Energy flux in viscoelastic anisotropic media

We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P , SV and SH waves.     

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.     

Effect of 3-D viscoelastic structure on post-seismic relaxation from the 2004 M= 9.2 Sumatra earthquake

The 2004 M = 9.2 Sumatra–Andaman earthquake profoundly altered the state of stress in a large volume surrounding the ∼1400 km long rupture. Induced mantle flow fields and coupled surface deformation are sensitive to the 3-D rheology structure. To predict the post-seismic motions from this earthquake, relaxation of a 3-D spherical viscoelastic earth model is simulated using the theory of coupled normal modes. The quasi-static deformation basis set and solution on the 3-D model is constructed using: a spherically stratified viscoelastic earth model with a linear stress–strain relation; an aspherical perturbation in viscoelastic structure; a 'static' mode basis set consisting of Earth's spheroidal and toroidal free oscillations; a "viscoelastic" mode basis set; and interaction kernels that describe the coupling among viscoelastic and static modes. Application to the 2004 Sumatra–Andaman earthquake illustrates the profound modification of the post-seismic flow field at depth by a slab structure and similarly large effects on the near-field post-seismic deformation field at Earth's surface. Comparison with post-seismic GPS observations illustrates the extent to which viscoelastic relaxation contributes to the regional post-seismic deformation.     

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.     

Discrepancies in energy calculations for inhomogeneous waves

Summary. Two apparently equivalent methods used in the literature for calculating the mean energy flux and other properties of harmonic plane waves in a viscoelastic medium are shown to give slightly different results when applied to inhomogeneous P/SV -waves. The discrepant results of the two methods have appeared in recent papers and textbooks. Hence, it is important to point out and explain them. It is also shown that the discrepancies occur even for perfect elasticity. The first method makes use of the familiar classical formulae for the energy flux and other quantities, which can be derived from first principles. However, in the second method, the formulae for these quantities are inferred from a specifically derived conservation relation_ The validity of the results of the first method cannot be questioned. It is shown that the errors in the results of the second method are due to the presence of a certain mathematical non-uniqueness.     

The traveltime perturbations for seismic body waves in factorized anisotropic inhomogeneous media

The traveltime perturbation equations for the quasi-compressional and the two quasi-shear waves propagating in a factorized anisotropic inhomogeneous (FAI) media are derived. The concept of FAI media simplifies considerably these equations. In the FAI medium, the density normalized elastic parameters a _ijkl ( X _i ) can be described by the relation a _ijkl ( X _i) = f ²( x _i ) A _ijkl, where A _ijkl are constants, independent of coordinates x _i and f ²( x _i) is a continuous smooth function of x _i . The types of anisotropy ( A _ijkl ) and inhomogeneity [ f ( x _i)] are not restricted. The traveltime perturbations of individual seismic body waves ( q ^P , qS 1 and qS 2) propagating in the FAI medium depend, of course, both on the structural pertubations [δ f ²( x _i)] and on the anisotropy perturbations (δ A _ijkl ), but both these effects are fully separated. The perturbation equations for the time delay between the two qS -waves propagating in the FAI medium are simplified even more. If the unperturbed (background) medium is isotropic, the perturbation of the time delay does not depend on the structural perturbations (δ f ²( x _i) at all. This striking result, valid of course only in the framework of first-order perturbation theory, will simplify considerably the interpretation of the time delay between the two split qS -waves in inhomogeneous anisotropic media. Numerical examples are presented.     

An alternative method for calculation of Rayleigh and Love wave phase velocities by using three-component records on a single circular array without a central station

A method for the computation of phase velocities of surface waves from microtremor waveforms is shown. The technique starts from simultaneous three-component records obtained in a circular array without a central station. Then, Fourier spectra of vertical, radial and tangential components of motion are calculated for each station and considered as complex-valuated functions of the azimuthal coordinate. A couple of intermediate real physical quantities, B and C , can be defined from the 0- and ±1-order coefficients of the Fourier series expansion of such functions. Finally, phase velocities of Rayleigh and Love waves can be retrieved from B and C by solving respective one-unknown equations. The basic assumption is the possibility of expanding the wavefield as a sum of plane surface waves with Rayleigh and Love wavenumbers being univocal functions of the circular frequency. The method is tested in synthetic ambient noise wavefields confirming its reliability and robustness for passive seismic surveying.     

First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.     

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.     

Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method

Summary. Several approaches to computing body wave seismograms in 2–D and 3–D laterally inhomogeneous layered structures are suggested. They are based on the Gaussian beam method, which has been recently applied to the evaluation of time-harmonic high-frequency wavefields in inhomogeneous media. Three variants are discussed in some detail: the spectral method, the convolutory method and the wave-packet method. The most promising seems to be the wave-packet approach. In this approach, the wavefield, generated by a source, is expanded into a system of wave packets, which propagate along rays from the source in all directions. The wave packets change their properties due to diffusion, spreading, reflections/transmissions, etc. The resulting seismogram at any point of the medium is then obtained as a superposition of those packets which propagate close to the point. The final expressions in all the three methods are regular even in regions, in which the ray method fails, e.g. in the vicinity of caustics, in the critical region, at boundaries between shadow and illuminated regions, etc. Moreover, they are not as sensitive to the minor details of the medium as the ray method and, what is more, they remove the time-consuming two-point ray tracing from computations. Numerical examples of synthetic seismograms computed by the wave-packet approach are presented.     

Frequency response modelling of seismic waves using finite difference time domain with phase sensitive detection (TD–PSD)

This paper describes an efficient approach for computing the frequency response of seismic waves propagating in 2- and 3-D earth models within which the magnitude and phase are required at many locations. The approach consists of running an explicit finite difference time domain (TD) code with a time harmonic source out to steady-state. The magnitudes and phases at locations in the model are computed using phase sensitive detection (PSD). PSD does not require storage of time-series (unlike a fast Fourier transform), reducing its memory requirements. Additionally, the response from multiple sources can be obtained from a single finite difference run by encoding each source with a different frequency. For 2-D models with many sources, this time domain phase sensitive detection (TD–PSD) approach has a higher arithmetic complexity than direct solution of the finite difference frequency domain (FD) equations using nested dissection re-ordering (FD–ND). The storage requirements for 2-D finite difference TD–PSD are lower than FD–ND. For 3-D finite difference models, TD–PSD has significantly lower arithmetic complexity and storage requirements than FD–ND, and therefore, may prove useful for computing the frequency response of large 3-D earth models.     

极隙区极光粒子沉降对电离层影响的模拟研究

利用一维自洽的时变极区电离层模型,研究极隙区极光粒子沉降对极区电离层F层电子密度影响的时变过程。假设一维时变电离层模型描述的磁流管中F层等离子体在对流作用下经过极隙区,随对流路径的不同,磁流管在极隙区经历的时间不同,以此考察极光粒子沉降作用下电离层随时间的演化过程。数值计算结果表明,当磁流管在极隙区停留的时间足够长,F层电子密度能显著增大。然而在磁流管经历极隙区实际时间较短的情况下,极隙区极光粒子沉降对F层电子密度的影响并不大。本文还给出了统计对流模型作用下磁流管在经历极隙区时,有沉降粒子作用和没有沉降粒子作用两种情况下,F层等离子体的时间变化过程的差异。     

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.     

A general correspondence principle for time-domain electromagnetic wave and diffusion fields

A general correspondence principle is presented that relates any time-domain electromagnetic diffusion field to an electromagnetic wavefield in a 'corresponding' configuration. The principle applies to arbitrarily inhomogeneous and anisotropic media and arbitrary transmitters and receivers. For the correspondence between the two types of electromagnetic fields to hold, the electric conductivity in the diffusive case and the permittivity in the wavefield case should have the same spatial variation, while the permeability distributions in space in the two cases are to be identical. Essential steps in the derivation of the correspondence principle are the use of the time Laplace transformation of causal signals, taken at real, positive values of the transform parameter, the Schouten-Van der Pol theorem in the theory of the Laplace transformation, and the reliance upon Lerch's theorem of the uniqueness of the interrelation between causal field quantities and their time-Laplace-transform representations at real, positive values of the transform parameter. Correspondence is then established between the tensorial Green's functions in the two cases, where the Green's functions are the point-receiver responses (either electric or magnetic field) to point-transmitter excitations (either electric- or magnetic-current source).
Through the correspondence principle, all transient electromagnetic wavefields (where losses are neglected) have as a counterpart a transient diffusive electromagnetic field (where the electric displacement current is neglected). The interrelation yields the tool to compare quantitatively the potentialities of the two types of fields in transient electromagnetic geophysical prospecting.
Finally, a general medium-parameter scaling law for time-domain electromagnetic wavefields is presented.     

Wavepath traveltime tomography

The elastic-wave equation is used to construct sensitivity kernels relating perturbations in elastic parameters to traveltime deviations. Computation of the functions requires a correlation of the forward-propagating seismic wavefield with a backward propagation of the residual wavefield. The computation of the wavefields is accomplished using a finite difference algorithm and is efficiently executed on a CM-2 parallel processor. The source and receiver locations have maximum sensitivity to velocity structure. The sensitivity kernels or wavepaths are well suited for transmission traveltime inversion such as cross-borehole tomography and vertical seismic profiling. Conventional ray tomography and wavepath tomography are applied to a set of P -wave arrival times, from a cross-borehole experiment at Kesterson, California. Because the wavepaths have increased sensitivity near the source and receiver there are differences in resolution of the velocity structure. Both techniques recover the same relative variations in velocity where the coverage is adequate. The wavepath solution is more laterally continuous and the dominant variation is vertical, as is expected for the layered sediments in this region.     

