Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions

Power-law distributions contain precious information about a large variety of processes in geoscience and elsewhere. Although there are sound theoretical grounds for these distributions, the empirical evidence in favor of power laws has been traditionally weak. Recently, Clauset et al. have proposed a systematic method to find over which range (if any) a certain distribution behaves as a power law. However, their method has been found to fail, in the sense that true (simulated) power-law tails are not recognized as such in some instances, and then the power-law hypothesis is rejected. Moreover, the method does not work well when extended to power-law distributions with an upper truncation. We explain in detail a similar but alternative procedure, valid for truncated as well as for non-truncated power-law distributions, based in maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and Monte Carlo simulations. An overview of the main concepts as well as a recipe for their practical implementation is provided. The performance of our method is put to test on several empirical data which were previously analyzed with less systematic approaches. The databases presented here include the half-lives of the radionuclides, the seismic moment of earthquakes in the whole world and in Southern California, a proxy for the energy dissipated by tropical cyclones elsewhere, the area burned by forest fires in Italy, and the waiting times calculated over different spatial subdivisions of Southern California. We find the functioning of the method very satisfactory.Comment: 26 pages, 9 figure

Testing Universality in Critical Exponents: the Case of Rainfall

One of the key clues to consider rainfall as a self-organized critical phenomenon is the existence of power-law distributions for rain-event sizes. We have studied the problem of universality in the exponents of these distributions by means of a suitable statistic whose distribution is inferred by several variations of a permutational test. In contrast to more common approaches, our procedure does not suffer from the difficulties of multiple testing and does not require the precise knowledge of the uncertainties associated to the power-law exponents. When applied to seven sites monitored by the Atmospheric Radiation Measurement Program the test lead to the rejection of the universality hypothesis, despite the fact that the exponents are rather close to each other

Data-Driven Prediction of Thresholded Time Series of Rainfall and SOC models

We study the occurrence of events, subject to threshold, in a representative SOC sandpile model and in high-resolution rainfall data. The predictability in both systems is analyzed by means of a decision variable sensitive to event clustering, and the quality of the predictions is evaluated by the receiver operating characteristics (ROC) method. In the case of the SOC sandpile model, the scaling of quiet-time distributions with increasing threshold leads to increased predictability of extreme events. A scaling theory allows us to understand all the details of the prediction procedure and to extrapolate the shape of the ROC curves for the most extreme events. For rainfall data, the quiet-time distributions do not scale for high thresholds, which means that the corresponding ROC curves cannot be straightforwardly related to those for lower thresholds.Comment: 19 pages, 10 figure

The perils of thresholding

The thresholding of time series of activity or intensity is frequently used to define and differentiate events. This is either implicit, for example due to resolution limits, or explicit, in order to filter certain small scale physics from the supposed true asymptotic events. Thresholding the birth-death process, however, introduces a scaling region into the event size distribution, which is characterised by an exponent that is unrelated to the actual asymptote and is rather an artefact of thresholding. As a result, numerical fits of simulation data produce a range of exponents, with the true asymptote visible only in the tail of the distribution. This tail is increasingly difficult to sample as the threshold is increased. In the present case, the exponents and the spurious nature of the scaling region can be determined analytically, thus demonstrating the way in which thresholding conceals the true asymptote. The analysis also suggests a procedure for detecting the influence of the threshold by means of a data collapse involving the threshold-imposed scale.Comment: 16 pages, 10 figure

Lack of Fire Has Limited Physiological Impact on Old-Growth Ponderosa Pine in Dry Montane Forests of North-Central Idaho

