History

The Institute was formed of the Department of Computational Geophysics at the Institute of Physics of the Earth (IPE), Russian Academy of Sciences (RAS). The Institute’s organizer and first Principal Manager (1990–1998) was the outstanding geophysicist Vladimir Isaakovich Keilis-Borok. The research at the Department and at the Institute aimed at applying modern mathematical methods to the study of critical (extreme) phenomena in the solid earth, earthquake prediction, the study of regional tectonics, and the analysis of seismicity and source regions of earthquakes. International recognition was awarded to the work conducted at the Department and the Institute in the identification of underground nuclear explosions, the development of methods to deal with forward and inverse problems in seismology, the analysis and simulation of nonlinear dynamics in application to systems of tectonic faults, the prediction of large earthquakes and social-economic impact, as well as studies in seismic risk and optimization of prevention measures.

A History of the Department of Computational Geophysics at the Institute of Physics of the Earth

The 1960s saw revolutionary events in science due to the rapid development of computer technologies. Simultaneously, the amount of data in geophysics began to increase due to the development of seismic networks. It became necessary and possible to advance essentially novel approaches to geophysical problems using mathematical and statistical methods with the help of computers. These challenges were met by Academician M.A. Sadovsky, who headed the IPE during over 30 years since 1960, and supervised research in the physics of the solid Earth in the USSR RAS by setting up the Department of Computational Geophysics at the IPE. V.I. Keilis-Borok, who headed the Department, published the paper Seismology and Logic in English in 1964 [1] (in Russian in 1968 [2]) where he formulated the principal problems that faced seismology under the new conditions, stressing the importance of mathematical methods and computers for the analysis of geophysical data. Essentially this paper defined a new line of research in geophysics, viz., computational seismology. The research conducted at the Department in forward and inverse problems of seismology has played the key role in the development of this line of research. Thanks to scientific contacts with leading geophysicists worldwide, which were made, in particular, during the Geneva talks on the nuclear test ban treaty, the research was pursued on a broad international basis at the very outset. In 1964 the International Committee on Geophysical Theory and Computers was set up (subsequently renamed the Committee on Mathematical Geophysics), and its Chair was invariably V.I. Keilis-Borok during 15 years. The exchange of ideas and joint research have played a major part in the development of this novel line of research.

The Main Areas of Research at the Department and the Institute

Theory and interpretation of seismic wave fields

Seismic risk assessment

Recognition of potential sites of large earthquakes

Earthquake prediction

Models for the dynamics of the lithosphere and seismicity

Computational geodynamics

The study of geophysical fields

The study and prediction of extreme events in complex systems

References

Theory and Interpretation of Seismic Wave Fields

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The analysis of dispersed signals, surface waves in the first place, is an important part among the various aspects of the vast problem of seismic signal analysis. By the late 1960s, a computer technology had been developed for extracting dispersion curves and other characteristics of surface waves from digital seismograms, or Multichannel Filtering. However, this method involved systematic errors in dispersion measurement, when the amplitude spectrum of a signal rapidly varied with frequency. A.V. Lander, A.L. Levshin, and V. F. Pisarenko [3, 4] put forward new methods to overcome these drawbacks, resulting in the development of a new procedure, Frequency-Time Analysis (FTAN). A.V. Lander [5, 6] gave a new impetus to the development of this method, with a detailed analysis of optimality for the FTAN being contained in the book [7]. The computer software that implements the method has found wide acceptance worldwide.

The problem of detecting hidden periodicities in observations of seismicity and other geophysical fields was addressed by V.F. Pisarenko who proposed a regular method for estimating the parameters of harmonics (amplitude, phase, and frequency) where it was not required to search for a minimum on a multidimensional space [8-10]. The method of high resolution spectral analysis put forward in these publications was subsequently generalized and refined, giving motivation for the development of new high resolution methods. A comparative quantitative analysis of the resolution attainable with these methods is due to G.M. Molchan [11-13].

