Fifty years of seismic anisotropy studies in Russiachichinina.rusiamexico.com/.../2014/...Anisotropic.pdf · (1949), and Rytov (1956) investigated anisotropy produced by alternating

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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 traveltimecurves in media with elliptical anisotropy), and N.I. Berdennikova (shear-wave velocity anisotropy). In the 1960s–1980s, there were two basicschools of thought: one of G.I. Petrashen’, with a more theoretical approach, and the other of N.N. Puzyrev dealing more with experimentalwork. 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 understoodowing to Klem-Musatov’s model of a subsurface with a system of aligned cracks. The problem was fully resolved after field data had beenprocessed with an algorithm by I.R. Obolentseva and S.B. Gorshkalev, for separating the total field of interfering shear waves of two typesinto 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 inRussia over fifty years between the 1940s and the 1980s. Thatwas the time of key discoveries in seismic anisotropy, both inRussia and worldwide. In this paper we focus on anisotropyof the uppermost crust and leave beyond the consideration theseismological issues of crustal and mantle anisotropy. Thechoice of topics is limited to the main highlights, all lines ofthe anisotropy research being impossible to cover. Theoverview is arranged chronologically.

First evidence of compressional (P-wave) and shear (S-wave) velocity anisotropy in Russia dates back to the earlyperiod of the 1940s to 1950s. Tarkhov (1940), Riznichenko(1949), and Rytov (1956) investigated anisotropy produced byalternating thin layers which already at that time was consid-ered to be the main kind of anisotropy in sediments. A.G. Tar-khov was the first geophysicist to formulate the theoreticalgrounds of the phenomenon and to check the theory withexperiments. Later Yu.V. Riznichenko, whose work is known

in Russia as well as abroad, solved the same problem, likewisewith a static approach, but in a more generalized formulation.Finally, S.M. Rytov proposed a dynamic solution, in which aperiodically stratified medium gave way to a transverselyisotropic one, and explored propagation of P, SV, and SHwaves in an arbitrary direction. Note that Rytov published hisstudy in 1956, i.e., almost simultaneously with the well knownpaper by Postma (1955). I.I. Gurvich was the first to proposetechniques for processing anisotropy-affected reflection andrefraction traveltime curves (Gurvich, 1940). The first Russianpublication on shear waves from directed controlled sourceswas that by Berdennikova (1959). Her data corroboratedearlier experimental results (Hagedoorn, 1954; Jolly, 1956;Uhrig and Van Meele, 1955; White et al., 1956) and provedvalid the theory by Rytov (1956) and Postma (1955).

The most important accomplishments in the Russian anisot-ropy research were in 1960 through 1980. It was then part ofa research program on P and S wave propagation in rocksunder the leadership of N.N. Puzyrev at the Institute ofGeology and Geophysics of the USSR Academy of Sciences,Siberian Branch (Novosibirsk), run in collaboration with theVNIIGeofizika Research Institute (L.Yu. Brodov, L.N. Khudo-bina, T.N. Kulichikhina, and others). The same line of research

Russian Geology and Geophysics 51 (2010) 1133–1146

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

doi:10.1016/j.rgg.2010.0 001068-7971/$ - see front matter D 2010, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved.V S. . Sabolev

9. 5

was followed at other academic and R&D institutions inMoscow and Leningrad, such as the Leningrad Section of theSteklov 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 numberof exploration survey teams (especially, the SibneftegeofizikaAssociation in 1971–1986, then the Siberian GeophysicalSurveys, which worked together with the Institute of Geologyand Geophysics in Novosibirsk; multicomponent seismicstudies were headed by G.V. Vedernikov).

The objectives of the studies focused on (i) pulse andvibration controlled sources of shear waves, techniques foracquisition and processing of shear- and converted-wave data(reflection profiling, vertical seismic profiling, refraction sur-veys, etc.); (ii) S-wave velocities, γ = vS / vP ratios and attenu-

ation for different rock types in different seismogeologicalconditions; (iii) polarization of shear and converted waves inmedia with plane and tilted interfaces, diffractors, etc.;(iv) theory and experiments on anisotropy and processingalgorithms for anisotropic media.

The latter objective was the top priority because solutionsto other relevant problems without due regard to anisotropywere ambiguous or even impossible. With the discovery ofazimuthal anisotropy, which seemed to be one of mostintriguing things in the 1960–1980s, anisotropy became themain target in multicomponent surveys. We give a historicbackground of this issue in Russia in comparison with that inthe US in the respective section of the paper.

