Elastic wave mode separation for TTI media

Upper mantle structure beneath the Mongolian region from multimode surface waves: Implications for the western margin of Amurian plate

Scattering of harmonic P1 and SV waves by a shallow lined circular tunnel in a poroelastic half-plane

P and SV velocity waveforms from the Sterling explosion of December 3, 1966, are analyzed from a point of view often taken in earthquake source studies. The detonation occurred in the cavity excavated by the Salmon explosion within the Tatum salt dome, Mississippi. Velocity recordings taken from nearby borehole strong motion instruments show the occurrence of significant SV energy (Perret, 1968a, b). Kinematic source models are constructed by considering constraints afforded by the preshot cavity, condition of the surrounding material, and aspects of the P and SV waveforms such as wave polarity and duration. Waveforms for these models are computed and compared directly with the data. Modeling results suggest that the SV waves were radiated primarily by induced normal faulting occurring beneath the cavity rather than by an asymmetric pressure distribution based on the preshot cavity geometry. Secondary arrivals within the P waveforms and SV amplitudes suggest that these normal faults also had a component of tensional displacement which was probably due to forceable injection of material from the blast. A measurement for Qβ of 35 is obtained for salt from the amplitude decay arid pulse width changes observed for SV waves along a profile taken at shot level. This value for Qβ is for the region away from the cracked area induced by the previous Salmon explosion and qualitatively explains the differing frequency content observed at the closest Sterling sensors between P and SV waves. The anomalous SV radiation has little to do with ‘tectonic’ stress release but is a result of the medium's geometry and anisotropy.

SUMMARYIn this paper, the 3-D theoretical seismogram at the free surface of an elastic half-space with ‘vertical transverse isotropy (VTI)’ and an ‘elliptic anisotropy‘ is synthesized via semi-analytical formulation. To this end, the time-domain solution is obtained via the Hankel and Laplace integral transforms and the Cagniard contour integration scheme. The formulations include complete Green's functions expressed in compact and elegant formulations in terms of some elementary line integrals over the finite interval (0, π/2), due to the general point force and point double-couple source of arbitrary orientation, varying with time as Heaviside step function. This solution is further implemented to model seismic waves generated from moment tensor source with a ramp-type slip function. The solution clearly delineates compressional P waves, shear SV waves and SH waves, diffracted SPwaves and surface Rayleigh waves, with two critical distances in which the mode conversion happens. The first marks the conversion of the total reflection of SV wave into the diffracted SPwave travelling on the free surface with the velocity of compressional wave. The second, however, marks a location where the order of arrival times of SP and SH waves is reversed. The interesting phenomenon of shear wave splitting by the apparent time lag between the arrivals of SV and SHwaves is further demonstrated. Studied also is the effect of non-double-couple (non-DC) components of the moment tensor of the shear fault. Particularly, it is shown that non-DC components may lead to large amplitude of the P wave in the presence of anisotropy, resulting in the changed polarity of P wave at the onset of the motion. It is found that for the elliptic anisotropy, and for oblique-thrust faults, the isotropic moment (ISO) is always negative if the elastic constants satisfy C11 > C33 and C44 > C66, whereas it is always positive if C11 < C33 and C44 < C66 (for oblique-reverse faults, the situation is reversed). It is shown that for some sedimentary rocks, the decomposition of moment tensor in the general VTI and in its elliptic approximation are close to each other, and that the moment tensor decomposition in VTI media is very sensitive to the elastic coefficients, such that a small change in the elastic constants may lead to a remarkable change in the moment tensor decomposition. Particularly, the effect of elastic constant C13 is shown to be significant. While the time-domain solution obtained in this paper can be also degenerated to the solution in the corresponding isotropic half-space, the Mathematica code of the solution provided can be served as benchmark to other numerical solutions, applicable for the computation of the theoretical seismogram in the involved media.

