Выпуск 24

Prediction of stationary sequence of rare events (large earthquakes) is considered. The prediction uses any information that is related in a stationary way to the sequence. The goal of prediction is stated as minimization of a loss function which is a function of two long-run errors: the rate of failures-to-predict (in percent to total number of strong earthquakes) and the total time of alarms (in percent to time-period considered). We describe the optimal prediction strategies for any convex loss function. Stability of prediction is analyzed within the framework of this loss-function approach. The results provide refinement of some earthquake prediction algorithms.

The problem of aftershock identification in earthquake catalogs is studied. Some empirical methods are considered and quantitatively analyzed. Game theory approach is applied to formulate the problem which allows for the whole class of optimal methods of aftershock identification. Each method is optimal dependently on the goals and gives the best trade off between the number of missed aftershocks and the number of failurely identified ones. Some illustrations of the new approach to aftershock identification problem are presented.

The model of a shallow focus seismicity is suggested which is based on a view on the Earth crust as a hierarchical structure of blocks. Qualitatively the seismicity is represented as a result of dynamical balance between two processes oppositely directed in the hierarchy of block structure: the consolidation of small blocks into the big ones and the destruction (deconsolidation) of big blocks resulted from aftershocks. The model is represented by the system of simple dynamical equations with random parameters. The results of numerical experiments are presented.

The dynamic equations for solid media under deformation are derived for the mean field stress model on the base of statistical mechanics of cracks. “Earthquakes” are expected to be a fast moving of boundaries of narrow (similar to cracks) areas of the high crack density phase.

Theoretical model of nonhomogeneous friction of nominally flat rough surfaces is formulated. Two types of nonhomogeneities are considered: random fluctuation of fractal relief of rough surfaces, and inhomogeneities of density of asperities. It is shown that interaction of inhomogeneities may lead to instability of slow sliding. A jump of displacements occurs under smooth relative movement of contacting surfaces. Rapid temporal rate of displacements appears as a precursor immediately before the instability time: \mid ∂u/∂t\mid \sim (t_c - t)^{-1/2}.

The two-dimensional problem on the elastic equilibrium of homogeneous isotropic half-plane under stress with a relief on the free surface and an internal fault has been solved by perturbation method. The small parameter is the ratio of a maximum relief amplitude to the minimum distance between the fault and the bounding free surface. The results show, that the influence of the relief should be taken into account for strain measurements.

The paper presents new results in the geometry of stable polynomials. Using these results a fast search algorithm is launched to obtain a solution with the widest waveguide in the one-dimensional kinematic inverse problem.

Sturm–Liouville’s matrix problem is described which is equivalent to a system of equations for Rayleigh surface waves. A potential of the system is generated by a symmetrical matrix. Under the assumption that the matrix equals a constant, we analytically construct elastic parameters and density as functions of depth. The paper describes this analytical construction which is especially simple in the case of a Poissonian body, i.e. if \lambda=\mu. The medium parameters are determined from the D-matrix, and the modal structure of the Rayleigh waves on characteristic frequencies is analyzed. Examples of media are given in which Rayleigh waves do not propagate for the characteristic frequencies.

A class of elastic heterogeneous media is constructed analytically for the non-Poissonian case. Simple description of Rayleigh waves propagating in the medium is presented. The choice of the class is based on the elementary type of the potential in the corresponding matrix Sturm–Liouville’s problem, provided that the symmetrical matrix D, which determines the potential, is assumed to be constant. This assumption allows us to derive a differential equation for shear modulus and to investigate its solutions. Remaining parameters of the medium can be expressed by shear modulus. Possibility of reconstructing the medium parameters from the potential is considered. Modal structure of Rayleigh waves on characteristic frequencies is analyzed.

Real in-field experiment with excitation of the stationary field of horizontal vibrations by a monochromatic horizontal vibrator is described. The purpose of the experiment was to check practical possibility of determining velocity dependence on depth with the help of the Love modes characteristics on the fixed frequency of the source. Results of the data processing are presented and agree with results of the standard methods in general.

A method for constructing the uniform asymptotic expansion of reflected and head wave fields in elastic medium, which is valid both in the vicinity and far apart from critical ray, is outlined. The formula for the main term of the uniform series, which describes the main part of the reflected P- or S-wave field, is derived. It appears to be of the same form as in the ray series, but with reflection coefficient dependent on frequency. In spite of apparent complexity of the formula for the reflection coefficient it is simple for calculations. For a set of frequencies the reflection coefficient of P-wave has been calculated for a contact of two homogeneous half-spaces as a function of the angle of incidence.

The boundary-value problem for such flow is considered. A new simple approach to the asymptotic decomposition of eigenvalues and eigenfunctions is described.

The paper is a continuation of the paper with the same title Computational Seismology in the previous issue. A complete asymptotic decomposition of induction operator eigenvalues and eigenvectors is constructed in the kinematic dynamo problem of magnetic field generation by a motion of conducting medium with internal scaling along three spatial variables in a sphere with the α-effect.

The work presents the numerical analysis of the gravitational instability in a two-layer viscoelastic model and the propagation of tectonic waves in lithosphere – asthenosphere. The analysis includes horizontal forces. Modes of maximum instability and rates of growth were computed for the case of density inversion. These rates increase several times with horizontal stresses and spatial wavelengths also increase approximately twice. In the absence of density inversion fast tectonic waves are found. The variation of this velocity is computed for various geometries and physical parameters, including horizontal forces.

This paper studies the theoretical performance of the MUSIC (Multiple Signal Classification) and MN (Minimum–Norm) algorithms in estimating the frequencies \{\omega_k\} of multiple sinusoids in noise. Both algorithms are based on principal components of signal correlation matrix of size m. An asymptotical analysis of frequency distribution is given under conditions, that the number of observation N\to \infty and m is fixed or it increases with N. Surprisingly, the frequency accuracy of order o(N^{-1}) is impossible for m \simeq N – c.

The problem of linear multidimensional system parameter estimation is discussed. Estimation is based on observations of input and output obscured by noise. In geophysics such problems arise in analysis of seismic response of media and structures using experimental data. The lower bounds of estimator variance are found and optimal estimation procedures are designed. Stability of estimator quality are examined, when statistical noise properties are varied.

In this work the Syntactic Pattern Recognition Scheme is constructed for the purpose of classification of multichannel seismic records in case of infinite cardinality of grammars, including the tools for verification of the classification reliability. It is applied for classification of three-component strong motion records from different geological regions.

Design and operational principles of a high resolution microbarograph with capacity transducer are described. The microbarograph can be used in investigation of seismic noise caused by local fluctuation of atmospheric pressure in problems of pressure noise compensation.