Research through the 1960–1980s continued the anisotropystudies that began since 1940. Anisotropy in finely layeredsedimentary rocks called later the “polar anisotropy” wasrather well documented (e.g., Lyakhovitskii and Nevskii, 1970,1971; Nevskii, 1974; Sibiryakov et al., 1980). Fractured mediawere 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 ananisotropic 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 bookson seismic anisotropy. A number of essential issues wereconsidered in (Martynov, 1986; Uspenskii and Ogurtsov,1962). Martynov and Mikhailenko (1979) elaborated numeri-cal methods for wavefield modeling. Babich (1961) suggestedthe ray method developed later in (Petrashen’, 1980; Pe-trashen’ and Kashtan, 1984). The papers by Kashtan andKovtun (1984), Kashtan et al. (1984), Obolentseva (1975),Obolentseva and Grechka (1988, 1989) dealt with ray algo-rithms and synthetic seismograms for stratified media. Nu-merical modeling of anisotropic shear and converted wavepropagation 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 infield data appeared in the 1930s (Beers, 1940; McCollum andSnell, 1932; Pirson, 1937; Weatherby et al., 1934; etc.). See,for instance, Uhrig and Van Meele (1955) for a brief overviewof P-wave anisotropy data collected in the 1930s and later inthe 1940s through 1950s. Thus, by the time of first Russiananisotropy publications (Tarkhov, 1940), P waves in stratifiedrocks had been known to travel more slowly in the verticaldirection than along oblique and horizontal rays. The ratios oflayer parallel-to-layer orthogonal P-wave velocities, for shal-low sediments, were estimated to be 1.4 for shale and 1.3 forchalk (McCollum and Snell, 1932; Weatherby et al., 1934).

Anisotropy in finely layered sedimentary formations:estimating elastic constants for an equivalent transverselyisotropic medium

A.G. Tarkhov’s static solution

Tarkhov (1940) raised the question why metamorphic andsedimentary rocks may produce seismic anisotropy. He notedthat alternation of thin layers with different elasticity constantswas not the only cause of anisotropy in sediments. A rock canbecome anisotropic if thin layers with the same elasticproperties are separated by planes which experienced othereffects than the background. Another cause Tarkhov invokedwas pressure that could change in magnitude and direction.Then he decided to estimate the magnitude of anisotropythrough finding Young’s moduli along and across layers.

Theoretical solution. Let the medium consist of alternatingthin layers 1 and 2 with Young’s moduli E1 and E2, thedensities ρ1 = ρ2, and the thicknesses h1 and h2, respectively.Relative volumetric contents of components 1 and 2 in themedium are n1 = h1 / (h1 + h2), n2 = h2 / (h1 + h2). Sought werethe elastic constants of the medium as a whole in the directionsparallel and orthogonal to the layer planes.

Tarkhov simulated the shortening of a prism consisting oftwo alternating layers 1 and 2 under the load of the pressurep on its top, the bottom being fixed. The solution was for twocases of the compression p directed orthogonally (case 1) andparallel (case 2) to the layers (Fig. 1). Then, according toHooke’s law,

E⊥ = E1E2

E2n1 + E1n2, E|| = E1n1 + E2n2, (1)

where the subscripts ⊥ and || denote the layer-orthogonal andlayer-parallel directions, respectively. With (1) one can calcu-late the magnitude of natural anisotropy.

From (1) Tarkhov derived simple equations for the anisot-ropy coefficient KP at n1 = n2 = 0.5, with the known equation

for isotropic P-wave velocity vP =

E(1 − σ)ρ (1 + σ) (1 − 2σ)

1 / 2

,

where σ is Poisson’s ratio and ρ is the density, assumingσ = 1 / 4 and, hence, vP = √6E / 5ρ :

1134 I.R. Obolentseva, T.I. Chichinina / Russian Geology and Geophysics 51 (2010) 1133–1146

KP = vP|| / vP⊥ = √E|| / E = E1 + E2

2√E1E2 =

1 + E2 / E1

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 thelinear one. At E2 / E1 ≤ 1.5 anisotropy is vanishing but be-

comes notable already at E2 / E1 = 2 (KP = 1.06) and large

since E2 / E1 = 2.5. Laboratory data. Young’s modulus was measured in labo-

ratory on 6 × 1 × 0.5 cm3 samples cut out of a single piece ofrock at 0o to 90o relative to the layer planes (Fig. 2). Themeasurements were in a static mode: a sample was placedhorizontally on two fixed prismatic stands and a load (weightsfrom 0 to 1.5 kg) was applied to its center. Measured was theforth and back change in loading-dependent flexure. The datawere averaged and then processed. Young’s modulus wasmeasured for three rock samples: jaspilite consisting of1–2 mm thick quartzite and magnetite layers, and limestoneand sandstone, both with weakly pronounced lamination.Ellipse approximation of the observed relationships ofYoung’s modulus gave the anisotropy coefficients 1.45 forjaspilite, 1.24 for limestone, and 1.19 for sandstone.

Yu.V. Riznichenko’s rigorous static solution

Riznichenko (1949) completed the solution for static defor-mation of finely layered media caused by layer-parallel andlayer-orthogonal loading. He obtained correct equations forstiffnesses (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, andthe density and thickness ratios of the alternating layers. Thenhe extended the solution to media of more than two compo-nents and to those with continuously changing layer-orthogo-nal velocity.

S.M. Rytov’s dynamic solution

Rytov (1956) finally had solved the problem for effectiveelastic constants of finely layered formations by simulatingthe propagation of plane harmonic waves in a periodicallystratified medium consisting of two alternating layers. In thathe proceeded from his experience in solving a similar problemfor electromagnetic properties (Rytov, 1955).