Reduced frequency of fire in historically fire-adapted ecosystems may have adverse effects on ecosystem structure, function, and resilience. Lack of fire increases stand density and promotes successional replacement of seral dominant trees by late-successional, more shade-tolerant species. These changes are thought to increase competition for limited resources among trees and to increase physiological stress of dominant, fire-adapted species. However, there has been little effort to directly investigate effects of lack of fire on the physiological status of old trees, especially in unlogged, protected forests. At four remote sites in the Selway-Bitterroot region of Idaho, we tested whether the physiological status of dominant old-growth ponderosa pine trees in repeatedly burned stands (three to four 20th-century wildfires at roughly historical fire frequency) differs from trees in paired stands not burned for at least 70 years. We hypothesized that trees in relatively unburned stands would exhibit signs of physiological stress due to increased competition for resources in higher-density stands. Needle chemistry and morphological variables, fine root production, mycorrhizal infection rates, depth of soil water resources, and recent basal area growth rates were measured as indictors of competition-induced stress. Contrary to predictions, needle carbon isotopic ratio (δ13C) and fine root production, variables related to water stress, were slightly higher in repeatedly burned stands driven by site-specific responses, and there were no significant biological differences between trees in repeatedly burned stands vs. stands unburned for at least 70 years in the remaining variables. Our results raise the possibility that dominant ponderosa pine trees in uneven-aged forests may be more resilient to increased stand density associated with the lack of fire than previously thought. If so, our results have implications for the management of uneven-aged, old-growth forests

Interactive Effects of Historical Logging and Fire Exclusion on Ponderosa Pine Forest Structure in the Northern Rockies

Increased forest density resulting from decades of fire exclusion is often perceived as the leading cause of historically aberrant, severe, contemporary wildfires and insect outbreaks documented in some fire-prone forests of the western United States. Based on this notion, current U. S. forest policy directs managers to reduce stand density and restore historical conditions in fire-excluded forests to help minimize high-severity disturbances. Historical logging, however, has also caused widespread change in forest vegetation conditions, but its long-term effects on vegetation structure and composition have never been adequately quantified. We document that fire-excluded ponderosa pine forests of the northern Rocky Mountains logged prior to 1960 have much higher average stand density, greater homogeneity of stand structure, more standing dead trees and increased abundance of fire-intolerant trees than paired fire-excluded, unlogged counterparts. Notably, the magnitude of the interactive effect of fire exclusion and historical logging substantially exceeds the effects of fire exclusion alone. These differences suggest that historically logged sites are more prone to severe wildfires and insect outbreaks than unlogged, fire-excluded forests and should be considered a high priority for fuels reduction treatments. Furthermore, we propose that ponderosa pine forests with these distinct management histories likely require distinct restoration approaches. We also highlight potential long-term risks of mechanical stand manipulation in unlogged forests and emphasize the need for a long-term view of fuels management

Power laws and scaling of rain events and dry spells in the Catalonia region

We analyze the statistics of rain-event sizes, rain-event durations, and dry-spell durations in a network of 20 rain gauges scattered in an area situated close to the NW Mediterranean coast. Power-law distributions emerge clearly for the dryspell durations, with an exponent around 1.50 ± 0.05, although for event sizes and durations the power-law ranges are rather limited, in some cases. Deviations from power-law behavior are attributed to finite-size effects. A scaling analysis helps to elucidate the situation, providing support for the existence of scale invariance in these distributions. It is remarkable that rain data of not very high resolution yield findings in agreement with self-organized critical phenomena

Assessing heat exposure to extreme temperatures in urban areas using the Local Climate Zone classification

Trends of extreme-temperature episodes in cities are increasing (in frequency, magnitude and duration) due to regional climate change in interaction with urban effects. Urban morphologies and thermal properties of the materials used to build them are factors that influence spatial and temporal climate variability and are one of the main reasons for the climatic singularity of cities. This paper presents a methodology to evaluate the urban and peri-urban effect on extreme-temperature exposure in Barcelona (Spain), using the Local Climate Zone (LCZ) classification as a basis, which allows a comparison with other cities of the world characterised using this criterion. LCZs were introduced as input of the high-resolution UrbClim model (100 m spatial resolution) to create daily temperature (median and maximum) series for summer (JJA) during the period 1987 to 2016, pixel by pixel, in order to create a cartography of extremes. Using the relationship between mortality due to high temperatures and temperature distribution, the heat exposure of each LCZ was obtained. Methodological results of the paper show the improvement obtained when LCZs were mapped through a combination of two techniques (land cover-land use maps and the World Urban Database and Access Portal Tools - WUDAPT - method), and the paper proposes a methodology to obtain the exposure to high temperatures of different LCZs in urban and peri-urban areas. In the case of Barcelona, the distribution of temperatures for the 90th percentile (about 3-4 ∘C above the average conditions) leads to an increase in the relative risk of mortality of 80 %