Выпуск 23

The prediction problem for a stochastic point process in terms of loss function \nu is considered. Losses depend on prediction errors: \stackrel{\circ}{n} – the rate of failures-to-predict, \stackrel{\circ}{\tau} – the rate of alarm time. (\stackrel{\circ}{n},\stackrel{\circ}{\tau}) characterizes prediction strategy and allows to compare prediction algorithms. \nu-optimal strategies of prediction for some functions \nu are outlined. The previously announced prediction of strong earthquakes for the eight segments of San Andreas fault is analyzed.

Zones of increased probability of large earthquakes within few years are identified by «overcritical» behavior of the frequency-of-occurrence law in several regions.

Formal definition of seismic quiescence is formally defined on the basis of non-linear transformation of the time scale into discrete sequence of seismic events. This allows a uniform analysis of normal and aftershock sequences. Corresponding algorithm of intermediate-term prediction was data-fitted for Kuril arc and applied to Kamchatka and to aftershocks.

TIPs are diagnosed retrospectively before five out of six strong earthquakes with alarm time from 15 to 31% in different regions.

Algorithm M8 for intermediate-term earthquake prediction and tested previously only for natural earthquake was applied retrospectively to induced earthquake, M = 4.9 near Koyna reservoir, India. No free parameters of algorithm were readapted. TIP is diagnosed nearly four years before the earthquake. Two strongest earthquakes of this area are also preceded by seismic activation 1.5 years in advance.

Algorithm for identification of location of incipient strong earthquakes by long range aftershocks (LRA) is adapted for Pamir–Tien-Shan region. In combination of seismic quiescence this algorithm retrospectively outlines twenty percent of time-space domain, which includes ten out of eleven strong earthquakes.

The failure properties of hierarchically organized bundles of fibers is considered by computational and analytic means using a renormalization approach. This problem also arises in the study of the strength of tectonic faults with hierarchically organized structure of asperities. A numerical experiment shows a universal scaling law which relates the failure threshold to the size of the system. This law does not depend either on the specific failure properties of an individual fiber (asperity) or on the specific hierarchical organization.

Using the insight derived from a computational investigation of the failure threshold of hierarchically organized fiber bundles, it is proved analytically that the threshold for failure of a hierarchical fiber bundle obeys a universal scaling law with respect to the size of the bundle, independent of the specific failure properties of an individual fiber and independent of the specific hierarchical organization employed.

The singular self-similar measures with almost everywhere zero derivative are connected with fractal sets. The relation between Fourier transform of the measure (characteristic function) and its generalized fractal dimension d₂ is established. This result is applied to spatial distribution of earthquake epicenters.

Self-similar probability measures or multifractals are investigated. A method for calculation of multifractal characteristic evaluation is suggested. Multifractal characteristics of spatial distribution of earthquake epicenters have been obtained for Caucasus, Pamir–Tien-Shan and California.

Initially the disc (z = 0 plane) and the fluid rigidly rotate about the z-axis with constant velocity. Then the additional velocity is added to the disc. It is assumed that the additional velocity changes along the disc radius. The stationary flow is calculated using Fourier–Bessel integrals. Two examples are computed.

The Rayleigh–Taylor problem for two-layer model of viscous compressible fluid is stated. This problem is reduced to the calculation of eigenvalues of a system of ordinary differential equations with boundary conditions. This problem, in its turn, leads to the calculation of complex zeros of an analytic function. The calculations show that in compressible fluids initial perturbations grow much faster and maximum instability modes are of much lower space frequencies than in the case of incompressible fluids.

The problem of gravitational instability of two viscoelastic compressible layers is reduced to an eigenvalue boundary problem for a system of ordinary differential equations. These eigenvalues fall into two groups: «high frequency» (corresponding to P and S waves) and «low frequency» one (corresponding to a slow growth of initial perturbations). The models involving a compressible medium differ by faster growth of perturbations and greater wavelength of maximum instability mode.

Generation of magnetic field by motion of a conducting medium with internal scale in an axisymmetric body is considered in the case of the α-effect. A complete asymptotic decomposition of induction operator eigenvalues and eigenvectors is constructed for this kinematic dynamo problem.

The connection is established between two classical problems: the one-dimensional kinematic inversion (seismology) and the problem of the description of stable polynomials (mathematics). It gives a new method for parametrization of seismic structure. New algorithms of inversion are suggested.

Inverse problem of seismology is considered for laterally inhomogeneous half-space. The wave velocity is to be constant along any straight line passing through a surface source and is increasing monotonously with the slope of this line.

The method for estimation of source parameters is suggested. It reduces determination of the depth and mechanism of an equivalent point source to solution of a system of linear equations with linear constraints. Influence of lateral inhomogeneity of the medium on the calculations is considered. The efficiency is demonstrated by numerical tests for different block structures.

Matrix Sturm–Liouville equation which determines Rayleigh wave velocities is considered. Potential of the equation is a symmetric matrix depending on the Lame parameters.

Syntactic pattern recognition of waveforms is applied to strong-motion accelerograms of the fore-, main- and after-shocks in California. Each type of earthquakes shows a tendency to cluster in the space of accelerogram parameters.

Optimal statistical algorithms are suggested for detection of waves scattered on sharp inhomogeneities within the Earth and recorded by a seismic array. The algorithms estimate the power of scattered radiation and suppress the regular seismic wave which constitutes a signal-generated noise in the problem considered. Synthetic records and real records of the NORSAR array are analyzed.

Places, where strong earthquakes (M ≥ 7.0) may occur, are recognized in the Mountain Belt of the Andes.