Ian G Main

The largest earthquake (M8½) in eastern China occurred near Tangcheng in 1668. Six M≥6 earthquakes occurred in this region in historical time. We re-examine the primary historical archives from this region, adding independent constraints such as maps of active faults, local topography, intensity attenuation laws for China, and relevant administrative practices in archiving data, to reassess source parameters (epicenter, magnitude, or maximum intensity). We use the seismic density index - a measure of the clustering of seismicity weighted by source magnitude – to identify candidate zones of present-day clustering of seismicity than may represent a long-term memory of past historical events in an intraplate setting. We update at least one of the source parameters for four of the seven events. The main shock and two aftershocks of the 1668-1672 M8½ Tancheng earthquake sequence locate along the strike of the Tan-Lu fault zone, each associated with significant and contiguous seismic dens...

criticality in global earthquake populations’’ ’ by Chien-chih Chen and Chun-Ling Chang, Geophys. Res. Lett., 31, L06609, doi:10.1029/2004GL019497. [1] We are grateful to Chen and Chang [2004] for their comment on our first paper [Main and Al-Kindy, 2002]. We share their concern regarding the definition of self-organised criticality as a concept, and welcome their brief review of earlier work on the suggestion that far-from-equilibrium systems may behave in similar ways to equilibrium ones [see also Egolf, 2000]. However, we disagree with their main conclusion, for reasons stated below. [2] Chen and Chang [2004] correctly point out an algebraic error in our appendix derivation of the relation between entropy and energy by integrating the probability

An accurate understanding of the interplay between random and deterministic processes in generating extreme events is of critical importance in many fields, from forecasting extreme meteorological events to the catastrophic failure of materials and in the Earth. Here we investigate the statistics of record-breaking events in the time series of crackling noise generated by local rupture events during the compressive failure of porous materials. The events are generated by computer simulations of the uniaxial compression of cylindrical samples in a discrete element model of sedimentary rocks that closely resemble those of real experiments. The number of records grows initially as a decelerating power law of the number of events, followed by an acceleration immediately prior to failure. The distribution of the size and lifetime of records are power laws with relatively low exponents. We demonstrate the existence of a characteristic record rank k^{*}, which separates the two regimes of ...

We have seen in the previous two chapters that the classical approach to earthquake physics provides an intuitively reasonable model of earthquake occurrence. This theory can be successfully applied to explain the propagation of seismic waves radiated by earthquake sources. However, the classical theory is not able to account globally for the following basic features of earthquake occurrence (cf. section 1.1), 1 Earthquakes are rare events. 2 Earthquakes are clustered in both space and time. 3 Earthquakes are rupture events which occur mostly on preexisting faults. 4 Earthquakes have a quasi-constant stress drop which is, on average, much smaller than ambient stress (Abercrombie and Leary, 1993). 5 The external forcing function, i.e. tectonic strain, is small and constant, inducing extremely low strain rates. 6 Fault traces are power law distributed in length. 7 Faults are rough surfaces, with power law distributed roughness. 8 The spatial distribution of hypocentral locations of earthquakes and laboratory acoustic emissions are power law distributed in both space and time (Kagan and Knopoff, 1980; Hirata et al., 1987). 9 Earthquakes are power law distributed in size (Gutenberg-Richter law). 10 Earthquakes have aftershock sequences that decay with a power law in time (Omori law). 11 Seismicity can be induced by stress perturbations smaller than the stress drop of individual events. These may be due to previous earthquakes occurring at relatively great distances (see section 2.6), or to changes in local pore fluid pressure through man-made activity.

