Выпуск 25

An inter disciplinary problem of earthquake prediction which involves economics is considered. It helps to understand prediction problem on the whole and reveals additional requirements on seismostatistics. The problem is formulated as an optimal control problem: having the possibility to declare several types of alarms it is necessary to find an optimal policy which minimizes the total expected losses. Alarms require special costs both for maintaining and for changing of the alarm type, each successful alarm prevents certain amount of losses; total expected losses are integrated on the half infinity time interval, taking into account time discount factor. Algorithmic and exact solutions are indicated.

The problem is formulated to recognize strongest earthquakes aftershock sequences which can contain strong aftershocks. The recognition is based on seismic activity data in the focal zone of a strong shock. Functions characterizing the seismic process were found using California earthquake catalog. The material for the recognition was represented by 17 aftershock sequences. The application of the preliminary solution rule resulted in 1 error. The algorithm was tested on independent data of 8 seismoactive regions of the world without additional data fitting. The results of the test can be considered as satisfactory, because errors do not exceed 15%. The statistical significance of the obtained result is 96%. Two successful advance predictions were done.

We study clusterisation properties of real and modeled flows of seismic events. The distribution of clusters by number of events is investigated in terms of multifractal technology. It is shown that clusterization parameters of real seismic flows can be well approximated by the relevant parameters of multifractal seismic flow models.

We consider a general model of defects developing in hierarchical selfsimilar media. We establish the existence of the critical value of defect generation rate on the lower level, which turned to be a root of the derivative of a certain function explicitly determined by the model.

Places where earthquakes with М\geq6.5 may occur, are recognized in the Himalayan region on the basis of morphostructural zoning scheme.

We apply the method of morphostructural zoning for plains and platform to northern part of India shield and the Himalayan foredeep. We compare morphostructures within conjunction zone between Indian shield and the Himalaya. We also determine locations of morphostructural knots. That is a base for a future recognition of some natural events (earthquakes, mineral deposits).

The three dimensional problem concerning elastic equilibrium of the stressed homogeneous isotropic half-space with a relief-free surface and an interior fault has been solved by the perturbation method. The small parameter is the ratio of the maximum relief amplitude to the minimum distance between the fault and the surface. The results show, that the relief should be considered in the interpretation of seismotectonic crustal movements data.

The paper contains further new results in the solution of the kinematic problem of magnetic field generation by a Beltrami flow in a sphere. We also present the corrections to the numerical results, reported previously in [1].

Fine structure of the Earth crust and upper mantle is not known in much detail. Only some average values, say, of density or wave velocity can be specified in various models. This work presents a study of the gravitational instability in a two-layer non-deterministic model with Newtonian rheology. The density in the upper layer is random; the density in the lower layer (half-space) is deterministic. It is supposed that the density depends only on depth. An inversion in the mean density is assumed. The integro-differential equation is derived for mean velocities, and the boundary conditions are usual for a Newtonian fluid. Using the Laplace transform techniques this problem was reduced to the calculation of roots of an analytic function. To compute the roots a program was constructed based on the principle of the argument. Variations of eigenvalues are computed for several sets of parameters. The existence of oscillatory instability is shown for some ranges of stochastic parameters.

This work presents the numerical analysis of thermoconvective cell layering for the case where the thickness of the upper mantle is three times less than that of the lower mantle. This is more adequate for the Earth than the case of the same thickness assumed in the analysis of Richter and Johnson (1974). They state that the system of two chemically distinct layers of the same thickness results in a two-layered cell structure if the density of the lower layer is slightly (even less than 1%) greater than the density of the upper layer. The present study produces a stability diagram on (Ra, R_ρ) plane where Ra is the Rayleigh number and R_ρ is the nondimensional density contrast. The diagram shows that in the considered case a four-layer cell structure is more preferable with one cell in the upper layer and three cells in the lower one.

We propose a mechanical model for earthquakes which, unlike the well-known Burridge–Knopoff model, takes into account the transversal vibrations against the background of displacements along a fault. It is shown that the energy of the moving fault can generate vertical displacements of the ground, as well as horizontal ones. The mechanism of parametric instability resulting in stochastic transversal vibrations is responsible for this energy transfer. Numerical experiments show that the amplitudes of transversal and longitudinal vibrations have the same order of magnitude.

The paper presents the study of evolution of hydrodynamic perturbation with initial vorticity being harmonic along the flow and initially localized in a thin layer. We suppose that there are two small parameters: the velocity profile curvature and viscosity. The solution of the Cauchy problem may be treated as the evolution of a continuous spectrum wave in the viscous flow. It is useful to introduce the “viscous time” t_ν ∼ ν ^-1/3 (kU′) ^-2/3, where ν is viscosity, k is wave number, U′ is z-gradient of the flow. The solutions to problems with viscous and ideal fluids are shown to be very close to each other up to the viscous time outside of neighborhoods of the critical layer. That neighborhood is growing in time as (νt)^1/2. In the case of a viscous fluid our approach is based on the detailed asymptotics for minors of the fundamental matrix for Orr–Sommerfeld’s equation. In some cases the leading terms of asymptotics for two-order minors depend on the higher terms of asymptotics for particular solutions.

This paper, being the third part of the studies, presents the proof of all the hypotheses formulated in the second part.

We generalize the τ-method for anisotropic horizontally homogeneous media with seismic signal velocity v depending on two variables у and α, where у is depth and α is the angle between the vertical and ray direction. We consider also the specific case where the velocity of seismic signal propagation in horizontal direction increases with depth. In that case we can evaluate the horizontal velocity and the velocity indicatrix by τ-functions of reflected and refracted waves.

Reflection and transmission of seismic waves at the second-order boundaries are treated in the framework of the ray theory, the first higher term of the ray series being taken into account. The method was applied for calculation of the wave field reflected from a transition layer between two homogeneous half-spaces. The results are compared with those obtained by Thomson–Haskell technique, when transition layer is approximated by a set of homogeneous layers. Inferences about applicability of the proposed method are made.

An unexpected behavior of Rayleigh waves is found in elastic half-spaces in which density and elastic parameters are proportional and increase exponentially with depth. Such a structure, which might approximate the behavior of shallow sediments, has a frequency band in which Rayleigh waves cannot propagate.

Seismic moment and focal mechanism of Spitak earthquake are determined by the analysis of long-period spectrum of surface waves. Graphic presentation of the results illustrates the discrimination quality of vector orientation for compression and dilatation. Analysis of surface waves in shorter period ranges yields integral estimates for source duration, source geometry and the rupture spreading process along the fault. The quality of those estimates is also discussed.

The paper illustrates the structure and information contents of the relational Strong Motion Data Base.