Bruce D. Malamud

Power-law correlations of landslide areas in central Italy

Abstract We have studied the frequency–area statistics of landslides in central Italy. We consider two data sets. Data set A contains 16 809 landslide areas in the Umbria–Marche area of central Italy; they represent a reconnaissance inventory of very old, old, and recent (modern) landslides. The noncumulative frequency–area distribution of these landslides correlates well with a power-law relation, exponent −2.5, over the range 0.03 km 2 A L 2 . Data set B contains 4233 landslides that were triggered by a sudden change in temperature on 1 January 1997, resulting in extensive melting of snow cover. An inventory of these snow-melt-triggered landslides was obtained from aerial photographs taken 3 months after the event. These landslides also correlate well with a power-law relation with exponent −2.5, over the range 0.001 km 2 A L 2 . We show that the correlation of data set B is essentially identical to the correlation of 11 000 landslides triggered by the 17 January 1994 Northridge, California earthquake. We attribute a rollover for small landslides in data set A to incompleteness of the record due to erosion and other processes, and to limitations in the reconnaissance mapping technique used to complete the inventory. On the other hand, we conclude that rollovers for small landslides in data set B and the California earthquake data are real and are associated with the surface morphology. We conclude that the power-law distribution is valid over a wide range of landslide areas and discuss possible reasons. We also discuss the contribution of the snow-melt- and earthquake-triggered landslide events to the total landslide inventory.

Scale-invariance of soil moisture variability and its implications for the frequency-size distribution of landslides

Abstract Power spectral analyses of soil moisture variability are carried out from scales of 100 m to 10 km on the microwave remotely-sensed data from the Washita experimental watershed during 1992. The power spectrum S(k) has an approximate power-law dependence on wave number k with the exponent −1.8. This behavior is consistent with the behavior of a stochastic differential equation for soil moisture at a point, and it has important consequences for the frequency-size distribution of landslides. We present the cumulative frequency-size distributions of landslides induced by precipitation in Japan and Bolivia as well as landslides triggered by the 1994 Northridge, California earthquake. Large landslides in these regions, despite being triggered by different mechanisms, have a cumulative frequency-size distribution with a power-law dependence on area with an exponent ranging from −1.5 to −2. We use a soil moisture field with the above statistics in conjunction with a slope stability analysis to model the frequency-size distribution of landslides. In our model, landslides occur when a threshold shear stress dependent on cohesion, pore pressure, internal friction and slope angle is exceeded. This implies a threshold dependence on soil moisture and slope angle since cohesion, pore pressure and internal friction are primarily dependent on soil moisture. The cumulative frequency-size distribution of domains of shear stress greater than a threshold value with soil moisture modeled as above and topography modeled as a Brownian walk is a power-law function of area with an exponent of −1.8 for large landslide areas. This distribution is similar to that observed for landslides. The effect of strong ground motion from earthquakes lowers the shear stress necessary for failure, but does not change the frequency-size distribution of failed areas. This is consistent with observations. This work suggests that remote sensing of soil moisture can be of great importance in monitoring landslide hazards and proposes a specific quantitative model for landslide hazard assessment.

Self-affine time series: measures of weak and strong persistence

Abstract In this paper, we examine self-affine time series and their persistence. Time series are defined to be self-affine if their power-spectral density scales as a power of their frequency. Persistence can be classified in terms of range, short or long range, and in terms of strength, weak or strong. Self-affine time series are scale-invariant, thus they always exhibit long-range persistence. Synthetic self-affine time series are generated using the Fourier power-spectral method. We generate fractional Gaussian noises (fGns), −1⩽β⩽1, where β is the power-spectral exponent. These are summed to give fractional Brownian motions (fBms), 1⩽β⩽3, where the series are self-affine fractals with fractal dimension 1⩽D⩽2; β=2 is a Brownian motion. With β>1, the time series are non-stationary and moments of the time series depend upon its length; with β 1 to have strong persistence and with β

Self-affine time series: I. Generation and analyses

Publisher Summary This chapter introduces the basic concepts of self-affine time series. In a self-affine time series, the power-spectral density is defined as a power-law function of the frequency. Time series are quantified by their statistical distribution of values and by their persistence or antipersistence. Persistence can be classified in terms of range, short-range or long-range, and in terms of strength, weak, or strong. The distribution of values is usually either Gaussian (normal) or log-normal. The basic characteristic of a self-affine time series is that the persistence is scale invariant. Thus, a self-affine time series has a long-range persistence and these are found in a wide variety of geophysical applications. The chapter examines a variety of techniques to quantify the strength of long-range persistence in self-affine time series. These include Fourier power-spectral analysis, semivariogram analysis, rescaled-range analysis, average extreme-event analysis, and wavelet variance analysis.

Self-Organized Criticality Applied to Natural Hazards

The concept of self-organizedcriticality evolved from studies of three simplecellular-automata models: the sand-pile, slider-block,and forest-fire models. In each case, there is asteady “input” and the “loss” is associated with afractal (power-law) distribution of “avalanches.” Each of the three models can be associated with animportant natural hazard: the sand-pile model withlandslides, the slider-block model with earthquakes,and the forest-fire model with forest fires. We showthat each of the three natural hazards havefrequency-size statistics that are well approximatedby power-law distributions. The model behaviorsuggests that the recurrence interval for a severeevent can be estimated by extrapolating the observedfrequency-size distribution of small and mediumevents. For example, the recurrence interval for amagnitude seven earthquake can be obtained directlyfrom the observed frequency of occurrence of magnitudefour earthquakes. This concept leads to thedefinition of a seismic intensity factor. Both globaland regional maps of this seismic intensity factor aregiven. In addition, the behavior of the modelssuggests that the risk of occurrence of large eventscan be substantially reduced if small events areencouraged. For example, if small forest fires areallowed to burn, the risk of a large forest fire issubstantially reduced.

