Observation and modelling of fault-zone fracture seismic anisotropy – II. P-wave polarization anomalies

Summary. In Part I of this paper we modelled shear-wave splitting observed in crystalline rock bordering an active, normal fault-zone at Oroville, California, with Červený's ray-tracing system applied to anisotropic heterogeneous media using Hudson's formulation of elastic constants for a medium containing aligned cracks. In Part II we use the ray-tracing results of Part I to quantitatively interpret P -wave polarization anomalies observed in the three-component seismograms recorded in the Oroville fault zone. We show that the eigenvectors of the first-order Christoffel tensor defined by the ray-tracing slowness vector and Hudson's first-order anisotropic corrections to the isotropic elastic tensor correctly account for P -wave first motion that deviates from the ray vector.     

Shear-wave polarizations on a curved wavefront at an isotropic free surface

Summary. We present polarization diagrams of the particle motions at the free surface of an isotropic half-space generated by incident shear waves from a local buried point source. The reflectivity technique is used to calculate synthetic seismograms from which the particle motions are plotted. The particle motions are examined over a range of epicentral distances in a uniform isotropic half-space for different source frequencies and polarization angles, and for different Poisson's ratios. The particle motions due to a curved wavefront possess different characteristics from those generated by plane wavefronts at corresponding incidence angles. A curved wavefront generates a local SP -phase: a P -headwave which propagates along the free surface, and arrives shortly before the direct S -wave. These two arrivals give rise to cruciform particle motions in the sagittal and horizontal planes, which could be misinterpreted as anisotropy-induced shear-wave splitting. An examination of the particle motion in the transverse plane, mutually orthogonal to the sagittal and horizontal planes, can be used to discriminate between isotropic and anisotropic interpretations. The amplitude of the SP -phase is enhanced when it propagates in a low-velocity surface layer overlying the source layer, and may then become the dominant phase on radial-component seismograms. The presence of even a single surface layer may introduce considerable complexity into the seismogram, and we examine the effects of layer thickness, velocity contrast, and source depth on the corresponding polarization diagrams. Reliable information on the source and propagation path characteristics of shear waves from a buried local point source can only be obtained from free-surface records if they are recorded within a very limited epicentral distance range.     

A numerical method to determine physical source parameters from free oscillation data

A method to determine physical source parameter using free oscillation data is presented. It is assumed that the geometry of the source is known, e.g. from P -wave data. The source is assumed to propagate in the horizontal direction, while unknown parameters to be determined are the azimuth and velocity of propagation, the distance over which the seismic source propagated and the source intensity as a function of propagation distance.
The method consists in the systematic search for the set of source parameters rendering phase corrections which maximize the spectral peak amplitudes within the excitation criterion scheme.
If there is no precursive motion, the average dislocation time function can be determined from the spectrum of the seismic moment and the space source intensity. The source intensity as a function of instantaneous source location is found independently of the P -wave origin time and source dislocation time function. The method does not require to correct the data for attenuation.     

Use of receiver functions in wide-angle controlled-source crustal data sets

First-arrival waveforms remain underutilized in crustal refraction–reflection seismology by mostly reducing them to traveltime picks. However, as in earthquake seismology, the waveforms also contain important information about shallow near-receiver structures. We illustrate the use of three-component waveform analysis on the records from the ACCRETE wide-angle data set (SE Alaska and British Columbia; 1994), apply the Receiver Function (RF) methodology to the codas of P -wave arrivals, and draw two important conclusions. First, the P -wave polarization azimuths are found to be controlled by the near-receiver structures and virtually unrelated to the source–receiver backazimuths, from which they deviate by up to ∼40°. This observation might be important for studies of anisotropy and also for earthquake RF studies. Second, after correcting for the polarization azimuths, clear P / S mode conversions are reliably detected within 80–400 ms following the primary arrivals. The conversions are interpreted as originating at the base of the sedimentary cover of the fjord channel. In most cases, imaging of the basement requires only several records; however, notable exceptions are also found and interpreted as caused by multipathing, localized scattering, and onsets of crustal and Moho reflections. The ACCRETE example shows that RF methodology could be useful for constraining sediment thickness and deriving P - and S -wave receiver statics in land refraction surveys where collocated reflection profiles are not available. In addition, RFs from repeatable controlled sources could be useful for testing and calibration of RF techniques.     

