Analysis of the Spatial Coherence of Short-Period Acoustic-Gravity Waves in the Atmosphere

Summary The coherence of atmospheric acoustic-gravity waves has been measured in the period range 10–100 s at the Large Aperture Microbarograph Array in south-eastern Montana. The acoustic-gravity waves observed were signals generated by presumed nuclear explosions. The decrease of coherence with increasing distance between pairs of microbarographs is less rapid in the direction of wave propagation than transverse to it. Variation of direction of arrival over a small range of azimuth (±5°) explains the spatial behaviour of coherence in the direction normal to the wave propagation; variation of phase velocity of ±10 ms^-1 explains the behaviour along the direction of wave propagation. Both effects may be due to inhomogeneities in the atmosphere; the velocity variation may be due to the presence in the signal of several normal modes of acoustic- gravity waves, each travelling at a slightly different phase velocity in the range 300–330 ms^-1.     

Existence of transverse waves in anisotropic poroelastic media

Out of the four waves in an anisotropic poroelastic medium, two are termed as quasi-transverse waves. The prefix 'quasi' refers to their polarizations being nearly, but not exactly, perpendicular to direction of propagation. In this composite medium, unlike perfectly elastic medium, the propagation of a longitudinal wave along a phase direction may not be accompanied by transverse waves. The existence of a transverse wave in anisotropic poroelastic media is ensured by the two equations restricting the choice of elastic coefficients of porous aggregate as well as fluid–solid coupling. Necessary and sufficient conditions for the existence of transverse waves along the coordinate axes and in the coordinate planes for general anisotropy are discussed. The discussion is extended to the case of orthotropic materials and existence for few specific phase directions is also explored. The conditions for the transverse waves decided on the basis of their apparent polarizations, that is, particle motion being perpendicular to ray direction, are also discussed. For a particular numerical model, the existence of these apparent transverse waves is solved numerically for phase directions in coordinate planes. For general directions of phase propagation, the existence of these transverse waves is checked graphically for the chosen numerical model.     

Gaussian beams for surface waves in laterally slowly-varying media

Summary. Asymptotic ray theory is applied to surface waves in a medium where the lateral variations of structure are very smooth. Using ray-centred coordinates, parabolic equations are obtained for lateral variations while vertical structural variations at a given point are specified by eigenfunctions of normal mode theory as for the laterally homogeneous case. Final results on wavefields close to a ray can be expressed by formulations similar to those for elastic body waves in 2-D laterally heterogeneous media, except that the vertical dependence is described by eigenfunctions of 'local' Love or Rayleigh waves. The transport equation is written in terms of geometrical-ray spreading, group velocity and an energy integral. For the horizontal components there are both principal and additional components to describe the curvature of rays along the surface, as in the case of elastic body waves. The vertical component is decoupled from the horizontal components. With complex parameters the solutions for the dynamic ray tracing system correspond to Gaussian beams: the amplitude distribution is bell-shaped along the direction perpendicular to the ray and the solution is regular everywhere, even at caustics. Most of the characteristics of Gaussian beams for 2-D elastic body waves are also applicable to the surface wave case. At each frequency the solution may be regarded as a set of eigenfunctions propagating over a 2-D surface according to the phase velocity mapping.     

Experimental study of the anisotropy of longitudinal and transverse waves from local earthquake records

Summary. The paper gives the results of a study of the anisotropy of seismic wave velocities within the Ashkhabad test field in Central Asia. The anisotropy was studied by analysing variations in the values of apparent velocities of first arrivals for epicentral distances ranging from 30 to 130 km and by analysing the delays (Δ t_s1-_s2 ) between the arrival times of shear waves with different polarizations.
The velocities of P -waves vary with azimuth from 5.3 to 6.27 km s^-1 and the velocities of S -waves vary from 3.15 to 3.5 km s^-1.
The delay times Δ t_S1 - _S2 depend on the direction of the propagation. The character of the variation of the propagation velocity of the longitudinal wave, the presence of two differently polarized shear waves S 1 and S 2 propagating at different velocities, and the character of the distribution of Δ t_S1 - _S2 on the stereogram suggest that the symmetry of the anisotropic medium is close to hexagonal with a nearly horizontal symmetry axis coinciding with the direction of maximal velocity. The azimuth of the symmetry axis of the medium is 140° and coincides with the direction of geological faults.     

