Institute of Earthquake Prediction Theory and Mathematical Geophysics

Abstract. We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction. Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems. Both these results have implications for the study and prediction of natural hazards and their human impacts.

Abstract The recent installation of six broadband digital IRIS/IDA seismic stations in the USSR has provided new opportunities for studying surface-wave propagation across Eurasia. Group velocities of fundamental Rayleigh and Love modes between epicenters and these stations were determined for 35 events that occurred since April 1989 to the middle of July 1990 near Eurasia. Differential phase velocities were found for the same arrivals along paths between several pairs of stations. Group and phase velocities were obtained in the period range from 15 to 300 sec. Frequency-time polarization analysis was used for studying polarization properties of surface waves. In some cases, significant anomalies in the particle motion for periods up to 100 sec were observed. They are attributed to surface-wave refraction and scattering due to lateral inhomogeneities at the boundaries and inside the Eurasia continent.

Fat-tail distributions of sizes abound in natural, physical, economic, and social systems. The lognormal and the power laws have historically competed for recognition with sometimes closely related generating processes and hard-to-distinguish tail properties. This state-of-affair is illustrated with the debate between Eeckhout [Amer. Econ. Rev. 94, 1429 (2004)] and Levy [Amer. Econ. Rev. 99, 1672 (2009)] on the validity of Zipf's law for US city sizes. By using a uniformly most powerful unbiased (UMPU) test between the lognormal and the power-laws, we show that conclusive results can be achieved to end this debate. We advocate the UMPU test as a systematic tool to address similar controversies in the literature of many disciplines involving power laws, scaling, "fat" or "heavy" tails. In order to demonstrate that our procedure works for data sets other than the US city size distribution, we also briefly present the results obtained for the power-law tail of the distribution of personal identity (ID) losses, which constitute one of the major emergent risks at the interface between cyberspace and reality.

T h e problem of aftershock identification in earthquake catalogues is studied. Some empirical methods are considered and quantitavely analysed.

A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent b close to 3. We develop a battery of new non-parametric and parametric tests to characterize the distributions of empirical returns of moderately large financial time series, with application to 100 years of daily returns of the Dow Jones Industrial Average, to 1 year of 5-min returns of the Nasdaq Composite index and to 17 years of 1-min returns of the Standard & Poor's 500. We propose a parametric representation of the tail of the distributions of returns encompassing both a regularly varying distribution in one limit of the parameters and rapidly varying distributions of the class of the stretched-exponential (SE) and the log-Weibull or Stretched Log-Exponential (SLE) distributions in other limits. Using the method of nested hypothesis testing (Wilks‘ theorem), we conclude that both the SE distributions and Pareto distributions provide reliable descriptions of the data but are hardly distinguishable for sufficiently high thresholds. Based on the discovery that the SE distribution tends to the Pareto distribution in a certain limit, we demonstrate that Wilks‘ test of nested hypothesis still works for the non-exactly nested comparison between the SE and Pareto distributions. The SE distribution is found to be significantly better over the whole quantile range but becomes unnecessary beyond the 95% quantiles compared with the Pareto law. Similar conclusions hold for the log-Weibull model with respect to the Pareto distribution, with a noticeable exception concerning the very-high-frequency data. Summing up all the evidence provided by our tests, it seems that the tails ultimately decay slower than any SE but probably faster than power laws with reasonable exponents. Thus, from a practical viewpoint, the log-Weibull model, which provides a smooth interpolation between SE and PD, can be considered as an appropriate approximation of the sample distributions.