Head waves in laterally heterogeneous media

Summary. A layer of constant thickness over a half-space is assumed, and the propagation of head waves is considered for the following two cases: (1) the P -wave velocity varies in the layer in the horizontal direction, and is constant in the half-space: (2) the P -wave velocity varies in the half-space in the horizontal direction, and is constant in the layer. In each case the horizontal velocity gradient is assumed to remain constant. The wave propagation is investigated in the direction of the gradient (direct profile), and opposite to it (reverse profile). Formulae for the travel times and the amplitudes are obtained on the basis of ray-theoretical considerations. Conditions are discussed for the discrimination in a field experiment between the case of a sloping boundary separating the homogeneous media, and the case of an intrinsic horizontal velocity gradient.     

Radiation Patterns For Explosively-Loaded Axisymmetric Cavities In an Elastic Medium: Analytic Approximations and Numerical Results

Summary. A combination of analytic solutions and numerical computations has been used to obtain the low-frequency, far-field P - and S -wave radiation patterns from Heaviside-loaded ellipsoid cavities of fixed volume and varying eccentricity (including the sphere and the finite cylinder) in an unbounded elastic solid. Results show that as the spherical cavity becomes a prolate ellipsoid, S -waves are radiated with strength proportional to the increasing aspect ratio ( A _R) (major axis/minor axis). With large aspect ratios ( A _R > 10) both P - and S -wave radiation strengths approach asymptotically a limit for which the maximum P -wave amplitude is twice as large as the maximum S -wave amplitude, irrespective of Poisson's ratio.
A representation of the radiation pattern is obtained whose mathematical form can be interpreted as due to the action of a centre of dilatation plus a vector linear dipole, both appropriately weighted. the weights are nonlinear functions of A _R. On the basis of numerical results, we have obtained an approximate formula that relates those weights to A _R. the radiation patterns thus obtained are in excellent agreement with the experimental results reported by Gupta & Kisslinger. the results may be of some importance to the investigation of explosions in non-spherical cavities and in the study of the origin and mechanisms of volcanic (explosive) earthquakes.     

A Hybrid Collocation Method For Calculating Complete Theoretical Seismograms In Vertically Varying Media

A numerical method is presented for calculating complete theoretical seismograms, under the assumption that the earth models have velocity, density and attenuation profiles which are arbitrary piece-wise continuous functions of depth only. Solutions for the stress-displacement vectors in the medium are expanded in terms of orthogonal cylindrical functions. Our method for solving the resulting two-point boundary value problems differs from that of other investigators in three ways. First, collocation is used in traditionally troublesome situations, e.g. for highly evanescent waves, at turning points, and in regions having large gradient in material properties. Second, in some situations (high frequencies and small gradients) P and S -waves decouple and we use a different solution method for each wave type, instead of trying to force a single method to find all solutions. For example, above the P - and S -waves turning points an approximate fundamental matrix may be used for each wave type. At the P -wave turning point, the fundamental matrix may be used for the S -wave components but collocation is used for the P -wave. Between the P - and S -wave turning points collocation is used for the evanescent P -wave and the fundamental matrix is used for the S -wave. At the S -wave turning point and below, collocation is used for both. Third, the computational algorithm chooses the appropriate solution method and depth domain upon which it is employed based upon a specified error tolerance and the known inaccuracies of the various approximations employed. Once solutions of the boundary value problems are obtained, a Fourier—Bessel transform is then applied to get back into the space-time domain.     

