

2006 year, number 5
A.M. Aizenberg, K.D. KlemMusatov, M.A. Aizenberg*, H.B. Helle**, and J. Pajchel**
Institute of Petroleum Geology and Geophysics, Siberian Branch of the RAS, 3 prosp. Akad. Koptyuga, Novosibirsk, 630090, Russia * Norwegian University of Science and Technology, 7491, Trondheim, Norway ** Norsk Hydro, O & E Research Centre, N5020, Bergen, Norway
Keywords: Tip wave superposition method, integral reflection operator, reflection coefficient, head waves
Pages: 532540
Abstract >>
The use of integral reflection and transmission operators to account for boundary conditions follows from the theory of wave scattering at curved interfaces in layered inhomogeneous media obtained in our previous studies. In this respect it appears practical to investigate the possibility of applying these operators instead of planewave reflection and transmission coefficients to describe highfrequency wavefields by Kirchhofftype surface singular integrals common in forward and inverse seismic problems. We compare the integral reflection operator and reflection coefficient approaches in numerical experiments for single scattering of a spherical wave at a curved interface. We show that the use of the integral reflection operator allows eliminating artefacts associated with the use of the reflection coefficient and ensures including head waves.

B.G. Mikhailenko and G.V. Reshetova
Institute of Computing Mathematics and Mathematical Geophysics, Siberian Branch of the RAS, 6 prosp. Akad. Lavrentieva, Novosibirsk, 630090, Russia
Keywords: Seismic waves, acousticgravity waves, Laguerre transform, finitedifference method
Pages: 541550
Abstract >>
A numericalanalytical solution for seismic and acousticgravity wave propagation is applied to a heterogeneous `Earthatmosphere' model. Seismic wave propagation in elastic earth is described by a system of firstorder dynamic equations of the elasticity theory and propagation of acousticgravity waves in the atmosphere by the linearized NavierStokes equations. The solution algorithm combines the timedomain integral Laguerre transform, the finite integral Bessel transform along the radial coordinate, and the FD solution of the reduced problem along the vertical coordinate. The suggested algorithm is tested with numerical experiments for the heterogeneous `Earthatmosphere' model for different source locations.

V. Cerveny and I. Psencik *
Department of Geophysics, Faculty of Mathematics and Physics, Charles University, 3 Ke Karlovu, 121 16 Praha 2, Czech Republic * Geophysical Institute, Academy of Sciences of Czech Republic, Bocni II, 141 31 Praha 4, Czech Republic
Keywords: Viscoelastic anisotropic medium, polarization, homogeneous and inhomogeneous plane waves
Pages: 551562
Abstract >>
Particle motion of homogeneous and inhomogeneous timeharmonic plane waves propagating in unbounded viscoelastic anisotropic media is generally elliptical. Exception is linear polarization of P and S waves propagating along some specific directions. A typical example is a linear polarization of SH waves propagating in a plane of symmetry of a viscoelastic anisotropic medium. Two most important characteristics of the particle motion are the orientation of the axes of the polarization ellipse and its eccentricity. They both usually vary considerably with the direction of wavefront propagation, and with varying strength of inhomogeneity of the considered plane wave. The orientation of the P wave polarization ellipse generally differs from the direction of wavefront propagation, and it is usually closer to the direction of the energy flux. The orientation of the polarization ellipses of S waves often differs from the direction perpendicular to the wavefront propagation, and it is usually closer to the direction perpendicular to the direction of the energy flux. The eccentricity of the polarization ellipse depends particularly strongly on the inhomogeneity of the plane wave. For homogeneous plane waves, the particle motion is usually nearly linear, i.e., polarization ellipses have large eccentricity, and the eccentricity decreases with increasing inhomogeneity of the wave. For strongly inhomogeneous plane waves, the polarization ellipse becomes nearly circular, eccentricity being very small. The eccentricity of the polarization ellipse usually also decreases in the vicinity of singular directions.

L.A. Tabarovsky and M.I. Epov*
Baker Atlas, 2001 Rankin Road, Houston, Texas, 77073, USA * Institute of Petroleum Geology and Geophysics, Siberian Branch of the RAS, 3 prosp. Akad. Koptyuga, Novosibirsk, 630090, Russia
Keywords: Equivalence, resolution, observation distribution, confidence intervals, geoelectric prospecting
Pages: 563573
Abstract >>
A new method for estimating the quality of geoelectrical data inversion implies formalization of the concepts for horizontal and vertical resolution and equivalence. Various aspects of the suggested approach are illustrated by examples.

