Fifty years of seismic anisotropy studies in Russia

Russian Geology and Geophysics 51 (2010) 1133–1146 www.elsevier.com/locate/rgg

Fifty years of seismic anisotropy studies in Russia I.R. Obolentseva a,*, T.I. Chichinina b a

A.A. Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, prosp. Akad. Koptyuga 3, Novosibirsk, 630090, Russia b Instituto Mexicano del Petroleo, Eje Central Lazaro Cardenas 152, 07730, Mexico D.F., Mexico Received 23 June 2009; accepted l4 August 2009

Abstract This is a historic overview of seismic anisotropy studies in Russia run as part of seismic exploration work in the 1940s through the 1980s, with a focus on main research lines. At the early stage in the 1940s through 1950s, most important contributions belonged to A.G. Tarkhov, Yu.V. Risnichenko, and S.M. Rytov (averaging the parameters of stratified media), I.I. Gurvich (processing reflection and refraction traveltime curves in media with elliptical anisotropy), and N.I. Berdennikova (shear-wave velocity anisotropy). In the 1960s–1980s, there were two basic schools of thought: one of G.I. Petrashen’, with a more theoretical approach, and the other of N.N. Puzyrev dealing more with experimental work. Most of experiments addressed a newly discovered phenomenon of azimuthal anisotropy. This anisotropy appearing as “anomalous” polarization of shear and converted waves was found out to result from vertical fractures in rocks. The unusual polarization became understood owing to Klem-Musatov’s model of a subsurface with a system of aligned cracks. The problem was fully resolved after field data had been processed with an algorithm by I.R. Obolentseva and S.B. Gorshkalev, for separating the total field of interfering shear waves of two types into fast and slow phases polarized in crack-parallel and crack-orthogonal directions, respectively. © 2010, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: multicomponent seismic surveys; polarization; anisotropy; azimuthal anisotropy

Introduction We overview the history of seismic anisotropy studies in Russia over fifty years between the 1940s and the 1980s. That was the time of key discoveries in seismic anisotropy, both in Russia and worldwide. In this paper we focus on anisotropy of the uppermost crust and leave beyond the consideration the seismological issues of crustal and mantle anisotropy. The choice of topics is limited to the main highlights, all lines of the anisotropy research being impossible to cover. The overview is arranged chronologically. First evidence of compressional (P-wave) and shear (Swave) velocity anisotropy in Russia dates back to the early period of the 1940s to 1950s. Tarkhov (1940), Riznichenko (1949), and Rytov (1956) investigated anisotropy produced by alternating thin layers which already at that time was considered to be the main kind of anisotropy in sediments. A.G. Tarkhov was the first geophysicist to formulate the theoretical grounds of the phenomenon and to check the theory with experiments. Later Yu.V. Riznichenko, whose work is known

* Corresponding author. E-mail address: [email protected] (I.R. Obolentseva)

in Russia as well as abroad, solved the same problem, likewise with a static approach, but in a more generalized formulation. Finally, S.M. Rytov proposed a dynamic solution, in which a periodically stratified medium gave way to a transversely isotropic one, and explored propagation of P, SV, and SH waves in an arbitrary direction. Note that Rytov published his study in 1956, i.e., almost simultaneously with the well known paper by Postma (1955). I.I. Gurvich was the first to propose techniques for processing anisotropy-affected reflection and refraction traveltime curves (Gurvich, 1940). The first Russian publication on shear waves from directed controlled sources was that by Berdennikova (1959). Her data corroborated earlier experimental results (Hagedoorn, 1954; Jolly, 1956; Uhrig and Van Meele, 1955; White et al., 1956) and proved valid the theory by Rytov (1956) and Postma (1955). The most important accomplishments in the Russian anisotropy research were in 1960 through 1980. It was then part of a research program on P and S wave propagation in rocks under the leadership of N.N. Puzyrev at the Institute of Geology and Geophysics of the USSR Academy of Sciences, Siberian Branch (Novosibirsk), run in collaboration with the VNIIGeofizika Research Institute (L.Yu. Brodov, L.N. Khudobina, T.N. Kulichikhina, and others). The same line of research

1068-7971/$ - see front matter D 2010, V . S. Sabolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.rgg.2010.09.005

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was followed at other academic and R&D institutions in Moscow and Leningrad, such as the Leningrad Section of the Steklov Mathematical Institute, Leningrad State University (G.I. Petrashen’, B.M. Kashtan, A.A. Kovtun, L.A. Molotkov), Institute of the Physics of the Earth in Moscow (A.L. Levshin, E.I. Gal’perin, E.M. Chesnokov), Moscow State University (F.M. Lyakhovitskii, M.V. Nevskii), etc., as well as a number of exploration survey teams (especially, the Sibneftegeofizika Association in 1971–1986, then the Siberian Geophysical Surveys, which worked together with the Institute of Geology and Geophysics in Novosibirsk; multicomponent seismic studies were headed by G.V. Vedernikov). The objectives of the studies focused on (i) pulse and vibration controlled sources of shear waves, techniques for acquisition and processing of shear- and converted-wave data (reflection profiling, vertical seismic profiling, refraction surveys, etc.); (ii) S-wave velocities, γ = vS / vP ratios and attenuation for different rock types in different seismogeological conditions; (iii) polarization of shear and converted waves in media with plane and tilted interfaces, diffractors, etc.; (iv) theory and experiments on anisotropy and processing algorithms for anisotropic media. The latter objective was the top priority because solutions to other relevant problems without due regard to anisotropy were ambiguous or even impossible. With the discovery of azimuthal anisotropy, which seemed to be one of most intriguing things in the 1960–1980s, anisotropy became the main target in multicomponent surveys. We give a historic background of this issue in Russia in comparison with that in the US in the respective section of the paper. Research through the 1960–1980s continued the anisotropy studies that began since 1940. Anisotropy in finely layered sedimentary rocks called later the “polar anisotropy” was rather well documented (e.g., Lyakhovitskii and Nevskii, 1970, 1971; Nevskii, 1974; Sibiryakov et al., 1980). Fractured media were explored in models by Klem-Musatov et al. (1973), Aizenberg et al. (1974), and Molotkov (Molotkov, 1979, 1991; Molotkov and Khilo, 1983, 1986). The basic theories of seismic wave propagation in an anisotropic media were discussed in the books by Fedorov (1965), Sirotin and Shaskolskaya (1979), and Petrashen’ (1980), which came in use also as reference and tutorial books on seismic anisotropy. A number of essential issues were considered in (Martynov, 1986; Uspenskii and Ogurtsov, 1962). Martynov and Mikhailenko (1979) elaborated numerical methods for wavefield modeling. Babich (1961) suggested the ray method developed later in (Petrashen’, 1980; Petrashen’ and Kashtan, 1984). The papers by Kashtan and Kovtun (1984), Kashtan et al. (1984), Obolentseva (1975), Obolentseva and Grechka (1988, 1989) dealt with ray algorithms and synthetic seismograms for stratified media. Numerical modeling of anisotropic shear and converted wave propagation was the subject of the papers (Brodov et al., 1986; Grechka and Obolentseva, 1987a,b; Obolentseva and Grechka, 1987).