Finite-frequency sensitivity of body waves to anisotropy based upon adjoint methods

We investigate the sensitivity of finite-frequency body-wave observables to mantle anisotropy based upon kernels calculated by combining adjoint methods and spectral-element modelling of seismic wave propagation. Anisotropy is described by 21 density-normalized elastic parameters naturally involved in asymptotic wave propagation in weakly anisotropic media. In a 1-D reference model, body-wave sensitivity to anisotropy is characterized by 'banana–doughnut' kernels which exhibit large, path-dependent variations and even sign changes. P -wave traveltimes appear much more sensitive to certain azimuthally anisotropic parameters than to the usual isotropic parameters, suggesting that isotropic P -wave tomography could be significantly biased by coherent anisotropic structures, such as slabs. Because of shear-wave splitting, the common cross-correlation traveltime anomaly is not an appropriate observable for S waves propagating in anisotropic media. We propose two new observables for shear waves. The first observable is a generalized cross-correlation traveltime anomaly, and the second a generalized 'splitting intensity'. Like P waves, S waves analysed based upon these observables are generally sensitive to a large number of the 21 anisotropic parameters and show significant path-dependent variations. The specific path-geometry of SKS waves results in favourable properties for imaging based upon the splitting intensity, because it is sensitive to a smaller number of anisotropic parameters, and the region which is sampled is mainly limited to the upper mantle beneath the receiver.     

A review of the effects of anisotropic layering on the propagation of seismic waves

Summary. This paper reviews recent work, much of it unpublished, on the effects of anisotropy on seismic waves, and lays the theoretical background for some of the other papers in this number of the Geophysical Journal .
The propagation of both body and surface waves in anisotropic media is fundamentally different from their propagation in isotropic media, although the differences in behaviour may be comparatively subtle and difficult to observe. One of the most diagnostic of these anomalies, which has been observed on some surface-wave trains, and should be evident in body-wave arrivals, is generalized, three-dimensional polarization, where the Rayleigh motion is coupled to the Love, and the P and SV motion is coupled to the SH . This coupling introduces polarization anomalies which may be used to investigate anisotropy within the Earth.     

Anisotropic tomography of P-wave traveltimes

The ray path of a P -wave is specified in terms of the ray parameter and three Euler angles. the P -wave traveltime depends only on the ray parameter for a spherically symmetric earth. If we introduce an aspherical perturbation, including general ani-sotropy, the dependence on Euler angles can be expanded in terms of the rotation matrix for a fixed ray parameter. If the perturbation is isotropic, the expansion coefficients satisfy certain relations which may be used to obtain definite evidence for anisotropy rather than isotropic lateral heterogeneity.     

Generation of the teleseismic P-wave coda from Aleutian earthquakes

We investigate large-amplitude phases arriving in the P -wave coda of broad-band seismograms from teleseisms recorded by the Gräfenberg array, the German Regional Seismic Network and the Global Seismic Network. The data set consists of all events m _b≤ 5.6 from the Aleutian arc between 1977 and 1992. Earthquakes with large-amplitude coda waves correlate with the presence of oceanic crust in the source region. The amplitudes sometimes approach those of the P wave, much larger than predicted by theory. Modelling indicates that phases in the P -wave coda cannot be P -wave multiples beneath the source and receiver, or underside reflections, which precede PP , from upper-mantle discontinuities. Among the events, seismograms are very similar, where the arrival times of the unusual phases agree approximately with the predicted times of S -to- P conversions from the upper-mantle discontinuities under the source. Because the large-amplitude phases in the P -wave coda have little, if any, dependence on event depth and have predominantly an SV -wave radiation pattern towards the receiver, we suggest that they originate as SV and/or Rayleigh waves and are enhanced by lateral heterogeneity and multipathing from the subducting Aleutian slab.     