The Ms:mb ratio, conventionally used for the discrimination of earthquakes and underground nuclear explosions, has been studied for a pure earthquake catalogue in an attempt to classify earthquakes into two categories, one related to extremely lowfrequency events of creep and the other to relatively high-frequency explosions. Ms and mb measured at 20 s and 1 s respectively are considered as estimates of source spectra at two separate points, thus allowing one to compare large samples of earthquakes from the spectral point of view. Creepex, or the deviation of (Ms, mfc)-points from the orthogonal regression of Ms on mb, is by definition a parameter independent of the size of seismic events. It has been estimated from NEIC magnitudes and is studied in relation to parameters of central moment tensor solutions of the Harvard group. For this purpose, the parameters of central moment tensors have been factorized to make classes of equivalence of events with the same dip and rake and arbitrary strike and size. The resulting triangle representation of source mechanisms has been used to correct the global distribution of creepex for the type of source mechanism. Dip-slip events have consistently lower creepex than strike-slip ones. However, the global distribution of creepex, either corrected or uncorrected for source mechanism, has a clear tectonic pattern: negative values in most of the subduction zones and positive ones in mid-ocean ridges. The dependence of creepex on focal depth is negligible according to empirical data for events shallower than 40 km. Limitation of the sample by considering only events in this depth range does not diminish considerably the amount of data and does not change the tectonic pattern of the creepex.

Accelerating displacements preceding some catastrophic landslides have been found to display a finite time singularity of the velocity v ∼1/( t c − t ) [ Voight , 1988a , 1988b ]. Here we provide a physical basis for this phenomenological law based on a slider block model using a state‐ and velocity‐dependent friction law established in the laboratory. This physical model accounts for and generalizes Voight's observation: Depending on the ratio B / A of two parameters of the rate and state friction law and on the initial frictional state of the sliding surfaces characterized by a reduced parameter X i , four possible regimes are found. Two regimes can account for an acceleration of the displacement. For B / A > 1 (velocity weakening) and X i < 1 the slider block exhibits an unstable acceleration leading to a finite time singularity of the displacement and of the velocity v ∼ 1/( t c − t ), thus rationalizing Voight's empirical law. An acceleration of the displacement can also be reproduced in the velocity‐strengthening regime for B / A < 1 and X i > 1. In this case, the acceleration of the displacement evolves toward a stable sliding with a constant sliding velocity. The two other cases ( B / A < 1 and X i < 1 and B / A > 1 and X i > 1) give a deceleration of the displacement. We use the slider block friction model to analyze quantitatively the displacement and velocity data preceding two landslides, Vaiont and La Clapière. The Vaiont landslide was the catastrophic culmination of an accelerated slope velocity. La Clapière landslide was characterized by a peak of slope acceleration that followed decades of ongoing accelerating displacements succeeded by a restabilization. Our inversion of the slider block model in these data sets shows good fits and suggests a classification of the Vaiont landslide as belonging to the unstable velocity‐weakening sliding regime and La Clapière landslide as belonging to the stable velocity‐strengthening regime.

Abstract The problem of statistical estimation of earthquake hazard parameters is considered. The emphasis is on estimation of the maximum regional magnitude, Mmax, and the maximum magnitude, Mmax(T), in a future time interval T and quantiles of its distribution. Two estimators are suggested: an unbiased estimator with the lowest possible variance and a Bayesian estimator. As an illustration, these methods are applied for the estimation of Mmax and related parameters in California and Italy.

Abstract For a general use of the frequency-magnitude (FM) relation in seismic risk assessment, we formulate a multi-scale approach that relies on the hypothesis that only the ensemble of events that are geometrically small, compared with the elements of the seismotectonic regionalization, can be described by a log-linear FM relation. It follows that the seismic zonation must be performed at several scales, depending upon the self-similarity conditions of the seismic events and the linearity of the log FM relation, in the magnitude range of interest. The analysis of worldwide seismicity, using the Harvard catalog, where the seismic moment is recorded as the earthquake size, corroborates the idea that a single FM relation is not universally applicable. The multi-scale model of the FM relation is tested in the Italian region.

Carbonatites are igneous rocks formed in the crust by fractional crystallization of carbonate-rich parental melts that are mostly mantle derived. They dominantly consist of carbonate minerals such as calcite, dolomite, and ankerite, as well as minor ...Read More

Validation is often defined as the process of determining the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. Validation is crucial as industries and governments depend increasingly on predictions by computer models to justify their decisions. We propose to formulate the validation of a given model as an iterative construction process that mimics the often implicit process occurring in the minds of scientists. We offer a formal representation of the progressive build-up of trust in the model. Thus, we replace static claims on the impossibility of validating a given model by a dynamic process of constructive approximation. This approach is better adapted to the fuzzy, coarse-grained nature of validation. Our procedure factors in the degree of redundancy versus novelty of the experiments used for validation as well as the degree to which the model predicts the observations. We illustrate the methodology first with the maturation of quantum mechanics as the arguably best established physics theory and then with several concrete examples drawn from some of our primary scientific interests: a cellular automaton model for earthquakes, a multifractal random walk model for financial time series, an anomalous diffusion model for solar radiation transport in the cloudy atmosphere, and a computational fluid dynamics code for the Richtmyer-Meshkov instability.