Early studies (1940–1950) The first reports of anisotropy in sediments discovered in field data appeared in the 1930s (Beers, 1940; McCollum and Snell, 1932; Pirson, 1937; Weatherby et al., 1934; etc.). See, for instance, Uhrig and Van Meele (1955) for a brief overview of P-wave anisotropy data collected in the 1930s and later in the 1940s through 1950s. Thus, by the time of first Russian anisotropy publications (Tarkhov, 1940), P waves in stratified rocks had been known to travel more slowly in the vertical direction than along oblique and horizontal rays. The ratios of layer parallel-to-layer orthogonal P-wave velocities, for shallow sediments, were estimated to be 1.4 for shale and 1.3 for chalk (McCollum and Snell, 1932; Weatherby et al., 1934). Anisotropy in finely layered sedimentary formations: estimating elastic constants for an equivalent transversely isotropic medium A.G. Tarkhov’s static solution Tarkhov (1940) raised the question why metamorphic and sedimentary rocks may produce seismic anisotropy. He noted that alternation of thin layers with different elasticity constants was not the only cause of anisotropy in sediments. A rock can become anisotropic if thin layers with the same elastic properties are separated by planes which experienced other effects than the background. Another cause Tarkhov invoked was pressure that could change in magnitude and direction. Then he decided to estimate the magnitude of anisotropy through finding Young’s moduli along and across layers. Theoretical solution. Let the medium consist of alternating thin layers 1 and 2 with Young’s moduli E1 and E2, the densities ρ1 = ρ2, and the thicknesses h1 and h2, respectively. Relative volumetric contents of components 1 and 2 in the medium are n1 = h1 / (h1 + h2), n2 = h2 / (h1 + h2). Sought were the elastic constants of the medium as a whole in the directions parallel and orthogonal to the layer planes. Tarkhov simulated the shortening of a prism consisting of two alternating layers 1 and 2 under the load of the pressure p on its top, the bottom being fixed. The solution was for two cases of the compression p directed orthogonally (case 1) and parallel (case 2) to the layers (Fig. 1). Then, according to Hooke’s law, E⊥ =

E1E2 E2n1 + E1n2

, E|| = E1n1 + E2n2,

(1)

where the subscripts ⊥ and || denote the layer-orthogonal and layer-parallel directions, respectively. With (1) one can calculate the magnitude of natural anisotropy. From (1) Tarkhov derived simple equations for the anisotropy coefficient KP at n1 = n2 = 0.5, with the known equation 1/2

  E(1 − σ) , for isotropic P-wave velocity vP =    ρ (1 + σ) (1 − 2σ)  where σ is Poisson’s ratio and ρ is the density, assuming  σ = 1 / 4 and, hence, vP = √ 6E / 5ρ :

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Fig. 1. A model, with height L, consisting of alternating thin layers 1 and 2 of thicknesses h1 and h2. Pressure p is applied from top orthogonally (a) and parallel (b) to layer planes. Bottom is fixed. After (Tarkhov, 1940).

 E|| / E = KP = vP|| / vP⊥ = √

E1 + E2 1 + E2 / E1 = . 2√   E1E2 2√ E2 / E1

(2)

Table 1 lists the KP values for some E2 / E1 ratios that give an idea of this relationship, which obviously approaches the linear one. At E2 / E1 ≤ 1.5 anisotropy is vanishing but becomes notable already at E2 / E1 = 2 (KP = 1.06) and large since E2 / E1 = 2.5. Laboratory data. Young’s modulus was measured in laboratory on 6 × 1 × 0.5 cm3 samples cut out of a single piece of rock at 0o to 90o relative to the layer planes (Fig. 2). The measurements were in a static mode: a sample was placed horizontally on two fixed prismatic stands and a load (weights from 0 to 1.5 kg) was applied to its center. Measured was the forth and back change in loading-dependent flexure. The data were averaged and then processed. Young’s modulus was measured for three rock samples: jaspilite consisting of 1–2 mm thick quartzite and magnetite layers, and limestone and sandstone, both with weakly pronounced lamination. Ellipse approximation of the observed relationships of Young’s modulus gave the anisotropy coefficients 1.45 for jaspilite, 1.24 for limestone, and 1.19 for sandstone. Yu.V. Riznichenko’s rigorous static solution Riznichenko (1949) completed the solution for static deformation of finely layered media caused by layer-parallel and layer-orthogonal loading. He obtained correct equations for stiffnesses (elastic constants) CP⊥, CP|| (CP⊥ ≡ c33, CP|| ≡ c11)   and P velocities vP⊥ = √ CP⊥ /ρ , vP|| = √ CP|| / ρ as a function of all layer parameters: Young’s moduli, Poisson’s ratios, and the density and thickness ratios of the alternating layers. Then he extended the solution to media of more than two components and to those with continuously changing layer-orthogonal velocity.

S.M. Rytov’s dynamic solution Rytov (1956) finally had solved the problem for effective elastic constants of finely layered formations by simulating the propagation of plane harmonic waves in a periodically stratified medium consisting of two alternating layers. In that he proceeded from his experience in solving a similar problem for electromagnetic properties (Rytov, 1955). The idea of the elastic solution is as follows. The general solution of the wave equation is expressed via partial solutions, namely, even and odd relative to the layer centers. At the layer boundaries, the conditions of continuity and periodicity fulfill for the displacement vectors and for the stress vectors normal to the interface. Substituting the partial solutions depending on unknown displacement amplitudes into the boundary conditions leads to a system of homogeneous equations with respect to these unknowns. Bringing the systems’ determinants to zero gives dispersion equations and, hence, the square P and S velocities. Thus it became possible to find the five elastic constants as a function of layer parameters. The transition from layers of an arbitrary thickness to thin layers was the final step in the solution. The formulas for the five elastic constants of a transversely isotropic medium, which is an effective model of stratified media (Rytov, 1956), coincided with the respective formulas from (Postma, 1955). Processing anisotropy-affected data I.I. Gurvich was the first to approach seismic anisotropy in terms of survey data accuracy and possible ways of taking it into account in processing (Gurvich, 1940). At that time anisotropy was known to show up as P-wave velocities being faster along and slower across sediment layers, and the correction was applied using a special template (Pirson, 1937). I.I. Gurvich developed processing techniques for anisotropy-affected reflection and refraction data assuming elliptic

