 1, 318–333.
M. Alvioli

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GOVERNING EQUATIONS
TRIGRS adopts the solution of Baum et al. (2008) for the steady and transient components of the problem for saturated soil conditions.
Excess water from the grid cells where rainfall intensity exceeds the local infiltration capacity is considered, adopting a simple runoff-routing scheme.
Within the saturated model, initial conditions are the most critical input of the model.
TRIGRS adopts the solution of Baum (2008) for the steady and transient components of the problem for saturated soil conditions. Short-time and long-time solution are selected as a function of a time scale, governed by slope and soil parameters. Excess water from the grid cells where rainfall intensity exceeds the local infiltration capacity is considered adopting a simple runoff-routing scheme. Within the saturated model, initial conditions are the most critical input of the model.
References:
Baum, R., Savage, W., Godt, J.W., TRIGRS - a Fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis, version 2.0., U.S. Geological Survey Open-file Report, 1159, 75.
M. Alvioli

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GOVERNING EQUATIONS
For unsaturated conditions, TRIGRS uses the linearized equations defined by Srivastava (1991), which is a linear diffusion equation.
Excess water from the grid cells where rainfall intensity exceeds the local infiltration capacity is considered, adopting a simple runoff-routing scheme.
Results are very sensitive to the initial conditions: water table depth and initial steady infiltration rate at each cell.
For unsaturated conditions, TRIGRS uses the linearized euqations defined by Srivastava (1991). An exponential model is used for the hydraulic conductivity Kz(ψ) and the water content θ, in the Richards equations. With the exponential model, the original equation reduces to a linear diffusion equation for which analytic solutions are available, once the boundary conditions are specified. In the model, due to vertical infiltration, the water table rises when the amount of infiltrating water exceeds the maximum amount that can be drained by gravity. Excess water from the grid cells where rainfall intensity exceeds the local infiltration capacity is considered adopting a simple runoff-routing scheme. Results are very sensitive to the initial conditions.
M. Alvioli

16. 16
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INPUT PARAMETERS
The difficulty in determining input parameters in TRIGRS is of practical and conceptual nature:
Extensive field surveys and laboratory measurements are needed.
The natural variability of geo-technical and hydrological soil properties makes them difficult to determine realistically.
The physically based approach requires an accurate definition of the input parameters, some of which may vary over orders of magnitude, especially the hydraulic conductivity and the diffusivity. The difficulty in determining input parameters is of practical and conceptual nature: extensive field surveys and laboratory measurements are needed; the natural variability of soil properties makes it difficult their realistic determination. The applicability of the model was limited by computational demand for large areas and repeated runs, and by the lack of detailed, spatially-distributed information. A reduction of the size of the input grids for improved accuracy should be reflected in comparable accuracy in knowledge of the input parameters.
M. Alvioli

17. 17
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INPUT PARAMETERS
We use a probabilistic approach for parameters, defining proper Probability Density Functions (PDF) from which their values are randomly sampled.
Each ξi is sampled from a suitable PDF.
Multiple runs with different parameters result in different model outputs.
The probabilistic approach proposed by Raia et al. (2013) in TRIGRS-P deals with the variability of input parameters by introducing the possibility of sampling their values randomly from suitable PDFs, on a cell-by-cell basis, at the beginning of each single model run. The transition from the TRIGRS code to the TRIGRS-P code can be viewed as a transition from a single-run, deterministic model to a multiple-run, probabilistic model. Results can be analyzed statistically.
References:
Raia, et al., Improving predictive power of physically based rainfall induced shallow landslides models: a probabilistic approach. Geoscientific Model Developement (Discussion Paper).
M. Alvioli

18. PROBABILITY DENSITY FUNCTIONS
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PROBABILITY DENSITY FUNCTIONS
TRIGRS-P implements two types of PDFs for sampling the modeling parameters (Raia et al., 2013):
the Uniform density function. Used for a global search of parameters within a broad range.
the Gaussian density function. Used when reasonable values of the parameters are known.
TRIGRS-P implements two types of PDFs for sampling the modeling parameters. The uniform distribution function is used in a global search to explore the values of the input parameters in a broad range. This option is useful in a preliminary set of runs, in order to exclude unrealistic values resulting in unrealistic values of the factor of safety when compared to known landslides. The Gaussian distribution function is used when reasonable values of the parameters are known and to simulate an additional variability within some range, estimated by proper means, used to determine the standard deviation of the (Gaussian) distribution function.
References:
Raia, et al., Improving predictive power of physically based rainfall induced shallow landslides models: a probabilistic approach. Geoscientific Model Developement (Discussion Paper).
M. Alvioli