Although Love waves are widely analyzed in earthquake seismology, there is much less attention on utilizing Love waves than Rayleigh waves by the near-surface community. Unlike incident P and Sv waves, a plane SH wave for a series of horizontal layers refracts and reflects only SH waves, which makes the shallow SH-wave refraction method more popular to define shear (S)-wave velocity in near-surface geologic applications. For this reason, abundant SH-wave refraction data have been acquired and S-wave velocities have been determined using their first arrivals. If signals are recorded long enough, Love-wave energy can be clearly observed on SH-wave refraction data. Wave conversion may occur in an SH-wave refraction and Love-wave analysis only results in SH-wave velocities, which suggest that we may benefit from revisiting the existing SH-refraction data using multichannel analysis of Love waves (MALW). We used numerical modeling results and real-world examples to demonstrate three advantages of revisiting SH-wave data using the MALW method. Owing to a long geophone spread commonly used in SH-wave refraction survey, a sharp image of Love-wave energy can be generated, allowing better pickings for the phase velocities of Love waves. Because Love waves are independent of P-wave velocity, “mode crossing” in an image of Love-wave energy is less common than in Rayleigh-wave images. Fewer unknowns in the method MALW make dispersion curves of Love waves are simpler, which leads to more stable inversion of Love waves and reduces the degree of nonuniqueness.

Summary We have derived anisotropic moveouts of seismic waves in HTI media by a new approach. In this approach, we use the idea of rotating VTI media into HTI media to study the behavior of the velocity and moveout of seismic waves in HTI media. This is achieved by developing a relationship between the incident angle and azimuth angel in HTI media and the angle to the symmetry axis in VTI media. In this approach, we firstly derive two general definitions which easily define the NMO velocity and the quadratic parameter in VTI media. Based on these definitions and the relationship between the angles of VTI and HTI media, we derived the exact formula of NMO velocities and anisotropy parameters of P, SV and SH waves in HTI media. The NMO velocities in HTI media obtained in this approach are exactly elliptical for any strength anisotropy. Based on the NMO velocity and anisotropy parameters, we also derived the exact format of moveout for P and SV waves. For weak HTI media, a unified moveout for P and SV waves is obtained.

The diffraction of P and S waves by various obstacles is studied theoretically, in order to evaluate frequency dependent corrections to ray theory for elastic waves which travel nearly along the Earth's core shadow boundary. Most of the properties of this scattering process are conveniently illustrated by a simple Earth model, which gives rise to a problem in plane strain. This model is an infinite homogeneous elastic solid in which a steady state plane body wave (of the type P, SV, or SH) is incident on a circular cylindrical cavity. A Poisson summation is used for the scattered elastic potentials, and contributions from waves diffracted at least once around the cylinder are neglected. Simple approximation formulae are developed to examine the behavior of P, SV, and SH waves on and near their geometrical shadow boundary behind the fluid. Computed numerical results are believed to be valid for frequencies above 0.03 Hz. The solution method, which may be regarded as a corrected Fresnel theory, is taken through four successive stages of generalization to study increasingly realistic Earth models: (i) diffraction of cylindrical waves from a line source. For this problem our solution is in excellent agreement with the results of an ultrasonic model experiment conducted by Teng and Wu (1968). (ii) Diffraction by a fluid cylinder of cylindrical waves from a line source. (iii) Diffraction by a spherical fluid of spherical waves from a point source. Here we find good agreement between numerical results from our approximate method, and computation of the exact Poisson line integral. The final stage of generalization, to study (iv) diffraction by a spherical fluid/solid discontinuity in a realistic radially heterogeneous Earth, is obtained by methods similar to (iii), but after an extensive revision of Hook's (1961) discussion of elastic potentials in general media. In our approach, we recognize that the designation of P and S displacements is somewhat arbitrary in heterogeneous elastic media, but becomes precise in the high frequency limit of ray theory (in which P and two S components are decoupled). These facts are used for radially heterogeneous isotropic Earth models to establish three potentials (P,S,T) with the properties (a) that T(_~r,t) is decoupled from P and S, and is a potential for SH motion, (b) the coupling of P and SV waves is reflected in a system of coupled scalar equations for P(_~r,t) and S(_~r,t), and (c) in the high frequency limit we have P(_~r,t) and S(_~r,t) satisfying canonical uncoupled wave equations with the respective velocities (λ+2µ/ρ)^(1/2),( μ/ρ)^1/2. Many possibilities are suggested by the coupled equations for P(_~r,t) and S(_~r,t), apart from their use in the solution of (iv) above. They lead to a statement of conditions on the Earth model under which P and SV waves can propagate independently (at any frequency). We also use them to obtain approximate reflection coefficients for upper mantle transition regions which generate observed precursors to the phase PKPPKP, finding that the extent of velocity gradient anomaly in such regions must be less than about 4 km, in order to observe short period (1 sec) reflections. Our numerical study of core diffraction provides an explanation for the observed polarization towards SH of diffracted S waves, and also shows that there is a slight dispersion effect dT/dΔ data, obtained for P in the range beyond 90°, which can and must be allowed for in accurate Herglotz-Wiechert inversion studies. The numerical methods developed for discussion of (iv) are expected to have wider applications in seismological studies of the Earth's core, mantle, and crust.