Long-range, stress-related and fault-related characteristics of correlations in fluctuations in flow-rates are explained conceptually in the context of the lithosphere's near-critical mechanical state and a strong feedback between deformation and local permeability. A more sophisticated statistical model, devised to extract a parsimonious set of flow-rate correlations, has shown similar characteristics. Coupled geomechanical-flow modeling was able to reproduce those characteristics for a generic pattern waterflood perturbed with random noise, but only when loaded to a near-critical state, hence providing strong support for the conceptual model. Coupled modeling of a cross-section representative of the Gullfaks field also demonstrated long-range influences. The matrix of empirical correlations between all well-pairs for a field can be decomposed in various ways. The principal components of the matrix, when interpolated with appropriate spatial correlation functions, have indicated the importance of particular faults in the rate fluctuation history; it is inferred that those faults are mechanically active during the development, and thus are potentially conductive features. Introduction It is usually assumed that geomechanical modelling coupled with reservoir simulation is only required in a minor subset of reservoirs that are termed 'stress-sensitive'. This subset is sometimes recognized a posteriori; i.e. during the course of field development, often through a general severe decline in permeability levels with depleting pressure; or perhaps a priori in the case of weak reservoir rocks where compaction drive is considered important to recovery (e.g. unconsolidated sands or weak chalks). Rarely outside such bounds is geomechanical modelling deemed necessary for reliable, unbiased reservoir performance predictions. However, some past analyses of preferred directions of flooding 1,2, and of the correlations in rate fluctuations 3 have suggested that geomechanics may be playing a commercially significant role in many secondary and tertiary floods, if not other development schemes. Key to the interpretation of this field data are the following concepts, which may be novel to many reservoir engineers or geoscientists:Much of the lithosphere is in a near-critical statee.g.4,5. This means that there are percolating paths of faults, fractures and incipient fractures that are near mechanical failure in the prevailing stress states. This concept is supported by direct measurements of stress state 6,7, observations of (micro-)seismicity induced by oilfield development 8, shear-wave splitting observed in most types of rock in the crust 9, and the observations of so-called '1/f' scaling of properties in well-logs 10. These observations are themselves underpinned by theoretical explanations of the evolution of the earth's lithosphere into a near-critical state. Such models include 'self-organized criticality' (SOC)4,11–15; or 'self-organized sub-criticality'16–19; or in the sense of a spinodal critical point 20. Whilst there is still ongoing discussion as to which of these models best fits lithospheric deformation, common characteristics of these theories, also observed in the real behaviour of rock, are:Strong susceptibility to small perturbation (metastability)Responses often at a distance from perturbing load (long-range correlation)Percolating paths of incipient failure (localization), oriented in association with the modern-day stress state.

Reservoir depletion/injection results in changes in effective stress, which may cause formation failure, propagation of existing fractures/faults and initialisation of new fractures/faults. A flow-deformation coupled reservoir geomechanical modelling approach has been applied to compute the change in reservoir effective stress and associated propagation of fractures and reactivation of faults. The computed inelastic strain change was used to predict the magnitudes of microseismic emissions associated with fault reactivations. The moment magnitude of microseismic events, consistent with the Richter scale, can be computed from the scalar value of their moment tensor. In a discontinuous model, the moment tensor can be calculated directly; in finite element models, the moment tensor is calculated by integrating the change in inelastic shear strain over the volume of the elements containing the failed fault. Coupled 3D geomechanical (deformation and fluid flow) simulations for Valhall fi...

Fluid flow in the Earth's crust can be accurately described by a diffusion equation, with highly anisotropic, heterogeneous and time dependent diffusivities. The so-called Bhatnagar-Gross-Krook (BGK) models provide a simple and efficient way to solve the scalar diffusion equation with complex boundary conditions. We show that it can be adapted to include an anisotropic, space and time dependent diffusivity, and a gravity body force, hence providing a suitable numerical model for realistic simulations of incompressible pore fluid diffusion in heterogeneous, anisotropic rocks in the presence of a gravity field. A stability condition limits in some cases the magnitude of anisotropy that can be simulated. Numerical results are in very good agreement with a known analytical solution.

A major key to rejuvenating mature oil and gas fields for improved recovery is the ability to characterize the reservoir between existing wells: currently practised methods for surveying inter-well properties are generally costly. A recent project has combined three technologies which utilize readily available fluid production histories to produce a process for identifying inter-well reservoir communications in producing oil or gas fields. Fluctuations in the well production and injection rate histories are analyzed in the context of coupled geomechanical-flow processes involving activated faults and fractures through (i) the Statistical Reservoir Model (SRM), (ii) conventional correlation analysis and a new technique for extracting rate diffusivity tensors, and (iii) coupled geomechechanical-flow modelling. These technologies have been applied to five oilfields located in the North Sea from the North Viking graben to the Central graben with the key results that (i) the general long...