The monitoring of underground nuclear explosions was in the forefront during the 1980s. It became therefore important to be able to detect low-amplitude seismic signals in observations at small seismic arrays. Teams of researchers where A.F. Kushnir was an important member, proposed [14-16] optimal algorithms for detection, identification, and estimation of parameters in signals recorded upon the background of spatially correlated noise. Some of these algorithms are now part of the software available at the International Data Center, Vienna where they are employed for nuclear test verification [17-20].

Since the processing of seismic data was becoming more and more automatic, it was necessary to develop methods for optimal identification of the seismic phases later than the first arrivals based on their onset times. An algorithm for such identification has been developed by a team with participation of V.I. Keilis-Borok and A.L. Levshin [21]. The software implementation of the algorithm that was carried out at the International Seismological Centre, UK in 1966 is still in use today.

Thanks to the studies of V.I. Keilis-Borok, A.L. Levshin and other scientists, an approach based on the theory of differential operators has been developed [22-24], which resulted in computational algorithms for simulation of surface wave fields in arbitrary vertically and radially varying media. For example, numerical simulation has provided a first-ever explanation of the origin of Lg, Rg, and Sa phases in wide use in seismology [25].

The passage from the theory of surface waves in a vertically varying earth to a similar theory for a radially varying sphere involved certain difficulties owing to more complex boundary problems. M.L. Gerver and D.A. Kazhdan analyzed the uniqueness of the velocity section to be found from knowledge of a Love wave dispersion curve to find that the passage from the spherical model to the plane model can be done exactly by a change of variables [26]. The cycle of research for a similar transformation in application to Rayleigh waves [27, 28) is due to V.M. Markushevich and his associates.

The ray method for describing body wave fields was in wide use during the 1960s. However, the method could not incorporate interference effects due to the existence of thin layers, thin-layered sequences, and transition zones. L.I. Ratnikova and T.B. Yanovskaya have developed algorithms for the simulation of wave fields in such media [29, 30]. The algorithms were widely used in seismic prospecting, as well as in general and engineering seismology.

A.L. Levshin and B.G. Bukchin took part in the research devoted to the passage of surface waves through a vertical and an inclined contact [31-33]. This has made it possible to find approximate solutions in the problem of surface wave propagation incorporating phenomena arising at boundaries of plates and major tectonic blocks. When combined with the ray method, this approach succeeded in describing the most important characteristics of surface wave fields in a realistic blocky earth model [7, 34].

The theory behind the interpretation of travel time curves had been, until the 1960s, based on mathematical results that are only applicable to velocity models where the velocity increases monotonically with increasing depth. B. Gutenberg’s discovery of the low velocity layer in the upper mantle required a revision of the methods used in the interpretation of travel times. The pioneering work of M.L. Gerver and V.M. Markushevich [35-37] provided a rigorous theory for the forward and inverse seismic problems, demonstrated and described in rigorous terms the unremovable nonuniqueness in travel time inversion, when waveguides are present. The graphic visualization of this nonuniqueness in velocity section construction in the form of a figure like a giraffe has become classical. The results of further work due to M.L. Gerver on this topic are reported in [38].

The studies of E.N. Bessonova and V.M. Fishman with colleagues initiated the development of the τ(p) method [39, 40]. These publications demonstrated an effective method for transforming travel time curves into a τ(p) function and converting it into a velocity function; the method is applicable both to refracted and reflected waves. It was used by seismologists all over the world and developed further for the processing of great amounts of seismic data in seismic prospecting, deep seismic sounding, and in seismology.

The studies of M.L. Gerver in inversion for the one-dimensional wave equation [41, 42] turned out to be needed for finding the velocity and density earth sections from the spectrum of free oscillations of the Earth, as well as for inversion of reflected wave fields into velocity sections in seismic prospecting.