Table 1. Anisotropy coefficient Kp = vP|| / vP⊥ as a function of Young’s modulus ratio E2, E1 in thin layers 2 and 1 E2/E1

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Kp

1.02

1.06

1.11

1.15

1.20

1.25

1.30

1.34

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Fig. 2. Samples of a stratified rock cut out of a single piece in a way that normal to the sample top is at 0°, 45°, and 90° to the layer plane normal. After (Tarkhov, 1940).

anisotropy, on the basis of the following theorem valid for an elliptical direction dependence of velocity. Theorem. The traveltime equation for the transversely isotropic case is identical to that for the isotropic case if the anisotropic coordinates (x, z) are converted into the isotropic ones (x′, z′): x′ = x, z′ = kz, where k = vx / vz = vp|| / vP⊥. Using that theorem, I.I. Gurvich derived the equations of reflection and refraction traveltimes for two cases of a tilted interface, with layers lying horizontally or parallel to the interface above the latter. He suggested formulas and calculated parameters to characterize the interface depth and dip, as well as the anisotropy coefficient k above the interface to correct for the relative errors associated with neglect of anisotropy. The anisotropy coefficient KP ≡ k was derived from measured traveltimes t along vertical and oblique rays between the source and the receiver: t2 = x2 / (k2v2) + z2 / v2, where z is the depth of one in-hole receiver, x is the source distance from the borehole, and v is the pre-estimated vertical velocity. Thus found k was used for anisotropy correction. First evidence of anisotropy in sediments from shear wave observations The first reports of shear-wave sources and seismograms date back to the 1950s (Jolly, 1956; White and Sengbush, 1953; White et al., 1956). The velocities of SH and SV phases recorded on the surface and in boreholes were equal in the vertical direction but the SH phase was faster than SV in the horizontal direction. N.I. Berdennikova contributed to the anisotropy community with her data on shallow Cambrian clay from the Leningrad region (Berdennikova, 1959) collected on the surface and in two boreholes using several different sources of shear waves (Fig. 3). The down-hole vertical S velocity was logged to 45 m and turned out to increase with depth from 170 m/s on the surface to 320 m/s at 45 m. The horizontal velocities were estimated along refracted rays through the third layer (Fig. 3) as vSV|| = 340 m/s and vSH|| = 480 m/s. The former value vSV|| = 340 m/s approaches the vertical velocity vSV⊥ = 320 m/s measured at 45 m, while the difference between vSV|| = 340 m/s and vSH|| = 480 m/s is evidence of transverse isotropy.

Fig. 3. Layout of field experiments investigating anisotropy of Cambrian clay in Leningrad region (Berdennikova, 1959). 1, receivers; 2, Z-, Y-strikes 100 kg; 3, Z-strike 400 kg; 4, Z-gun; 5, explosion in air.

Fig. 4. A typical in-hole log of P, SH, and SV waves. Borehole 2, h = 24 m (for location see Fig. 3). Two horizontal mutually orthogonal seismometers (“Hor 1” and “Hor 2”) are oriented arbitrarily at each depth; “Vert” is vertical seismometer (Berdennikova, 1959).

The wave pattern in oblique rays looked as in Fig. 4: the SH phase arrived the first (recorded by two mutually orthogonal horizontal seismometers), and the secondary was SV being prominent in the z component but interfering with SH in the horizontal components. Thus, the data reported in (Berdennikova, 1956) were consistent with the known experiments (Jolly, 1956; Hagedoorn, 1954; Uhrig and Van Meele, 1955; White et al., 1956) and proved valid the theory by G.W. Postma and S.M. Rytov (Postma, 1955; Rytov, 1956).

Experimental work in 1960 through 1980 A puzzle of large “accessory” displacement components in shear and converted waves Systematic studies of shear and converted waves in Russia began in the late 1950s–early 1960s, by two research teams. The Leningrad team leaded by G.I. Petrashen’ focused on theoretical issues that arose after experiments of the 1950s, including those by Berdennikova (1959). The Novosibirsk team, with N.N. Puzyrev at the head, in collaboration with the Moscow team (L.Yu. Brodov and others), held experimental surveys at sites with different seismogeological conditions, mostly in the Caspian basin.

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First shear-wave surveys were with waves traveling from the source along X or Y and being received along the same directions: along x from an X-directed source (Xx scheme) and along y from a Y-directed source (Yy scheme); converted PS phases were recorded along the x and z components. That choice of the source-receiver configuration stemmed from the a priori idea of vertical transversely isotropic (VTI) behavior of a stratified earth according to theoretical and experimental data of the 1940s–1950s. The experiments were run in different parts of the European USSR, as well as in West and East Siberia, and did indicate moderate to strong VTI anisotropy in reflection, refraction, and VSP data. Anisotropy showed up as difference in S traveltimes and in slopes of traveltime curves observed in the Xx and Yy schemes. The experimental velocities of two shear phases as a function of wave-propagation direction turned out to fit well the theoretical relationships for SV and SH waves. The apprehension that the VTI model was far from being a universal one to account for the elastic properties of rocks appeared after the single-component (y, with a Y source), and two-component (x, z, with X and Z sources) recording had given way to three-component (x, y, z) surveys. It became evident that the observed displacements were beyond the available knowledge of anisotropy in a real earth. There were “accessory” displacement components (y from an X source and x from a Y source for SS phases and y components for PS phases) often as large as the “main” components. Obvious misfit between the observed and expected threecomponent records was discovered in survey data from salt domes in the Caspian basin. For the observation period from 1963 through 1978 it became clear that the “accessory” components could be large enough and that their amplitude ratios relative to the main components were larger than expected in the case of tilted interfaces in an isotropic earth or in a transversely isotropic earth with a symmetry axis normal to layer planes. Already in the 1960s, circular surveys on a slope of the Terkobai salt dome showed the ratio of the “accessory” tangential components to the “main” radial components in a PS reflection from a 10º tilted interface to vary between 0.5 and 1.5. The 1963–1978 data from salt domes in the Caspian basin are discussed in more detail below. Anomalous polarization of PS and SS waves Unusual polarization of shear and converted waves was discovered in 1963 at the Terkobai salt dome site. The PS phase reflected from the base of Cretaceous sediments (reflector III) was recorded in two circular profiles of the radii r = 300 m and r = 500 m, respectively. One source (in-hole shot) was at the circle center (see Figs. 5 and 6), and the horizontal receivers (Fig. 6) at each station were oriented along the radius (x) and along the tangent (y). The depth along the normal to reflector III under the shot point was h ≈ 400 m and the reflector’s dip was ϕ ≈ 10º. Two-component (x, y) PS seismograms of the r = 300 m profile are given in Fig. 7. Two columns in the left-hand panel

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Fig. 5. Terkobai salt dome. Section in reflector rize–dip plane, southeastern dome slope, site of circular profiles (Bakharevskaya et al., 1967).