How many plumes are there

The objective of this paper is to quantitatively assess the role of mantle plumes in transporting heat to the base of the lithosphere. We first review the mechanisms responsible for mantle heat flow. We take the total global surface heat flow to be 4.43×1013 W. Of this, we attribute 0.68×1013 W (15%) to radiogenic heat production in the continental crust and 3.75×1013 W (85%) to heat loss from the mantle. Of the heat loss from the mantle, 2.17×1013 W (58%) is attributed to the subduction of the oceanic lithosphere and the remainder, 1.58×1013 W (42%), heats the base of the oceanic and continental lithosphere. Prior buoyancy studies of plumes give a plume heat flux of 0.24×1013 W, which is only 15% of the total heat flux associated with basal heating of the lithosphere. Thus, the amount that remains unaccounted for is a basal heat flux of 1.34×1013 W. The missing heat flux can be attributed either to plumes that do not have a significant surface expression, or to secondary mantle convection beneath the plates. We show that the cumulative frequency-size distribution of the large and intermediate size plume fluxes can be reasonably well approximated by a power-law distribution. We then extrapolate this distribution to smaller plumes in order to estimate a total plume heat flux. This requires about 5200 plumes, with the smallest plume fluxes about 109 W. This compares with 12×109 W for the smallest plume fluxes previously reported in the literature. We suggest that the large number of seamounts represents surface evidence for small plumes, and conclude that it is reasonable to attribute the entire basal heat flux to plumes.

Reviewing and visualizing the interactions of natural hazards

This paper presents a broad overview, characterization, and visualization of the interaction relationships between 21 natural hazards, drawn from six hazard groups (geophysical, hydrological, shallow Earth, atmospheric, biophysical, and space hazards). A synthesis is presented of the identified interaction relationships between these hazards, using an accessible visual format particularly suited to end users. Interactions considered are primarily those where a primary hazard triggers or increases the probability of secondary hazards occurring. In this paper we do the following: (i) identify, through a wide-ranging review of grey- and peer-review literature, 90 interactions; (ii) subdivide the interactions into three levels, based on how well we can characterize secondary hazards, given information about the primary hazard; (iii) determine the spatial overlap and temporal likelihood of the triggering relationships occurring; and (iv) examine the relationship between primary and secondary hazard intensities for each identified hazard interaction and group these into five possible categories. In this study we have synthesized, using accessible visualization techniques, large amounts of information drawn from many scientific disciplines. We outline the importance of constraining hazard interactions and reinforce the importance of a holistic (or multihazard) approach to natural hazard assessment. This approach allows those undertaking research into single hazards to place their work within the context of other hazards. It also communicates important aspects of hazard interactions, facilitating an effective analysis by those working on reducing and managing disaster risk within both the policy and practitioner communities.

Modelling Mediterranean landscape succession-disturbance dynamics: A landscape fire-succession model

We present a spatially explicit Landscape Fire-Succession Model (LFSM) developed to represent Mediterranean Basin landscapes and capable of integrating modules and functions that explicitly represent human activity. Plant-functional types are used to represent spatial and temporal competition for resources (water and light) in a rule-based modelling framework. Vegetation dynamics are represented using a rule-based community-level modelling approach that considers multiple succession pathways and vegetation climax states. Wildfire behaviour is represented using a cellular-automata model of fire spread that accounts for land-cover flammability, slope, wind and vegetation moisture. Results show that wildfire spread parameters have the greatest influence on two aspects of the model: land-cover change and the wildfire regime. This sensitivity highlights the importance of accurately parameterising this type of grid-based model for representing landscape-level processes. We use a pattern-oriented modelling approach in conjunction with wildfire power-law frequency-area scaling exponent @b to calibrate the model. Parameters describing the role of soil moisture on vegetation dynamics are also found to significantly influence land-cover change. Recent improvements in understanding the role of soil moisture and wildfire fuel loads at the landscape-level will drive advances in Mediterranean LFSMs.

Cellular-automata models applied to natural hazards

The concept of self-organized criticality evolved from studies of three simple cellular automata models: the forest-fire, slider-block, and sandpile models. Each model is associated with natural hazards which have frequency-size statistics that are well approximated by power law distributions. These distributions have important implications for probabilistic hazard assessments.

An inverse-cascade model for self-organized critical behavior

We introduce an inverse-cascade model to explain self-organized critical behavior. This model is motivated by the forest-fire model. In the forest-fire model trees are randomly planted on a grid, sparks are also dropped on the grid resulting in fires in which trees are lost. In the inverse-cascade model single trees are introduced and these combine to form larger and larger clusters. This is the inverse cascade and gives a power-law (fractal) frequency-size distribution of clusters. Model fires eliminate trees from all cluster sizes but significant numbers of trees are lost only from the largest clusters and this loss terminates the power-law scaling. Finally, our model illustrates important differences between critical and self-organized critical behavior.