The idea of the elastic solution is as follows. The generalsolution of the wave equation is expressed via partial solutions,namely, even and odd relative to the layer centers. At the layerboundaries, the conditions of continuity and periodicity fulfillfor the displacement vectors and for the stress vectors normalto the interface. Substituting the partial solutions dependingon unknown displacement amplitudes into the boundaryconditions leads to a system of homogeneous equations withrespect to these unknowns.

Bringing the systems’ determinants to zero gives dispersionequations and, hence, the square P and S velocities. Thus itbecame possible to find the five elastic constants as a functionof layer parameters. The transition from layers of an arbitrarythickness to thin layers was the final step in the solution. Theformulas for the five elastic constants of a transverselyisotropic medium, which is an effective model of stratifiedmedia (Rytov, 1956), coincided with the respective formulasfrom (Postma, 1955).

Processing anisotropy-affected data

I.I. Gurvich was the first to approach seismic anisotropy interms of survey data accuracy and possible ways of taking itinto account in processing (Gurvich, 1940). At that timeanisotropy was known to show up as P-wave velocities beingfaster along and slower across sediment layers, and thecorrection was applied using a special template (Pirson, 1937).

I.I. Gurvich developed processing techniques for anisot-ropy-affected reflection and refraction data assuming elliptic

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).

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

I.R. Obolentseva, T.I. Chichinina / Russian Geology and Geophysics 51 (2010) 1133–1146 1135

anisotropy, on the basis of the following theorem valid for anelliptical direction dependence of velocity.

Theorem. The traveltime equation for the transverselyisotropic case is identical to that for the isotropic case if theanisotropic coordinates (x, z) are converted into the isotropicones (x′, z′): x′ = x, z′ = kz, where k = vx / vz = vp|| / vP⊥.

Using that theorem, I.I. Gurvich derived the equations ofreflection and refraction traveltimes for two cases of a tiltedinterface, with layers lying horizontally or parallel to theinterface above the latter.

He suggested formulas and calculated parameters to char-acterize the interface depth and dip, as well as the anisotropycoefficient k above the interface to correct for the relativeerrors associated with neglect of anisotropy.

The anisotropy coefficient KP ≡ k was derived from meas-ured 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 fromthe borehole, and v is the pre-estimated vertical velocity. Thusfound k was used for anisotropy correction.

First evidence of anisotropy in sediments from shear wave observations

The first reports of shear-wave sources and seismogramsdate back to the 1950s (Jolly, 1956; White and Sengbush,1953; White et al., 1956). The velocities of SH and SV phasesrecorded on the surface and in boreholes were equal in thevertical direction but the SH phase was faster than SV in thehorizontal direction. N.I. Berdennikova contributed to theanisotropy community with her data on shallow Cambrian clayfrom the Leningrad region (Berdennikova, 1959) collected onthe surface and in two boreholes using several differentsources of shear waves (Fig. 3).

The down-hole vertical S velocity was logged to 45 m andturned out to increase with depth from 170 m/s on the surfaceto 320 m/s at 45 m. The horizontal velocities were estimatedalong refracted rays through the third layer (Fig. 3) asvSV|| = 340 m/s and vSH|| = 480 m/s. The former value vSV|| =340 m/s approaches the vertical velocity vSV⊥ = 320 m/smeasured at 45 m, while the difference between vSV|| =340 m/s and vSH|| = 480 m/s is evidence of transverse isotropy.

The wave pattern in oblique rays looked as in Fig. 4: theSH phase arrived the first (recorded by two mutually orthogo-nal horizontal seismometers), and the secondary was SV beingprominent in the z component but interfering with SH in thehorizontal components. Thus, the data reported in (Berdennik-ova, 1956) were consistent with the known experiments (Jolly,1956; Hagedoorn, 1954; Uhrig and Van Meele, 1955; Whiteet al., 1956) and proved valid the theory by G.W. Postma andS.M. Rytov (Postma, 1955; Rytov, 1956).

Experimental work in 1960 through 1980

A puzzle of large “accessory” displacement componentsin shear and converted waves

Systematic studies of shear and converted waves in Russiabegan in the late 1950s–early 1960s, by two research teams.The Leningrad team leaded by G.I. Petrashen’ focused ontheoretical issues that arose after experiments of the 1950s,including those by Berdennikova (1959). The Novosibirskteam, with N.N. Puzyrev at the head, in collaboration with theMoscow team (L.Yu. Brodov and others), held experimentalsurveys at sites with different seismogeological conditions,mostly in the Caspian basin.

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

Fig. 3. Layout of field experiments investigating anisotropy of Cambrian clay inLeningrad 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 (forlocation see Fig. 3). Two horizontal mutually orthogonal seismometers (“Hor 1”and “Hor 2”) are oriented arbitrarily at each depth; “Vert” is vertical seismome-ter (Berdennikova, 1959).


