The theory of travel time and residence time distributions is reworked from the point of view of ... more The theory of travel time and residence time distributions is reworked from the point of view of the hydrological storages and fluxes involved. The forward and backward travel time distribution functions are defined in terms of conditional probabilities. Previous approaches that used fixed travel time distributions are not consistent with our new derivation. We explain Niemi's formula and show how it can be interpreted as an expression of the Bayes theorem. Some connections between this theory and population theory are identified by introducing an expression which connects life expectancy with travel times. The theory can be applied to conservative solutes, including a method of estimating the storage selection functions. An example, based on the Nash hydrograph, illustrates some key aspects of the theory. Generalization to an arbitrary number of reservoirs is presented.
Friction coefficient of faults inferred from earthquake focal mechanisms
ABSTRACT In earthquake mechanics and structural geology the static friction coefficient is usuall... more ABSTRACT In earthquake mechanics and structural geology the static friction coefficient is usually assumed to have the laboratory value (μ = 0.6-0.8) according to the Coulomb-Byerlee's law. Estimates from deep boreholes and/or natural faults generally confirm this hypothesis but in some cases friction coefficients can be significantly lower, suggesting the existence of weak faults able to be activated by lower effective stress than theoretically expected. We apply a modified version of the method proposed by Yin and Ranalli (1995, Journal of Structural Geology, vol. 17, pp. 1327-1335), where the average friction coefficient of a set of n faults is estimated. This method is based on minimization of the sum of squares of the misfit ratios, where the misfit ratio of each fault is given dividing the misfit stress difference (i.e. the misfit between normalized stress difference and average normalized stress difference) by the average normalized stress difference. The normalized stress difference is defined as the critical stress difference divided by the effective overburden pressure, while the average stress difference is obtained considering the entire fault dataset. Input data are (i) the orientation of faults, (ii) the stress field orientation, and (iii) the stress ratio. The latter two must be independently estimated. A uniform stress field and a similar normalized critical stress difference for the fault dataset are assumed. The procedure has been extended to apply to fault plane solutions by considering both nodal planes of a set of n focal mechanisms and estimating the range of acceptable average friction coefficients from all possible combination of planes (2n number of combinations). The amount of calculation can be considerably reduced if independent information makes it possible to select which one of the nodal planes of each focal mechanism is the true fault plane (for example when aftershocks delineate the fault geometry at depth), resulting in only n combinations. We present an initial application of this method by applying it to a set of focal mechanisms from the Giudicarie region in northern Italy, and analyzing various subsets. The data were obtained from literature data and earthquake fault plane solutions computed by seismological agencies. Preliminary results show that best-fitting average friction coefficients are sometimes within the expected range, but occasionally well below it (μ about 0.4 or less).
This paper develops a new framework to analyze the relevance of hillslope with respect to channel... more This paper develops a new framework to analyze the relevance of hillslope with respect to channel processes in the generation of the hydrologic response at different scales. The dependence between basin response and geomorphic features of channels and hillslopes is analyzed through the mean and variance of travel times under simple but realistic dynamical hypotheses and within the framework of the rescaled width function. The distribution of travel times is also studied in association with variations of the fraction of saturated areas: with small values of the saturated fraction, the channel network controls most of the mean and variance of travel time distribution. Vice versa, when the basin is saturated, the properties of this distribution may be significantly affected by hillslope processes. These properties depend also on the basin size, with the role of hillslope processes becoming more important in the smaller basins. Furthermore, the average residence time is related to contributing area and drainage density through a simple expression which is shown to be consistent with previous empirical laws.
Four Decades of Progress in Monitoring and Modeling of Processes in the Soil-Plant-Atmosphere System: Applications and Challenges, 2013
We examine the role of vegetation on the stability of shallow soils under unsaturated transient r... more We examine the role of vegetation on the stability of shallow soils under unsaturated transient regime. Two main positive effects of the vegetation on slope stability are discussed: i) a geo-mechanical effect, i.e., the reinforcement of soil by plant roots; ii) a soil-hydrological effect, i.e., the soil suction regime affected by root water uptake. The root distribution is assessed by an eco-hydrological model, which predicts the root density as function of local climatic conditions in growing season and soil hydrological properties. The predicted root distribution is employed for assessing the vertical variability of both the apparent soil cohesion provided by roots and the root water uptake. A one-dimensional model of vertical soil water dynamics is employed for simulating soil suction regime, assumed representative of well-drained soils on steep forested plane slopes. The geo-mechanical and the soil-hydrological effects on slope stability are examined with an infinite slope stability model, generalized for unsaturated conditions. We show that in the case of a loamy-sand soil under a Mediterranean climatic regime, the geo-mechanical effect tends to be more relevant than the soil-hydrological effect during the rainy season, within depths up to twice the average root depth.