Formalism for traveltime inversion with finite frequency effects

A simple modification of the waveform inversion formula, based on the normal mode perturbation theory, is shown to lead to a formula for traveltime anomalies. The kernel which is derived can be used for traveltime inversion with automatic inclusion of finite frequency effects. Inversion for Earth structure with such kernels will lead to better resolution estimates than ray-theoretical traveltime inversion. Examples of kernels for transverse component seismograms are shown for direct S waves, ScS , Love waves and diffracted S waves. A measure of finite frequency effects is also proposed by comparing our formula with the one from ray theory. A quantity which should be 1 in the case of ray theory is computed for the finite frequency kernels and is shown to have deviations up to about 30 per cent from 1. Therefore, the use of ray theory for long-period body waves applies incorrect weight along a ray path and may introduce a small bias to an earth model.     

Finite-frequency vectorial tomography: a new method for high-resolution imaging of upper mantle anisotropy

Amplitude measurements of the transverse component of SKS waves, the so-called splitting intensity, can be used to formulate a non-linear inverse problem to image the 3-D variations of upper mantle anisotropy. Assuming transverse isotropy (or hexagonal symmetry), one can parametrize anisotropy by two anisotropic parameters and two angles describing the orientation of the symmetry axis. These can also be written as two collinear pseudo-vectors. The tomographic process consists of retrieving the spatial distribution of these pseudo-vectors, and thus resembles surface wave vectorial tomography. Spatial resolution results from the sensitivity of low-frequency SKS waves to seismic anisotropy off the ray path. The expressions for the 3-D sensitivity kernels for splitting intensity are derived, including the near-field contributions, and validated by comparison with a full wave equation solution based upon the finite element method. These sensitivity kernels are valid for any orientation of the symmetry axis, and thus generalize previous results that were only valid for a horizontal symmetry axis. It is shown that both lateral and vertical subwavelength variations of anisotropy can be retrieved with a dense array of broad-band stations, even in the case of vertically propagating SKS waves.     

Finite-frequency sensitivity kernels for head waves

Head waves are extremely important in determining the structure of the predominantly layered Earth. While several recent studies have shown the diffractive nature and the 3-D Fréchet kernels of finite-frequency turning waves, analogues of head waves in a continuous velocity structure, the finite-frequency effects and sensitivity kernels of head waves are yet to be carefully examined. We present the results of a numerical study focusing on the finite-frequency effects of head waves. Our model has a low-velocity layer over a high-velocity half-space and a cylindrical-shaped velocity perturbation placed beneath the interface at different locations. A 3-D finite-difference method is used to calculate synthetic waveforms. Traveltime and amplitude anomalies are measured by the cross-correlation of synthetic seismograms from models with and without the velocity perturbation and are compared to the 3-D sensitivity kernels constructed from full waveform simulations. The results show that the head wave arrival-time and amplitude are influenced by the velocity structure surrounding the ray path in a pattern that is consistent with the Fresnel zones. Unlike the 'banana–doughnut' traveltime sensitivity kernels of turning waves, the traveltime sensitivity of the head wave along the ray path below the interface is weak, but non-zero. Below the ray path, the traveltime sensitivity reaches the maximum (absolute value) at a depth that depends on the wavelength and propagation distance. The sensitivity kernels vary with the vertical velocity gradient in the lower layer, but the variation is relatively small at short propagation distances when the vertical velocity gradient is within the range of the commonly accepted values. Finally, the depression or shoaling of the interface results in increased or decreased sensitivities, respectively, beneath the interface topography.     

On the computation of shear waves by the ray method

Summary. The ray series solution of the elastodynamic equation of motion for shear waves propagating through a laterally inhomogeneous three-dimensional medium can be simplified by the use of a particular coordinate system that accompanies the wave front along the ray of investigation. The system is entirely determined by parameters that are obtainable from the ray. The transport equations for the principal shear wave components are then no longer coupled, but reduce to the same type of equation which determines the principal compressional wave component.     

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.     