First shear-wave surveys were with waves traveling fromthe source along X or Y and being received along the samedirections: along x from an X-directed source (Xx scheme) andalong y from a Y-directed source (Yy scheme); converted PSphases were recorded along the x and z components. Thatchoice of the source-receiver configuration stemmed from thea priori idea of vertical transversely isotropic (VTI) behaviorof a stratified earth according to theoretical and experimentaldata of the 1940s–1950s.

The experiments were run in different parts of the EuropeanUSSR, as well as in West and East Siberia, and did indicatemoderate to strong VTI anisotropy in reflection, refraction,and VSP data. Anisotropy showed up as difference in Straveltimes and in slopes of traveltime curves observed in theXx and Yy schemes. The experimental velocities of two shearphases as a function of wave-propagation direction turned outto fit well the theoretical relationships for SV and SH waves.

The apprehension that the VTI model was far from beinga universal one to account for the elastic properties of rocksappeared after the single-component (y, with a Y source), andtwo-component (x, z, with X and Z sources) recording hadgiven way to three-component (x, y, z) surveys. It becameevident that the observed displacements were beyond theavailable knowledge of anisotropy in a real earth. There were“accessory” displacement components (y from an X source andx from a Y source for SS phases and y components for PSphases) often as large as the “main” components.

Obvious misfit between the observed and expected three-component records was discovered in survey data from saltdomes in the Caspian basin. For the observation period from1963 through 1978 it became clear that the “accessory”components could be large enough and that their amplituderatios relative to the main components were larger thanexpected in the case of tilted interfaces in an isotropic earthor in a transversely isotropic earth with a symmetry axisnormal to layer planes.

Already in the 1960s, circular surveys on a slope of theTerkobai salt dome showed the ratio of the “accessory”tangential components to the “main” radial components in aPS reflection from a 10º tilted interface to vary between 0.5and 1.5. The 1963–1978 data from salt domes in the Caspianbasin are discussed in more detail below.

Anomalous polarization of PS and SS waves

Unusual polarization of shear and converted waves wasdiscovered in 1963 at the Terkobai salt dome site. The PSphase reflected from the base of Cretaceous sediments (reflec-tor III) was recorded in two circular profiles of the radiir = 300 m and r = 500 m, respectively. One source (in-holeshot) was at the circle center (see Figs. 5 and 6), and thehorizontal receivers (Fig. 6) at each station were oriented alongthe radius (x) and along the tangent (y). The depth along thenormal to reflector III under the shot point was h ≈ 400 m andthe reflector’s dip was ϕ ≈ 10º.

Two-component (x, y) PS seismograms of the r = 300 mprofile are given in Fig. 7. Two columns in the left-hand panel

are the x-, y-component records from the first half of theprofile and two other columns, in the right-hand panel, arethose from the other half (Fig. 7). The minimum traveltimescorrespond to the rising segment of the reflector and themaximum ones to its dipping segment. The position of thetwo segments can be determined from both time extremes andfrom polarity changes of arrivals in the y component. The dipis more prominent, with one or two traces (straight line inFig. 7), than the rise within ten traces (brace).

The large magnitude of y components was striking andunclear (at that time). In the pattern of the x and y componentsand their ratios (Fig. 8), the tangential y components have theiramplitudes commensurate to those of the radial x componentsor 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 they components should be no more than two or three percent ofthe x component amplitudes. See Fig. 9 for the x and y PScomponents calculated with the ray method (Alekseev andGel’chinskii, 1959).

In Figure 9 the x and y components of the displacementvectors are plotted against the azimuth of the observation point

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

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


in a circular profile counted from the direction of thereflector’s rise (0º), calculated for three ratios of the circleradius 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 betweenthose for r / h = 0.6 and r / h = 1.0. Thus, the Uy/Ux ratios arethe largest (within 0.02–0.03) at azimuths between the reflec-tor’s strike (90º) and dip (180º) but are minor between thestrike (90º) and the rise (0º). The theoretical curves fit the bestthe experimental data at ϕ = 26º, with a very large rms error,as it was expected from the above analysis of the behavior oftheoretical curves (Fig. 9) and their comparison with theobservations (Fig. 8).

Table 2 synthesizes the data on observed PS and SSreflections on salt dome slopes in the Caspian basin. There is

no correlation between the amplitude ratios of the twocomponents (Uaccess /Umain) with the tilt ϕ, i.e., the model ofa tilted reflector between two isotropic layers is not the oneto account for the observed anomalous polarization of shearand converted waves.

An anisotropic model with an arbitrary position of thesymmetry axis (i.e., arbitrary polar and azimuthal angles thatcharacterize the position of this axis) was a critical stepforward in understating the anomalous polarization. Thisanisotropy cannot result from sedimentary lamination (exceptfor few cases of vertical layers), but is rather due mainly toaligned cracks: either long fractures in parallel planes or shortpenny-shaped cracks produced by variations in crustal hori-zontal stress. Tarkhov (1940) already mentioned those causeswhen wrote about a medium consisting of identical thin layers

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) andtangential (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 (Bakharevskayaet al., 1967).


separated by planes which arose during deposition changes.His experiments on static loading of limestone and sandstoneshowed that the anisotropy of that kind could be quite strong.Later Klem-Musatov et al. (1973) suggested a transverselyisotropic model based on the linear slip theory to simulate thepresence of cracks in rock.