The March 11, 2011 megathrust on the Pacific coast of the Tohoku Region, Japan, and its consequences once again confirmed the presence of evident problems in the conventional methodology of risk and earthquake loss evaluation. A systematic analysis shows that the results of the Global Seismic Hazard Assessment Program (GSHAP, 1992–1999) contradict the actual occurrence of strong earthquakes. In particular, since the publication of the GSHAP final results in 1999, all 60 earthquakes with magnitudes of 7.5 or higher were “surprises” for the GSHAP maps. Moreover, in half of the cases they were “big surprises,” when instead of the expected “light” or “moderate,” “significant” or even “total” destruction took place. All twelve of the deadliest earthquakes happened in 2000–2011 (total number of deaths exceeded 700000 people) prove that the GSHAP results, as well as underlying methodologies, are deeply flawed and, evidently, unacceptable for any critical risk assessments entitled to prevent disasters caused by earthquakes.

We compare the aftershock decay rate in natural data with predictions from a stochastic analytical model based on a Markov process with stationary transition rates. These transition rates vary according to the magnitude of a scalar representing the state of stress and defined as the overload. Thus, the aftershock decay rate in the model is a sum of independent exponential decay functions with different characteristic times. From different shapes of the overload distribution and different expressions of the transition rates, we discuss the magnitude of the exponent of the power law aftershock decay rate and the time interval over which we can expect to observe this regime. Before and after this time interval, we show that the decay is linear and exponential, respectively. From our analytical solutions, we deduce a model of aftershock decay rate in which a power law scaling exponent and two characteristic rates emerge. One rate is a short‐term linear decrease before the onset of the power law decay to account for a finite number of events at zero time, and the other one can be interpreted as an inverse correlation time, after which aftershocks no longer occur. Then, we interpret the empirical modified Omori law (MOL) and its parameters in the framework of our theoretical model. We suggest a technique to systematically estimate and interpret the temporal limits of the power law aftershock decay rate in real sequences. We approximate these temporal limits from data available from several well‐known aftershock sequences and show from an Akaike Information Criteria (AIC) that, in almost all cases examined here, our model fits better the aftershock decay rate than the MOL despite a quantitative penalty for the extra parameter required. From this work, we conclude that the time delay before the onset of the power law decay may be related to the recurrence time of an earthquake. Finally, we suggest that the power law decay rates extend over longer times according to the concentration of the deformation along dominant major faults.

Abstract Pricaspian basin geology is reviewed in the light of 500,000 km of seismic profiles and several thousand wells. We focus on how hydrocarbons from three sources accumulated in relation to the 1800 salt structures in a basin that changed little in planform from the Devonian to the Paleogene. Riphean to Carboniferous shelf sedimentary strata are still flat lying between a poorly known crystalline basement and a base of salt now 10 km deep. Slow and almost continuous sedimentation in the basin center downbuilt huge massifs in Permian salt initially 4.5 km thick. Basin sediments are flat lying or backtilted between down-to-basin growth faults along northern and western margins starved of sediments. By contrast, progradation of Permian sediments from the Urals, Triassic sediments from the South Emba shear zone, and Jurassic sediments from the Dombass-Tuarkyr fold belt downbuilt successive waves of salt structures basinward from margins in the east, southeast and then the south. A zone of salt overhangs records extrusion that starved basin-marginal salt structures, particularly during a basinwide hiatus in the Early Jurassic. Salt diapirs along polygonal normal faults rooting to the crests of still-potent salt structures through Cretaceous–Paleogene strata indicate that salt upbuilt back to the surface and resumed downbuilding. Coarse clastic fans infill deep canyons incised across the basin by rivers draining to the Caspian in Pliocene times.