The long-term temporal evolution of soil thickness in hollows depends on the processes controllin... more The long-term temporal evolution of soil thickness in hollows depends on the processes controlling the rates of colluvium accumulation and erosion. Accumulation is due to soil creep and mass-wasting processes from the adjacent slopes, while erosion of colluvial deposits is mainly due to debris flow and landsliding. An analysis of the longterm evolution of colluvial deposits is developed through a stochastic model of soil mass balance at a point accounting for colluvium infilling, expressed as a deterministic function of the deposit thickness, and soil erosion by shallow landslides, modeled as a random (Poisson) process. Landsliding is related to the characteristics of the triggering precipitation through an infinite-slope stability analysis, a kinematic model of hollow response to rainfall, and the intensity-duration-frequency curves characterizing the regime of extreme precipitation. This analysis provides a probabilistic representation of the long-term dynamics at a point of colluvium thickness as a function of the timescale of hollow infilling and of the frequency of triggering rainfalls. The model is solved both numerically and (under simplified conditions) analytically, showing the existence of different regimes in the temporal evolution of soil thickness. In the case of steep slopes (i.e., with slope angles, b, greater than the soil repose angle, f) the hollow can be either in a supply-limited state or in event-limited conditions, depending on whether the dynamics are limited by the supply of sediment from the adjacent slopes or by the occurrence of rainstorms able to trigger landslides. Nevertheless, since the likelihood of landslide occurrence increases with increasing values of deposit thickness, colluvium accretion always leads to conditions favorable to landsliding. Vice versa, in the case of gentle slopes (i.e., b < f) the probability of landsliding decreases with increasing values of soil thickness, and event-limited conditions may evolve into unconditionally stable states.
This paper examines the effect of hyetograph shape on the potential for landsliding in soil-mantl... more This paper examines the effect of hyetograph shape on the potential for landsliding in soil-mantled landscapes. An existing pore pressure response model (Iverson, 2000) is used to study the effects of unsteady rainfall infiltration in hillslopes, and the effect of slope and convergent topography is expressed using a steady state model of slope-parallel subsurface flow. Slope stability is assessed using an infinite slope analysis. This theoretical framework is coupled with simple hyetograph models and to intensity-duration-frequency functions to determine the return period of landslide-triggering rainfall. Results also show that hyetographs with a peak at the end of a rainfall event have a stronger destabilizing effect than hyetographs with a constant rainfall or with a peak at the beginning of a storm. Thus the variability of hyetograph shapes adds uncertainty to the assessment of landsliding triggered by rainfall.
Topographic index-based hydrological models have gained wide use to describe the hydrological con... more Topographic index-based hydrological models have gained wide use to describe the hydrological control on the triggering of rainfall-induced shallow landslides at the catchment scale. A common assumption in these models is that a spatially continuous water table occurs simultaneously across the catchment. However, during a rainfall event isolated patches of subsurface saturation form above an impeding layer and their hydrological connectivity is a necessary condition for lateral flow initiation at a point on the hillslope. Here, a new hydrological model is presented, which allows us to account for the concept of hydrological connectivity while keeping the simplicity of the topographic index approach. A dynamic topographic index is used to describe the transient lateral flow that is established at a hillslope element when the rainfall amount exceeds a threshold value allowing for (a) development of a perched water table above an impeding layer, and (b) hydrological connectivity between the hillslope element and its own upslope contributing area. A spatially variable soil depth is the main control of hydrological connectivity in the model. The hydrological model is coupled with the infinite slope stability model and with a scaling model for the rainfall frequency-duration relationship to determine the return period of the critical rainfall needed to cause instability on three catchments located in the Italian Alps, where a survey of soil depth spatial distribution is available. The model is compared with a quasi-dynamic model in which the dynamic nature of the hydrological connectivity is neglected. The results show a better performance of the new model in predicting observed shallow landslides, implying that soil depth spatial variability and connectivity bear a significant control on shallow landsliding.