Note that anisotropy of rocks with a system of verticalcracks produces wave patterns quite different from those inthe case of thin layers that differ in elastic constants. In thesection below we describe briefly the model (Klem-Musatovet al., 1973; Aizenberg et al., 1974) and the attempts ofapplying it to anomalous polarization of converted (PS) andshear (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 withnonwelded contact conditions on their boundaries. For non-

stationary oscillations, these boundary conditions are (Klem-Musatov et al., 1973)

σq (M0) = limM → M0

σq (M),

dUq (M0)dt

= limM → M0

dUq (M)dt

+ ∫ 0

t

σq (M)τΛq (t − τ) dτ

, (3)

Λq (t) = 1π ∫

0

ω

eiωt

Zq (ω) dω, q = p, s,

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 ina layer boundary plane, and M is the variable point inside alayer. The impedances Zq (q = p, s) can be found in differentways, and the choice influences the behavior and the magni-tude of anisotropy. Anisotropy appears on transition to thelimit n → ∞ in the solution for a discrete stratified medium (nis the number of thin layers in the model).

In a homogeneous anisotropic medium, assumed instead ofthe stratified one, the P, SV and SH velocities depend on the

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

Fig. 9. Modeled (with ray method) amplitude curves of x and y components ofdisplacement U of PS reflection in an isotropic model as a function of angle ψbetween 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. Velocityratio above reflector is γ = vS / vP = 0.31. After (Puzyrev and Obolentseva,1967).

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 reflectortilt

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


source direction defined by the angle θ between the wavenormal and the symmetry axis z as

vP (θ) = cP 1 + gP (1 − 2γ 2 sin2 θ)2 + gS γ 2 sin2 2θ

−1

,

vSV (θ) = cS 1 + gP γ 2 sin2 2θ2 + gS cos2 2θ

−1

, (4)

vSH (θ) = cS 1 + gS cos2 θ −1

, γ = cS / cP,

where cP, cS are the P and S velocities in unfractured rocksand 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), whichcame into broad use though appeared later than that ofKlem-Musatov et al. (1973) and Aizenberg et al. (1974), thephase velocity equations are (Schoenberg and Douma, 1988)

vP (θ) = √(λb + 2µb) / ρb 1 − EN (1 − 2γb sin2 θ)2 −

ET γb sin2 2θ 1 / 2

,(5)

vSV (θ) = √µb / ρb 1 − EN γb sin2 2θ − ET cos2 2θ 1 / 2

,

vSH (θ) = √µb / ρb 1 − ET cos2 θ 1 / 2

,

where λb, µb are the Lamé constants in the isotropic back-

ground, 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 sys-tem.

Equations (5) agree with equations (4). In order to see that,one has to expand the right-hand sides of the equations intobinomial series with respect to the small parameters: gP, gSfor (4) and EN, ET for (5), because both are much below 1,and to hold the linear terms only. When comparing theobtained equations, one arrives at the following relationshipbetween the model parameters:

cP = √(λb + 2µb)ρb−1 , cS = √µb ρb

−1 , ρ = ρb,

γ 2 = cS 2 / cP

2 = γb, 2gP = EN, 2gS = ET.

A transversely isotropic subsurface, an effective model ofa medium with nonwelded contact layer boundaries, isdescribed with four (c11, c33, c44, c66) instead of five elasticconstants (see (4) and (5)). That is why the knowledge of Pand S velocities on the symmetry axis (θ = 0º) and in theisotropy plane (θ = 90º) is, in principle, enough to find allelastic constants:

vSH⊥ = VSV⊥ = VSV|| = cS (1 − gS) =

√µb / ρb (1 − ET / 2) = √c44 /ρ ,

vSH|| = cS = √µb / ρ = √c66 /ρ ,

vP⊥ = cP (1 − gP) = √(λb + 2µb) / ρb (1 − EN / 2) = √c33 / ρ ,

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 − c132 = 2c44 (c11 +

c13) (Schoenberg and Sayers, 1995).

First steps in processing x and y seismograms using themodel by Klem-Musatov et al.

The anisotropic model by Klem-Musatov et al. (1973) andAizenberg et al. (1974) was used to obtain synthetic seismo-grams to fit the best the field data (Figs. 7, 8) from theTerkobai salt dome. The axis of symmetry was assumed to bevertical rather than normal to the reflector, in order to amplifythe tangential components. The anisotropy coefficientKP = vP|| / vP⊥ was allowed to vary within 1.0–1.1; the valuesabove 1.1 were avoided because the PSV – PSH arrival timedifference at KP > 1.1 would exceed 1.5 periods and result ina respective extension of the total wave PSV + PSH record,which was never observed in the field. However, no satisfac-tory fit between the theoretical and field seismograms wasachieved (see Fig. 6 in Klem-Musatov et al., 1973), especiallyon azimuths between the rise and the strike of the reflector(0o, 90o). Therefore, search according to the symmetry axisposition was apparently required as well.