It has been suggested that the finite width of the seismogenic lithosphere can have a strong effect on the frequency-moment relation for large earthquakes. Theories have been proposed in which large earthquakes have either a shallower or a steeper power-law slope in the incremental frequency-moment distribution, at char-acteristic seismic moments corresponding to earthquakes that rupture the entire seis-mogenic depth. However, many authors have applied the predicted double-slope distribution, which requires five independent parameters, to cumulative frequency data, and used the location of the break of slope to make inferences on characteristic size effects in the earthquake source in different seismotectonic regions. Here we examine the problem in a forward modeling mode by adding a realistic degree of statistical scatter to ideal incremental frequency-moment distributions of various commonly used forms. Adopting a priori the assumption of a piecewise linear dis-tribution, we find ...

W e conducted triaxial deformation experiments on large (0.1-m; 0.33-ft)-diameter cores of four sandstones from the Moab area to investigate the effect of total axial strain and effective confining pressure on the evolution of bulk permeability of faulted samples. Sandstones with low bulk porosities (Dewey Bridge and Slickrock Subkha) exhibited an increase in permeability with increasing inelastic axial strain at low effective confining pressures, whereas those with high porosity (Navajo and Slickrock Aeolian) showed a decline in permeability. However, all samples showed permeability decline with increasing inelastic axial strain at high effective confining pressures. Meanwhile, microstructural observations revealed no systematic dependence of the width of the shear zone and the number of deformation bands on either strain or effective confining pressure, although grain-size reduction was more intense at high effective confining pressures. A new geometric model has 13 Kwon, O., B. T...

A modified Griffith criterion for a fractal ensemble of cracks is applied to the interpretation of Acoustic Emission (AE) statistics during the compressional deformation of intact and artificially pre-cut rock specimens in the laboratory. A mean energy release rate per unit crack surface area ( G) is recovered from the observed A E event rate N and the seismic b value, by calculating an inferred mean crack length (c) and measuring the differential stress a for a range of experimental conditions. Temporal variations in ( G) under compressive deformation show very similar trends to those predicted by a synoptic model determined by direct extrapolation from observations of subcritical crack growth under tension. (In the tensile case, deformation is centred on a dominant macrocrack and the stress intensity K, which scales as the square root of G, is the relevant measured variable.) The three independent variables measured during the tests (a, N, b) are reduced to points that map out a p...

To a great extent, it is the nearly ubiquitous presence of fractures that makes the mechanical behavior of rock masses different from that of most engineering ma-terials. These fractures also cause the behavior of rock masses to differ from that of small laboratory-size rock samples. Most laboratory tests on rock samples are

Abstract. An accurate predictive model for the long-term strength of the continental lithosphere is important in a range of geophysical and geodynamic problems. While laboratory experiments are consistent with Mohr-Coulomb brittle faulting in the cold, upper continental crust, there is increasing evidence that time-dependent processes may also be important in these rocks, even at low temperature. However, there is some ambiguity as to the exact form of the constitutive law for describing time-dependent behavior of upper crustal rocks. Here we present results of room temperature creep experiments on a suite of water-saturated sandstones spanning a range of petrophysical and rheological properties and underlying deformation mechanisms. On physical and microstructural grounds our analysis suggests that a modified power law creep, of the form ̇ A(d f) , where d is the differential stress and f is the long-term failure (fundamental) strength, provides a more complete description of the e...

Abstract: This paper describes the new concept of a ‘Statistical Reservoir Model ’ to determine significant well-pair correlations. We solve this conceptual problem using a predictive error filter, combined with Bayesian methods that identify those well pairs that are related to each other with statistical significance, for the Gullfaks reservoir in the North Sea. Significant, long-range, corre-lations in the whole field are found at an optimal time lag of one month. The correlation function for significantly-correlated well pairs, after normalization for the distribution of available wells, shows a long-range power-law decay that is consistent with a critical-point response at the reser-voir scale. A principal component analysis shows a strong correlation with the location and orien-tation of faults that intersect the main producing horizon. A predictive experiment shows that the model performs very well both in history matching and predictive mode on a time scale of about one mont...

This paper presents the experimental characterisation of hydraulic and mechanical properties of two fine-grained materials from the Aigion Fault Zone, on the southern shore of the Gulf of Corinth, Greece. One is a calcilutite, which commonly caps the basal limestone units, and the other is siliceous active fault gouge from 760 m depth in the borehole. Oedometric compaction/consolidation tests of unconsolidated cylindrical samples at variable stresses were conducted under drained conditions, and then permeability measured by a constant flow technique. The axial strain measurements show consolidation curves with a decelerating strain rate that can be approximated by a power-law function. The permeability is negatively and linearly correlated to the stress, and ranges from 9 to 15 × 10−18 m2 for the calcilutite and 0.9 to 2 × 10−18 m2 for the fault gouge with varied vertical stresses. To cite this article: I. Song et al., C. R. Geoscience 336 (2004). 2004 Académie des sciences. Publish...