Fig. 6. Ray scheme for PS reflection from a tilted interface observed in a circular profile. O is shot point, M is reflection and conversion point, S is observation point with polar coordinates (r, ψ), where ψ is angle to direction of reflector’s rise. PS phase arrives at point S with displacement vector U0 and, after being reflected from the Earth’s surface, is recorded, together with all reflections, by a three-component (x, y, z) station, in which x and y components are oriented radially (x) and tangentially (y). After (Puzyrev and Obolentseva, 1967).

are the x-, y-component records from the first half of the profile and two other columns, in the right-hand panel, are those from the other half (Fig. 7). The minimum traveltimes correspond to the rising segment of the reflector and the maximum ones to its dipping segment. The position of the two segments can be determined from both time extremes and from polarity changes of arrivals in the y component. The dip is more prominent, with one or two traces (straight line in Fig. 7), than the rise within ten traces (brace). The large magnitude of y components was striking and unclear (at that time). In the pattern of the x and y components and their ratios (Fig. 8), the tangential y components have their amplitudes commensurate to those of the radial x components or even exceed the latter in some azimuths. As it was shown theoretically (Puzyrev and Obolentseva, 1967), had the y components been due to the reflector’s tilt, they would have been smaller than in Fig. 8. At ϕ = 10º the y components should be no more than two or three percent of the x component amplitudes. See Fig. 9 for the x and y PS components calculated with the ray method (Alekseev and Gel’chinskii, 1959). In Figure 9 the x and y components of the displacement vectors are plotted against the azimuth of the observation point

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Fig. 7. Two-component (x, y) seismogram of a PS reflection from a tilted interface on a slope of Terkobai salt dome recorded on a circular profile along radial (x) and tangential (y) components. Gain in y component is two times that in x component. Shown are directions of reflector’s rise (0º) and dip (180º). After (Bakharevskaya et al., 1967).

in a circular profile counted from the direction of the reflector’s rise (0º), calculated for three ratios of the circle radius to the reflector depth at the circle center: r / h = 0.2, 0.6, and 1.0. For the experimental amplitude curves (Fig. 8), r / h = 0.75, i.e., the respective theoretical curves are between those for r / h = 0.6 and r / h = 1.0. Thus, the Uy/Ux ratios are the largest (within 0.02–0.03) at azimuths between the reflector’s strike (90º) and dip (180º) but are minor between the strike (90º) and the rise (0º). The theoretical curves fit the best the experimental data at ϕ = 26º, with a very large rms error, as it was expected from the above analysis of the behavior of theoretical curves (Fig. 9) and their comparison with the observations (Fig. 8). Table 2 synthesizes the data on observed PS and SS reflections on salt dome slopes in the Caspian basin. There is

no correlation between the amplitude ratios of the two components (Uaccess /Umain) with the tilt ϕ, i.e., the model of a tilted reflector between two isotropic layers is not the one to account for the observed anomalous polarization of shear and converted waves. An anisotropic model with an arbitrary position of the symmetry axis (i.e., arbitrary polar and azimuthal angles that characterize the position of this axis) was a critical step forward in understating the anomalous polarization. This anisotropy cannot result from sedimentary lamination (except for few cases of vertical layers), but is rather due mainly to aligned cracks: either long fractures in parallel planes or short penny-shaped cracks produced by variations in crustal horizontal stress. Tarkhov (1940) already mentioned those causes when wrote about a medium consisting of identical thin layers

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Fig. 9. Modeled (with ray method) amplitude curves of x and y components of displacement U of PS reflection in an isotropic model as a function of angle ψ between source–receiver line and reflector’s rise. Solid line is for r / h = 0.2, dashed line is for r / h = 0.6, and chain line is for r / h = 1.0. ϕ = 10º. Velocity ratio above reflector is γ = vS / vP = 0.31. After (Puzyrev and Obolentseva, 1967).

Fig. 8. Amplitudes of Ux, Uy components and their ratios Uy/Ux as a function of azimuth ψ in a circular profile on a slope of Terkobai salt dome. Solid line is primary survey, dashed line is repeated survey. After (Bakharevskaya et al., 1967).

separated by planes which arose during deposition changes. His experiments on static loading of limestone and sandstone showed that the anisotropy of that kind could be quite strong. Later Klem-Musatov et al. (1973) suggested a transversely isotropic model based on the linear slip theory to simulate the presence of cracks in rock. Note that anisotropy of rocks with a system of vertical cracks produces wave patterns quite different from those in the case of thin layers that differ in elastic constants. In the section below we describe briefly the model (Klem-Musatov et al., 1973; Aizenberg et al., 1974) and the attempts of applying it to anomalous polarization of converted (PS) and shear (SS) waves observed since 1963. Anisotropy as a possible cause of unusual polarization Model by Klem-Musatov et al. Let the medium consist of identical parallel thin layers with nonwelded contact conditions on their boundaries. For non-

stationary oscillations, these boundary conditions are (KlemMusatov et al., 1973) σq (M0) = lim σq (M), M → M0

  dU (M) t dUq (M0)   q = lim  + ∫ σq (M)τΛq (t − τ) dτ , dt dt  M → M0  0   1 Λq (t) = π

ω

∫ 0

(3)

eiωt dω, q = p, s, Zq (ω)

where σq is the stress and Uq is the normal (q = p) and tangential (q = s) displacement; Zq (ω) is the longitudinal (q = p) and transversal (q = s) impedance; M0 is the point in a layer boundary plane, and M is the variable point inside a layer. The impedances Zq (q = p, s) can be found in different ways, and the choice influences the behavior and the magnitude of anisotropy. Anisotropy appears on transition to the limit n → ∞ in the solution for a discrete stratified medium (n is the number of thin layers in the model). In a homogeneous anisotropic medium, assumed instead of the stratified one, the P, SV and SH velocities depend on the