19. TRIGRS-P OUTPUT: Factor of Safety (Fs) Maps
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TRIGRS-P OUTPUT: Factor of Safety (Fs) Maps
The maps show the geographical distributions of the minimum (left), maximum (center) and the standard deviation (right) of the factor of safety for a set of 16 runs using the TRIGRS-P code (Raia et al., 2013). The calculations were performed for the Frontignano study area, Italy, using uniform PDFs for all the input parameters. Black polygons show the known rainfall-induced landslides.
References:
Raia, S., Alvioli, M., Rossi, M., Baum, R., Godt, J.W., Guzzetti, F., Improving predictive power of physically based rainfall induced shallow landslides models: a probabilistic approach. Geoscientific Model Developement (Discussion Paper).
A: minimum FS; B: maximum FS; C: standard deviation
M. Alvioli

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MODEL PERFORMANCE
The model output can be validated against known rainfall events that have caused shallow landslides.
The original model is deterministic, i.e., it provides a single output. Our probabilistic approach provides several outputs.
Statistical analyses are performed on multiple results of the probabilistic runs, initialized with a random set of parameters sampled from the specified PDF.
The TRIGRS-P code was tested in two well known study areas: the Frontignano, Italy, and Mukilteo, USA, areas. For both areas, rainfall and landslide landslide information was available. The original model of TRIGRS is a deterministic one, and produces a single map for the factor of safety. Our probabilistic approach produces several outputs. Statistical analyses are performed on a set of 16 results from probabilistic runs, initialized with different sets of input parameters sampled from the uniform Probability Density Function (PDF). The central values and the width of the PDFs were varied in order to assess the model performance.
M. Alvioli

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MODEL PERFORMANCE
Model performance can be assessed by different evaluation criteria.
Tools for validation are correct assignments (True Positives, True Negatives) and model errors (False Positives, False Negatives) in a contingency table.
Model performance can be assessed using a large set of evaluation criteria. Tools for validation are correct assignments (True Positives, True Negatives) and model errors (False Positives, False Negatives) in a contingency table. Users may ascribe different relevance to correct assigments, model errors, or a combination of them. User discretion is required to asses whether a less conservative evaluation of the model is to be preferred, to fulfil the need of a general evaluation of landslide probability and location in a given area, or a more conservative approach is needed where, for example, a single run or a series of runs are used in an alert system.
M. Alvioli

22. MODEL PERFORMANCE Mukilteo, Seattle, USA Frontignano, Umbria, Italy
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MODEL PERFORMANCE
Mukilteo, Seattle, USA Frontignano, Umbria, Italy
The plots show Receiving Operating Characteristics (ROC) curves, which are defined by the false alarm rate (FPR) plotted vs. the hit rate (TPR). In the ROC space, a point located in the upper left corner represents a perfect prediction (FPR=0, TPR=1). Points along the diagonal TPR=FPR represent random predictions. An acceptable prediction requires TPR>FPR. In the plots, two separate points show the predicted performance of the original TRIGRS model, for saturated (squares) and unsaturated (circles) conditions. Multiple model outputs with the TRIGRS-P code allowed to quantitatively measure the performance of the probabilistic approach using the Area Under the Curve (AUC). The probability thresholds required to build the ROC curves were varied between 0.1 and 0.9 with 0.1 steps.
M. Alvioli

23. 23
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MODEL PERFORMANCE
Variation of the parameter controlling the mean value
Smaller mean variation
Larger mean variation
The four-fold plots (FFP) show results for the Frontignano study area, Italy. The figure shows the proportion of corrects assignments (True Positive, TP; True Negative, TN) and model errors (False Positive, FP; False Negative, FN). A uniform PDF was used in the calculations.
In these figures, the central values of the intervals from which the random values of all the parameters are sampled (mean of the PDF) were changed. The three FFP correspond to three different sets where the extent of variation of the mean from the starting value is changed.
M. Alvioli

24. 24
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MODEL PERFORMANCE
Variation of the parameter controlling the range of variation
Smaller range
Larger range
The four-fold plots (FFP) show results for the Frontignano study area, Italy. The figure shows the proportion of corrects assignments (True Positive, TP; True Negative, TN) and model errors (False Positive, FP; False Negative, FN). A uniform PDF was used in the calculations.
In these figures, the widths of the intervals from which the random values of all the parameters are sampled (width of the PDF) are changed. The three FFP correspond to three different sets of runs where the extent of variation of the width from the starting value is changed.
M. Alvioli

