## 1. Introduction

[2] Landslides are natural geomorphic processes shaping mountainous regions and redistributing mass on steep slopes. Expansion of human activity and infrastructure to steep mountainous regions increases exposure to this ubiquitous natural hazard. *Sidle and Ochiai* [2006] estimated the direct costs associated with rebuilding or replacing infrastructure at several billion dollars per year, not considering indirect costs related to construction of engineered barriers and temporary loss of site functionality. Among the different types of mass movements (e.g., falls, topples, slides, and flows) and cover materials involved (rock, debris, and soil) this study focuses on rapid shallow translational landslides in soils with relatively low cohesive strength [*Varnes*, 1978], most frequently induced by intense and/or prolonged rainfall [*Sidle and Ochiai*, 2006; *Schuster and Wieczorek*, 2002].

### 1.1. Hydrologic Triggering of Shallow Landslides: Mechanisms and Modeling

[3] Physically based models attempt to link dynamics of hydrologic status and pathways with the mechanical state of a hillslope and the onset of failure. Soil mass on steep slopes is stabilized by frictional and capillary forces and by reinforcing elements and cohesion. Water infiltration during rainfall may increase the total weight of the soil mass and associated downslope forces [*Chigira and Yokoyama*, 2005] as well as reduce internal soil strength [*Mitchell and Soga*, 2005; *Lu et al.*, 2010]. Moreover, based on local hydrologic conditions (e.g., infiltration, initial water content, and hydraulic conductivity), a perched water table or increased groundwater table may occur resulting in positive water pressure and reduced effective stress [*Iverson*, 2000]. In extreme cases, rapid increase in soil water content may result in deformation of soil volumes until they become liquefied and are released in an abrupt motion [*Iverson et al.*, 1997]. During deformation, soil porosity and shear strength may attain values characteristic of the so-called critical state [*Gabet and Mudd*, 2006] attainable from compaction of loose soil or dilation of initially densely packed soil. In either case, when shearing rate exceeds soil drainage capacity, water pressure may peak and cause liquefaction of the deforming soil mass [*Iverson et al.*, 2000].

[4] For most applications, details of soil failure and liquefaction are seldom considered and mass release is attributed to a simple factor of safety (FOS) calculation, i.e., the ratio between driving and resistive forces often computed based on an infinite slope approach [*O'Loughlin and Pearce*, 1976; *Wu et al.*, 1979; *Casadei et al.*, 2003]. Some models express FOS as a function of seepage flow (transmissivity) and rainfall rate [*Montgomery and Dietrich*, 1994; *Fernandes et al.*, 2004]. Others have considered effects of pore pressure on landslide initiation based on numerical solution of the Richards equation [*Iverson*, 2000; *Tsai and Yang*, 2006], including attempts to resolve unsaturated flow processes over an entire catchment [*Simoni et al.*, 2008]. To consider effects of soil volumes on stability in up- and downslope directions, FOS can be formulated for slopes consisting of soil slices with force equilibrium for each slide and between neighbored slices. While *Bishop* [1955] and *Janbu* [1973] considered normal forces between adjacent slices but ignored interslice-shear force, both force components were accounted for by *Morgenstern and Price* [1965]. To describe impact of local deformations at smaller scale that are not considered in FOS analysis, finite element methods have been studied [*Griffith and Lane*, 1999; *Zheng et al.*, 2005; *Eichenberger et al.*, 2010]. Many of the numerical methods rely on continuum representation of key mechanical processes where landslides slowly and continuously deform. Such a continuous deformation may be in agreement with processes in deep-seated slow landslides but often fail to capture the abruptness characterizing hazardous shallow landslides. As abruptness we define in this study the release of large soil masses at once with no apparent indicators for the imminence of mass release.

### 1.2. Landslide Scaling Laws and Criticality

[5] Occurrence of shallow landslides is not always preceded by detectable surface deformation, but may be associated with internal changes such as build-up of positive pressure related to subsurface flows [*Huang et al.*, 2008] that may relax close to failure [*Germer and Braun*, 2011], or marked by increased acoustic emissions (release of elastic waves) [*Cadman and Goodman*, 1967] or gradual internal shearing of soil material [*Iverson et al.*, 2000]. *Petley et al.* [2005] attributed the resulting abrupt failure (characterized by very short time from first indicators to mass release) to a gradual process of progressive failure occurring at smaller scales where small failures events (e.g., cracks) gradually form and abruptly coalesce to a continuous failure plane. While *Petley et al.* [2005] discussed the concept of progressive failure in the context of microcracks in deforming cohesive soil, progressive failure may occur also in noncohesive soils by reduced internal strength and liquefaction We associate local failure of mechanical elements as indicators of “progressive failure” that may ultimately propagate across the entire system marking attainment of criticality [*Jensen*, 1998; *Ramos*, 2010]. Note that this definition of criticality or critical state where a local perturbation can lead to release of large masses is different than the term used in critical state mechanics marking transition of a sheared soil to “frictional fluid” [*Schofield and Wroth*, 1968].