Table 2. Experiments on polarization of PS and SS reflections over slopes of salt domes in Caspian basin Site and year of observations

Wave type, scheme

Uaccess/Umain ratio

Correlation with reflector tilt

Terkobai, southeastern slope, 1963, 1964

PS, ϕ = 10º, circular profiles

0.5–1.5

No

Terkobai, eastern slope, 1971

PS, ϕ = 21º, ϕ = 25º, circular profiles

0.3–0.5

No

Station 2, 1970

PS, ϕ = 26º, circular profiles

0.5–2.0

Yes

Shukat, 1976

Two SS waves: 1, ϕ = 10º; 2, ϕ = 30º; CMP

0.2–1.0

Yes for 1, No for 2

Tanatar, southern slope, 1978

PS, SS from Y source, ϕ = 5−6º, linear, radial, and circular profiles, CMP

0–5

No

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source direction defined by the angle θ between the wave normal and the symmetry axis z as vP (θ) = cP 1 + gP (1 − 2γ 2 sin2 θ)2 + gS γ 2 sin2 2θ   vSV (θ) = cS 1 + gP γ 2 sin2 2θ2 + gS cos2 2θ  

−1

,

−1

,

(4)

−1

vSH (θ) = cS 1 + gS cos2 θ , γ = cS / cP,   where cP, cS are the P and S velocities in unfractured rocks and gP, gS are the constants that represent the decrease of cP, cS due to the presence of cracks. At gP = gS = 0, the medium is isotropic and vP = cP, vS = cS. In a similar model by Schoenberg (1980, 1983), which came into broad use though appeared later than that of Klem-Musatov et al. (1973) and Aizenberg et al. (1974), the phase velocity equations are (Schoenberg and Douma, 1988)  (λb + 2µb) / ρb 1 − EN (1 − 2γb sin2 θ)2 − vP (θ) = √  ET γb sin2 2θ 

1/2

,

(5)

 µb / ρb 1 − EN γb sin2 2θ − ET cos2 2θ vSV (θ) = √  

1/2

,

1/2

 µb / ρb 1 − ET cos2 θ , vSH (θ) = √   where λb, µb are the Lamé constants in the isotropic background, ρb is the density of the isotropic background, γb = µb / (λb + 2µb), and EN, ET are the normal (N) and tangential (T) weaknesses that characterize the fracture system. Equations (5) agree with equations (4). In order to see that, one has to expand the right-hand sides of the equations into binomial series with respect to the small parameters: gP, gS for (4) and EN, ET for (5), because both are much below 1, and to hold the linear terms only. When comparing the obtained equations, one arrives at the following relationship between the model parameters: (λb + 2µb)ρ−1 µb ρ−1   cP = √ b , cS = √ b , ρ = ρb, γ 2 = c2S / c2P = γb, 2gP = EN, 2gS = ET. A transversely isotropic subsurface, an effective model of a medium with nonwelded contact layer boundaries, is described with four (c11, c33, c44, c66) instead of five elastic constants (see (4) and (5)). That is why the knowledge of P and S velocities on the symmetry axis (θ = 0º) and in the isotropy plane (θ = 90º) is, in principle, enough to find all elastic constants: vSH⊥ = VSV⊥ = VSV|| = cS (1 − gS) =   √ µb / ρb (1 − ET / 2) = √ c44 /ρ ,  µb / ρ = √  c66 /ρ , vSH|| = cS = √   (λb + 2µb) / ρb (1 − EN / 2) = √ c33 /ρ , vP⊥ = cP (1 − gP) = √

vP|| = cP 1 − (1 − 2γ 2)2 gP =      √ (λb + 2µb) / ρb 1 − (1 − 2γb)2 EN / 2 = √ c11 /ρ .   As for the fifth constant of the transversely isotropic medium (c13), one can find it from c11 c33 − c213 = 2c44 (c11 + c13) (Schoenberg and Sayers, 1995). First steps in processing x and y seismograms using the model by Klem-Musatov et al. The anisotropic model by Klem-Musatov et al. (1973) and Aizenberg et al. (1974) was used to obtain synthetic seismograms to fit the best the field data (Figs. 7, 8) from the Terkobai salt dome. The axis of symmetry was assumed to be vertical rather than normal to the reflector, in order to amplify the tangential components. The anisotropy coefficient KP = vP|| / vP⊥ was allowed to vary within 1.0–1.1; the values above 1.1 were avoided because the PSV – PSH arrival time difference at KP > 1.1 would exceed 1.5 periods and result in a respective extension of the total wave PSV + PSH record, which was never observed in the field. However, no satisfactory fit between the theoretical and field seismograms was achieved (see Fig. 6 in Klem-Musatov et al., 1973), especially on azimuths between the rise and the strike of the reflector (0o, 90o). Therefore, search according to the symmetry axis position was apparently required as well. Another example of synthetic seismograms approximating the field data (Klem-Musatov et al., 1973) was given in (Grechka and Obolentseva, 1987b). For the case of the Tanatar salt dome experiment (Table 2) reported in (Puzyrev et al., 1983), the position of the symmetry axis was found in the space (from the polar (β) and azimuthal (α) angles) with anisotropy parameters being allowed to vary in broad ranges. The best fit was achieved with the symmetry axis at 10º to the reflector’s normal, for both angles (β and α). The two angles measured the given direction, respectively, to the downward z axis (β) and to the x axis along the reflector rise (α). Thus, the symmetry axis was found to be at βsa = 20º, αsa = 10º, while the normal to the reflector was at βnr = 10º, αnr = 0º. A series of synthetic seismograms were calculated specially to understand how the PS and SS displacement component ratios in circular and linear profiles depended on the position of the reflector relative to the symmetry axis of the overlying transversely isotropic earth. The number of variables being too great, the calculations had to be confined to the combination of parameters which, according to field data (Bakharevskaya et al., 1967; Puzyrev, 1985; Puzyrev et al., 1983), appeared more realistic and had some general implications. Figure 10 shows the calculated ratios of the accessory-tomain displacement components (Uaccess/Umain) for PS and SS reflections from a ϕ = 10º tilted interface, which were observed in the profile to be at 45º to the line of the reflector’s rise and dip. The symmetry axis of the medium over the reflector is in the rise–dip plane of the latter and is at the angle β (0º, 90º) to the vertical. The choice of this reflector orientation, symmetry axis, and profile direction proceeded