L.T. Santos and M. Tygel
Department of Applied Mathematics, IMECC, State University of Campinas (UNICAMP), CP 6065, 13081970, Campinas (SP), Brazil
Keywords: Reflection wave, impedance, inversion, AVO, AVA, AVP
Pages: 574581
Abstract >>
Amplitude versus offset (AVO) or amplitude versus angle (AVA) curves are nowadays regularly extracted from seismic data for various purposes of reservoir studies and characterization. After adequate processing, these curves represent the variation of the reflection coefficients with respect of offset or angle at points of key target reflectors, such as, for example, the top of a reservoir. Besides the information that can be obtained merely from their shape, AVO/AVA curves can also be used to invert more quantitative attributes, such as the intercept and the gradient of the reflection coefficient or, even better, the elasticparameter contrasts. For a commonmidpoint gather, the AVA curve is generally derived from its AVO counterpart by means of a wellknown expression that relates the reflection angle to offset. It is to be noted that a successful inversion of the soughtfor attributes is strongly dependent on the approximations of the reflection coefficient that are considered. The recently introduced reflection impedance concept provides an attractive approximation of the elastic PPreflection coefficient as a function of the ray parameter. In this sense, that approximation can be of value when amplitude versus ray parameter (AVP) curves are available from seismic data. It is to be noted that an AVP curve tend to be more reliable that its AVA counterpart. This is because the ray parameter, as a function of offset, depends on the RMSvelocity only, as opposed to the incidence angle, which also depends on the (more unstable) interval velocity. In this paper, we propose an algorithm to invert elasticparameter contrasts from AVP curves using the reflection impedance approximation of the PPreflection coefficient. First results shown on synthetic data indicate that the procedure may offer a promising alternative to existing methods of inverting reservoir attributes from AVO/AVA curves.

A. Stovas and B. Ursin
Norwegian University of Science and Technology (NTNU), Department of Petroleum Engineering and Applied Geophysics, S.P.Andersensvei 15A, NO7491 Trondheim, Norway
Keywords: Horizontally layered medium, velocity gradient, layer boundaries, layer velocity, Dix equation
Pages: 582586
Abstract >>
For a horizontally layered medium with isotropic layers with constant velocity gradient, it is possible to estimate the velocity function (gradient and velocity at the top of the layer) and thickness of each layer. From largeoffset PP seismic reflections one can estimate three traveltime parameters: the zerooffset twoway traveltime, the NMO velocity and a heterogeneity coefficient, using the shifted hyperbola approximation or a fractional approximation. From the estimated traveltime parameters at the top and bottom of a layer, it is possible to compute the thickness and velocity function of the layer. It is necessary to solve a simple nonlinear equation which also can be solved approximately. There are two solutions, corresponding to a positive and negative velocity gradient. Therefore, the sign of the velocity gradient must be chosen. When there is one solution, the velocity gradient in the layer is zero, and the result is the standard Dix equations.

E.A. Blias
75100 Signature Way, SW, Calgary, Alberta, T3H 2X9, Canada
Keywords: Reflection traveltimes, NMO velocity, traveltime inversion, shallow velocity anomalies, Dixtype inversion
Pages: 587603
Abstract >>
The paper addresses traveltime processing for 2D models with laterally inhomogeneous layers and curved interfaces in shallow subsurface. Analytical equations are derived to approximate the relation between shallow velocity anomalies and NMO velocity as a basis for traveltime inversion to obtain an initialapproximation velocity model. The new approach is applied to real field data from regions with shallow velocity anomalies.

G.A. Sobolev, A.V. Ponomarev, A.V. Kol'tsov, A.A. Kruglov, V.A. Lutsky, and Yu.V. Tsyvinskaya
Institute of the Earth's Physics, Russian Academy of Sciences, 10 ul. Bol. Gruzinskaya, Moscow, 123810, Russia
Keywords: Seismic activity, acoustic signal, water injection
Pages: 604617
Abstract >>
A model composed of silicate sand, crushed granite, and cement was subjected to biaxial compression, and the resulting deformations have been described. During the experiment lasting for several months, with a quasistatic level of applied stresses, water was repeatedly injected into the model. It has been established that when the volume of water injected into a zone of active fractures is small as compared with the volume of the model, the acoustic emission drastically increases. The shape of seismograms and curves of repeatedness of the acoustic signals recorded before and after water injection do not differ significantly, which implies that the effect is of trigger nature. The time succession of injectioncaused acoustic events is qualitatively similar to the swarm and, in some case, aftershock activity of earthquakes. Hence, this factor may have an effect on seismicity. Different types of time succession of induced acoustic activity are described by an equation known from the kinetic theory of strength provided that the parameters of activation energy and working stresses depend on time.

