A new method for determination of most likely landslide initiation points and the evaluation of digital terrain model scale in terrain stability mapping

This paper introduces a new approach for determining the most likely initiation points for landslides from potential instability mapped using a terrain stability model. This approach identifies the location with critical stability index from a terrain stability model on each downslope path from ridge to valley. Any measure of terrain stability may be used with this approach, which here is illustrated using results from SINMAP, and from simply taking slope as an index of potential instability. The relative density of most likely landslide initiation points within and outside mapped landslide scars provides a way to evaluate the effectiveness of a terrain stability measure, even when mapped landslide scars include run out zones, rather than just initiation locations. This relative density was used to evaluate the utility of high resolution terrain data derived from airborne laser altimetry (LIDAR) for a small basin located in the Northeastern Region of Italy. Digital Terrain Models were derived from the LIDAR data for a range of grid cell sizes (from 2 to 50 m). We found appreciable differences between the density of most likely landslide initiation points within and outside mapped landslides with ratios as large as three or more with the highest ratios for a digital terrain model grid cell size of 10 m. This leads to two conclusions: (1) The relative density from a most likely landslide initiation point approach is useful for quantifying the effectiveness of a terrain stability map when mapped landslides do not or can not differentiate between initiation, runout, and depositional areas; and (2) in this study area, where landslides occurred in complexes that were sometimes more than 100 m wide, a digital terrain model scale of 10 m is optimal. Digital terrain model scales larger than 10 m result in loss of resolution that degrades the results, while for digital terrain model scales smaller than 10 m the physical processes responsible for triggering landslides are obscured by smaller scale terrain variability

An Analytical and Numerical Study of Optimal Channel Networks

We analyze the Optimal Channel Network model for river networks using both analytical and numerical approaches. This is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations. The fractal character of river networks is reflected in the power law behaviour of various quantities characterising the morphology of the basin. In the context of a finite size scaling Ansatz, the exponents describing the power law behaviour are calculated exactly and show mean field behaviour, except for two limiting values of a parameter characterizing the dissipated energy, for which the system belongs to different universality classes. Two modified versions of the model, incorporating quenched disorder are considered: the first simulates heterogeneities in the local properties of the soil, the second considers the effects of a non-uniform rainfall. In the region of mean field behaviour, the model is shown to be robust to both kinds of perturbations. In the two limiting cases the random rainfall is still irrelevant, whereas the heterogeneity in the soil properties leads to new universality classes. Results of a numerical analysis of the model are reported that confirm and complement the theoretical analysis of the global minimum. The statistics of the local minima are found to more strongly resemble observational data on real rivers.Comment: 27 pages, ps-file, 11 Postscript figure

Unified View of Scaling Laws for River Networks

Scaling laws that describe the structure of river networks are shown to follow from three simple assumptions. These assumptions are: (1) river networks are structurally self-similar, (2) single channels are self-affine, and (3) overland flow into channels occurs over a characteristic distance (drainage density is uniform). We obtain a complete set of scaling relations connecting the exponents of these scaling laws and find that only two of these exponents are independent. We further demonstrate that the two predominant descriptions of network structure (Tokunaga's law and Horton's laws) are equivalent in the case of landscapes with uniform drainage density. The results are tested with data from both real landscapes and a special class of random networks.Comment: 14 pages, 9 figures, 4 tables (converted to Revtex4, PRE ref added

Geometry of River Networks II: Distributions of Component Size and Number

The structure of a river network may be seen as a discrete set of nested sub-networks built out of individual stream segments. These network components are assigned an integral stream order via a hierarchical and discrete ordering method. Exponential relationships, known as Horton's laws, between stream order and ensemble-averaged quantities pertaining to network components are observed. We extend these observations to incorporate fluctuations and all higher moments by developing functional relationships between distributions. The relationships determined are drawn from a combination of theoretical analysis, analysis of real river networks including the Mississippi, Amazon and Nile, and numerical simulations on a model of directed, random networks. Underlying distributions of stream segment lengths are identified as exponential. Combinations of these distributions form single-humped distributions with exponential tails, the sums of which are in turn shown to give power law distributions of stream lengths. Distributions of basin area and stream segment frequency are also addressed. The calculations identify a single length-scale as a measure of size fluctuations in network components. This article is the second in a series of three addressing the geometry of river networks.Comment: 16 pages, 13 figures, 4 tables, Revtex4, submitted to PR

Basins of attraction on random topography

We investigate the consequences of fluid flowing on a continuous surface upon the geometric and statistical distribution of the flow. We find that the ability of a surface to collect water by its mere geometrical shape is proportional to the curvature of the contour line divided by the local slope. Consequently, rivers tend to lie in locations of high curvature and flat slopes. Gaussian surfaces are introduced as a model of random topography. For Gaussian surfaces the relation between convergence and slope is obtained analytically. The convergence of flow lines correlates positively with drainage area, so that lower slopes are associated with larger basins. As a consequence, we explain the observed relation between the local slope of a landscape and the area of the drainage basin geometrically. To some extent, the slope-area relation comes about not because of fluvial erosion of the landscape, but because of the way rivers choose their path. Our results are supported by numerically generated surfaces as well as by real landscapes