Another example of synthetic seismograms approximatingthe field data (Klem-Musatov et al., 1973) was given in(Grechka and Obolentseva, 1987b). For the case of the Tanatarsalt dome experiment (Table 2) reported in (Puzyrev et al.,1983), the position of the symmetry axis was found in thespace (from the polar (β) and azimuthal (α) angles) withanisotropy parameters being allowed to vary in broad ranges.The best fit was achieved with the symmetry axis at 10º tothe reflector’s normal, for both angles (β and α). The twoangles measured the given direction, respectively, to thedownward 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 speciallyto understand how the PS and SS displacement componentratios in circular and linear profiles depended on the positionof the reflector relative to the symmetry axis of the overlyingtransversely isotropic earth. The number of variables being toogreat, the calculations had to be confined to the combinationof parameters which, according to field data (Bakharevskayaet al., 1967; Puzyrev, 1985; Puzyrev et al., 1983), appearedmore realistic and had some general implications.

Figure 10 shows the calculated ratios of the accessory-to-main displacement components (Uaccess/Umain) for PS and SSreflections from a ϕ = 10º tilted interface, which were ob-served in the profile to be at 45º to the line of the reflector’srise and dip. The symmetry axis of the medium over thereflector is in the rise–dip plane of the latter and is at theangle β (0º, 90º) to the vertical. The choice of this reflectororientation, symmetry axis, and profile direction proceeded


from experimental data and preliminary calculations showingthat the Uaccess/Umain (α, β) ratios were maximum at α about45º, and the objective was to explore the Uaccess/Umain ratiosas a function of β. Large anisotropy parameters were selectedto provide the highest Uaccess/Umain ratios. The anisotropycoefficients of the P, SH, and SV phases found as thefastest-to-slowest phase velocity ratios were kP = kSV = 1.1,kSH = 1.2 at the phase velocities along the symmetry axisvP⊥ = 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 meansover l = 1.5h long profiles, where h is the subsource depth ofthe reflector measured along the normal.

The calculations were performed in a zero approximationof the ray method. The sources were the pressure center forthe 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 Hzand 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 isotropicmodel) when the symmetry axis is along the normal to thereflector (β = ϕ). As its departure from the normal grows,Uaccess/Umain (β) increase to 0.5–1 at β < ϕ and to ~1 atβ > ϕ. In the latter case, the increase results from shear-wavesplitting at a horizontal (or quasi-horizontal) symmetry axis.

Above we wrote about the search for the anisotropycoefficient kP at a fixed β (β = 0º) from circular profile PSdata at the Terkobai site. As one can see in Fig. 10,kP was estimated on the left-hand branch (β < ϕ), whereUaccess/Umain (β) in PS cannot exceed 0.5. In another exampleof synthetic PS seismograms fitted to the data from the Tanatardome, β = 20º, i.e., it was found in the right-hand branch ofthe Uaccess/Umain (β) curve. Note, however, that search in thecase of Tanatar was simultaneously for two angles α and β (αwas found to be 10º) while the curves in Fig. 10 were forα = 0º.

Thus, although Figure 10 provides an idea of possibleUaccess/Umain ratios in PS and SS, it fails to cover all practicalcases and to show the Uaccess/Umain (β) behavior at α differentfrom 10º. Furthermore, the anisotropy parameters can havedifferent values with a boundless number of various combi-nations. That is why approximation of field data by theoreticalfunctions is a tough problem that has no unique solution. Thus,the anomalous polarization of PS and SS phases has to beaccounted for in some other way.

Algorithm to separate interfering S1 and S2 waves

The idea of possibility to pick pure S1 and S2 waves (SVand SH in transversely isotropic media) out of two- orthree-component field seismograms occurred to I.R. Obolen-tseva and S.B. Gorshkalev in the late 1970s–early 1980s andwas realized at that very time. The algorithm for separatingtwo interfering S1 and S2 waves and an application examplewere reported in (Obolentseva and Gorshkalev, 1986). Thatpublication was coeval with those of Alford (1986) andNaville (1986) in which the algorithms were designed forsolving the same problem for CMP and VSP data, respec-tively.

The algorithm by Obolentseva and Gorshkalev (1986)handles each two-component record (Ux(t), Uy(t)) at anobservation point, where the S ray direction should be known.The rays of two shear waves were assumed to coincide andto be wave normal equivalent. Strictly speaking, the assump-tion is valid in the case of weak anisotropy. Let the raydirection be defined by the vector l0 (hereafter the superscript“0” denotes the unit vector).

The first step of the algorithm consists in constructing thematrix 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 nextstep, (Ux(t), Uy(t)) is transposed from the xyz coordinates intox′y′z′ using the matrix M. The following steps are to convertthe 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) andUy′ (t − τ) is calculated at each step and saved in the memory.The final step of the algorithm is to find max

α max

τ K(τ). The

found α defines the position of the x′ and y′ axes of the“natural” coordinates, while τ is the lag of the slow phasebehind the fast one (S1 and S2, respectively).