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CONCLUSIONS
We know how to forecast landslides using empirical rainfall thresholds.
Physically based models present limitations due to poorly known soil characteristics.
A new probabilistic approach was recently proposed to reduce the impact of poorly known quantities.
The probabilistic approach improves the predicting power of simulations.
1) We know how to forecast landslides using empirical rainfall thresholds: they can be specialized on a given area, climate, lithological type and other carachteristics (see presentation by M. T. Brunetti and S. Peruccacci)
2) Physically based models present limitations due to unknown or poorly known soil geo-technical and hydrological characteristics, initial conditions, soil depth and others.
3) A new, probabilistic approach was recently proposed to reduce the impact of poorly known quantities: the model input parameters are sampled from suitable Probability Distribution Functions with a Monte Carlo procedure, in analogy with other fields of natural sciences.
4) It was shown that the probabilistic approach improves the predicting power of the physically based model we considered.
M. Alvioli

26. Massimiliano.Alvioli@irpi.cnr.it ICL Landslide Teaching Tools
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27. REFERENCES 1/3 ICL Landslide Teaching Tools PPT-tool 3.039-1.2 (27) 27
Physically based landslide modeling
REFERENCES 1/3
Baum, R., Savage, W., Godt, J.W., TRIGRS - a Fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis., U.S. Geological Survey Open- file Report, 424, 61.
Baum, R., Savage, W., Godt, J.W., TRIGRS - a Fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis, version 2.0., U.S. Geological Survey Open-file Report, 1159, 75.
Cruden, D.M., Varnes, D.J., Landslide types and processes. In: Turner, A.K., Schuster, R.L. (Eds.), Landslides, Investigation and Mitigation, Special Report 247. Transportation Research Board, Washington D.C., pp. 36–75. ISSN: X, ISBN: X.
DeRose, R., Relationships between slope morphology, regolith depth, and the incidence of shallow landslides in eastern Taranaki hill country, Zeitschrift fur Geomorphologie Supplementband, 105, 49–60.
Guzzetti F., Peruccacci S., Rossi M., Stark C.P., Rainfall thresholds for the initiation of landslides in Central and Southern Europe. Meteorology and Atmospheric Physics, 98,
M. Alvioli

28. REFERENCES 2/3 ICL Landslide Teaching Tools PPT-tool 3.039-1.2 (28) 28
Physically based landslide modeling
REFERENCES 2/3
Iverson, R.M, Water Resources Research, 36(7), 1897.
Neteler, M., Hamish Bowman, M., Landa, M., Metz, M., GRASS GIS: a multi-purpose open source GIS. Environ. Modell. Softw –130.
Raia, S., Alvioli, M., Rossi, M., Baum, R., Godt, J.W., Guzzetti, F., Improving predictive power of physically based rainfall induced shallow landslides models: a probabilistic approach. Geoscientific Model Development (Discussion Paper).
Richards L.A., Capillary conduction of liquids through porous mediums. Journal of Applied Physics, 1, 318–333.
Rossi M., Peruccacci S., Brunetti M. T., Marchesini I., Luciani S., Ardizzone F., Balducci V., Bianchi C., Cardinali M., Fiorucci F., Mondini A.C., Reichenbach P., Salvati P., Santangelo M., Bartolini D., Gariano S. L., Palladino M., Vessia G., Viero A., Antronico L., Borselli L., Deganutti A. M., Iovine G., Luino F., Parise M., Polemio M., Guzzetti F., SANF: a national warning system for rainfall-induced landslides in Italy. Proceedings 11th International & 2nd North American Symposium on Landslides, June 2-8, 2012, Banff, Alberta, Canada.
M. Alvioli

29. REFERENCES 3/3 ICL Landslide Teaching Tools PPT-tool 3.039-1.2 (29) 29
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REFERENCES 3/3
Srivastava, R., Yeh, T.-C.J., Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils, Water Resources Research, 27(5), 753–762.
Varnes, D.J., Slope movements: types and processes. In: Schuster, R.L., Krizek, R.J. (Eds.), Landslide Analysis and Control, National Academy of Sciences, Special Re- port Transportation Research Board, Washington D.C., pp. 11–33.
M. Alvioli