Efficient global matrix approach to the computation of synthetic seismograms

Summary. A numerically efficient global matrix approach to the solution of the wave equation in horizontally stratified environments is presented. The field in each layer is expressed as a superposition of the field produced by the sources within the layer and an unknown field satisfying the homogeneous wave equations, both expressed as integral representations in the horizontal wavenumber. The boundary conditions to be satisfied at each interface then yield a linear system of equations in the unknown wavefield amplitudes, to be satisfied at each horizontal wavenumber. As an alternative to the traditional propagator matrix approaches, the solution technique presented here yields both improved efficiency and versatility. Its global nature makes it well suited to problems involving many receivers in range as well as depth and to calculations of both stresses and particle velocities. The global solution technique is developed in close analogy to the finite element method, thereby reducing the number of arithmetic operations to a minimum and making the resulting computer code very efficient in terms of computation time. These features are illustrated by a number of numerical examples from both crustal and exploration seismology.     

Stress transfer relations among the earthquakes that occurred in Kerman province, southern Iran since 1981

We explore the possible stress triggering relationship of the   M ≥ 6.4  earthquakes that occurred in Kerman Province, southern Iran since 1981. We calculated stress changes due to both coseismic sudden movement in the upper crust and the time-dependent viscous relaxation of the lower crust and/or upper mantle following the event. Four events of   M ≥ 6.4  occurred between 1981 and 2005, on and close to the Gowk fault, show a clear Coulomb stress load to failure relationship. The  2003 M = 6.5  Bam earthquake, however, which occurred approximately 95 km SW of the closest Gowk event, shows a very weak stress relation to preceding earthquakes. The coseismic static stress change at the hypocentre of the Bam earthquake is quite small (∼0.006 bars). The time-dependent post-seismic stress change could be 26 times larger or 7 times lower than that of coseismic static stress alone depending on the choice of viscoelastic crustal model and the effective coefficient of friction. Given the uncertainties in the viscoelastic earth models and the effective coefficient of friction, we cannot confidently conclude that the 2003 Bam event was brought closer to failure through coseismic or post-seismic stress loading. Interestingly, the southern Gowk segment with a similar strike to that of the Bam fault, experienced a stress load of up to 8.3 bars between 1981 and 2003, and is yet to have a damaging earthquake.     

Small-scale heterogeneities below the Gräfenberg array, Germany from seismic wavefield fluctuations of Hindu Kush events

Small-scale elastic heterogeneities (<5  km) are found in the upper lithosphere underneath the Gräfenberg array, southeast Germany. The results are based on the analysis of broadband recordings of 17 intermediate-depth (201–272  km) events from the Hindu Kush region. The wavefront of the first P arrival and the following 40  s coda are separated into coherent and incoherent (scattered) parts in the frequency range from 0.05 to 5  Hz. The frequency-dependent intensities of the mean and fluctuation wavefields are used to describe the scattering characteristics of the lithosphere underneath the receivers. It is possible to discriminate a weak-fluctuation regime of the wavefield in the frequency range below approximately 1.5–2.5  Hz and a strong-fluctuation regime starting at 2.0–2.5  Hz and continuing to higher frequencies. In order to explain the observed wavefield fluctuations, an approach with seismic scattering at random media-type structures is proposed. The preferred model contains heterogeneities with 3–7 per cent perturbations in seismic velocity and correlation lengths of 0.6–4.8  km in the crust. This is compatible with models from active seismic experiments. Scattering in the lithospheric mantle is not required, but cannot be excluded at weak velocity contrasts (<3 per cent).     

黄土高原不同气候区裸地水、热特征对比

为了更好地理解陆地表面水、热通量的空间差异,利用2009年和2010年1月黄土高原陆面过程观测试验（LOPEX）的陆面过程资料,对比分析了不同气候区主要地表水、热通量特征。结果表明：半干旱区的定西在近地层气象要素、土壤湿度、动量通量、地表能量及CO2通量等主要陆面特征物理量与半湿润区的平凉、庆阳差异明显;气候背景相接近的平凉、庆阳的主要陆面特征量比较接近。虽然地形、地势、海拔、下垫面影响因子也是影响陆面特征的重要因素,但是气候背景差异愈明显,陆面特征差异也愈大,即一个地区的气候背景是陆面过程特征的主要决定因子,对陆面特征产生综合影响,而地形、地貌只对一个或几个陆面物理量有影响。