We revisit the 1976 Friuli earthquake sequence by combining hypocenters relocation, long period surface wave inversion, field geology and strong motion modelling. We show that fault‐related folding is the main active deformation by which the seismic energy was released during the main shock (Ms=6.5) and that some of the surface effects reported in 1976 correspond to widespread bedding planes displacements induced by flexural‐slip folding. The fault evolved from blind to semi‐blind along strike showing the control of the inherited structural geology on the fault surface break and rupture arrest. Our fault model produces waveforms that fit the accelerograms recorded in the area.

An earthquake of magnitude M and linear source dimension L(M) is preceded within a few years by certain patterns of seismicity in the magnitude range down to about (M - 3) in an area of linear dimension about 5L-10L. Prediction algorithms based on such patterns may allow one to predict approximately 80% of strong earthquakes with alarms occupying altogether 20-30% of the time-space considered. An area of alarm can be narrowed down to 2L-3L when observations include lower magnitudes, down to about (M - 4). In spite of their limited accuracy, such predictions open a possibility to prevent considerable damage. The following findings may provide for further development of prediction methods: (i) long-range correlations in fault system dynamics and accordingly large size of the areas over which different observed fields could be averaged and analyzed jointly, (ii) specific symptoms of an approaching strong earthquake, (iii) the partial similarity of these symptoms worldwide, (iv) the fact that some of them are not Earth specific: we probably encountered in seismicity the symptoms of instability common for a wide class of nonlinear systems.

Abstract It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here, we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous) nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. To prove these results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a prize of the French Academy. This formulation leads to simple recurrence relations among the time-Taylor coefficients of the Lagrangian map from initial to current fluid particle positions; the coefficients can then be bounded using elementary methods. We first consider various classes of incompressible fluid flow, governed by the Euler equations, and then turn to highly compressible flow, governed by the Euler–Poisson equations, a case of cosmological relevance. The recurrence relations associated with the Lagrangian formulation of these incompressible and compressible problems are so closely related that the proofs of time-analyticity are basically identical.

Characterising the state of stress in the brittle upper-crust is essential in mechanics of faulting, industrial production processes, and operational earthquake forecasting. Nevertheless, unresolved questions concern the variation of pore-fluid with depth and the absolute strength on tectonically active faults. Here we show that, along the San Andreas fault system, the time-delay before the onset of the power-law aftershock decay rate (the c-value) varies by three orders of magnitude in the first 20 km below the surface. Despite the influence of the lithostatic stress, there is no continuous change in c-value with depth. Instead, two decay phases are separated by an abrupt increase at an intermediate depth range of 2-5 km. This transitional regime is the only one observed in fluid-injection-induced seismic areas. This provides strong evidence for the role of fluid and a porosity reduction mechanism at depth of few kilometres in active fault zones. Aftershock statistics can then be used to predict changes in differential shear stress with depth until the brittle-ductile transition is reached.

We discuss the possibility of applying some standard statistical methods (the least-square method, the maximum likelihood method, and the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A "segmentation fitting" maximum likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value x(1) considered as an additional unknown parameter. The segmentation fitting method, called "piece-wise" ML, is similar in spirit but simpler and has smaller bias than the "multiple shooting" previously proposed. Comparisons with different previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size N and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).

The prototype of a 4He pumped vector magnetometer is presented. Large auxiliary coils systems used in previously developed apparatus to allow vector measurements from a scalar (atomic or nuclear resonance) sensor are replaced by a light triaxial modulation system associated with advanced techniques of signal processing. The performances of the helium scalar sensor are first briefly recalled; then the principle of the vector measurement, obtained by adding three (approximately) orthogonal modulations of different frequencies (all of the order of 10 Hz) is explained. Afterwards a second part of the paper is devoted to the calibration process, and a first estimate of the performances of the vector magnetometer is obtained. They confirm that this instrument could be a good candidate for an automatic absolute magnetic observatory: after the calibration process completion and a proper installation, it would provide by itself the absolute value of three orthogonal components of the field. In addition to that, the 4He vector magnetometer appears to be also promising for space applications.