Seismic body waves in anisotropic media: reflection and refraction at a plane interface

Summary. A formulation is derived for calculating the energy division among waves generated by plane waves incident on a boundary between generally anisotropic media. A comprehensive account is presented for P, SV and SH waves incident from an isotropic half-space on an orthorhombic olivine half-space, where the interface is parallel to a plane of elastic symmetry. For comparison, a less anisotropic medium having transverse isotropy with a horizontal axis of symmetry is also considered. The particle motion polarizations of waves in anisotropic medium differ greatly from the polarizations in isotropic media, and are an important diagnostic of the presence of anisotropy. Incident P and SV waves generate quasi- SH waves, and incident SH waves generate quasi- P and quasi- SV waves, often of considerable relative magnitude. The direction of energy transport diverges from the propagation direction.     

Reply to comment by J. L. Mateos, O. Novaro, T. H. Seligman and J. Flores on 'Can horizontal P waves be trapped and resonate in a shallow sedimentary basin?'

Using a complete mathematical formulation, we show that the trapping of horizontal P waves in a very soft shallow alluvial layer is a minor effect. These waves do not have a stable way of propagation since in order to exist they require an incident wave and are therefore incapable of resonating in the lateral direction when confined in a basin of limited extent.     

Geometrical structure of shear wave surfaces near singularity directions in anisotropic media

There are three types of surfaces which are used for studying wave propagation in anisotropic media: normal surfaces, slowness surfaces and wave surfaces. Normal surfaces and slowness surfaces have been researched in detail. Wave surfaces are the most complicated and comparatively poorly known compared with the other two. Areas of complicated geometrical structure of the wave surfaces are located in the vicinity of conical acoustic axes. There is an elliptical hole on the quick shear wave surface and complicated folds and cusps on the slow shear wave surface. Decomposition of the slow shear wave surface into smooth sheets is used for the study of its geometrical structure. Complexity of shear wave surfaces can be expressed by the number of waves corresponding to a fixed ray. An original approach to the calculation of wave normals depending on ray direction is presented.     

Quasi-isotropic approximation of ray theory for anisotropic media

Wave propagation in weakly anisotropic inhomogeneous media is studied by the quasi-isotropic approximation of ray theory. The approach is based on the ray-tracing and dynamic ray-tracing differential equations for an isotropic background medium. In addition, it requires the integration of a system of two complex coupled differential equations along the isotropic ray.
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails.     

Core-log integration studies in hole-A of Taiwan Chelungpu-fault Drilling Project

Taiwan Chelungpu-fault Drilling Project (TCDP) was initiated to understand the physical mechanisms involved in the large displacements of the 1999 Taiwan Chi-Chi earthquake. Continuous measurements of cores (including laboratory work) and a suite of geophysical downhole logs, including P - and S -wave sonic velocity, gamma ray, electrical resistivity, density, temperature, electrical borehole images and dipole-shear sonic imager, were acquired in Hole-A over the depth of 500–2003 m. Integrated studies of cores and logs facilitate qualitative and quantitative comparison of subsurface structures and physical properties of rocks. A total of 10 subunits were divided on the basis of geophysical characteristics. Generally, formation velocity and temperature increase with depth as a result of the overburden and thermal gradient, respectively. Gamma ray, resistivity, formation density, shear velocity anisotropy and density-derived porosity are primarily dependent on the lithology. Zones with changes of percentage of shear wave anisotropy and the fast shear polarization azimuth deduced from Dipole Shear-Imager (DSI) are associated with the appearance of fractures, steep bedding and shear zones. The fast shear wave azimuth is in good agreement with overall dip of the bedding (approximately 30° towards SE) and maximum horizontal compressional direction, particularly in the Kueichulin Formation showing strong shear wave velocity anisotropy. Bedding-parallel fractures are prevalent within cores, whereas minor sets of high-angle, NNW–SSE trending with N- and S-dipping fractures are sporadically distributed. The fault zone at depth 1111 m (FZA1111) is the Chi-Chi earthquake slip zone and could be a fluid conduit after the earthquake. The drastic change in fast shear wave polarization direction across the underlying, non-active Sanyi thrust at depth 1710 m reflects changes in stratigraphy, physical properties and structural geometry.     