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Thus, although Figure 10 provides an idea of possible Uaccess/Umain ratios in PS and SS, it fails to cover all practical cases and to show the Uaccess/Umain (β) behavior at α different from 10º. Furthermore, the anisotropy parameters can have different values with a boundless number of various combinations. That is why approximation of field data by theoretical functions is a tough problem that has no unique solution. Thus, the anomalous polarization of PS and SS phases has to be accounted for in some other way. Algorithm to separate interfering S1 and S2 waves Fig. 10. Ratios uy/ux of tangential components to radial ones for PS reflections and ux/uy for SS reflections (y-directed source) as a function of polar angle of symmetry axis measured from vertical direction. ϕ = 10º. The uy/ux and ux/uy ratios are average for relative distances r / h = 0−1.25 on profiles with azimuth 45º to direction of reflector’s rise. Seismograms are calculated by ray method. Main frequencies are f = 20 Hz (solid line) and f = 40 Hz (dashed line) for PS waves and f = 10 Hz (solid line) and f = 20 Hz (dashed line) for SS waves. After (Grechka and Obolentseva, 1987b).

from experimental data and preliminary calculations showing that the Uaccess/Umain (α, β) ratios were maximum at α about 45º, and the objective was to explore the Uaccess/Umain ratios as a function of β. Large anisotropy parameters were selected to provide the highest Uaccess/Umain ratios. The anisotropy coefficients of the P, SH, and SV phases found as the fastest-to-slowest phase velocity ratios were kP = kSV = 1.1, kSH = 1.2 at the phase velocities along the symmetry axis vP⊥ = 2 km/s; vS⊥ = 0.6 km/s (γ⊥ = 0.3). The Uaccess/Umain ratios along the y axis in the plots of Fig. 10 are the means over l = 1.5h long profiles, where h is the subsource depth of the reflector measured along the normal. The calculations were performed in a zero approximation of the ray method. The sources were the pressure center for the PS waves and the force Y (normal to the profile direction) for SS. Excitation was by a 2.5/f Berlage wavelet at f = 20 Hz and 40 Hz for PS and f = 10 Hz and 20 Hz for SS. According to the plots in Fig. 10, the Uaccess/Umain (β) ratios are the lowest (zero as in the respective isotropic model) when the symmetry axis is along the normal to the reflector (β = ϕ). As its departure from the normal grows, Uaccess/Umain (β) increase to 0.5–1 at β < ϕ and to ~1 at β > ϕ. In the latter case, the increase results from shear-wave splitting at a horizontal (or quasi-horizontal) symmetry axis. Above we wrote about the search for the anisotropy coefficient kP at a fixed β (β = 0º) from circular profile PS data at the Terkobai site. As one can see in Fig. 10, kP was estimated on the left-hand branch (β < ϕ), where Uaccess/Umain (β) in PS cannot exceed 0.5. In another example of synthetic PS seismograms fitted to the data from the Tanatar dome, β = 20º, i.e., it was found in the right-hand branch of the Uaccess/Umain (β) curve. Note, however, that search in the case of Tanatar was simultaneously for two angles α and β (α was found to be 10º) while the curves in Fig. 10 were for α = 0º.

The idea of possibility to pick pure S1 and S2 waves (SV and SH in transversely isotropic media) out of two- or three-component field seismograms occurred to I.R. Obolentseva and S.B. Gorshkalev in the late 1970s–early 1980s and was realized at that very time. The algorithm for separating two interfering S1 and S2 waves and an application example were reported in (Obolentseva and Gorshkalev, 1986). That publication was coeval with those of Alford (1986) and Naville (1986) in which the algorithms were designed for solving the same problem for CMP and VSP data, respectively. The algorithm by Obolentseva and Gorshkalev (1986) handles each two-component record (Ux(t), Uy(t)) at an observation point, where the S ray direction should be known. The rays of two shear waves were assumed to coincide and to be wave normal equivalent. Strictly speaking, the assumption is valid in the case of weak anisotropy. Let the ray direction be defined by the vector l0 (hereafter the superscript “0” denotes the unit vector). The first step of the algorithm consists in constructing the matrix M for transposition from the xyz coordinates to the new system x′y′z′, in which z′0 = l0 (the vector l0 is defined, in the general case, by the polar and azimuthal angles) and the axes x′ and y′ are in the plane normal to z′0; the position of the x′ axis is chosen arbitrary while y′0 ⊥ x′0. As the next step, (Ux(t), Uy(t)) is transposed from the xyz coordinates into x′y′z′ using the matrix M. The following steps are to convert the x-, y-component records into those of the x′ and y′ components by rotating through the angles α about the z′ axis. The function K(τ) of correlation between Ux′ (t) and Uy′ (t − τ) is calculated at each step and saved in the memory. The final step of the algorithm is to find max max K(τ). The α

τ

found α defines the position of the x′ and y′ axes of the “natural” coordinates, while τ is the lag of the slow phase behind the fast one (S1 and S2, respectively). Figure 11 illustrates the application of the algorithm to in-hole data from the Dossor salt dome (Gorshkalev, 2002) where an oriented three-component seismograph was placed at a depth of 100 m in Albian clay. An X-type source acted on the surface at 45º to the symmetry plane. Figure 11 shows the Ux(t), Uy(t), Uz(t) records for the source distances L from the borehole in the left-hand panel and the Ux′ (t), Uy′ (t) records for S1 and S2 in the right-hand panel. The algorithm

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Fig. 11. Example of conversion of x, y, z records (from X-directed source) into x′, y′, z′ records of S1 (fast) and S2 (slow) waves: original seismograms from borehole in Albian sediments at 100 m depth at Dossor dome site, at surface source moving 45º to symmetry plane for distances L (a); polarization processing result (Gorshkalev, 2002) (b).

also solved for the angles that characterize the position of the symmetry axis and the S1 to S2 time delay. Nikolskii (1987, 1992) developed further the algorithm of Obolentseva and Gorshkalev (1986) and applied the modified version to two-component (x, y) PS data from the Tanatar dome collected in 1982. Having converted those records into SV and SH, he obtained a map of polarization vectors of PS-wave split into PSV and PSH phases (Nikolskii and Shitov, 1992). Overlapping that map on the surface of reflector III showed polarization of SV and SH mostly in the planes of the reflector’s dip and strike. Note that a similar pattern was observed at the Dossor site (Trigubov and Gorshkalev, 1988). The polarization changed in zones of faults, whereby the displacement vectors became more scattered. The algorithms for separating the interfering S1 and S2 waves are applicable also to media with other symmetries, besides the transversely isotropic media (Trigubov and Gorshkalev, 1988). The application of the separation algorithm to the Dossor data yielded the indicatrices of the S1 and S2 ray

velocities lying in three symmetry planes, which led to revealing of a monoclinic symmetry.