M.A. Brodsky and A.L. Levshin [43] have generalized studies where the asymptotic theory of free oscillations of the Earth based on the WKB approximation was first developed. The theory made it possible to combine the mode approach and the ray approaches for a description of free oscillations into a general theory and to reformulate the inverse problem in the determination of a velocity section from the frequencies of free oscillations in terms of the τ(p) method. A.L. Levshin has explained these results in terms of constructive interference between elementary body waves [44].

In the mid-20th century, the inverse problems of seismology were not formulated rigorously enough. Accordingly, the methods dealt with particular cases. The pioneering work of a research team with participation of V.I. Keilis-Borok [45] formulated the inverse problem of seismology in application to travel times and amplitudes of body waves, and proposed a solution by Monte Carlo. V.P. Valyus has developed a more effective algorithm to search for an inversion that was given the picturesque name of Hedgehog [46]. This algorithm combined a random search for possible inversions with a deterministic investigation of subregions around emerging solutions. The method was applied by V.P. Valyus, V.I. Keilis-Borok, and A.L Levshin to a set of diverse seismological data: travel times and amplitude curves of body waves, and to dispersion curves of surface waves [47, 48]. Subsequently the approach was widely used in crustal studies using body and surface waves.

B.G. Bukchin and A.L. Levshin have developed a method for determining the earthquake source parameters [49-51] based on a joint use of compressional wave polarization and surface wave spectra incorporating the lateral variation of earth structure. A search in the space of source parameters estimates the seismic moment tensor, depth of focus, and second-order moments, which characterize the spatial and temporal dimensions of the source, the direction and velocity of rupture. The method was implemented by A.A. Egorkin and A.Z. Mostinsky as a program package that was used in studies of source mechanisms for large earthquakes and for identification of underground nuclear explosions [52-58].

The northeasternmost Asia had long been one of the few large seismic regions where no acceptable plate tectonic explanation of the origin of seismicity could be found. Based on available data, it looked as if the long Chukchi seismic belt did not have closure on the west, so that one could not succeed in reliably identifying the hypothetical Beringia plate. A.V. Lander used more recent seismic data to identify the Koryak seismic belt, which confirmed the existence of the plate [59]. He developed, in collaboration with M.N. Shapiro, a model that explained significant differences of the Kamchatka segment of the present-day Kuril-Kamchatka subduction zone from its southern Kuril segment, appealing to a recent collision between the Kronotsky arc terrane and the rest of Kamchatka. This led to the decay of the northern segment of the older subduction zone (where the Sredinnyi Range volcanic belt is at present) and to the origination of the present-day zone and of the Eastern Volcanic Belt [60].

A.V. Lander has carried out a study of systematic errors in the operation of regional seismograph networks and developed a procedure to remove them. He made a table of station corrections to travel times by a procedure that performed a joint determination of hypocenters and station corrections. The corrections thus determined were then used to find the aftershock hypocenters of the Olyutorsk earthquake [61].

One important characteristic of a seismic event is the moment tensor, which determines its magnitude and rupture mechanism. The moment tensors for large earthquakes are determined using only records of long period surface waves, because the records of body waves involve high noise levels. The global CMT catalog for earthquakes with magnitude 7.5 or greater contains over 30% of such events. At the same time, it is a known fact that the moment tensor of a shallow seismic event cannot in principle be determined uniquely based on records of surface waves alone. B.G. Bukchin and A.Z. Mostinsky studied this nonuniqueness working in collaboration with French colleagues [62]. They have derived the requirement for the existence of equivalent double couples and a complete description of these. They have shown that the “best double couple” as reported by world seismic agencies cannot provide an adequate description of earthquake ruptures within the approximation of plane rupture, and that the observed significant difference between the deviatoric moment tensor and the double couple can also be caused by the nonuniqueness in question. Methods and associated software have been developed to estimate the nonuniqueness and to reduce it by incorporating long period polarities of direct P waves.