A model-based method is proposed for improving upon existing threshold relationships which define... more A model-based method is proposed for improving upon existing threshold relationships which define the rainfall conditions for triggering shallow landslides but do not allow the magnitude of landsliding (i.e. the number of landslides) to be determined. The SHETRAN catchment-scale shallow landslide model is used to quantify the magnitude of landsliding as a function of rainfall return period, for focus sites of 180 and 45 km 2 in the Italian Southern Alps and the central Spanish Pyrenees. Rainfall events with intensities of different return period are generated for a range of durations (1-day to 5-day) and applied to the model to give the number of landslides triggered and the resulting sediment yield for each event. For a given event duration, simulated numbers of landslides become progressively less sensitive to return period as return period increases. Similarly, for an event of given return period, landslide magnitude becomes less sensitive to event duration as duration increases. The temporal distribution of rainfall within an event is shown to have a significant impact on the number of landslides and the timing of their occurrence. The contribution of shallow landsliding to catchment sediment yield is similarly quantified as a function of the rainfall characteristics. Rainfall intensity-duration curves are presented which define different levels of landsliding magnitude and which advance our predictive capability beyond, but are generally consistent with, published threshold curves. The magnitude curves are relevant to the development of guidelines for landslide hazard assessment and forecasting.
Many current conceptual rainfall-runoff and shallow landslide stability models are based on the t... more Many current conceptual rainfall-runoff and shallow landslide stability models are based on the topographic index concept derived from the steady-state assumption for subsurface water flow dynamics and the hypothesis that the surface gradient is a good approximation for the gradient of the total hydraulic head. However, increasing field evidence from sites around the world has shown poor correlations between the topographic index and the patterns of soil water storage. Here we present a new, smoothed, dynamic topographic index and test the ability of this index to reproduce spatial patterns of wetness areas and storage as provided by a distributed, physically based, Boussinesq equation (BEq) solver. Our results show that the new smoothed dynamic topographic index outperforms previous, locally computed indices in the estimation of storage dynamics, resulting in less fragmented and disconnected spatial patterns of storage. Our new dynamic index is able to capture both the upslope and downslope controls on water flow and approximates storage dynamics across scales. The new index is compatible with highresolution topographic data. We encourage the use of our smoothed dynamic topographic index to describe the lateral subsurface flow component in landslide generation models and conceptual rainfall-runoff models, especially when high-resolution digital elevation models are available.
This paper explores the effect of hillslope hydrological behavior on slope stability in the conte... more This paper explores the effect of hillslope hydrological behavior on slope stability in the context of transient subsurface saturation development and landslide triggering. We perform a series of virtual experiments to address how subsurface topography affects the location and spatial pattern of slip surface development and pore pressure dynamics. We use a 3D Darcy-Richards equation solver (Hydrus 3-D) combined with a cellular automata slope stability model to simulate the spatial propagation of the destabilized area. Our results showed that the soil-bedrock interface and in particular, bedrock depressions, played a key role in pore pressure dynamics, acting as an impedance for the downslope drainage of perched water. Filling and spilling of depressions in the bedrock surface microtopography induced localized zones of increased pressure head such that the development of porepressure fields-not predictable by surface topography-lead to rapid landslide propagation. Our work suggests that landslide models should consider the subsurface topography in order to include a connectivity component in the mathematical description of hydrological processes operating at the hillslope scale.
Getting the Rescaled Width Function and the Derived WGIUH
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It is shown how rescaled width functions (Rinaldo et al., 1995; D'Odorico et al., 1996) are ... more It is shown how rescaled width functions (Rinaldo et al., 1995; D'Odorico et al., 1996) are built by means of Grass and some other custom programs. From them the width function based instantaneous unit hydrograph, WGIUH, is obtained. In the end, it is discussed the ...
Traditional methods of estimating the potential of landsliding and mass movements in mountains ar... more Traditional methods of estimating the potential of landsliding and mass movements in mountains are based on surveys which estimate the quaternary depositions and the volumes of sediments available in areas whose geomorphic characters warn for pos- sible landslide. These methods use also geo-lithological and land-use information to finally produce maps of the location of landslides and debris flow initiaton. To
A Probabilistic Approach to Transient Hydrology and Landslide Triggering
We present results from a 3D, transient coupled simulation of hillslope hydrology and slope stabi... more We present results from a 3D, transient coupled simulation of hillslope hydrology and slope stability for a steep alpine catchment that experienced shallow landslides during an intense period of rain. These results reproduce observed spatial and temporal distributions of rainfall-induced landslides. The distributed coupled hydrological-geotechnical model, GEOtop-SF, describes processes related to slope hydrology and slope stability. It combines a 3D
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Papers by Riccardo Rigon