T.B. Yanovskaya and V.M. Kozhevnikov*
St. Petersburg State University, Institute of Physics, 1 ul. Ul'yanovskaya, St. Petersburg, Petrodvorets, Russia * Institute of the Earth's Crust, Siberian Branch of the RAS, 128 ul. Lermontova, 664 033, Irkutsk, Russia
Keywords: Surface waves, group velocity, upper mantle, Asia, anisotropy
Pages: 618625
Abstract >>
Upper mantle anisotropy in Asia was investigated by inversion of Love and Rayleigh wave group velocity dispersion curves obtained by the FTAN procedure at periods between 10 and 150 s along paths that traverse Central Asia and Siberia. Our data were supplemented with data from the Center for Imaging the Earth's Interior (Boulder, Colorado), and 2500 to 4000 paths, depending on period, were used altogether. Locally averaged group velocity dispersion curves for Love and Rayleigh waves obtained by 2D seismic tomography for different periods were inverted to velocitydepth profiles, jointly and separately for SV (Rayleigh) and SH (Love) waves. SH waves have higher velocities than SV waves in the crust and upper mantle to a depth of ~300 km. The velocity profiles were used to estimate the mean anisotropy coefficient over depth intervals from the Moho to 100 km, from 100 to 200 km, and from 200 to 300 km. Anisotropy is the most prominent between 100 and 200 km, i.e., in the asthenosphere, and in anticorrelation with asthenospheric velocity: the lower the velocity the higher the anisotropy coefficient. Anisotropy is the greatest beneath active deforming regions of orogeny and vanishing in stable cratonic areas.

N.I. Pavlenkova
Institute of the Earth's Physics, Russian Academy of Sciences, 10 ul. Bol'shaya Gruzinskaya, Moscow, 123810, Russia
Keywords: Upper mantle, seismic profiles, 2D inverse problem, Siberian Platform
Pages: 626641
Abstract >>
Several longrange profiles were shot with nuclear explosions within the Siberian Platform. The seismic records of both chemical and nuclearexplosion profiles have revealed new peculiarities of the uppermantle structure in the region. Seismic sections were constructed by ray tracing. To build the most informative starting model, the τ (V, x) method was applied, which implies construction of τtime sections using the refraction and wideangle reflection travel times for several velocity lines V = const. The modeling has revealed abnormally high (up to 8.48.5 km/s) uppermantle velocities beneath the ancient Siberian Platform and low ones beneath the young West Siberian Plate. The upper mantle is shown to be of layered structure, with reflecting boundaries at depths of about 100, 150, 240, and 320 km. The boundary N1 at depths of 70120 km is the most persistent; it often underlies lowvelocity zones. The existence of velocity inversion zones and reflecting boundaries might be accounted for by fluid concentration at particular depths of the lithosphere, which suggests its rheological layering. The asthenospherelithosphere boundary has not been recognized in the velocity crosssections.

V.N. Nikolaevskiy
Institute of the Earth's Physics, Russian Academy of Sciences, 10 ul. Bol. Gruzinskaya, Moscow, 123810, Russia
Keywords: Crust, faults, tectonics, geotherm, triaxial testing, strength limit, dilatancy, seismic interfaces, upper mantle, radon, helium, permeability
Pages: 642654
Abstract >>
A universal theory that accounts for the structure of fractured crust is suggested using data on strength and finite failure of rocks in highpressure and hightemperature triaxial deformation tests. Correlation of the experimental results to the seismic cross section of the crust and mathematical modeling data indicates that the conditions at the Moho correspond to closure of the cracks network and annihilation of its hydraulic permeability. Therefore, upper mantle is rather dry than watersaturated, which means that the existing views of rock physics need a revision. Variations in crustal thickness are controlled by the trend of the geotherm and, besides, by olivinetoserpentinite transition by the Hess reaction in oceanic crust. The Conrad discontinuity fits the horizontal foot of listric faults, waveguides (lowvelocity zones) correspond to the crackedporous inclusions and the lower crust to cataclastic state of rocks. The crust cut into faultbounded blocks is modeled in the context of rotation tectonic waves that follow the sinGordon equation.