Seismic Risk Assessment

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Seismic risk assessment was also studied at the Department during the 1960s. Earthquake hazard was previously quantified as maximum intensity of possible shaking at a site. The frequency of occurrence for maximum intensity could vary by a few orders of magnitude at different sites within a zone of uniform intensity. However, this did not affect the regulations for construction of engineering facilities that differed in critical importance, service term, and payback. The magnitude of earthquake impact could be so large, and the insurance practices were so imperfect, that no country over the world, no matter what was the economic system concerned, could approach the assessment of earthquake hazard in economic terms. The introduction of standardized mass construction in the USSR during the 1960s has stimulated economic approaches to the efficiency assessment of earthquake prevention measures. A method was proposed for finding shakeability (this was a new parameter to characterize earthquake hazard, meaning recurrence of shaking) and the (average) economic effect of earthquake prevention measures [63]. For example, seismic regionalization gave rise to a new discipline, seismic risk, which combined seismology, tectonics, and strong seismic motions with economics and demography. The purely seismological aspect of risk, shakeability, has long been the subject of detailed studies by USSR AS Corresponding Member Yu.V. Riznichenko and his school. As concerns applications, it has become clear that the shakeability concept allows assessment of the probability distribution of shaking at a site for a fixed time interval assuming the Poisson hypothesis of earthquake occurrence and the deterministic nature of felt effects. Shakeability therefore has made it possible to find earthquake-induced loss at a site instead of average loss, which is important for most engineering purposes.

However, the history of construction on the Baikal—Amur Railway (BAR) has also revealed some drawbacks in this methodology. The BAR project was in the approval phase in 1968. The railway path 980 km in length lay in a zone of high earthquake hazard, and this required an economic feasibility study to determine whether the BAR railway should be strengthened against earthquake shaking. In view of the special status of this “construction project of the century”, global average assessment of the losses, as well as point distributions of the damage, were inappropriate. What was required consisted in assessment of low-probability total damage in relation to earthquake-resistant protection measures and the period of use. A problem like this was unique in engineering practice. M.A. Sadovsky understood its importance and assigned it to the Department of Computational Geophysics. A team of mathematicians and programmers was set up with the key role played by G.M. Molchan. The economic issues that arose were dealt with in consultation with the outstanding mathematician Academician L.V. Kantorovich, who subsequently became a Nobel Laureate for economics, while seismological questions were assigned to I.L. Nersesov. The problem was solved in 1969, and the final conclusion was paradoxical for the time: in spite of serious palaeoseismological evidence for past catastrophic earthquakes in the construction path, hence in spite of construction regulations, any earthquake-resistant strengthening of the railway was to a high probability questionable owing to its being unprofitable.

These new approaches were later transformed to become a general probabilistic conception of seismic risk that was suitable for the analysis of many economic and social problems related to seismicity [64-67]. The methodology has considerably extended the widespread risk methodologies. It has been tried for use in Italy and the Caucasus [68, 69] and was recommended by the UNESCO for use to analyze seismic risk that earthquakes pose to people living in major cities worldwide [70, 71], as well as being employed in optimization of insurance payments for seismic regions [72]. G.M. Molchan has developed an hierarchical approach to the description of spatial seismicity models for purposes of seismic risk assessment [73, 74]. It can be asserted today that the originally cautious, occasionally downright adverse, attitude to seismic risk has come to be replaced with the recognition of necessary economic probabilistic analysis of earthquake hazard.

The work of V.F. Pisarenko in statistical estimation of maximum possible earthquakes [75-78] and his studies [79-91] in collaboration with M.V. Rodkin, has contributed significantly to the development of approaches to predicting the possible damage caused by natural disasters.

Macroseismic data are actively employed in risk analyses. G.M. Molchan and T.L. Kronrod proposed [92-95] an original method for the analysis of isoseismal shapes and have proved that these are related to source mechanisms. The method was used to develop an atlas of macroseismic data for Europe.