[6] Critical systems exhibit no characteristic length scale and often yield a power law of failure size distribution [*Sornette*, 2004]. Criticality associated with landslides is indeed manifested at larger scales in the form of power law scaling exhibited in the number, areas, or volumes of landslides [*Stark and Hovius*, 2001]. For landslides larger than volume or area the frequency of a landslide as a function of landslide volume or area can be expressed as probability density or [*Stark and Guzzetti*, 2009]:

with power law exponents and . The entire range of landslide size (including small landslides that may be less frequent) is represented by a unimodal distribution [*Malamud et al.*, 2004; *Guzzetti et al.*, 2002; *Stark and Hovius*, 2001]. For the power law tail of such unimodal probability density with the standard constraints that (1) the area under the PDF is 1 and (2) maximum landslide size is not limited (infinite), the probability density can be expressed as Pareto distribution [*Stark and Hovius*, 2001] with exponent and a proportionality factor (corresponding expressions for landslide area). For a Pareto distribution the exponent corresponds to the power law exponent of the complementary cumulative probability distribution function, integrating the probability density from a landslide size to infinity. Note that a finite value of such integration is only fulfilled for and , respectively. Smaller exponent values require definition of an upper cutoff (defining largest landslide size) for validity of power law.

[7] For measured or modeled landslide distribution with total number of landslides, we estimated the probability density as [*Malamud et al.*, 2004]

with the number of landslides within a range of volumes that is increasing with increasing landslide volume. The frequent use of aerial images to estimate landslide properties favors representation of landslide area of the scar (volume is more difficult to obtain) [*Guzzetti et al.*, 2002; *Piegari et al.*, 2009; *Van Den Eeckhaut et al.*, 2007; *Turcotte*, 1999] than analysis of landslide volumes [*Dai and Lee*, 2001; *Imaizumi and Sidle*, 2007]. The exponents of power law characterizing relationship between landslide area and frequency range between 1.4 and 3.5 for [*Van Den Eeckhaut et al.*, 2007], and between 1.0 and 1.9 for landslide volumes [*Brunetti et al.*, 2009]. The average values deduced from the studies above are = 2.4 and = 1.4, indicating that exponent values are smaller for volume statistics. The smaller absolute exponent value can be justified by a power law relationship between landslide volume and area with exponent values determined as ranging from 1.1 to 1.3 [*Larsen et al.*, 2010] or as 1.5 [*Guzzetti et al.*, 2009] using inventory data, and 1.3 to 1.4 based on limit equilibrium principles [*Klar et al.*, 2011]. Note that these reported values may involve different types of landslides and triggering mechanisms not limited to shallow rapid translational slides that are in the focus of this study. In addition, in most cases the landslides included in the statistics are not triggered by the same rainfall event. To isolate potential effects of rainfall characteristics on the power law, we present in Figure 1 data from landslide inventories obtained from the Prealps region of Switzerland [*Rickli and Graf*, 2009; *Rickli*, 2001; *Rickli and Bucher*, 2003; *Rickli et al.*, 2008] where the triggered landslides were attributed to a single rainfall event.

### 1.3. Model Concepts of Abrupt Failure and Criticality

[8] Abruptness and criticality of landslides considered in this study arise from mechanical interactions among many soil columns in which mechanical thresholds resembling concepts of self-organized criticality (SOC) were embedded within a hydromechanical framework. We emphasize that complete representation of landslides as SOC systems would require geological time scales for the full spectrum of SOC events to be manifested; hence we limit application of SOC principles to two primary aspects, namely (1) progressive failure preceding a landslide and (2) emergence of power law size-frequency scaling relationships resulting from interactions of many elements with threshold mechanics. A key ingredient in delineating progressive failures is the application of formalism based on the fiber bundle model (FBM), representing all soil strength-imparting components (capillarity, cements, friction, and roots). To clarify the links between the hydromechanical triggering model and the concepts of SOC and FBM, we introduce next the key elements of these two concepts.