I.A. Garagash
Institute of the Earth's Physics, Russian Academy of Sciences, 10 ul. Bol. Gruzinskaya, Moscow, 123810, Russia
Keywords: Bifurcation, variation condition, shear bands, compaction bands, ultimate yield, dilatancy, internal friction, ultimate condition
Pages: 655666
Abstract >>
The variation principle was applied to the bifurcated deformation of an inelastic body to clear up under what conditions a regular system of shear and compaction bands can form. It has been established that a regular system of bands where plastic deformations are localized should form most likely in materials with a distinct yield point. Internal friction and dilatancy promoting the development of instability of the material in the regime of strainhardening contribute much to this phenomenon.

G.G. Kocharyan, A.A. Kulyukin, and D.V. Pavlov
Institute for Dynamics of Geospheres, Russian Academy of Sciences, 38 Leninsky prosp., korp. 1, Moscow, 119334, Russia
Keywords: Fault, block structure, earthquake, afterslip, rheology
Pages: 667681
Abstract >>
Results of laboratory experiments and observation data on postseismic deformations in different regions were used to detect the regularities of formation of slow relative block displacements provoked by dynamic events. It is shown that under a gradual change of the stressstrain state of a rock massif, the asymptotic form of displacement with time is close to the law of a quasistatic flow, and under a drastic change in deformation rate, the relative block displacements are controlled mainly by the laws of friction force change during shear. The results obtained show that the dynamics of forces resisting to shears along block boundaries must be taken into account when constructing geomechanical models of different scales.

B.P. Sibiryakov and M.Yu. Podberezhny
Institute of Petroleum Geology and Geophysics, Siberian Branch of the RAS, 3 prosp. Akad. Koptyuga, Novosibirsk, 630090, Russia
Keywords: Specific surface, continuum operator, parametric resonance, catastrophe
Pages: 682693
Abstract >>
The paper presents an approach implying constructing continuum images of liquid or gassaturated porous and cracked solids. The hypothesis of classical continuum mechanics that closely positioned points of a solid occur in similar physical conditions breaks down in the case of porous media in which fluid and matrix material have strongly different properties. We suggest to represent a real microheterogeneous body by its continuum image using continuum operators, whereby the conservation laws are applied rather to the image than to its prototype. The continuum image is described by the standard equation of motion and the respective real solid by infiniteorder equations. The latter include velocities of elastic waves as well as infinitely low velocities of anomalies due to a great number of the degrees of freedom in microheterogeneous media consisting of separate elementary blocks.

V.Yu. Zaitsev and L.A. Matveev*
Institute of Applied Physics, 46 ul. Ul'yanova, Nizhny Novgorod, 603600, Russia * Nizhny Novgorod University, 23 prosp. Gagarina, Nizhny Novgorod, 603950, Russia
Keywords: Microinhomogeneous medium, elastic nonlinearity, linear dissipation, amplitude dependent dissipation
Pages: 694709
Abstract >>
We discuss a mechanism of nonhysteretic strainamplitude dependent dissipation of elastic waves in microinhomogeneous media containing linearly dissipative and nonlinear elastic soft defects. The combined effect of these factors can cause wellpronounced essentially dissipative nonlinearity of elastic waves. The mechanism presumably works in rocks and other materials with microstructure where cracks and grain contacts act as soft inclusions. Unlike homogeneous materials where strain dependent variations of dissipation and elasticity are commonly of the same order, dissipation variations in microinhomogeneous materials can be times as great as the respective elasticity variations. A slight (a few percent) change in Young's modulus can correspond to a manifold dissipation change at moderate strain typical of acoustic and seismic applications. Moreover, in the case of variable frequency, this nonlinear mechanism can increase or decrease Young's modulus or reduce its variation to zero at certain frequencies while dissipation variations remain quite large. The nonhysteretic mechanism may act concurrently with hysteretic mechanisms, which are due to friction and adhesion effects and are commonly invoked for materials with microstructure, and influence the relationship between strain dependent dissipation and elasticity observed in experiments on nonlinear dissipative effects.

