Title of research project: Conversion of R&D in missile navigation to mathematical problems of ecology and natural hazards.

Reference number of the grant: ISTC-008-94

Period of performance: 1994 to 1997

Project Manager: Dr. A.A. Soloviev


The technical approach and methodology of the project are based on the following branches of applied mathematics which have applications both in navigation and control of missiles and in mathematical geophysics: - analysis of behavior of dynamical systems described by differential equations (ordinary and with partial derivatives) and inverse problems for them; - pattern recognition and scene analysis; - computer analysis of geological and geophysical data.

The following six main directions of investigations are selected for joint researches of the scientists from the participating institutes.

I. Development of the theory of dynamical systems described by ordinary differential equations and differential equations with partial derivatives, with special attention to inverse problems.

II. Investigation of the dynamics of the lithosphere by modeling of its behavior using differential equations.

III. Investigation of problems of geophysical hydrodynamics by computational methods.

IV. Inverse problems for determination of stress fields in the lithosphere by using seismological data and development of the complex valued polarization analysis of surface waves in frequency-time domain. V. Application of methods used in inverse problems of seismology for processing of data of double-frequency radio sounding of the ionosphere by two satellites. Application of the obtained results for investigation of ozone holes.

VI. Development of automatic methods based on scene analysis and pattern recognition for processing of data on geophysical fields with a view to detect tectonic structures interesting for investigation of ecology, natural hazards and mineral deposits.

The main result of the Project execution is that the former weapons scientists from the Institute of Mathematics and Mechanics, Uralian Branch of the Russian Academy of Sciences, assimilated in the process of joint work with the scientists from the International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, the subjects of the non-military research in mathematical geophysics, oriented to the problems of ecology and natural disasters. The participation of the large group of the scientists from IMM possessing the high mathematical level in the mathematical geophysics studies allowed to attain the important progress in solving of the problems under consideration. The following studies were made and the results were obtained within the framework of the selected main directions of investigations listed above.

Direction I. The stable to informational and computational errors algorithms of identification of inputs of dynamical systems by using inaccurate measurements of their outputs were developed. The problem of reconstruction of extremum intensities of forces acting on a system described by equations of parabolic and hyperbolic type with distributed parameters was considered. Such equations are used for instance to describe the processes taking place in the lithosphere. Three methods for reconstruction of intensity of disturbing actions from results of inaccurate measurements of a part of coordinates of phase trajectories were developed for systems described by nonlinear differential equations. The stable to informational and computational errors algorithms of solving of the problem of reconstruction of intensity of impact sources by using the data of sensory observations were obtained. The new mathematical method for prediction of development of a non-linear random process was developed. The correct expansion of the problems of achievableness of space elements was constructed under conditions of perturbation of integral restrictions. Several versions of application of this expansion in applied problems were developed including the elements of statements typical for analysis of seismological observations.

Direction II. The software for modelling of block structure dynamics with 2D movements of blocks was developed and the user's guide was prepared. Study of this software and the results of researches carried out by using it were one of the three main parts of the program of the Third Workshop on Non-Linear Dynamics and Earthquake Prediction held by the International Centre for Theoretical Physics UNESCO-IAEA 6 - 17 November 1995. After termination of the Workshop this software was distributed among the participants of the Workshop for using in their own studies. The algorithm and the software were developed for simulation of block structure dynamics in the case of 3D movements of blocks. The simulation of dynamics was carried out for different types of block structures including the abstract structures, which were used to study dependence of features of the synthetic earthquake flow on the structure separateness and the type of the specified movement of its outer boundaries, and the structures approximating the lithosphere structure of the specific regions for instance the Vrancea (Romania) region. It was found that the synthetic earthquake flow possessed the features of observed seismic flows: linearity of the logarithmic frequency-magnitude relation (Gutenberg-Richter curve), clustering of events, long- range interactions. The mathematical algorithm of the spherical model of the lithosphere dynamics which is intended for global simulation of dynamics of the lithosphere plates was developed. This algorithm with the simplifying assumptions was realised as a code for the multiprocessor computational complex; the test calculations were made. The new approach was developed for modelling of the lithosphere dynamics. This approach is based on consideration of the 3D massive of a large number (up to several hundreds of thousands) of diverse deformable elements. It allows to consider in the model the fields of stress, displacements, failures, temperature, and other characteristics describing the thermomechanics of the processes in the lithosphere. The algorithms and the software were developed to realise this approach on the multiprocessor computational complex. The calculations were made to determine the parameters of the model, to study the features of the synthetic earthquake flow, and to test the linearity of the logarithmic frequency- magnitude relation.