Abstract: This paper describes the new concept of a ‘Statistical Reservoir Model ’ to determine significant well-pair correlations. We solve this conceptual problem using a predictive error filter, combined with Bayesian methods that identify those well pairs that are related to each other with statistical significance, for the Gullfaks reservoir in the North Sea. Significant, long-range, corre-lations in the whole field are found at an optimal time lag of one month. The correlation function for significantly-correlated well pairs, after normalization for the distribution of available wells, shows a long-range power-law decay that is consistent with a critical-point response at the reser-voir scale. A principal component analysis shows a strong correlation with the location and orien-tation of faults that intersect the main producing horizon. A predictive experiment shows that the model performs very well both in history matching and predictive mode on a time scale of about one mont...

In a previous paper, Main and Burton (1984a) applied Information Theory to the problem of seismic hazard estimation using an earthquake magnitude catalog and the long-term seismic moment release rate (or slip rate) of geologically observed faults. In this letter, we address the problem of estimating long-term earthquake recurrence rates and discuss some of the implications of the distribution in more detail, with the important addition of equations for the propagation of uncertainties in each of the parameters. A slightly more general derivation of the distribution is also given, because uncertainties in one of the parameters of the magnitude moment distribution can now be numerically accounted for. Several authors have noted that the superposition of different earthquake populations can greatly affect estimates of long-term recurrence rates of large earthquakes from short-term catalogs, e.g., B~th (1981), Singh et al. (1983), and Main and Burton (1984b). This is especially true in ...

One of the primary goals of volcanology is to develop methods to forecast future eruptive activity. Good forecasts have the potential to increase the effectiveness of hazard mitigation plans, allowing well-timed evacuations or improved management of a return to exclusion zones. Volcanic earthquakes and seismicity are key datasets that form the basis for eruption forecasts. Seismometers are amongst most widely installed volcano monitoring equipment. Increased seismicity is a primary indicator of volcanic unrest, and eruptions are frequently preceded by changes in the rate, location, magnitude, and type of earthquakes. In combination with geodetic data, they can provide indications as to the volumes and locations of magma. Earthquakes provide an insight into otherwise hidden processes occurring at volcanoes and that may have a causal link to eruptive activity, such as edifice deformation, magma migration, and magma failure. Consequently, changes in the seismic activity at a volcano ca...

Reservoir depletion/injection results in changes in effective stress, which may cause formation failure, propagation of existing fractures/faults and initialisation of new fractures/faults. A flow-deformation coupled reservoir geomechanical modelling approach has been applied to compute the change in reservoir effective stress and associated propagation of fractures and reactivation of faults. The computed inelastic strain change was used to predict the magnitudes of microseismic emissions associated with fault reactivations. The moment magnitude of microseismic events, consistent with the Richter scale, can be computed from the scalar value of their moment tensor. In a discontinuous model, the moment tensor can be calculated directly; in finite element models, the moment tensor is calculated by integrating the change in inelastic shear strain over the volume of the elements containing the failed fault. Coupled 3D geomechanical (deformation and fluid flow) simulations for Valhall field were conducted. Well rate and reservoir pressure histories were used as inputs to simulate reservoir depletion/injection. The 99 faults visible on seismic interpretations and 22 subregional fracture sets were included in the model. An elasto-inelastic cap model including water-weakening mechanism was used to model the chalk reservoir; the failure of faults and fractures were simulated by Mohr-Coulomb criterion. Prior to performing production simulation, preproduction stress modelling was carried out to reach an equilibrium stress state that was consistent with the in-situ condition, in terms of magnitude and orientation. Then, oil production at 112 wells and water injection at 15 wells for the period from 1982 to 2006 were simulated, during which the simulated microseismic events were in a good agreement with observations. The computed formation deformation in the reservoir and overburden was related to the in situ stress and faulting structure and was correlated over long distances. These characteristics are indicative of a near-critical process (and are also observed in flow rate correlations in the field). The coupled 3D geomechanical simulation provides a tool for validating modelled reservoir geomechanical effects against specific field data, and so enhances confidence in the implications predicted for drilling operations and reservoir management. Introduction Under most circumstances, an oil or gas field may be described as a system of rock blocks with fluids, partially separated by faults and fractures. Field evidence has been growing that the interaction between reservoir rock deformation and fluid flow is an intrinsic component of the system behaviour over the development life of a reservoir. In previous studies (Zhang et al. 2007a, 2007b), three key features—long-range correlation, stress-relationship, and fault-relationship—were observed in correlations in fluctuations of well flow rates when the stress state in a field was at or close to a critical state. This is because when field stress state prior to depletion/injection is at or close to a critical state, a relatively small change in effective stress due to depletion or injection is likely to trigger the sliding of the existing fractures or faults or to initialise new fractures. In this study, 3D coupled geomechanical deformation and fluid-flow modeling was applied to the Valhall field to understand the impact of the mechanical-hydraulic interactions on reservoir and overburden deformation, particularly the potential fracture and fault movements in the overburden induced by reservoir compaction. Valhall field is located in the southern part of the Norwegian North Sea. The field was discovered in 1975. The water depth in Valhall field is about 70 m. The reservoir depth varies from 2400 to 2600 m. The primary reservoirs are in the Tor and lower Hod formations with the Lista formation forming the caprock. Since production started in October 1982, the highly porous, weak chalk reservoir has undergone significant reservoir compaction, exceeding more than 10 m in some locations in the reservoir, which has been associated with subsidence of the seafloor exceeding 6 m below the central platform complex