In this article the reflection phenomena of plane harmonic elastic waves in a homogeneous, prestressed, orthotropic half-space are discussed. It has been observed that the reflection phenomena of pure mode of P and SV waves in such a medium are different from that of the classical isotropic elastic medium. Apart from the directions parallel to the coordinate axes in the medium there exists a specific direction along which pure modes of P and SV waves may propagate through the medium. It is seen that under certain conditions the P wave propagating in a specific direction will give rise to a reflected wave which is either the superposition of both P and SV waves or an SV wave only, obeying the simple laws of reflection, but it cannot give rise to a pure P mode only. [This phenomenon, i.e., case 3 (as discussed in the paper) may be possible when the denominator of R2 is zero.] Similar phenomena are also observed in the case of an SV wave. Since initial stress generally induces orthotropy in an isotropic medium, similar reflection phenomena should also be observed in a stressfree orthotropic medium. But the reflection pehnomenon of an SH wave is similar to that in the classical elastic medium. Moreover, under equal normal initial stresses, the medium behaves like a classical isotropic one and hence the reflection phenomena are similar to that of the classical case.

Summary Using ray perturbation theory, a linear relationship is derived that describes the first-order perturbation in slowness vectors due to elastic parameter and interface perturbations as well as to source and receiver position perturbations. For layered media. a propagator matrix, which has symplectic properties, is defined in Cartesian coordinates. The method is valid for heterogeneous isotropic or anisotropic media with interfaces, and allows the efficient calculation of Frechet derivatives of the polarization vectors. This makes it possible to use polarization data in tomographic inversion. Explicit integral expressions of the Frechet derivatives of the traveltime and the polarization vector are given for the qP wave in a transverse isotropic medium. The reference medium is a heterogeneous medium with elliptical anisotropy. A sensitivity study shows that the qP-wave polarization vectors provide information on the isotropic component of the slowness.

Summary Sedimentary rocks encountered in exploration can be regarded as transversely isotropic (TI) media having only 5 independent elastic constants and a single symmetrical axis. Transversely isotropic phase velocity expressions exist for P-, SH- and SV-waves. Approximate explicit analytical Pwave ray velocity functions in transversely isotropic (TI) media have now been developed (Zhang and Uren, 2001 a, b). These ray velocity functions include several dimensionless constants. Unfortunately, no explicit exact analytical ray velocity functions have been formulated for P- or SV-waves propagating in TI media. In exploration seismology, explicit ray velocity functions are needed for efficient migration, velocity analysis and other seismic processing procedures. The formulae for calculating the travel times between any two points in homogeneous transversely isotropic (TI) media and for the reflection travel times from a single horizontal reflector overlain by transversely isotropic media with a vertical symmetrical axis (VTI) have been derived (Zhang and Uren, 2001 a, b). New traveltime expressions for HTI media are presented here. A new technique for velocity semblance analysis is presented to estimate the dimensionless parameters in the new NMO equation, using surface seismic survey data. The results of NMO corrections with these parameters flatten events over a large range of offsets.

This study uses ultrasonic physical modelling to test the accuracies of numerical calculations of traveltimes and conversion-point (CP) positions for P-SV wave propagation in a horizontal transversely isotropic (TI) medium. Study results show that the traveltimes and CP positions for P-SV wave propagation on the isotropic plane of a TI medium computed using Fermat’s minimum-time principle are the same as those of using the isotropic non-hyperbolic moveout equation and the isotropic CP equation. However, for P-SV wave propagation on the symmetry-axis plane of a TI medium, the arrival times and CP positions of SV-waves are difficult to determine by any ray methods when the propagation directions of SV-waves are within the cuspoidal SV-wave group velocities zone. But the first arrival times and the propagation of the dominant energy of P-SV waves can still be analysed by ray methods. Based on the calculation of Fermat’s minimum-time principle, if the source-receiver offset is greater than a critical distance, the reflection angles of the converted SV-waves are fixed at a specific angle with a local maximum SV-wave group velocity of the neighbourhood area. This is because the converted SV-waves prefer to propagate along the cuspoidal directions with larger amplitude and higher velocity. Verified by the physical modelling, the Fermat’s minimum-time principle used to calculate traveltimes of P-SV waves is better than the anisotropic non-hyperbolic moveout equation. The physical modelling for the CP position experiment can give a clearer visualisation of the variations of CP positions in the profile, and the feasibility of using Fermat’s minimum-time principle to determine CP positions is also better than that of the anisotropic CP equations. Therefore, in the seismic data processing, Fermat’s minimum-time method is recommended to accurately determine the arrival times and CP positions of P-SV wave propagation in TI media.