Recognition of Possible Sites of Future Earthquakes

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In the late 1960s to early 1970s the Department began work in prediction of possible sites for future earthquakes. The formulation and solution to this problem have required a combined effort on the part of scientists of many different specialties. Along with researchers of the Department itself (A.D. Gvishiani, A.M. Gabrielov, I.M. Rotvain, V.G. Kosobokov, A.A. Soloviev, A.I. Gorshkov and others), the people who took an active part in this research included Academician I.M. Gelfand and a team of mathematicians he headed [96, 97], as well as the geographer and geomorphologist E.Ya. Rantsman¸who was of the school of Academician I.P. Gerasimov [98]. The analysis involved formulating new concepts like morphostructural lineaments, intersections, and generally, a scheme of morphostructural regionalization (MSR) for a region. The principles of the MSR were formalized with participation of mathematicians [99], thus ensuring control and reproducibility of results in an area that seemed almost a humanitarian one at first. Comparative analysis of seismicity and MSR schemes gave rise to the hypothesis that the epicenters of great earthquakes in a region are confined to morphostructural intersections. A special procedure was applied to the data to corroborate the hypothesis for most seismic regions worldwide. A new problem arose, one in pattern recognition, viz., classifying all morphostructural lineaments into highly seismic ones where large earthquakes can occur and low seismic intersections where only smaller earthquakes are possible. The recognition work was conducted all over the world, so foreign scientists also took part. The result for California was derived in collaboration with outstanding American geophysicists, professors F. Press and L. Knopoff [97]. The research procedure was reliable enough, as evidenced by the following fact. Since the completion of the work (1976), 15 large earthquakes have occurred in California with magnitudes above 6.4 (which is a threshold value chosen to define large California earthquakes). The epicenters of all these earthquakes turned out to lie in the vicinities of the intersections that had previously been recognized as highly seismic ones.

Earthquake Prediction

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The successful work in the recognition of future earthquake locations created prerequisites for attacking the problem of predicting the time of occurrence for an earthquake and inspired some hope that the problem can be solved as well. The work along these lines started with the search for earthquake precursors assuming the following necessary requirements to be fulfilled: rigorous formalization and the use of reliable raw data that would be accessible and homogeneous over different seismic regions. This explains the choice of earthquake catalogs as the basis for precursors. One of the first precursors thus identified was the Sigma precursor as detected by V.I. Keilis-Borok and L.N. Malinovskaya, which showed the growth of seismic activity before a large earthquake [100]. Subsequently, V.I. Keilis-Borok and I.M. Rotvain in collaboration with L. Knopoff formulated and formalized the aftershock burst precursor [101]. A modification of that precursor in order to be able to make use of catalogs with undetermined magnitudes of moderate and small earthquakes [102] has yielded one of the first successful forecasts, namely, the forecast of the 1980 Irpinia, Italy earthquake. A.G. Prozorov formulated an algorithm of earthquake prediction based on remote aftershock data [103, 104].

Along with the search for precursors, the problem of a theoretical basis for earthquake prediction was raised for solution. V.I. Keilis-Borok proposed considering a large earthquake as a critical phenomenon in a nonlinear dynamic system, viz., the Earth’s lithosphere. The prediction problem was formulated as the search for symptoms that could be diagnosed as an instability arising in a nonlinear dynamic system [105]. The instability symptoms must be largely universal with respect to a broad class of dynamic systems. This approach has significantly expanded the prediction area, enabling the scarcity of historical observation to be compensated for by the universality of the symptoms. When dealing with the case of large earthquakes, the symptoms include the following characteristics of background seismicity: activity, expressed as the rate of main shocks; the departure of activity level from the long-term trend; the concentration of epicenters; the rate of aftershocks following the main shocks of moderate-sized earthquakes. A family of algorithms has been developed for intermediate-term earthquake prediction, namely, the M8 [106-108] and CN [109, 110]. The M8 algorithm was developed by V.G. Kosobokov for prediction of great (magnitude 8.0 or greater) earthquakes worldwide using integral estimates of the four characteristics listed above. The CN algorithm was developed by I.M. Rotvain for prediction of large earthquakes on the regional level. It is being used for seismicity monitoring in over twenty seismic regions.