Theoretical and modeling work in 1960 through 1980 There were quite many publications concerning the theory of seismic anisotropy and techniques for modeling wavefields in different kinds of anisotropic media. Below we cite only those that were published in Russia and were pioneering in the discussed line of research. Effective models of finely layered and fractured media Publications on anisotropy of thin layered media appeared in the late 1960s–early 1970s (Loktsik, 1969, 1970a,b). Of special interest are the studies by Lyakhovitskii and Nevskii (1970, 1971) on anisotropy produced by thin thin layering, which were synthesized later in (Nevskii, 1974). Having

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investigated the direction-dependent velocities in rocks consisting of alternating thin layers 1 and 2 with different ratios of velocities (vS1 / vS2, vP1 / vP2), densities (ρ1 / ρ2), and thicknesses (h1 / h2), the authors suggested a classification of thin layered media with four types of anisotropy. Theoretical and experimental (physical modeling with sheet models) studies of periodical layered structures were described by Sibiryakov et al. (1980) who showed the homogeneous transversely isotropic approximation to be valid at the layer thickness-to-wavelength ratio h / λ < 0.1−0.2. The anisotropic model of a fractured medium was first proposed in (Klem-Musatov et al., 1973; Aizenberg et al., 1974). We wrote about it in the previous section because discussion of 1960–1980 experiments required mentioning the respective theoretical background. Models of thin layered media, including fractured ones, were reported in (Molotkov, 1979, 1991; Molotkov and Khilo, 1983, 1986; and others) and in the overview by Bakulin and Molotkov (1998). Fundamentals of the theory of elastic wave propagation in anisotropic media There have been few books and tutorials published in Russia on elastic wave propagation in anisotropic media. The best one is the book of Fedorov (1965), which though being intended for physicists of crystals provides an excellent outlook of symmetry systems, propagation of plane elastic waves in media with different symmetries, as well as of the respective problems and solutions. Another important book appeared ten years later (Sirotin and Shaskolskaya, 1975, 1979) and became an indispensable handbook on many issues of symmetry and physical properties of solids. Later publications of this kind were those by Petrashen’ (1980, 1984). The latter is a collection of three papers by Petrashen’ and Kashtan, Kashtan, Kovtun, and Petrashen’, and Kashtan and Kovtun concerning the theory and algorithms. Computing methods and algorithms The numerical method and its applications to modeling wave propagation in inhomogeneous anisotropic solids for the halfspace and sphere cases were detailed in (Martynov and Mikhailenko, 1979). Point sources in a transversely isotropic elastic medium were first discussed in (Uspenskii and Ogurtsov, 1962), and a method for imaging wavefields from point sources in transversely isotropic media was suggested in (Martynov, 1986). The ray method was first applied to anisotropic (inhomogeneous) media by Babich (1961). It was also the subject of Chapter 5 of the book (Cerveny′ et al., 1977) written with participation of L.A. Molotkov. Later the ray method and the respective algorithms were reported in (Kashtan, 1982; Kashtan and Kovtun, 1984; Kashtan et al., 1984). In the 1970s–1980s, the issues of modeling and ray method for 3D anisotropic layered media were treated in a series of publications by I.R. Obolentseva and V.Yu. Grechka. The earliest publication concerned an algorithm for rays that travel v

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in a transversely isotropic layered medium from a source to given receivers (Obolentseva, 1975). The algorithm was similar to one published a year before for isotropic media (Obolentseva, 1974). In those studies, the coordinates of points where the ray crossed the reflector were found using directly the Fermat principle, i.e., from the stationary traveltime conditions; the obtained system of equations was solved by means of iteration. Later there appeared limitation-free nonlinear algorithms with the use of derivatives reported in (Obolentseva, 1980a) for isotropic media. Obolentseva and Grechka (1988) presented an optimization algorithm for ray paths in anisotropic media of any symmetry, and in a later paper (Obolentseva and Grechka, 1992), compared different inversion techniques for layered anisotropic media following a similar comparison for isotropic cases (Obolentseva, 1980b). The book (Obolentseva and Grechka, 1989) encompassed all earlier algorithms by the authors for anisotropic media, including ray modeling and respective synthetic seismograms. The latter issues were the subject of two first chapters of the book, and the third chapter contained algorithms for shear wave surfaces in the vicinity of singular directions and techniques for separating n-valued wave surfaces into n single-valued domains amenable to the ray method. Chapter 4 concerned algorithms for rays and displacements in anisotropic gyrotropic media. A series of papers published in 1987 (Grechka and Obolentseva, 1987a,b; Obolentseva and Grechka, 1987) concerned with numerical modeling of shear and converted waves in anisotropic media on the basis of our earlier algorithms and explained how the anomalous PS and SS polarization may arise in media with azimuthal anisotropy, with theoretical examples of PS and SS reflections from the top and the bottom of an anisotropic layer for cases of different symmetry axis orientations. The tangential components of displacement were shown to be small in the reflections from the layer top but almost as large as the radial components in the layer bottom reflections. Brodov et al. (1986) reported results of wavefield simulation for different cases of a transversely isotropic medium which were obtained using algorithms from (Petrashen’, 1984). There were also other works in Russia that influenced the anisotropy research. It was, for instance, the book of Lapin (1980) which became popular with seismic survey people of that time. Slightly later works by E.A. Blyas (Blyas, 1983, 1987, etc.) addressed approximate techniques for estimating PP and PS reflection traveltimes in anisotropic layered media. Seleznev et al. (1986) dealt with inversion of compressional and converted waves data. Of course, the above citations of studies on elastic wave propagation in anisotropic media cannot cover all publications of the respective period. We have selected first of all the highlights in the research of azimuthal anisotropy as a factor of unusual polarization of shear waves, and other features observed in three-component wavefield data. A special focus has been on the studies that were published in Russia but remained unfamiliar to English-speaking readers, in order to

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make them known due to the English version of this paper in Russian Geology and Geophysics.

new line of seismic studies in which seismic anisotropy is an important subject.