Summary The standard acoustic approximation in transversely isotropic (TI) media results in an approximate P-wave phase velocity expression with a complicated square root term. Thus, the corresponding wavefields suffer from the unstable SV-wave artifacts. Many subsequent pure P-wave simulation methods were proposed to eliminate the SV-wave artifacts at the expense of decreasing the accuracy or increasing the computing cost. In this abstract, we propose an optimized acoustic approximation for P-wave in TI media to overcome the defect of the standard acoustic approximation. Since the P-wave propagation is weakly dependent on the vertical S-wave velocity, we construct a function of the vertical S-wave velocity squared to approximate the P-wave phase velocity. The corresponding expression is quite concise, without square roots and complicated fractions. Then, we derive a pure P-wave equation in tilted transversely isotropic (TTI) media. We compare and analyze the wavefields of several simulation methods in the homogeneous and complex TTI models. The wavefields of the proposed method only involve the P-wave propagation with high accuracy. And the computing cost is acceptable. So, this optimized acoustic approximation and the corresponding pure P-wave simulation method can be further used for the reverse time migration and full waveform inversion in TI media.

Summary Salt plays an important role in the Gulf of Mexico and other areas of the world. In the previous laboratory measurements, it has shown cubic anisotropy on halite (rock salt). For salt formation in the Gulf of Mexico, the deformation and the flow of evaporates could lead to the realignment of constituent crystals which also generates anisotropic aggregates (Raymer et al., 2000). In this paper, we investigate and quantify anisotropy of pure halite. With respect to all polarization in the pure halite, Vp changes from 4.44 to 4.76 km/s and Vs changes from 2.47 to 2.92 km/s. The difference of the one-way travel time caused by this cubic anisotropy for a 1km thick 3D numerical model is up to 0.02s, 0.06s and 0.03s for P wave, SV wave and SH wave, respectively.

In the process of dipole-source acoustic far-detection logging, the azimuth of the fracture outside the borehole can be determined with the assumption that the SH–SH wave is stronger than the SV–SV wave. However, in slow formations, the considerable borehole modulation highly complicates the dipole-source radiation of SH and SV waves. A 3D finite-difference time-domain method is used to investigate the responses of the dipole-source reflected shear wave (S–S) in slow formations and explain the relationships between the azimuth characteristics of the S–S wave and the source–receiver offset and the dip angle of the fracture outside the borehole. Results indicate that the SH–SH and SV–SV waves cannot be effectively distinguished by amplitude at some offset ranges under low- and high-fracture dip angle conditions, and the offset ranges are related to formation properties and fracture dip angle. In these cases, the fracture azimuth determined by the amplitude of the S–S wave not only has a $180^\circ $ uncertainty but may also have a $90^\circ $ difference from the actual value. Under these situations, the P–P, S–P and S–S waves can be combined to solve the problem of the $90^\circ $ difference in the azimuth determination of fractures outside the borehole, especially for a low-dip-angle fracture.

Studies of seismic anisotropy in situ can help to discriminate between different rock types for the lower crust. In this context we investigate the sensitivity of an iterative linearized 3-D travel-time inversion scheme for transversely isotropic media with respect to two types of systematic errors: wrong velocities and interface topography of the hanging wall of the lower crust. The computations simulate realistic field conditions such as found for the Variscan crust at the Urach geothermal anomaly. The study focusses on the possible information content of split S M S arrivals observed along two orthogonal expanding spread profiles. It ensues that an imperfect knowledge of the layer geometry is of minor importance compared to errors in the velocities of the hanging wall. In particular, upper crust anisotropy has to be considered carefully. Generally, the anisotropy of transversely polarized shear waves (SH waves) was recovered with higher accuracy than the anisotropy of vertically polarized shear waves (SV waves).Key wordsTravel-time inversionseismic anisotropylower crustshear wavesUrach