#### 1.3.1. Self-Organized Criticality (SOC)

[9] A system driven to a critical state by internal self-organization of many interacting elements is denoted as a SOC system [*Bak*, 1996; *Jensen*, 1998; *Turcotte*, 1999; *Hergarten*, 2002; *Christensen and Moloney*, 2005]. The spontaneous organization of system components may result in a so-called marginally stable state corresponding to a dynamic critical state. *Sornette* [2004] and *Koslowski et al.* [2004] characterize SOC systems by the following properties: (1) power law frequency/magnitude relations result without any apparent external tuning, (2) external driving rates are significantly slower than internal relaxation processes, (3) the presence of a marginally stable state, and (4) a large number of interacting entities that show a highly nonlinear (threshold) response. Among the many SOC models we introduce two types that were previously used to simulate mass movements.

##### 1.3.1.1. Sandpile Cellular Automaton

[10] To represent dynamics of avalanches of grains in growing dry sandpile, *Bak et al.* [1987] have introduced a cellular automaton consisting of a two-dimensional regular grid (denoted as the BTW model). The grid cells in the model may store a prescribed number of “mass units” (sand grains). At each time step, a mass unit is added to the system at a random position until at some cell the number of sand grains reaches its threshold capacity (often set equal to the number of nearest neighbors). Subsequently, that content of cell is redistributed instantaneously by sending one grain to each of the neighbors which in turn may exceed their own capacity and redistribute their mass further. The number of cells that redistribute their grains in response to addition of a single sand grain to the system is denoted as an event or *an avalanche*. It can be shown that avalanche size and frequency follow a power law with exponent of close to −1 [*Hergarten*, 2002]. The notion that addition of a single grain may induce avalanches with a wide range of size distributions have been confirmed in laboratory experiments using sand or rice grains [*Yoshioka*, 2003; *Held et al.*, 1990; *Lörincz and Wijngaarden*, 2007], however, the resulting values of power laws were typically higher (between 1.1 and 2.5) than predicted by the theoretical BTW-sandpile model. This difference in power law values indicates that the frequency of large avalanches is lower in real systems and the redistribution mechanism in the sandpile model is too simplified, mainly due to frictional forces reducing stress release in real sandpiles [*Hergarten*, 2003].

[11] Nevertheless, the BTW model has been proposed as an analog for landslide triggering and dynamics [*Turcotte*, 1999; *Hergarten*, 2002]. The model was adapted to model gravity-driven failure in a slope by *Piegari et al.* [2006] considering slope dependent stress distribution when a force threshold (expressed as factor of safety) was exceeded for a random loading process. A similar model with randomly distributed stress thresholds was previously proposed by *Katz* [1986] for modeling brittle material failure.

##### 1.3.1.2. Spring-Block Models

[12] In the original BTW model no mechanical properties between grid cells were defined and mass redistribution was not restricted. This limitation was rectified by the *spring-block model* proposed by *Olami et al.* [1992] (denoted as the OFC model) that considers blocks of mass interconnected by springs. Block motion is controlled by friction at the block base and mechanical interactions between neighboring blocks. When forces acting on a block overcome friction at its base, the block slips and forces are instantaneously redistributed along mechanical bonds which may trigger additional displacement (the number of displaced blocks determines avalanche size). The resulting power law exponent for the initial OFC model was −1.23 for conditions where the force of a displaced block was entirely transferred to neighboring elements [*Hergarten*, 2003]. The OFC model was recently modified by *Faillettaz et al.* [2010] considering gravity-driven motion of blocks and introducing reduced friction and progressive damage of mechanical bonds between neighboring blocks.

#### 1.3.2. Fiber Bundle Model (FBM)

[13] The third ingredient for critical failure in soil covered hillsopes is provided by the fiber bundle model (FBM) originally introduced by *Peirce* [1926] and *Daniels* [1945]. The model considers a large number of equal-length fibers whose strengths obey a prescribed distribution connected to stiff plates at both ends. Incremental loading of the bundle of fibers results in distribution of the load uniformly among all intact fibers, until the weakest fiber breaks and its load is redistributed to remaining intact fibers. The load redistribution may trigger breakage of additional fibers due to the added load increment (released from the broken fiber). The process may cascade and result in an avalanche-like failure of a certain number of fibers. Early models have used fibers representing cotton yarns, however, the formalism is broader and may be applied to various load bearing elements composed of many discrete units such as ice crystals in weak snow layers [*Reiweger et al.*, 2009], roots in hillslopes [*Schwarz et al.*, 2010a, *Cohen et al.*, 2011], and soil material at various scales [*Cohen et al.*, 2009]. In addition to obvious “fiber like” elements such as cementing agents, roots, and capillary water in soil (grains interconnected by bond of water retained by capillary forces), *Cohen et al.* [2009] discussed frictional and compressive force chains [*Majmudar and Behringer*, 2005] that may also be represented as fiber-like elements. Because fibers can break long before mass is released, FBMs are a tool to quantify so-called precursor events. Precursors are defined here as any failure events at small scale that may indicate imminent mass release.

