Scale in structure and dynamics

Ian Main, Bruce Malamud, Chris Bean and John McCloskey summarize the presentations and lively debate at the British Geophysical Association's annual British Discussion Meeting on Scale-Invariance and Scale-Dependence in Earth Structure and Dynamics.     

Associations between elevated atmospheric temperature and human mortality: a critical review of the literature

The effects of the anomalously warm European summer of 2003 highlighted the importance of understanding the relationship between elevated atmospheric temperature and human mortality. This review is an extension of the brief evidence examining this relationship provided in the IPCC’s Assessment Reports. A comprehensive and critical review of the literature is presented, which highlights avenues for further research, and the respective merits and limitations of the methods used to analyse the relationships. In contrast to previous reviews that concentrate on the epidemiological evidence, this review acknowledges the inter-disciplinary nature of the topic and examines the evidence presented in epidemiological, environmental health, and climatological journals. As such, present temperature–mortality relationships are reviewed, followed by a discussion of how these are likely to change under climate change scenarios. The importance of uncertainty, and methods to include it in future work, are also considered.     

Quantification of Long-Range Persistence in Geophysical Time Series: Conventional and Benchmark-Based Improvement Techniques

Time series in the Earth Sciences are often characterized as self-affine long-range persistent, where the power spectral density, S, exhibits a power-law dependence on frequency, f, S(f) ~ f ^?β , with β the persistence strength. For modelling purposes, it is important to determine the strength of self-affine long-range persistence β as precisely as possible and to quantify the uncertainty of this estimate. After an extensive review and discussion of asymptotic and the more specific case of self-affine long-range persistence, we compare four common analysis techniques for quantifying self-affine long-range persistence: (a) rescaled range (R/S) analysis, (b) semivariogram analysis, (c) detrended fluctuation analysis, and (d) power spectral analysis. To evaluate these methods, we construct ensembles of synthetic self-affine noises and motions with different (1) time series lengths N = 64, 128, 256, …, 131,072, (2) modelled persistence strengths β _model = ?1.0, ?0.8, ?0.6, …, 4.0, and (3) one-point probability distributions (Gaussian, log-normal: coefficient of variation c _v = 0.0 to 2.0, Levy: tail parameter a = 1.0 to 2.0) and evaluate the four techniques by statistically comparing their performance. Over 17,000 sets of parameters are produced, each characterizing a given process; for each process type, 100 realizations are created. The four techniques give the following results in terms of systematic error (bias = average performance test results for β over 100 realizations minus modelled β) and random error (standard deviation of measured β over 100 realizations): (1) Hurst rescaled range (R/S) analysis is not recommended to use due to large systematic errors. (2) Semivariogram analysis shows no systematic errors but large random errors for self-affine noises with 1.2 ≤ β ≤ 2.8. (3) Detrended fluctuation analysis is well suited for time series with thin-tailed probability distributions and for persistence strengths of β ≥ 0.0. (4) Spectral techniques perform the best of all four techniques: for self-affine noises with positive persistence (β ≥ 0.0) and symmetric one-point distributions, they have no systematic errors and, compared to the other three techniques, small random errors; for anti-persistent self-affine noises (β < 0.0) and asymmetric one-point probability distributions, spectral techniques have small systematic and random errors. For quantifying the strength of long-range persistence of a time series, benchmark-based improvements to the estimator predicated on the performance for self-affine noises with the same time series length and one-point probability distribution are proposed. This scheme adjusts for the systematic errors of the considered technique and results in realistic 95 % confidence intervals for the estimated strength of persistence. We finish this paper by quantifying long-range persistence (and corresponding uncertainties) of three geophysical time series—palaeotemperature, river discharge, and Auroral electrojet index—with the three representing three different types of probability distribution—Gaussian, log-normal, and Levy, respectively.     

