Modelling fundamental waiting time distributions for earthquake sequences

The distribution of waiting times between time-neighbouring events for a time series obeying the Omori law is examined theoretically and numerically with the aim of understanding the characteristics of these distributions, how these characteristics change (e.g. scale) with the parameters of the Omori series, and thus how empirical waiting time data may be correctly interpreted. It is found that the waiting time distribution, for a single Omori aftershock sequence, consists in general of two power law segments followed by a rapid decay at larger waiting times. The analyses are illustrated using real data from the SIL network on Iceland. This data often shows characteristics predominantly consistent with the Omori law, but there are significant exceptions. We conclude that waiting time distributions and related statistical analysis has meaningful potential for the analysis of earthquake data sets, as a step towards developing physical models of the earthquake process.     

Corrigendum

Corrigendum

Corrigendum     

Integrative Monitoring Analysis Aiming at the Detection of Spatial and Temporal Trends of Metal Accumulation in Mosses

Goal and Scope. The UNECE heavy metals in Mosses Surveys provide data on the accumulation of metals in naturally growing mosses throughout Europe. Using Germany as an example, this article concentrates on the elaboration and application of methods of data analysis that are necessary for a comprehensive interpretation of spatial and temporal trends in metal accumulation.Methods and Results. The sampling of mosses, and the chemical analysis of Cr, Cu, Fe, Ni, Pb, Ti, V and Zn in 1990, 1995 and 2000 are summarized briefly. The variogram analyses detect distinct autocorrelation structures in the sample data which may consequently be estimated for all sites in Germany without monitoring metal accumulation by means of ordinary kriging. A procedure for the geostatistical detection of spatial outliers was developed and applied and, after the elimination of the spatial outliers, several measurements were seen to indicate an adequate quality of the geostatistical estimations. The cluster analyses of the z-transformed estimation data result in a nominal multi-element index that indicates regional metal accumulation types over time. Percentile statistics serve for computation of an ordinal, scaled, multi-element accumulation index which is spatially differentiated over time in terms of multivariate, statistically defined ecoregions. The integrative statistical analysis reveals, from 1990 to 2000, that the metal accumulation declines up to 80% in some of the ecoregions. Hot spots of metal accumulation are mapped and interpreted by means of metadata analysis.Conclusions. Dot maps depict the spatial structure of the metal accumulation without spatial bias. This information, detailed with respect to metal species and space, should be generalized for better supporting the interpretation. The combination of geostatistical analysis and estimation, percentile and multivariate statistics is suitable for the calculation of indices that serve for a comprehensive mapping of metal accumulation in the ecoregions over time, for quantifying the bias of the surface estimation, and for mapping spatial outliers and hot spots of metal accumulation.     

Practical visualization of rock structure

The conventionally applied method for visualizing the structure of a rock mass is limited to plotting the orientation of individual discontinuities by using the Schmidt stereographic net. Their spacing is not illustrated and other structural features, such as undulation and twisting, cannot be interpreted from such sterographic views. A new mode of visual presentation of rock structural features of one or several different types in rock volumes of any size that makes all this possible is proposed in the present report. The primary purpose is to quantify, for all elements of a rock mass where the structure has been evaluated by borehole investigations or mapping in the course of underground excavation, the orientation and average spacing of defined discontinuities like hydraulically and mechanically active fractures. These local structural patterns, which are plotted on three of the six faces of cubical boxes with defined edge length and with one edge oriented N−S can be combined by applying the mathematical tool for connecting faces with which certain visualization codes like Alias Sketch are equipped. This combines different local rock structures to form regional patterns that can reveal undulation and twisting and important variations in spacing of major discrete discontinuities. Visualization can be given in true perspective or orthographic form and larger discontinuities that have been identified in the field, such as fracture zones, can be introduced in the model for correlation with smaller features. Furthermore, the stress situation at all sites where rock stress measurements have been made can be visualized in the model in the form of principal planes. The proposed models can be visualized from any “camera angle” and also shown in Quick Time VR form, i.e. as being slowly rotated around any predetermined axis. This facilitates definition and selection of the rock structure, local or regional, for numerical calculations.     

Zero velocity hypersurfaces for the general three-dimensional three-body problem

The equation of zero velocity surfaces for the general three-body problem can be derived from Sundman's inequality. The geometry of those surfaces was studied by Bozis in the planar case and by Marchal and Saari in the three-dimensional case. More recently, Saari, using a geometrical approach, has found an inequality stronger than Sundman's. Using Bozis' algebraic method, and a rotating frame which does not take into account entirely the rotation of the three-body system, we also find an inequality stronger than Sundman's. The comparison with Saari's inequality is more difficult. However, our result can be expressed in four-dimensional space and the regions where motion is allowed can be seen (numerically) to lie inside those obtained by the use of Sundman's inequality.Agrégé de Faculté.     

Seismic hazard evaluation for dams in northern Colorado, U.S.A.

A seismic hazard evaluation for three dams in the Rocky Mountains of northern Colorado is based on a study of the historical seismicity. To model earthquake occurrence as a random process utilizing a maximum likelihood method, the catalog must exhibit random space-time characteristics. This was achieved using a declustering procedure and correction for completeness of recording. On the basis of the resulting a- and b-values, probabilistic epicentral distances for a 2 × 10^–5 annual probability were calculated. For a random earthquake of magnitude M _L 6.0–6.5, this distance is 15 km. Suggested ground motion parameters were estimated using a probabilistic seismic hazard analysis. Critical peak horizontal accelerations at the dams are 0.22g if median values are assumed and 0.39g if variable attenuation and seismicity rates are taken into account. For structural analysis of the dams, synthetic acceleration time series were calculated to match the empirical response spectra. In addition, existing horizontal strong motion records from two Mammoth Lakes, California earthquakes were selected and scaled to fit the target horizontal acceleration response spectra.