The research conducted at the Department of Computational Geophysics of the IPE was recognized as important, resulting in the decree of the Soviet Government to set up the International Institute of Earthquake Prediction Theory and Mathematical Geophysics on the basis of the Department. The decree was influenced by the successful forecast of the 1989 Loma Prieta earthquake in the United States [111] with attention afforded this forecast on the part of Presidents of the USSR and the United States, M.S. Gorbachev and R. Reagan. The Institute has existed since January 1, 1990. V.I. Keilis-Borok has done enormous organizational work to advance the institute; he invoked the help of a new team of mathematicians headed by Academician Ya.G. Sinai. The team produces fruitful results in the theory and methodology of predicting critical phenomena in nonlinear chaotic systems.

The Institute began in 1992 work in collaboration with the US Geological Survey to conduct a monitoring, unique as to its duration, of regions where great earthquakes can occur (those with magnitudes M ≥ 8.0). The prediction is based on the use of the M8 algorithm [112], and its results are at once reported to over one hundred scientists and experts worldwide. The experiment has succeeded in predicting a number of great earthquakes, including the October 4, 1994 M = 8.1 Shikotan earthquake and the December 3, 1995 M = 8.0 Iturup earthquake in Russia; the February 27, 2010 M = 8.8 Chile earthquake, the March 11, 2011 M = 9.0 Japanese earthquake, and the April 11, 2012 M 8.7 Sumatra earthquake. Summing up the results of this experiment, one can safely assert that it has proved the feasibility of prediction of great earthquakes in principle: the M8 alarms capture 30% of target events randomly, while actually predicting over 50% of these events. The results also corroborate one important postulate underlying the M8 algorithm, viz., that the precursory processes of great earthquakes occur over areas about 1500 km across. This statement was viewed with skepticism by the seismologists. However, the realities of the 1992 Landers, US earthquake turned out to be in complete agreement with remote interaction on this scale.

The work of G.M. Molchan on the statistical analysis of earthquake prediction results [113-117] and on the strategies of response to prediction [118-123] were most important for assessment of earthquake prediction algorithms and the approaches to a rational use of prediction results.

V.I. Keilis-Borok has reviewed the work in the development and application of intermediate-term earthquake prediction to formulate four paradigms in earthquake prediction [124]: the main types of precursory phenomena; remote interaction; similarity; a dual nature (universality and concrete occurrences in different regions) of precursory phenomena.

An important contribution to the development of approaches to the problem of short-term earthquake prediction is due to some recent work of P.N. Shebalin in the development of the RTP algorithm [125, 126] and an algorithm [127, 128] based on the initial part of aftershock sequences. The RTP algorithm analyzes a short-term (a few months before the main earthquake) precursor that reflects an increasing distance of correlation between small earthquakes along with intermediate-term (a few years) precursors; the analysis proceeds in the reverse order to their actual occurrence. The algorithm is based on the conception of self-organization in a fault system, with the self-organization attaining its maximum by the time of the earthquake; the algorithm can detect precursors that remain invisible for straightforward analysis. The algorithm has been tested and has predicted, among other events, also an earthquake in southeastern Hokkaido (September 25, 2003) with magnitude 8.1 and two earthquakes that occurred around Simushir Island of the Middle Kuriles (November 15, 2006 and January 13, 2007) whose magnitudes were 8.3 and 8.2.

Models for the Dynamics of the Lithosphere and of Seismicity

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Since the 1980s a simulation approach has been actively developed in the study of the dynamics of the lithosphere as a nonlinear system. The goal of this work was to simulate those symptoms of instability that underlie the prediction algorithms in use and to study the limits of predictability.