Temporal correlations and clustering of landslides

This paper examines temporal correlations and temporal clustering of a proxy historical landslide time series, 2255 reported landslides 1951–2002, for an area in the Emilia‐Romagna Region, Italy. Landslide intensity is measured by the number of reported landslides in a day (D_L) and in an ‘event’ (S_event) of consecutive days with landsliding. The non‐zero values in both time series D_L and S_event are unequally spaced in time, and have heavy‐tailed frequency‐size distributions. To examine temporal correlations, we use power‐spectral analysis (Lomb periodogram) and surrogate data analysis, confronting our original D_L and S_event time series with 1000 shuffled (uncorrelated) versions. We conclude that the landslide intensity series D_L has strong temporal correlations and S_event has likely temporal correlations. To examine temporal clustering in D_L and S_event, we consider extremes over different landslide intensity thresholds. We first examine the statistical distribution of interextreme occurrence times, τ, and find Weibull distributions with parameter γ << 1·0 [D_L] and γ < 1·0 [S_event]; thus D_L and S_event each have temporal correlations, but S_event to a lesser degree. We next examine correlations between successive interextreme occurrence times, τ. Using autocorrelation analysis applied to τ, combined with surrogate data analysis, we find for D_L linear correlations in τ, but for S_event inconclusive results. However, using Kendall's rank correlation analysis we find for both D_L and S_event the series of τ are strongly correlated. Finally, we apply Fano Factor analysis, finding for both D_L and S_event the timings of extremes over a given threshold exhibit a fractal structure and are clustered in time. In this paper, we provide a framework for examining time series where the non‐zero values are strongly unequally spaced and heavy‐tailed, particularly important in the Earth Sciences due to their common occurrence, and find that landslide intensity time series exhibit temporal correlations and clustering. Many landslide models currently are designed under the assumption that landslides are uncorrelated in time, which we show is false. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Landslide inventories and their statistical properties

Landslides are generally associated with a trigger, such as an earthquake, a rapid snowmelt or a large storm. The landslide event can include a single landslide or many thousands. The frequency–area (or volume) distribution of a landslide event quanti?es the number of landslides that occur at different sizes. We examine three well‐documented landslide events, from Italy, Guatemala and the USA, each with a different triggering mechanism, and ?nd that the landslide areas for all three are well approximated by the same three‐parameter inverse‐gamma distribution. For small landslide areas this distribution has an exponential ‘roll‐over’ and for medium and large landslide areas decays as a power‐law with exponent ‐2·40. One implication of this landslide distribution is that the mean area of landslides in the distribution is independent of the size of the event. We also introduce a landslide‐event magnitude scale m_L = log(N_LT), with N_LT the total number of landslides associated with a trigger. If a landslide‐event inventory is incomplete (i.e. smaller landslides are not included), the partial inventory can be compared with our landslide probability distribution, and the corresponding landslide‐event magnitude inferred. This technique can be applied to inventories of historical landslides, inferring the total number of landslides that occurred over geologic time, and how many of these have been erased by erosion, vegetation, and human activity. We have also considered three rockfall‐dominated inventories, and ?nd that the frequency–size distributions differ substantially from those associated with other landslide types. We suggest that our proposed frequency–size distribution for landslides (excluding rockfalls) will be useful in quantifying the severity of landslide events and the contribution of landslides to erosion. Copyright © 2004 John Wiley & Sons, Ltd.     

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

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.     

Analysis of historical landslide time series in the Emilia‐Romagna region,northern Italy

A catalogue of historical landslides, 1951–2002, for three provinces in the Emilia‐Romagna region of northern Italy is presented and its statistical properties studied. The catalogue consists of 2255 reported landslides and is based on historical archives and chronicles. We use two measures for the intensity of landsliding over time: (i) the number of reported landslides in a day (D_L) and (ii) the number of reported landslides in an event (S_event), where an event is one or more consecutive days with landsliding. From 1951–2002 in our study area there were 1057 days with 1 ≤ D_L ≤?45 landslides per day, and 596 events with 1 ≤ S_event ≤ 129 landslides per event. In the first set of analyses, we find that the probability density of landslide intensities in the time series are power‐law distributed over at least two‐orders of magnitude, with exponent of about ?2·0. Although our data is a proxy for landsliding built from newspaper reports, it is the first tentative evidence that the frequency‐size of triggered landslide events over time (not just the landslides in a given triggered event), like earthquakes, scale as a power‐law or other heavy‐tailed distributions. If confirmed, this could have important implications for risk assessment and erosion modelling in a given area. In our second set of analyses, we find that for short antecedent rainfall periods, the minimum amount of rainfall necessary to trigger landslides varies considerably with the intensity of the landsliding (D_L and S_event); whereas for long antecedent periods the magnitude is largely independent of the cumulative amount of rainfall, and the largest values of landslide intensity are always preceded by abundant rainfall. Further, the analysis of the rainfall trend suggests that the trigger of landslides in the study area is related to seasonal rainfall. Copyright © 2010 John Wiley & Sons, Ltd.