Nationality, Citizenship Law, and Questions of Scale

2016 Barringer Medal for Keith Holsapple

In most earthquake hazard models, the b -value of the Gutenberg-Richter law plays a central role in forecasting future seismicity based on the observed history. The cumulative earthquake-size distribution is commonly described by a power law: log10( N ) = a - bM , where N is the cumulative number of earthquakes of magnitude M or greater, a is the earthquake productivity of a volume, and b is the relative size distribution (Gutenberg and Richter 1944; Ishimoto and Iida 1939). The slope of this power law, the b -value, is a critical parameter in seismology that describes the size distribution of events. A high b -value indicates a relatively larger proportion of small events, and vice versa. In earthquake forecasting projects such as the source-related probabilistic seismic hazard assessment (PSHA), an underlying fundamental question is: What do the (numerous) smaller earthquakes tell us about the (infrequent) larger ones? Embedded into this question is also the question of stationarity: Can one trust recent small earthquakes to convey accurate information about infrequent large ones? It is these questions of scaling and stationarity that our model fundamentally addresses. In PSHA projects, the b -value is either chosen as a regional constant or varies spatially between local zones. However, there is currently no obvious scientific basis for choosing either of these approaches. The model we propose, the Asperity-based Likelihood Model (ALM), assumes that the earthquake-size distribution, and specifically the b -value of recent micro-earthquakes, is the most important information for forecasting future events of M 5+. Below we first briefly review the evidence that leads us to propose the ALM model, and then we describe the actual steps involved in deriving the model parameters. Most of the evidence in support of ALM stems from observational data from a variety of tectonic regimes as well …

Thomas Lux comments on the paper by Blake LeBaron, on page 621 of this issue, by recalling related findings of spurious scaling properties and questions whether we can distinguish between true and spurious scaling laws in finite data series.

Ecosystems and city systems often form hierarchically structured landscapes whose spatial pattern is scale dependent. While trends in the upper tail of national city-size distributions leave the impression that fractal-scaling laws such as Zipf’s law or the rank-size rule truly represent the essence of the system, the linearity depicted at aggregate scale actually obscures variation and discontinuity in the urban size-density function, including multimodalities evident in regional data sets. Tracing individual city trajectories through these hierarchical patterns reveals structural resilience at macroscopic scale, the punctuated growth of individual cities of differing sizes, the persistence and self-reinforcing character of spatial agglomeration, and a general need for further empirical investigation of the relationship between city size and growth. It also raises questions for future exploration, including the meaning of persistent departures from the power laws of traditional urban systems theory. Interpretation of such departures in the context of questions of jurisdictional scale in environmental management and “smart growth” policy adds a practical dimension to the research agenda.

The problem of scaling is at the heart of ecological theory, the essence of understanding and of the development of a predictive capability. The description of any system depends on the spatial, temporal, and organizational perspective chosen; hence it is essential to understand not only how patterns and dynamics vary with scale, but also how patterns at one scale are manifestations of processes operating at other scales. Evolution has shaped the characteristics of species in ways that result in scale displacement: Each species experiences the environment at its own unique set of spatial and temporal scales and interfaces the biota through unique assemblages of phenotypes. In this way, coexistence becomes possible, and biodiversity is enhanced. By averaging over space, time, and biological interactions, a genotype filters variation at fine scales and selects the arena in which it will face the vicissitudes of nature. Variation at finer scales is then noise, of minor importance to the survival and dynamics of the species, and consequently of minor importance in any attempt at description. In attempting to model ecological interactions in space, contributors throughout this book have struggled with a trade-off between simplification and realistic complexity and detail. Although the challenge of simplification is widely recognized in ecology, less appreciated is the intertwining of scaling questions and scaling laws with the process of simplification. In the context of this chapter simplification will in general mean the use of spatial or ensemble means and low-order moments to capture more detailed interactions by integrating over given areas. In this way, one can derive descriptions of the system at different spatial scales, which provides the essentials for the extraction of scaling laws by examination of how system properties vary with scale.

Knowledge of police and court procedures and respect for the law a survey