Plane wave propagation in nonlinear-elastic anisotropic media

Summary. Kelvin-Christoffel equations describing plane wave propagation in anisotropic media are generalized to account for the effects of nonlinear elasticity. The polarization and waveform of nonlinear distortions of a transient plane wave are investigated by means of perturbation theory. Detailed analysis for an anisotropic medium with hexagonal symmetry shows that for "pure" shear-waves the polarization vector of the nonlinear component is always perpendicular to that of the linear wave. In the case of a high-amplitude excitation (for instance, in the vicinity of large earthquakes) the influence of nonlinearity may cause distortions of shear-wave polarization, which contains the most reliable information on the presence and characteristics of anisotropy. The solutions presented in this paper make it possible to solve reflection-transmission problems in nonlinear-elastic anisotropic media.     

Travel-time anomalies in the mantle under the North Atlantic

Summary. Lateral heterogeneity exists in the Earth's mantle, and may result in seismic velocity anomalies up to several per cent. If convection cells and plumes extend down to the core, then these features may be associated with local inhomogeneities observed in the lower mantle.
Published data for direct and core-reflected P -wave residuals are used to delineate velocity anomalies in the middle—lower mantle under the North Atlantic. Differential ( PcP — P ) residuals indicate travel-time anomalies near the core—mantle transition, and may be due to core topography or lateral variations in velocity. It is assumed that the anomalies occur near the midpoints of the ray paths. The main source of error in the data set may arise from phases which have been identified incorrectly. Hence trend-surfaces are fitted to the residual data to show only the large-scale trends in anomaly values, with wavelengths of the order of 1000 km.
The Azores and Colorado hot spots occur in a region covered by the data. A possible interpretation of the trend maps is that an anomalous zone extends from a relatively fast region at the core boundary at 35° N, 50° W up to these hot spots, at about 30 degrees from the vertical. This may agree with the suggestion of Anderson that plumes are chemical rather than thermal in origin. If inclⁱned plumes do exist, the deviation from the ideal vertical plume or convection cell boundary may imply that lateral shear or other distortion effects exist in the mantle.     

Polarization and amplitude attributes of reflected plane and spherical waves

The characteristics of a reflected spherical wave at a free surface are investigated by numerical methods; in particular, the polarization angles and amplitude coefficients of a reflected spherical wave are studied. The classical case of the reflection of a plane P wave from a free surface is revisited in order to establish our terminology, and the classical results are recast in a way which is more suited for the study undertaken. The polarization angle of a plane P wave, for a given angle of incidence, is shown to be 90° minus twice the angle of reflection of the reflected S wave. For a Poisson's ratio less than 1/3, there is a non-normal incident angle for which both amplification coefficients are 2 precisely; for this incident angle the direction of the particle motion at the free surface is also the direction of the incident wave. For a wave emanating from a spherical source, the polarization angle, for all angles of incidence, is always less than, or equal to, the polarization angle of a plane P wave. The vector amplification coefficient of a spherical wave, for all angles of incidence, is always greater than the vector amplification coefficient of a plane P wave. As expected, the results for a spherical wave approach the results for a plane P wave in the far field. Furthermore, there was a good agreement between the theoretical modelling and the numerical modelling using the dynamic finite element method (DFEM).     

Three-dimensional seismic structure beneath the Australasian region from refracted wave observations

The earthquakes in the seismicity belt extending through Indonesia, New Guinea, Vanuatu and Fiji to the Tonga–Kermadec subduction zone recorded at the 65 portable broad-band stations deployed during the Skippy experiment from 1993–1996 provide good coverage of the lithosphere and mantle under the Australian continent, Coral Sea and Tasman Sea.
The variation in structure in the upper part of the mantle is characterized by deter-mining a suite of 1-D structures from stacked record sections utilizing clear P and S arrivals, prepared for all propagation paths lying within a 10° azimuth band. The azimuth of these bands is rotated by 20° steps with four parallel corridors for each azimuth. This gives 26 separate azimuthal corridors for which 15 independent 1-D seismic velocity structures have been derived, which show significant variation in P and S structure.
The set of 1-D structures is combined to produce a 3-D representation by projecting the velocity values along the ray path using a turning point approximation and stacking into 3-D cells (5° by 50 km in depth). Even though this procedure will tend to underestimate wave-speed perturbations, S -velocity deviations from the ak135 reference model exceed 6 per cent in the lithosphere.
In the uppermost mantle the results display complex features and very high S -wave speeds beneath the Precambrian shields with a significant low-velocity zone beneath. High velocities are also found towards the base of the transition zone, with high S -wave speeds beneath the continent and high P -wave speeds beneath the ocean. The wave-speed patterns agree well with independent surface wave studies and delay time tomography studies in the zones of common coverage.     