Cellular Models for River Networks

A cellular model introduced for the evolution of the fluvial landscape is revisited using extensive numerical and scaling analyses. The basic network shapes and their recurrence especially in the aggregation structure are then addressed. The roles of boundary and initial conditions are carefully analyzed as well as the key effect of quenched disorder embedded in random pinning of the landscape surface. It is found that the above features strongly affect the scaling behavior of key morphological quantities. In particular, we conclude that randomly pinned regions (whose structural disorder bears much physical meaning mimicking uneven landscape-forming rainfall events, geological diversity or heterogeneity in surficial properties like vegetation, soil cover or type) play a key role for the robust emergence of aggregation patterns bearing much resemblance to real river networks.Comment: 7 pages, revtex style, 14 figure

An Integrated Modeling System for Estimating Glacier and Snow Melt Driven Streamflow from Remote Sensing and Earth System Data Products in the Himalayas

Quantification of the contribution of the hydrologic components (snow, ice and rain) to river discharge in the Hindu Kush Himalayan (HKH) region is important for decision-making in water sensitive sectors, and for water resources management and flood risk reduction. In this area, access to and monitoring of the glaciers and their melt outflow is challenging due to difficult access, thus modeling based on remote sensing offers the potential for providing information to improve water resources management and decision making. This paper describes an integrated modeling system developed using downscaled NASA satellite based and earth system data products coupled with in-situ hydrologic data to assess the contribution of snow and glaciers to the flows of the rivers in the HKH region. Snow and glacier melt was estimated using the Utah Energy Balance (UEB) model, further enhanced to accommodate glacier ice melt over clean and debris-covered tongues, then meltwater was input into the USGS Geospatial Stream Flow Model (Geo- SFM). The two model components were integrated into Better Assessment Science Integrating point and Nonpoint Sources modeling framework (BASINS) as a user-friendly open source system and was made available to countries in high Asia. Here we present a case study from the Langtang Khola watershed in the monsoon-influenced Nepal Himalaya, used to validate our energy balance approach and to test the applicability of our modeling system. The snow and glacier melt model predicts that for the eight years used for model evaluation (October 2003-September 2010), the total surface water input over the basin was 9.43 m, originating as 62% from glacier melt, 30% from snowmelt and 8% from rainfall. Measured streamflow for those years were 5.02 m, reflecting a runoff coefficient of 0.53. GeoSFM simulated streamflow was 5.31 m indicating reasonable correspondence between measured and model confirming the capability of the integrated system to provide a quantification of water availability

Topological self-similarity on the random binary-tree model

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented

Geometry of River Networks I: Scaling, Fluctuations, and Deviations

This article is the first in a series of three papers investigating the detailed geometry of river networks. Large-scale river networks mark an important class of two-dimensional branching networks, being not only of intrinsic interest but also a pervasive natural phenomenon. In the description of river network structure, scaling laws are uniformly observed. Reported values of scaling exponents vary suggesting that no unique set of scaling exponents exists. To improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure, we report here a theoretical and empirical study of fluctuations about and deviations from scaling. We examine data for continent-scale river networks such as the Mississippi and the Amazon and draw inspiration from a simple model of directed, random networks. We center our investigations on the scaling of the length of sub-basin's dominant stream with its area, a characterization of basin shape known as Hack's law. We generalize this relationship to a joint probability density and show that fluctuations about scaling are substantial. We find strong deviations from scaling at small scales which can be explained by the existence of linear network structure. At intermediate scales, we find slow drifts in exponent values indicating that scaling is only approximately obeyed and that universality remains indeterminate. At large scales, we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape. The extent of approximate scaling is significantly restricted by these deviations and will not be improved by increases in network resolution.Comment: 16 pages, 13 figures, Revtex4, submitted to PR

Evidence for surface uplift of the Atlas Mountains and the surrounding peripheral plateaux: Combining apatite fission-track results and geomorphic indicators in the Western Moroccan Meseta (coastal Variscan Paleozoic basement)

This work represents an initial attempt to link the evolution of the topography in relation to the general tectonic framework of western Morocco. For this purpose, in a section of the Western Moroccan Meseta different tools are combined in order to attain the general objective. Apatite fission-track (AFT) data of granitic rocks of the Rabat–Khenifra area give ages around 200 Ma with track length distributions which are compatible with the thermal models already established for the area. An inverse correlation between AFT ages and elevation is observed which is compatible with previous models indicating northward tilting of the whole Western Moroccan Meseta which is younger than 20–25 Ma. In order to test this possibility a detailed analysis of the topography at different scales in the Western Moroccan Meseta has been performed. Results indicate that two open folds with different amplitudes are recognized and that the one with wider wavelength could correspond to a lithospheric fold as previously stated by other authors on the basis of independent geological arguments. The northward tilting proposed based on the AFT data agrees with the results obtained in the analysis of the topography which reinforces the presence of a very open fold with a wavelength of 200–300 km in the north-western limb of the Western Moroccan Meseta