Figure 11 illustrates the application of the algorithm toin-hole data from the Dossor salt dome (Gorshkalev, 2002)where an oriented three-component seismograph was placedat a depth of 100 m in Albian clay. An X-type source actedon the surface at 45º to the symmetry plane. Figure 11 showsthe Ux(t), Uy(t), Uz(t) records for the source distances L fromthe borehole in the left-hand panel and the Ux′ (t), Uy′ (t)records for S1 and S2 in the right-hand panel. The algorithm

Fig. 10. Ratios uy/ux of tangential components to radial ones for PS reflectionsand ux/uy for SS reflections (y-directed source) as a function of polar angle ofsymmetry 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 azimuth45º 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 PSwaves and f = 10 Hz (solid line) and f = 20 Hz (dashed line) for SS waves. After(Grechka and Obolentseva, 1987b).


also solved for the angles that characterize the position of thesymmetry axis and the S1 to S2 time delay.

Nikolskii (1987, 1992) developed further the algorithm ofObolentseva and Gorshkalev (1986) and applied the modifiedversion to two-component (x, y) PS data from the Tanatardome collected in 1982. Having converted those records intoSV and SH, he obtained a map of polarization vectors ofPS-wave split into PSV and PSH phases (Nikolskii and Shitov,1992). Overlapping that map on the surface of reflector IIIshowed polarization of SV and SH mostly in the planes of thereflector’s dip and strike. Note that a similar pattern wasobserved at the Dossor site (Trigubov and Gorshkalev, 1988).The polarization changed in zones of faults, whereby thedisplacement vectors became more scattered.

The algorithms for separating the interfering S1 and S2waves are applicable also to media with other symmetries,besides the transversely isotropic media (Trigubov and Gor-shkalev, 1988). The application of the separation algorithm tothe Dossor data yielded the indicatrices of the S1 and S2 ray

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

Theoretical and modeling work in 1960 through 1980

There were quite many publications concerning the theoryof seismic anisotropy and techniques for modeling wavefieldsin different kinds of anisotropic media. Below we cite onlythose that were published in Russia and were pioneering inthe discussed line of research.

Effective models of finely layered and fractured media

Publications on anisotropy of thin layered media appearedin the late 1960s–early 1970s (Loktsik, 1969, 1970a,b). Ofspecial 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

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 boreholein 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).


investigated the direction-dependent velocities in rocks con-sisting of alternating thin layers 1 and 2 with different ratiosof velocities (vS1 / vS2

, vP1 / vP2), densities (ρ1 / ρ2), and thick-

nesses (h1 / h2), the authors suggested a classification of thinlayered media with four types of anisotropy.

Theoretical and experimental (physical modeling with sheetmodels) studies of periodical layered structures were describedby Sibiryakov et al. (1980) who showed the homogeneoustransversely isotropic approximation to be valid at the layerthickness-to-wavelength ratio h / λ < 0.1−0.2.

The anisotropic model of a fractured medium was firstproposed in (Klem-Musatov et al., 1973; Aizenberg et al.,1974). We wrote about it in the previous section becausediscussion of 1960–1980 experiments required mentioning therespective theoretical background. Models of thin layeredmedia, including fractured ones, were reported in (Molotkov,1979, 1991; Molotkov and Khilo, 1983, 1986; and others) andin the overview by Bakulin and Molotkov (1998).

Fundamentals of the theory of elastic wave propagationin anisotropic media

There have been few books and tutorials published inRussia on elastic wave propagation in anisotropic media.

The best one is the book of Fedorov (1965), which thoughbeing intended for physicists of crystals provides an excellentoutlook of symmetry systems, propagation of plane elasticwaves in media with different symmetries, as well as of therespective problems and solutions. Another important bookappeared ten years later (Sirotin and Shaskolskaya, 1975,1979) and became an indispensable handbook on many issuesof symmetry and physical properties of solids. Later publica-tions of this kind were those by Petrashen’ (1980, 1984). Thelatter is a collection of three papers by Petrashen’ and Kashtan,Kashtan, Kovtun, and Petrashen’, and Kashtan and Kovtunconcerning the theory and algorithms.

Computing methods and algorithms

The numerical method and its applications to modelingwave propagation in inhomogeneous anisotropic solids for thehalfspace and sphere cases were detailed in (Martynov andMikhailenko, 1979). Point sources in a transversely isotropicelastic medium were first discussed in (Uspenskii and Ogurt-sov, 1962), and a method for imaging wavefields from pointsources in transversely isotropic media was suggested in(Martynov, 1986).

The ray method was first applied to anisotropic (inhomo-geneous) media by Babich (1961). It was also the subject ofChapter 5 of the book (C

v

erveny′ et al., 1977) written withparticipation of L.A. Molotkov. Later the ray method and therespective algorithms were reported in (Kashtan, 1982; Kash-tan and Kovtun, 1984; Kashtan et al., 1984).