Conclusions

References

Modeling a phenomenon is equivalent to understanding it. The model of a transversely isotropic medium with a vertical axis of symmetry (known today as VTI) had been the only model of anisotropic media from the beginning of seismic anisotropy studies to the mid-1960s. It was designed to explain the velocity difference observed since the 1930s along vertical and oblique rays (slower velocities along the latter). The explanation was found in 1940 through the mid-1950s: the velocity difference resulted from thin layering if the layer is much thinner than the wavelength. A model was obtained to relate the parameters of thin layers to those of a transversely isotropic medium with a symmetry axis normal to layering. The next step was the discovery, in the 1960s, that the VTI model most often failed to account for shear wave propagation in media where y components of displacement in SS waves (especially of large amplitudes) never appear from a y-directed source, neither the y components of PS waves. Furthermore, the VTI model turned out to be inapplicable to large delays at intersections of profiles of different azimuths. TI (transversely isotropic) media should have a horizontal (H) or tilted (T) axis of symmetry, i.e., the medium should be HTI or TTI, or rhombic or even monoclinic. In this case the question arises of a possible physical cause for this kind of symmetry. In the 1970s, there appeared a model based on the theory of linear interfacial slip, which is a kind of a nonwelded layer boundary. The model implied that a rock looking homogeneous can be cut by a network of parallel cracks (horizontal or tilted) which make the rock transversely isotropic of types HTI or TTI, respectively. In that model, tangential displacement components are a reasonable consequence because displacement is recorded in the coordinates that are arbitrary relative to the natural coordinate system. (“Natural coordinates” are the xyz system in which two shear phases with their fixed mutually orthogonal directions in the xy plane and different velocities correspond to a given direction of the wave normal z). Transition from arbitrary coordinates to the natural system was suggested in order to avoid the tangential components, and the respective algorithm was obtained in the 1980s. Field, theoretical, and numerical experiments were carried out simultaneously and concertedly thus making for ever better understanding and employing the survey data. Progressively, all anisotropy parameters (traveltimes and amplitudes of P, S1, and S2 waves depending on reflection and attenuation coefficients) have come into use for structural modeling and estimating the physical properties of rocks. Seismic anisotropy studies in Russia till the early 1990s were run parallel with similar research in other countries and leaded the latter in many cases (especially before the 1980s). Some of those studies have been the subject of our overview. The paper is dedicated to the memory of Nikolai Nikitovich Puzyrev, the pioneer of multicomponent surveys in Russia, a

Aizenberg, A.M., Klem-Musatov, K.D., Landa, E.I., 1974. A model of an anisotropic seismic medium, in: Puzyrev, N.N., Klem-Musatov, K.D. (Eds.), Seismic Waves in Structurally Complex Media [in Russian]. Nauka, Novosibirsk, pp. 64–110. Alekseev, A.S., Gelchinskii, B.Ya., 1959. The ray method for calculating wavefields and waves in inhomogeneous media with curved interfaces, in: Problems of the Dynamic Theory of Seismic Wave Propagation [in Russian]. LGU Press, Leningrad, Transactions of Leningrad University, Issue 3, pp. 107–161. Alford, R.M., 1986. Shear data in the presence of azimuthal anisotropy: Dilley, Texas. Proc. the 56th Annual SEG Meeting. Houston, pp. 476–479. Babich, V.M., 1961. The ray method for modeling wave fronts in the case of an elastic inhomogeneous anisotropic medium, in: Problems of the Dynamic Theory of Seismic Wave Propagation [in Russian]. LGU Press, Leningrad, Transactions of Leningrad University, Issue 5, pp. 36–46. Bakharevskaya, T.M., Brodov, L.Yu., Obolentseva, I.R., Puzyrev, N.N., 1967. Experimental studies of polarization of PS reflections from a tilted interface, in: Puzyrev, N.N. (Ed.), Shear and Converted Waves in Seismic Surveys [in Russian]. Nedra, Moscow, pp. 203–209. Bakulin, A.V., Molotkov, L.A., 1998. Effective Seismic Models of Fractured and Porous Solids [in Russian]. St. Petersburg Press, Saint Petersburg. Beers, R.F., 1940. Velocity stratification as an aid to correlation. Geophysics 5 (1), 15–21. Berdennikova, N.I., 1959. On some evidences of anisotropy in finely layered rocks in shear-wave studies, in: Problems of the Dynamic Theory of Seismic Wave Propagation [in Russian]. LGU Press, Leningrad, Transactions of Leningrad University, Issue 2, pp. 187–196. Blyas, E.A., 1983. Traveltime curves of reflected waves in a horizontally layered medium with transversely isotropic layers and their interpretation. Geologiya i Geofizika (Soviet Geology and Geophysics) 24 (2), 91–95 (88–92). Blyas, E.A., 1987. Traveltime curves of converted reflections in stratified media with transversely isotropic layers, in: Puzyrev N.N. (Ed.), Algorithmic Problems of Seismic Data Processing [in Russian]. IGiG, Novosibirsk, pp. 79–87. Brodov, L.Yu., Kovtun, A.A., Tikhonov, A.A., 1986. Some results of numerical modeling for a transversely isotropic medium. Izv. AN SSSR. Fizika Zemli, No. 11, 48–57. v Cerveny′ , V., Molotkov, L., Psencik, I., 1977. Ray method for anisotropic media. Ray method in seismology. Univerzita Karlova, Praha, pp. 159– 172. Fedorov, F.I., 1965. The Theory of Elastic Waves in Crystals [in Russian]. Nauka, Moscow. Gorshkalev, S.B., 2002. Developing Methods for Experimental Investigation into Anisotropy of Sediments in Multi-Component Seismic Surveys [in Russian]. Candidate Thesis. Novosibirsk. Grechka, V.Yu., Obolentseva, I.R., 1987a. Mathematical modeling of reflected waves in stratified-homogeneous transversely isotropic media. Izv. AN SSSR. Fizika Zemli, No. 7, 50–59. Grechka, V.Yu., Obolentseva, I.R., 1987b. Results of mathematical modeling for 3D displacement fields of PS and SS reflections in transversely isotropic media, in: Puzyrev, N.N. (Ed.), Multicomponent Seismic Surveys [in Russian]. Nauka, Novosibirsk, pp. 20–25. Gurvich, I.I., 1940. Interpretation of seismic data in the case of an anisotropic medium. Izv. AN SSSR, Ser. Geogr. Geophys., No. 5, 108–116. Hagedoorn, J.G., 1954. A practical example of an anisotropic velocity layer. Geophysical Prospecting 2 (1), 52–60. Jolly, R. N., 1956. Investigation of shear waves. Geophysics 21 (4), 905–938. Kashtan, B.M., 1982. Calculating geometrical spreading in piecewise homogeneous anisotropic media, in: Problems of the Dynamic Theory of v

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Editorial responsibility: V.S. Seleznev