Direction III. Within the framework of modeling of formation and development of sedimentary basins a three-dimensional model of viscous flow induced by the subsidence of heavy bodies in the asthenosphere with boundary conditions of ideal sliding and adherence was formulated in strict terms. A numerical algorithm based on the Galerkin approach and tricubic spline approximation of physical parameters was developed for calculation of density, velocity, and the subsidence of the upper model surface as functions of time and spatial coordinates. The software was constructed for making calculations on a single processor and on parallel processors. The algorithm and the program of calculation of topography of the surface of crystal basin base for the 3D case were developed. The experimental calculations were made for the descent of a heavy sphere in the 3D rectangular model. Simulation of the descent in the viscous liquid of heavy thin eclogite lenses of different shapes were carried out for different locations of the lenses using graphics for visualisation of the process passing in time. A case of the 3D model of an intracratonic sedimentary basin evolution was computed. Surface topography was calculated and compared with real subsidence data. The problem about currents of a rotating fluid in a spherical layer was considered; this rotating fluid is a model of the Earth's outer fluid core. In the case of the ideal fluid the structure of the point spectrum of the problem was studied. In the case of the viscous fluid the completeness of the system of the eigen and adjoint eigen functions of the relevant Poincare's operator was proved. The measure of the vector field oscillatoriness was suggested. The bases that are ordered by the degree of the oscillatoriness were constructed in the space of solenoid vector fields of two types: the fields with a normal component equal to zero at the boundary of the layer ("sliding") and the fields which are zero at the boundary ("adherence"). In accordance with the Galerkin's method the approximations of the eigenmodes are constructed on the basis of these bases. The approach based on the theory of weak solutions of the elliptic equations and the Ritz's method was suggested for the problem of the characteristic SV-oscillations of the Earth. The algorithms and the software designed for a general case were tested by the simple examples. The case of the spherical symmetric Earth was studied. The presence of two types of eigenoscillations (torsional and spheroidal) was shown. The boundary problem for the elliptic equation with a small parameter, which vanishes in a part of a domain, at a highest-order derivative (laplasian) was studied. The asymptotic of the solutions of these problem was obtained.

Direction IV. The statistical techniques for reconstruction of tectonic stress and seismogenious strain fields were developed. These techniques can be applied without using advance computed source mechanisms. The software which realize these techniques was created. The new methods for reconstruction of seismic source moment tensor, depth, and source time function from the spectra of body waves and for reconstruction of the spatio-temporal moments of a seismic source from records of surface waves were developed. These methods were realized as MS Windows application. The stable to calculation errors algorithm, based on the Backus-Gilbert method, and the relevant software were developed for solution of the problem of 2D ray seismic tomography. The mathematical analysis of the ray model of seismic wave propagation in the inhomogeneous anisotropic media was made, and the algorithm and the software were developed for calculation of rays. The approach to solution of the inverse kinematics problem of reconstruction of characteristics of the inhomogeneous anisotropic media from seismic wave arrival times was formulated. The method is based on solving of the special problem of optimal control. The MS DOS and UNIX versions of the software for frequency-time and polarization analysis were developed and passed to Edinburgh Anisotropy Project of the British Geological Survey. Models of seismic source evolution have been constructed by using the methods developed for the following earthquakes: Khailino earthquake (1991), Suusamir earthquake (Kyrgizia, 1992), Barisaho earthquake (Georgia, 1992), Chile earthquake (1995), Thienzhu earthquake (China, 1996). Field investigations were carried out in the Central Koryakia during which the tectonic structure of Ivtigin uplift where Khailino earthquake, the greatest earthquake in the region, occurred in 1991 was studied. The earthquake catalog for the regions of Northern-Eastern Asia and Bering sea has been collected including all global and regional data up to 1994. The evidence of existence of a new seismic belt (Koryak belt) has been obtained and the modern lithospheric plate Beringia was discriminated. The parameters of the Beringia motion were estimated.