The azimuthal variation of teleseismic P-wave travel times

Summary. Results of earlier studies of P -wave travel-time anisotropy are compared with the P -wave velocity-anisotropy obtained by other authors for different regions of the Earth. The azimuthal relationship of P -wave travel times from surface sources on the Siberian platform is investigated. It is found that the direction of the minimal travel time of P -waves for the Siberian platform, as well as that for European regions, is close to north–south.     

Detailed S-wave structure in the Dublin Basin and its northern margin

Summary. The seismic structure has been measured to a depth of about 3 km along a 30 km seismic profile in east central Ireland. This profile is unusual in that it is the S -wave velocity—depth structure that has been measured to a degree of precision more normally associated with P -wave results. One reason for this is that the sources used were quarry blasts which generated strong S -waves and short-period surface waves but rather weak P -waves.
The results show a layer of Carboniferous limestone with shear velocity 2.65 km⁻¹ s overlying a layer with a velocity of 3.06 km s⁻¹. This second layer was interpreted as Lower Palaeozoic strata (Silurian/Ordovician) since this velocity was evident in an inlier seen at the surface at the northern end of the line. A third refraction horizon, shear velocity 3.45 km s⁻¹ and displaying a basinal structure, was also recognized. This may be Cambrian or Precambrian basement.     

Reflection coefficients for weak anisotropic media

The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy.
Classical perturbation theory is applied on a 6times6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S -wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6times6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lamé parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.     

Seismic imaging of the laterally varying D" region beneath the Cocos Plate

We use an axisymmetric, spherical Earth finite difference algorithm to model SH -wave propagation through cross-sections of laterally varying lower mantle models beneath the Cocos Plate derived from recent data analyses. Synthetic seismograms with dominant periods as short as 4 s are computed for several models: (1) a D" reflector 264 km above the core–mantle boundary with laterally varying S -wave velocity increases of 0.9–2.6 per cent, based on localized structures from a 1-D double-array stacking method; (2) an undulating D" reflector with large topography and uniform velocity increase obtained using a 3-D migration method and (3) cross-sections through the 3-D mantle S -wave velocity tomography model TXBW. We apply double-array stacking to assess model predictions of data. Of the models explored, the S -wave tomography model TXBW displays the best overall agreement with data. The undulating reflector produces a double Scd arrival that may be useful in future studies for distinguishing between D" volumetric heterogeneity and D" discontinuity topography. Synthetics for the laterally varying models show waveform variability not observed in 1-D model predictions. It is challenging to predict 3-D structure based on localized 1-D models when lateral structural variations are on the order of a few wavelengths of the energy used, particularly for the grazing geometry of our data. Iterative approaches of computing synthetic seismograms and adjusting model characteristics by considering path integral effects are necessary to accurately model fine-scale D" structure.     

An analysis of P-wave amplitudes recorded by seismological stations in the USSR

Summary. Short period P -wave amplitudes at Soviet seismic stations are analysed to determine an amplitude–distance curve in the range 0°–180° from the USSR. Station amplitude terms, corrected for crustal amplification effects, are determined from seismic sources located between 30° and 90°. A relationship between these station terms and regional P_n wave speed and regional heat flow is determined and is found to be in close agreement with similar observations in North America. Station amplitude terms were found to be reasonably uniform across the continental portion of the USSR but the Baikal Rift valley is characterized by very low P -wave amplitudes and is the most significant feature discemible from the P -wave amplitude terms.