The model developed by A.M. Gabrielov and A.A. Soloviev for the dynamics of lithosphere blocks [129] has enabled them to formulate the inverse problem, viz., the determining the tectonic driving forces (including mantle flows) based on the surface distribution of observed seismicity in a fault system [130]. Explanation of the key features in plate tectonics was sought for in a global model simulating interaction between tectonic movements with incorporation of mantle flows and earth sphericity, in a model simulating the influence of fluids on the dynamics of a fault system, and in a generalized model for an arcuate subduction zone [131-133]. Dynamic models have been developed for specific regions: Vrancea, Roumania; the Near East; the western Alps; The Sunda island arc; the Apennines; Tibet and the Himalaya [130, 134-137]. These studies were conducted in collaboration with scientists affiliated with the Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences and colleagues abroad.

M.G. Shnirman and his team considered hierarchical models of defect generation [138]; they succeeded in obtaining phenomena like self-organized criticality that are observed recently in several areas of research ranging from physics to economics and social sciences. This allows us to explain the appearance of power law relations in the distribution of seismicity as a natural reflection of global natural laws, and to apply the knowledge obtained in other branches of science to geophysics. These models are of interest for the analysis of predictability of large events based on the data of background activity [139-145] and for the analysis of possible prediction of earthquake times [146]. This line of research is closely related to the work of I.V. Kuznetsov and I.M. Rotvain in seismicity modeling based on two-dimensional cell automata [140, 147-152] of the Bak-Tang-Wiesenfeld model (the BTW model).

A.M. Gabrielova, I.V. Zalyapin, and V.I. Keilis-Borok collaborated with their colleagues abroad to obtain a cascade model of seismicity based on the meeting dynamics of direct and inverse cascades [153, 154]. This is different from the case of turbulence where only the direct cascade occurs. The model was used to reproduce the main properties of actual seismicity, viz., the seismic cycle, the Gutenberg-Richter relation, clustering of events, and remote interaction, as well as several statistically significant precursors of large earthquakes based on small earthquakes.

Computational Geodynamics

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B.M. Naimark initiated work in computational geodynamics. He studied [155-164] gravitational stability of a vertically varying viscous liquid; instability and growth of initial disturbances in a two-layered viscous incompressible liquid; the existence and uniqueness of the solution to the Rayleigh-Taylor and Rayleigh-Bénard problem; simulation of mantle flows; and the inverse problem of gravitational instability. These kinds of problem are related to determination of crustal and mantle structure, as well as to the development of models to simulate the kinematic dynamo.

These studies were followed up by joint publications of B.M. Naimark and A.T. Ismail-Zade [165-176] which were concerned, among other things, with convection in the mantle and the formation of sedimentary basins. Considerable contribution into these studies is due to the fruitful cooperation with scientists at the Institute of Mathematics and Mechanics, Ural Branch RAS; the results include algorithms and software for calculations of 3-D motions on parallel computers incorporating thermal and gravitational action [177-180]. A.T. Ismail-Zade has preserved this cooperation and continued his research in geophysical hydrodynamics in cooperation with several Russian and foreign colleagues. In particular, they have found the shapes of mature salt diapirs in sedimentary basins for the 3-D case and the shapes of thermal diapirs that are produced when hot material rises from mantle interiors [181-186]. They have developed a method for numerical solution to the retrospective problem of gravitational advection (the Rayleigh–Taylor inverse problem) for the case of three spatial variables. For the first time in world practice, quantitative approaches have been proposed to the recovery of the evolution of thermal mantle structures (and of palaeotemperatures) based on recent movements and present-day temperatures [187-195]. The results were presented in a monograph [196].