The use of underground geological repositories, such as in radioactive waste disposal (RWD) and in carbon capture (widely known as Carbon Capture and Storage; CCS), constitutes a key environmental priority for the 21st century. Based on the identification of key scientific questions relating to the geophysics, geochemistry and geobiology of geodisposal of wastes, this paper describes the possibility of technology transfer from high-technology areas of the space exploration sector, including astrobiology, planetary sciences, astronomy, and also particle and nuclear physics, into geodisposal. Synergies exist between high technology used in the space sector and in the characterization of underground environments such as repositories, because of common objectives with respect to instrument miniaturization, low power requirements, durability under extreme conditions (in temperature and mechanical loads) and operation in remote or otherwise difficult to access environments.

Time-dependent brittle deformation is a fundamental and pervasive process operating in the Earth's upper crust. Its characterization is a pre-requisite to understanding and unraveling the complexities of crustal evolution and dynamics. The preferential chemical interaction between pore fluids and strained atomic bonds at crack tips, a mechanism known as stress corrosion, allows rock to fail under a constant stress that is well below its short-term strength over an extended period of time: a process known as brittle creep. Here we present the first experimental measurements of brittle creep in a basic igneous rock (a basalt from Mt. Etna volcano) under triaxial stress conditions. Results from conventional creep experiments show that creep strain rates are highly dependent on the level of applied stress (and can be equally well fit by a power law or an exponential law): with a 20% increase in stress producing close to three orders of magnitude increase in creep strain rate. Result...

Concrete bridges in the United Kingdom represent a major legacy that is starting to show signs of distress. Therefore, the need for monitoring them is an urgent task. The acoustic emission ~AE! technique was proposed as a valid method for monitoring these bridges but more study is needed to develop methods of analyzing the data recorded during the monitoring. The writers would like to propose a b-value analysis as a possible way to process AE data obtained during a local monitoring. The b-value is defined as the log-linear slope of the frequency-magnitude distribution of acoustic emissions. This paper presents the results of a b-value analysis carried out on data recorded during a laboratory test on a reinforced concrete beam designed as representative of a bridge beam. During the experiment, the specimen was loaded cyclically and it was continuously monitored with an AE system. The data obtained were processed and a b-value analysis was carried out. The b-value was compared with th...

An accurate predictive model for the long-term strength of the continental lithosphere is important in a range of geophysical and geodynamic problems. While laboratory experiments are consistent with Mohr-Coulomb brittle faulting in the cold, upper continental crust, there is increasing evidence that time-dependent processes may also be important in these rocks, even at low temperature. However, there is some ambiguity as to the exact form of the constitutive law for describing time-dependent behavior of upper crustal rocks. Here we present results of room temperature creep experiments on a suite of water-saturated sandstones spanning a range of petrophysical and rheological properties and underlying deformation mechanisms. On physical and microstructural grounds our analysis suggests that a modified power law creep, of the form ˙ A(d f) , where d is the differential stress and f is the long-term failure (fundamental) strength, provides a more complete description of the experimenta...