[14] As for the sandpile cellular automaton and spring-block models, the statistics of avalanches (the number of fibers failing during a load increment) follow a power law with theoretically established exponent with value of −5/2 [*Hemmer and Hansen*, 1992]. However, as a bundle approaches its global failure (i.e., the entire bundle breaks) the power law exponent converges to a value of −3/2 [*Pradhan et al.*, 2006] reflecting transition in which large events become more frequent. In contrast with the BTW and OFC models presented in previous subsections, a marginally stable critical state (fulfilling conditions of SOC systems) is not possible for FBM unless fibers can “heal” after failure [*Moreno et al.*, 1999]. The FBM formalism provides unique modeling capabilities that not only accommodate gradual loss of mechanical strength prior to abrupt failure, but also capture discrete mechanical events that may be measured independently by acoustic or electrical sensors [*Amitrano et al.*, 2005] and thus provide a means for noninvasive monitoring of system mechanical state.

### 1.4. Objectives

[15] The primary objective of this study was to develop a new hydromechanical modeling framework for simulating rainfall-induced rapid and shallow landslides that explicitly considers the following processes and characteristics: (1) describe hydrologic pathways and resulting spatial and temporal water content distributions, (2) quantify evolution of soil mechanical properties affected by hydrology, (3) represent and track local failure events occurring prior to landslide release, (4) capture abruptness (large soil masses released at once) and criticality (local perturbations cascade and lead to mobilization of soil material), and (5) reproduce the resulting power law distribution of landslide sizes at a catchment scale. The proposed modeling framework (denoted as landslide hydromechanical triggering model, LHT) differs from other physically based landslide models in its capability to seamlessly link physically based hydromechanical hillslope behavior with processes leading to abruptness and criticality associated with hydrologic triggering of a landslide. These traits emerge from implementation of SOC and FBM ingredients within the hydromechanical model. In short, we will describe a hillslope as a system of interconnected soil columns taking into account that mechanical soil properties are defined by small scale phenomena like root reinforcement, frictional grain contacts, and bonds of water bound by capillary forces. Table 1 summarizes key traits of the various models maintaining a separation between the operational hillslope scale of the model with many interacting soil columns, and local failure and precursor events at smaller scale of an FBM mechanical bond.

Model Type | Scale/Size | System Composites | Abruptness | Criticality | Marginal Stable State | |||
---|---|---|---|---|---|---|---|---|

Local Failure Condition | Relaxation | Recovery | Avalanche Definition | |ζ| | ||||

- a
While the sandpile cellular automaton [ *Bak et al.*, 1987] and the spring-block model [*Olami et al.*, 1992] are examples for self-organized criticality (SOC) systems, the fiber bundle model (FBM) [*Hemmer and Hansen*, 1992] and the landslide hydromechanical triggering model LHT introduced in this study (combining elements of SOC and FBM) do not evolve towards a sustainable critical state.
| ||||||||

Sandpile cellular automaton | Sandpile 10^{−1} – 10^{0} m | Stack of sand particles | Height of stack ≥ threshold | All particles redistributed to neighbors | Height threshold recovered | No. of stacks of sand with redistribution | ∼1 | Yes |

Spring-block model | Hillslope 10^{1} – 10^{3} m | Rigid soil blocks | Force at base > friction | Force redistributed along bonds | New random strength | No. of blocks with changed position | 1.23 | Yes |

Mechanical bonds (soil strength) | Load on bond > strength | Load transfer to other bonds | No | |||||

Fiber bundle model | Soil sample 10^{−4} – 10^{−1} m | Roots, water bonds, force chain | Load per fiber > fiber strength | Load redistributed to neighbors | No | No. of breaking fibers | 5/2 | No |

Hydromechanical triggering model LHT | Hillslope 10^{1} – 10^{2} m | Rigid soil block | Load at base > friction | Load redistributed along bonds | No | Volume of released mass | 1.0–2.2 | No |

Load > compress. strength | Mass removed | No | ||||||

Mechanical bonds (soil strength) | Load on bond > tensile strength | Load redistributed to other bonds | No | |||||

Roots, water bonds, force chain | Load per fiber > fiber strength | Load redistributed to neighbors | No |

[16] Moreover, by considering results from different realizations of similar hillslopes, we may draw general conclusions regarding larger catchment scale relationships. In section 2 we introduce aspects of progressive failure, followed by details of the hydromechanical hillslope model in section 3 (and the Appendix). In sections 4 and 5 simulation results of the triggering model are presented and discussed, and the main findings of the study are summarized in section 6.