Frequency Area Distribution Research Articles

An inverse cascade explanation for the power-law frequency-area statistics of earthquakes, landslides and wildfires

Abstract Frequency-magnitude statistics for natural hazards can greatly help in probabilistic hazard assessments. An example is the case of earthquakes, where the generality of a power-law (fractal) frequency-rupture area correlation is a major feature in seismic risk mapping. Other examples of this power-law frequency-size behaviour are landslides and wildfires. In previous studies, authors have made the potential association of the hazard statistics with a simple cellular-automata model that also has robust power-law statistics: earthquakes with slider-block models, landslides with sandpile models, and wildfires with forest-fire models. A potential explanation for the robust power-law behaviour of both the models and natural hazards can be made in terms of an inverse-cascade of metastable regions. A metastable region is the region over which an ‘avalanche’ spreads once triggered. Clusters grow primarily by coalescence. Growth dominates over losses except for the very largest clusters. The cascade of cluster growth is self-similar and the frequency of cluster areas exhibits power-law scaling. We show how the power-law exponent of the frequency-area distribution of clusters is related to the fractal dimension of cluster shapes.

An inverse cascade model for self-organized complexity and natural hazards

The concept of self-organized complexity evolved from the scaling behaviour of several cellular automata models, examples include the sandpile, slider-block and forest-fire models. Each of these systems has a large number of degrees of freedom and shows a power-law frequency-area distribution of avalanches with N∝A−α and α ≈ 1. Actual landslides, earthquakes and forest fires exhibit a similar behaviour. This behaviour can be attributed to an inverse cascade of metastable regions. The metastable regions grow by coalescence which is self-similar and gives power-law scaling. Avalanches sample the distribution of smaller clusters and, at the same time, remove the largest clusters. In this paper we build on earlier work (Gabrielov et al.) and show that the coalescence of clusters in the inverse cascade is identical to the formation of fractal drainage networks. This is shown analytically and demonstrated using simulations of the forest-fire model.

Forest fire burn areas in Western Canada modeled as self-similar criticality

Forest fire burn areas in the western Canadian provinces of Alberta and British Columbia have cumulative frequency-area distributions that are well described by a power law or an upper-truncated power law. The power law scaling extends over as many as five orders of magnitude and is observed for different geographical regions and for time intervals ranging from 1 to 40 years. The observed scaling exponent varies both geographically within and between provinces and temporally between annual records. The temporal variability decreases at the decadal scale, suggesting that decadal distributions may be useful for long term fire control planning within a geographical region. For example, for all of Alberta, based on the scaling parameters that describe the 1961–2000 record, we expect approximately 24 fires per year of 100 ha or larger. Unlike the original self-organized criticality (SOC) forest fire model that produces a single scaling exponent, the self-similar criticality (SSC) model replicates the range of scaling exponents observed for cumulative frequency-area distributions of natural forest fires.

Log-periodic behavior in a forest-fire model

Abstract. This paper explores log-periodicity in a forest-fire cellular-automata model. At each time step of this model a tree is dropped on a randomly chosen site; if the site is unoccupied, the tree is planted. Then, for a given sparking frequency, matches are dropped on a randomly chosen site; if the site is occupied by a tree, the tree ignites and an "instantaneous" model fire consumes that tree and all adjacent trees. The resultant frequency-area distribution for the small and medium model fires is a power-law. However, if we consider very small sparking frequencies, the large model fires that span the square grid are dominant, and we find that the peaks in the frequency-area distribution of these large fires satisfy log-periodic scaling to a good approximation. This behavior can be examined using a simple mean-field model, where in time, the density of trees on the grid exponentially approaches unity. This exponential behavior coupled with a periodic or near-periodic sparking frequency also generates a sequence of peaks in the frequency-area distribution of large fires that satisfy log-periodic scaling. We conclude that the forest-fire model might provide a relatively simple explanation for the log-periodic behavior often seen in nature.

Power-law correlations of landslide areas in central Italy

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<4 km 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<0.1 km 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.

Applications of statistical mechanics to natural hazards and landforms

The concept of self-organized criticality was introduced to explain the behavior of the cellular-automata sandpile model. A variety of multiple slider-block and forest-fire models have been introduced which are also said to exhibit self-organized critical behavior. It has been argued that earthquakes, landslides, forest-fires, and extinctions are examples of self-organized criticality in nature. The basic forest-fire model is particularly interesting in terms of its relation to the critical-point behavior of the site-percolation model. In the basic forest-fire model trees are randomly planted on a grid of points, periodically sparks are randomly dropped on the grid and if a spark drops on a tree that tree and the adjacent trees burn in a model fire. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi-steady-state cascade gives a power-law frequency–area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees. The earth's topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.

Forest fires: An example of self-organized critical behavior

Despite the many complexities concerning their initiation and propagation, forest fires exhibit power-law frequency-area statistics over many orders of magnitude. A simple forest fire model, which is an example of self-organized criticality, exhibits similar behavior. One practical implication of this result is that the frequency-area distribution of small and medium fires can be used to quantify the risk of large fires, as is routinely done for earthquakes.

Kardar-Parisi-Zhang Scaling of the Height of the Convective Boundary Layer and Fractal Structure of Cumulus Cloud Fields

We present the cumulative frequency-area distribution of tropical cumulus clouds as observed from satellite and space shuttle images from scales of 0.1 to 1000 km. The distribution is a power-law function of area with exponent $\ensuremath{-}0.8$. We show that this result and the fractal dimension of cloud perimeters implies that the top of the convective boundary layer (CBL) is a self-affine interface with roughness exponent or Hausdorff measure $H\ensuremath{\approx}0.4$, the same value as that of the Kardar-Parisi-Zhang equation in $2+1$ dimensions. In addition, we identify dynamic scaling in a time series of the local altitude of the top of the CBL as measured with FM/CW radar backscatter intensity. A possible growth model is discussed.