O.M. Podvigina has derived analytical and combined numeric—analytical solutions to several problems on the stability of flows near the convection threshold in a plane horizontal layer [197-210]. She considered linear and weakly nonlinear stability for flows in the case of free horizontal interfaces in the presence or absence of rotation, as well as in the case of rigid interfaces in the presence or absence of rotation and magnetic fields. These results have considerably contributed to the understanding of how convective systems behave, including the influence of rotation, an external magnetic field and Prandtl numbers on the type and stability of convective flows near the convection threshold. This is important for a qualitative explanation of the processes that occur in segments of the mantle and the Earth’s core.

The Study of Geophysical Fields

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An explanation of the existence of magnetic fields on the Earth, planets, and stars requires a study of the magnetic dynamo problem for different velocity fields of a conductive fluid. The complete set of magnetic hydrodynamic equations was studied by M.M. Vishik [211-213] where, among other things, he constructed explicit solutions on a special Riemann multifold to show that the velocity fields considered can act as a magnetic dynamo. These results can have relationships to descriptions of the evolution of the Earth’s magnetic field. He developed [214-216] a theory of a spatial periodic dynamo in a flow with an internal scale and found a numerical solution to the spectral problem for the induction operator in the flow.

V.A. Zheligovsky has carried out a numerical analysis of the kinematic problem dealing with the generation of a magnetic field due to the Beltrami flow of a conductive fluid in a sphere [217-220], studied the excitation of a magnetic field due to the motion of a conductive material with an internal scale in an axisymmetric volume and in a sphere in the presence of the α-effect, and derived the complete asymptotic expansion of eigenvalues and eigenvectors of the induction operator [221-223]. He also carried out a cycle of studies to develop a mathematical theory for the stability of magnetohydrodynamic regimes under long-scale disturbances in the linear and weakly nonlinear cases [224-231].

A.A. Soloviev [232-241] discovered possible generation of a magnetic field by the Couette– Poiseuille flow in a conductive fluid and determined the requirements for a magnetic dynamo to occur.

A .V. Khokhlov studied the fundamental problem of recovering the spatial description of the Earth’s observed magnetic field based on its partial measurements at a surface [242-248]. He considered two cases widely known in practice, namely, when measurements of a magnetic field at a point result in either the total vector only, which arises when geomagnetic observations are conducted by satellites, or when the vector direction only is known, as is the case for paleomagnetic data. Several theorems have been proved showing that the recovery is unique.

The Study and Prediction of Extreme Events in Complex Systems

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Relying on the universality of instability symptoms, V.I. Keilis-Borok developed an algorithm in 1981 to predict the outcome of the US presidential elections in collaboration with the American scientist in political sciences professor A. Lichtman [249]. The same methods of pattern recognition were used as those involved in developing the M8 prediction algorithm. A. Lichtman published his electoral forecasts in 1984, 1988, 1992, 1996, 2000, 2004, and 2008 beforehand, and the forecasts came true. A similar algorithm was developed to predict the outcome of senatorial elections in the US [250]. The pattern recognition technique was also applied to the development of algorithms to predict the beginning and end of economic recessions in the US [251, 252], the beginning of rising unemployment [253], and the beginning of periods when the murder rate in a megacity was liable to increase [254]. The algorithm was used to predict [255] the beginning of unemployment growth in the US in December 2006, which was precursory to an economic recession in the US followed by a worldwide economic crisis. The IEPT RAS researchers conducted this work in cooperation with foreign economists and sociologists, namely, professor D. Stock, US, Foreign Member of the RAS professor M. Intriligator, US, and professor C. Allègre, France. The results for algorithms that were designed to predict extreme events in socio-economic systems are reviewed in [256].

Approaches to prediction of crime dynamics in Russian cities are being developed by I.V. Kuznetsov and associates [257].

1111Литература

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A.I. Gorshkov, A.A. Soloviev, P.N. Shebalin – “To the 30th anniversary of ITPZ RAS.

A.L. Levshin – “Some history.”