Anisotropy of the mantle inferred from observations of P to S converted waves

Summary. Seismic anisotropy has been previously studied at depths usually not exceeding 100 or 150 km. In this paper we present a method of analysis of seismic records which is very sensitive to azimuthal anisotropy and is applicable at almost any depth range. The idea of the method is to detect and analyse the SH -component of the waves, converted from P to S in the mantle. The procedure of record processing includes frequency filtering, axis rotation, transformation of the record to a standard form, stacking the standardized SH -component records of many seismic events, and the harmonic analysis of amplitude as a function of the direction of wave propagation. When applied to the long-period records of NORSAR the procedure detected a converted wave with the properties implying the possibility of its propagation in a transversely isotropic medium with a horizontal axis of symmetry . Our preferred model postulates anisotropy of ∼ 1 per cent in a layer 50 km thick at the base of the upper mantle.     

Elastodynamic and elastostatic Green tensors for homogeneous weak transversely isotropic media

An explicit analytical formula for the complete elastodynamic Green tensor for homogeneous unbounded weak transversely isotropic media is presented. The formula was derived by analytical calculations of higher-order approximations of the ray series. The ray series is finite and consists of seven non-zero terms. The formula for the Green tensor is complete and correct for the whole frequency range, thus it describes correctly the wavefield at all distances and at all directions including the shear-wave singularity direction. The Green tensor consists of P, SV and SH far-field waves and four coupling waves. Three of them couple P and SV waves, and the fourth wave couples the SV and SH waves. The P-SV coupling waves behave similarly to the near-field waves in isotropy. However, the SV-SH coupling wave, which is called 'shear-wave coupling', behaves exceptionally and it has no analogy in the Green tensor for isotropy. The formula for the elastostatic Green tensor is also derived.     

A parabolic approximation for surface waves

Summary. A parabolic approximation to the equation of motion of elastic waves as a sum of surface modes and discovering a parabolic approximation be applied directly to surface waves. The approximation depends on the material properties varying slowly within a wavelength, whereas surface waves may travel in a surface wave guide whose depth is of the same order of magnitude as a wavelength. This difficulty is overcome by representing the waves as a sum of surface modes and discovering a parabolic approximation for the amplitudes as a function of position on the surface. The theory is applicable to the propagation of Love or Rayleigh waves in a structure which is vertically stratified in an arbitrary way, but varies slowly in any horizontal direction.     

横向沙丘气流平均速度变化规律的风洞模拟

在沙丘动力系统中,存在沙丘形态、气流、沙粒运移三者之间复杂的相互作用。通过风洞实验的方法,针对不同形态的6组横向沙丘模型,采用粒子图像测速系统,测量了模型沙丘周围气流水平速度和垂直速度的变化规律。实验结果表明,横向沙丘迎风坡水平气流存在1.28～1.89之间的加速率,垂直气流存在上扬趋势,这二者均有随沙丘迎风坡坡度增大而增大的趋势。在横向沙丘背风坡,由于气流的分离,水平气流速度减小并出现反向,其大小约为自由风速的17％;垂直气流速度存在下沉趋势,其最大沉速出现在气流重附点附近;背风坡气流速度的变化受沙丘迎风坡坡度影响较小,受自由风速的影响较大。沙丘对气流速度的改变在近地层较为显著,随着高度的增加地形影响逐渐减小。     

Synthetic logs of multipole sources in boreholes based on the Kelvin–Voigt stress–strain relation

We design a numerical algorithm for wave simulation in a borehole due to multipole sources. The stress–strain relation of the formation is based on the Kelvin–Voigt mechanical model to describe the attenuation. The modelling, which requires two anelastic parameters and twice the spatial derivatives of the lossless case, simulates 3-D waves in an axisymmetric medium by using the Fourier and Chebyshev methods to compute the spatial derivatives along the vertical and horizontal directions, respectively. Instabilities of the Chebyshev differential operator due to the implementation of the fluid–solid boundary conditions are solved with a characteristic approach, where the characteristic variables are evaluated at the source central frequency. The algorithm uses two meshes to model the fluid and the solid. The presence of the logging tool is modelled by imposing rigid boundary conditions at the inner surface of the fluid mesh. Examples illustrating the propagation of waves are presented, namely, by using monopoles, dipoles and a quadrupoles as sources in hard and soft formations. Moreover, the presence of casing and layers is considered. The modelling correctly simulates the features—traveltime and attenuation—of the wave modes observed in sonic logs, namely, the P and S body waves, the Stoneley wave, and the dispersive S waves in the case of multipole sources.     

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.