Direction V. The valley problem was studied which usually arises, when the inverse problem of determination of the function of density of electrons in the ionosphere is solved, and is caused by the presence of zones with low electron density in the ionosphere. These zones are called waveguides, or valleys. The usual procedure of inversion based on the theory of Abel's integral is no longer valid for solving of this problem. The valley problem was solved in the particular case when a radio signal propagates along the magnetic latitude. The methods developed in seismology to overcome the similar difficulty of presence of waveguides and the specific for the cold plasma dependence of the velocity of signal propagation on the frequency of radiation, namely, the Appleton - Hartree dispersion law were used. The scheme of the ionosphere sounding from the Earth surface with registration of the returning signal on the Earth surface was considered in details. A signal is transmitted on several fixed frequencies and one of two parameters is measured: the Doppler shift of the frequency or the travel time of the signal. The results of computer modeling show that the electronic density can be found with reasonable accuracy above the well pronounced waveguide. It was proved that the density cannot be determined inside the waveguide. The analysis of the problem in the case of radiosounding by using of two artificial Earth satellites was made. In this case it is possible to determine the electronic density inside the waveguide if the density satisfies some additional conditions which are not very restrictive. The method of determination of the electronic density developed for this case is applicable for any direction of sounding, not only for that along the latitude.

Direction VI. The mathematical criteria of extracting of the morphostructural territorial units by using the data of automatized processing of images of topographic maps, space photos, and digital maps of the relief were formulated for solving of the problem of automatized processing of geomorphological information for the purpose of morphostructural zoning. The program package MOST (MOrphoSTructure) was developed. It realizes the algorithms of preliminary processing and colour filtration of images of topographic and tectonic maps, the methods of vectoring of point images of linear objects, the algorithm of extraction of broken linear segments of the river system which correspond to morphostructural lineaments. The algorithms for noise filtration and extraction of linear structures on space photos of mountain regions were developed. The algorithms of processing of the field of relief altitudes were constructed to extract axial lines of mountain ridges and borders between plain and mountain parts of a region. These algorithms were tested by extracting of the axial line of the Main Caucasus ridge and the borders between it and the surrounding plains by using the real geophysical data. The principals of the construction of the informational and system providing of the program complex for automation of morphostructural zoning were formulated. The software was developed for visualization and processing of digital images of large volume. The numerical experiments were made to test efficiency of the developed algorithms and software. The initial information on several regions of the Caucasus was prepared for this purpose. The numerical images of the topographic maps (scale 1:1000000), the tectonic map (scale 1:1000000), and the space photo (scale 1:3000000) were prepared and referred to the same coordinate system. The parts of the digital map of relief were extracted. The experiments showed the possibility of automation of the main stages of morphostructural zoning. The method of fractal modeling of seismicity was developed. It is based on construction of the system of the compressing transformations. The attractor of the system approximates the initial set of epicentres. The program package REFRACT was developed to construct and to study the fractal model of the set of epicentres of the Southern California. The algorithms of processing of large arrays of points and continuos geophysical fields were formulated to storage economically the data and to reconstruct them promptly. The algorithms were tested on the examples of the field of epicentres of the Southern California and the field of altitudes of the mountain region and showed efficiency.