In the 1970s–1980s, the issues of modeling and ray methodfor 3D anisotropic layered media were treated in a series ofpublications by I.R. Obolentseva and V.Yu. Grechka. Theearliest publication concerned an algorithm for rays that travel

in a transversely isotropic layered medium from a source togiven receivers (Obolentseva, 1975). The algorithm wassimilar to one published a year before for isotropic media(Obolentseva, 1974). In those studies, the coordinates of pointswhere the ray crossed the reflector were found using directlythe Fermat principle, i.e., from the stationary traveltimeconditions; the obtained system of equations was solved bymeans of iteration. Later there appeared limitation-free non-linear algorithms with the use of derivatives reported in(Obolentseva, 1980a) for isotropic media. Obolentseva andGrechka (1988) presented an optimization algorithm for raypaths in anisotropic media of any symmetry, and in a laterpaper (Obolentseva and Grechka, 1992), compared differentinversion techniques for layered anisotropic media followinga similar comparison for isotropic cases (Obolentseva, 1980b).

The book (Obolentseva and Grechka, 1989) encompassedall 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 thebook, and the third chapter contained algorithms for shearwave surfaces in the vicinity of singular directions andtechniques for separating n-valued wave surfaces into nsingle-valued domains amenable to the ray method. Chapter4 concerned algorithms for rays and displacements in anisot-ropic gyrotropic media.

A series of papers published in 1987 (Grechka andObolentseva, 1987a,b; Obolentseva and Grechka, 1987) con-cerned with numerical modeling of shear and converted wavesin anisotropic media on the basis of our earlier algorithms andexplained how the anomalous PS and SS polarization mayarise in media with azimuthal anisotropy, with theoreticalexamples of PS and SS reflections from the top and the bottomof an anisotropic layer for cases of different symmetry axisorientations. The tangential components of displacement wereshown to be small in the reflections from the layer top butalmost as large as the radial components in the layer bottomreflections.

Brodov et al. (1986) reported results of wavefield simula-tion for different cases of a transversely isotropic mediumwhich were obtained using algorithms from (Petrashen’,1984).

There were also other works in Russia that influenced theanisotropy research. It was, for instance, the book of Lapin(1980) which became popular with seismic survey people ofthat time. Slightly later works by E.A. Blyas (Blyas, 1983,1987, etc.) addressed approximate techniques for estimatingPP and PS reflection traveltimes in anisotropic layered media.Seleznev et al. (1986) dealt with inversion of compressionaland converted waves data.

Of course, the above citations of studies on elastic wavepropagation in anisotropic media cannot cover all publicationsof the respective period. We have selected first of all thehighlights in the research of azimuthal anisotropy as a factorof unusual polarization of shear waves, and other featuresobserved in three-component wavefield data. A special focushas been on the studies that were published in Russia butremained unfamiliar to English-speaking readers, in order to


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

Conclusions

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 theonly model of anisotropic media from the beginning of seismicanisotropy studies to the mid-1960s. It was designed to explainthe velocity difference observed since the 1930s along verticaland oblique rays (slower velocities along the latter). Theexplanation was found in 1940 through the mid-1950s: thevelocity difference resulted from thin layering if the layer ismuch thinner than the wavelength. A model was obtained torelate the parameters of thin layers to those of a transverselyisotropic medium with a symmetry axis normal to layering.

The next step was the discovery, in the 1960s, that the VTImodel most often failed to account for shear wave propagationin media where y components of displacement in SS waves(especially of large amplitudes) never appear from a y-directedsource, neither the y components of PS waves. Furthermore,the VTI model turned out to be inapplicable to large delaysat intersections of profiles of different azimuths. TI (trans-versely 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 arisesof a possible physical cause for this kind of symmetry.

In the 1970s, there appeared a model based on the theoryof linear interfacial slip, which is a kind of a nonwelded layerboundary. The model implied that a rock looking homogene-ous can be cut by a network of parallel cracks (horizontal ortilted) which make the rock transversely isotropic of types HTIor TTI, respectively. In that model, tangential displacementcomponents are a reasonable consequence because displace-ment is recorded in the coordinates that are arbitrary relativeto the natural coordinate system. (“Natural coordinates” arethe xyz system in which two shear phases with their fixedmutually orthogonal directions in the xy plane and differentvelocities correspond to a given direction of the wavenormal z). Transition from arbitrary coordinates to the naturalsystem was suggested in order to avoid the tangential compo-nents, and the respective algorithm was obtained in the 1980s.

Field, theoretical, and numerical experiments were carriedout simultaneously and concertedly thus making for ever betterunderstanding and employing the survey data. Progressively,all anisotropy parameters (traveltimes and amplitudes of P, S1,and S2 waves depending on reflection and attenuation coeffi-cients) have come into use for structural modeling andestimating the physical properties of rocks. Seismic anisotropystudies in Russia till the early 1990s were run parallel withsimilar research in other countries and leaded the latter inmany cases (especially before the 1980s). Some of thosestudies have been the subject of our overview.

The paper is dedicated to the memory of Nikolai NikitovichPuzyrev, the pioneer of multicomponent surveys in Russia, a

new line of seismic studies in which seismic anisotropy is animportant subject.

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Editorial responsibility: V.S. Seleznev


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Fifty years of seismic anisotropy studies in Russiachichinina.rusiamexico.com/.../2014/...Anisotropic.pdf · (1949), and Rytov (1956) investigated anisotropy produced by alternating