SUMMARY The goodness of fit for competing statistical models with different numbers of degrees of freedom cannot be assessed solely by the residual sum of squares, because more complex models will naturally have lower residuals. A standard approach to hypothesis testing for large data sets is to use the objective Bayesian information criterion (BIC), which penalizes models with larger numbers of free parameters appropriately. We apply this method to the analysis of time delays from data on seismic shear-wave splitting in SW Iceland. The same data set has previously been used to estimate the time at which stress-modified micro-cracking reaches an inferred state of fracture criticality. The method does not forecast the location of the event, however, the time and magnitude were consistent with the actual occurrence of an M = 5 earthquake in the region. The forecast was based on a multi-line model with 17 degrees of freedom (five straight lines, each with a start and end point, plus th...

The spontaneous emergence of localized cooperative deformation is an important phenomenon in the development of shear faults in porous media. It can be studied by empirical observation, by laboratory experiment or by numerical simulation. Here we investigate the evolution of damage and fragmentation leading up to and including systemsized failure in a numerical model of a porous rock, using discrete element simulations of the strain-controlled uni-axial compression of cylindrical samples of different finite size.

SUMMARY Earthquake populations have recently been shown to have many similarities with critical-point phenomena, with fractal scaling of source sizes (energy or seismic moment) corresponding to the observed Gutenberg-Richter (G-R) frequency-magnitude law holding at low magnitudes. At high magnitudes, the form of the distribution depends on the seismic moment release rate Mand the maximum magnitude m max . The G-R law requires a sharp truncation at an absolute maximum magnitude for finite M ˙ .I n contrast, the gamma distribution has an exponential tail which allows a soft or 'credible' maximum to be determined by negligible contribution to the total seismic moment release. Here we apply both distributions to seismic hazard in the mainland UK and its immediate continental shelf, constrained by a mixture of instrumental, historical and neotectonic data. Tectonic moment release rates for the seismogenic part of the lithosphere are calculated from a flexural-plate model for glac...

The scaling of displacement as a function of length is important for a variety of applications which depend on the mechanical and hydraulic properties of faults and fractures. Recently it has been suggested that the power-law exponent ν which has been found to characterise this relationship may change significantly at a characteristic length for a variety of reasons, for example when cracks begin to interact, or when faults grow to a length comparable to a characteristic size in the brittle layer. Such a break of slope requires a second straight line, requiring two extra model parameters. Here we present a new method for analysing such data, which penalises the extra parameters using a modified form of Schwarz's Information Criterion, and a Bayesian approach which represents uncertainty in the unknown parameters. We apply the method to data from the Krafla fissure zone in the north of Iceland, and find a significant break of slope, from ν ≈ 3/2 to ν ≈ 2/3, at a characteristic le...

This chapter discusses the hydromechanical behavior of fractured rocks. The hydromechanical behavior of rock fractures can be studied on the scale of a single fracture and also on the scale of a fractured rock mass that contains many fractures. The behavior of single fractures must be thoroughly understood before the behavior of fractured rock masses can be understood. If a fracture is located in a rock mass with a given ambient state of stress, the traction acting across the fracture plane can be resolved into a normal component and a shear component. The normal traction gives rise to a normal closure of the fracture. The shear component of the traction causes the two rock faces to undergo a relative deformation parallel to the nominal fracture plane, referred to as “shear deformation.” A tangential traction also typically causes the mean aperture to increase, in which case the fracture dilates. Dilation arises because the asperities of one fracture surface must by necessity ride u...

We address the generic problem of testing for scale-invariance in extreme events, i.e. are the biggest events in a population simply a scaled model of those of smaller size, or are they in some way different? Are large earthquakes for example ‘characteristic’, do they ‘know’ how big they will be before the event nucleates, or is the size of the event determined only in the avalanche-like process of rupture? In either case what are the implications for estimates of time-dependent seismic hazard? One way of testing for departures from scale invariance is to examine the frequency-size statistics, commonly used as a bench mark in a number of applications in Earth and Environmental sciences. Using frequency data however introduces a number of problems in data analysis. The inevitably small number of data points for extreme events and more generally the non-Gaussian statistical properties strongly affect the validity of prior assumptions about the nature of uncertainties in the data. The ...