Выпуск 21

Statistical significance of a premonitory seismicity pattern called «burst of aftershocks» is estimated in 13 regions of the world. This is the first earthquake precursor for which the statistical significance is high. The precomputed graphs are given which allow one to estimate statistical significance of the success in earthquake prediction using the following data: number of earthquakes for which prediction is attempted; number of alarms identified; time of a single alarm.

The paper presents the results of monitoring the long-term pattern «burst of aftershocks» for 12 regions of the world during the last 3–10 years. The results are satisfactory. A burst of aftershocks is diagnosed before 9 of a total of 13 large earthquakes with only one false alarm.

The results of retrospective identification of the time of increased probability for earthquakes with magnitude M ≥ 7.5 in South America are presented. Algorithm M8 with fixed a priori parameters reduces by factor 8 the uncertainty in expectation of these earthquakes. Nevertheless, time of increased probability is diagnosed for each of the 3 earthquakes with M ≥ 7.5 which occurred in South America during 1975–1986.

Hypothesis of global correlation of tidal gravity anomalies, heat flow and seismicity are treated. Significance level of the hypothesis is estimated.

Time of increased probability of strong earthquake (TIP) is diagnosed separately within the time – intervals of different levels of seismic activity. This simplifies the algorithm of the diagnosis.

A method for studying spatio-temporal seismicity variations is put forward that does not require separating the study area into fixed regions, and which can be used with small numbers of events in time intervals to be compared.

The magnetic dynamo problem for the Couette-Poiseuille flow is reduced to existence of an eigenvalue with a positive real part for some boundary – value problem. It is shown that, for small values of the magnetic Reynolds number R_m , the real part of the eigenvalue of this boundary – value problem may be represented as a series in even powers of R_m , and the values of the first term of this series are obtained for some values of flow parameters.

In this work discrete hierarchical multiscale schemes of percolation theory and failure theory are studied. The necessary and sufficient conditions for percolation in multiscale models are obtained.

We consider a hierarchical dynamic model of defect formation. Exponential convergence of the system to the stationary state is proved. We study dependence of the system evolution on the rate of defect production at an elementary level; existence and uniqueness of the critical point is proved.

The problem under consideration is to determine the position of static equilibrium for a system of blocks with elastic interaction. The model of «Rigid and Soft Blocks» is introduced. This model assumes that some blocks are absolutely rigid and others are deformable. In the special case where we have soft blocks as narrow layers between rigid blocks, we obtain «Layered Blocks Model». This model can be used for studying failures in rocks and engineering structures, as well as for studying the interaction of lithospheric blocks. It is shown that the calculation of equilibrium position for layered blocks is a convex programming problem. Such an approach enables us to consider nonlinear and nonuniform systems (i.e. Hooke’s low may not hold and elastic parameters of layers may change from point to point).

Theorems of existence and uniqueness for the solution of the Rayleigh–Benard problem are proved. The problem is described by equations of conservation of momentum, conservation of energy and transport of physical properties. Newtonian rheology is assumed. Initial distributions of parameters (density, viscosity, conductivity, internal heat generation, etc.) are arbitrary. The method of proof is based on a priori evaluations of growth of solutions, on their smoothness properties and their representation in the form of multiplicative integrals.

The possibility is considered of taking into account a laterally varying Earth’s structure when space-time parameters of source are estimated. A numerical model of a source is suggested. It is a superposition of point sources with a given time function. This model is used to estimate the influence of lateral inhomogeneity of the medium on the calculations of earthquake source parameters.

Determination of layered structure by analysis of surface wave seismograms at two stations is considered as a problem of statistical estimation for frequency response parameters. A statistically optimal estimate is proposed which takes into account the power spectrum of noise background in seismograms at both stations. The statistical nonconsistency of the conventional least squares estimate is proved. The covariance matrix of the optimal estimate and its sensitivity to prior information on noise power spectrum is investigated.

An algorithm is considered for determining parameters of the medium between stations from surface wave spectra. The algorithm is based on a parametric approach to the identification of the frequency response of linear system from observed input and output signals. The results of numerical modeling are presented.

Records of surface waves, propagating across the continental shelf of Laptev’s sea, are used for determination of the structure of the crust.

The examples of vertically inhomogeneous elastic media are given by explicit formulae using wave-number and ration of horizontal and vertical components of the only Rayleigh – type wave excited by the monochromatic source.

The wave equation for axisymmetrical harmonic vibrations of the Rayleigh type in a nonuniform spherically symmetrical body has been reduced to Sturm–Liouville problem on a half-axis. The matrix potential of the latter problem is shown to decrease exponentially at infinity after the spectral parameter has been redefined. The Lame parameters and density of the body are determined from the vibrations at it surface and the source that has excited them.

Rayleigh-type harmonical vibrations of a very light plate with heavy surfaces are considered. Although the dispersion equation for the plate is the product of two quadratic equations, it retains all spectral features for these vibrations, i.e. existence of complex and double real points. It is shown that a plate underlain by a light and almost liquid half-space has the same features.

We obtained bounds for main characteristics of motion of absolutely elastic non-linear single-degree-of-freedom systems in terms of parameters of the seismic load.