Compressibility,core dynamics and the subseismic wave equation

It is shown that in the dynamics of a deep fluid of planetary scale such as the Earth's core, compressibility, stratification and self-gravitation are all important as well as rotation. The existing proof of Cowling's theorem prohibiting non-stationary axisymmetric dynamos, and the application of the Proudman-Taylor theorem to core flows, both based on the assumption of solenoidal flow, need to be reconsidered. For sufficiently small (subacoustic) frequencies or reciprocal time scales, an approximation which neglects the effect of flow pressure on the density is valid. We call this the “subseismic approximation” and show that it leads to a new second-order partial differential equation in a single scalar variable describing the low frequency dynamical behaviour. The new “subseismic wave equation” allows a direct connection to be made between the various possible physical regimes of core structure and its dynamics.     

Transient elastodynamic analysis by time domain bem in cylindrical coordinates

A time domain boundary element in a cylindrical coordinate system is developed for the analysis of wave propagation in a half space. The integral formulation is based on Graffi's dynamic reciprocal theorem and Stokes' fundamental solutions. The field quantities (displacements and tractions) are expressed as products of Fourier series in the tangential direction and linear polynomials in the other spatial directions. Gaussian integration is used to integrate the non-singular parts of the integral equations, whereas the integration of the singular components, which are either of order 1/r or 1/r², is handled by special numerical schemes. In the time marching aspect, the field quantities are assumed to vary linearly in the temporal direction as well. Examples for wave propagation due to various forms of surface excitations are reported to demonstrate the accuracy of the method.     

Verification of the reciprocal theorem on polar high-frequency radio paths during magnetic disturbances

Two high-latitude high-frequency (HF) paths of oblique-incidence ionospheric sounding were selected: Gor’kovskaya (GRK)–Dixon Island (DIK) and GRK–Lovozero (LOZ).The following propagation parameters were studied: F2MOF and F2MUF for the main signal, F2MOFl for lateral signals, EsMOF for a signal reflected from the sporadic ionospheric layer, and MOFm for the propagation modes of the M (or N) type. Here MOF and MUF are the maximum observed and usable frequencies, respectively. It was found that the reciprocal theorem is satisfied on two radio paths depending on the studied propagation parameter: the path length and location, time of day, and degree of magnetospheric disturbance. It was found that the reciprocal theorem is satisfied in 78, 50, and 30% of cases on the first path for the F2MUF values in the daytime under moderately disturbed conditions, in the morning, and at night, respectively. For the MOFm and moderately disturbed conditions on this path, the reciprocal theorem is satisfied in 40, 35, and 20% of cases in the daytime, morning, and at night, respectively. For the F2MUF values on the second path, reciprocity is satisfied in 73% of cases in the daytime and under moderately disturbed conditions and in 30% of cases at night during a quiet period. On this path, reciprocity is satisfied in 33% of all cases at night under strongly disturbed conditions. An unusual fact was also detected when ionograms of oblique-incidence ionospheric sounding were analyzed: the F2MUF values in the LOZ–GRK direction are almost always slightly (by 1–4%) larger than in the GRK–LOZ direction. A similar difference in the F2MUF values in the forward and backward directions can apparently be explained by the steep horizontal gradient in the electron density distribution from north to south in the reflection region of this path. Such a situation is not observed on the DIK–GRK line.     

Wavefield extrapolation and prestack depth migration in anelastic inhomogeneous media

Wavefield depth extrapolation and prestack depth migration in complex anelastic media are studied. Kjartansson's frequency‐independent Q law is used to describe the absorption of seismic energy. The macromodel used is analogous to the macromodel used for current migration schemes except that an additional frequency‐independent Q macromodel needs to be provided. Absorption in the forward one‐way propagator is introduced by assuming a complex phase velocity, and the inverse one‐way propagator is obtained using the reciprocity theorem for one‐way wavefields in dissipative media. The stability of the inverse propagator is achieved by limiting the angle of propagation of wavefields. A table‐driven explicit operator scheme for imaging complex 2D anelastic media is presented. High‐accuracy, short convolution operators are designed by the weighted least‐squares method, and two kinds of imaging conditions are proposed. Numerical examples of depth extrapolation in laterally varying media, the migration of a spatial impulse with dispersion as well as shot record depth migration demonstrate the potential of the proposed explicit forward operator, the explicit inverse operator and the prestack depth migration scheme, respectively.     

Motion of foundation on a layered soil medium - II. Response analysis

The objective of the present approach is to determine the structural response to external force excitations and earthquake excitations with consideration of soil-structure interaction. The physical model concerned is a flexible structure resting on a rigid or flexible foundation embedded in a layered soil medium. The vibration of the structure is first analyzed using the wave propagation approach without reference to the interaction. The interaction effect is recovered by the impedance relationship developed in the companion paper. In an attempt to use the free field ground motion as the input to the system, the Maxwell's reciprocal theorem is applied to remedy the difference between the free field and real ground motion in the presence of the structure. As a result, the structural response at any location is written as a linear function of external excitations and the seismic source. Such a solution form is convenient for statistical analysis.     

Polarization analysis in the stationary wavelet packet domain: Application to HVSR method

Microzonation studies using ambient noise measurements have become very popular in the last years for local soil characteristics determination. For defining the capabilities of the ambient noise based methods, a physical understanding of the nature of noise is essential. In this way, many studies have shown that the noise wavefield composition is heterogeneous, comprising both body and surface waves with fundamental and upper modes.In this work, a recently developed method of time-frequency polarization analysis, based on the stationary wavelet packet transform, is used to study how the waves’ polarization influences in the application of the HVSR method. For this purpose, each recorded signal is first decomposed into a set of sub-signals, according to their reciprocal ellipticity. After that, the H/V analysis is carried out on the raw recorded signal, as well as on the obtained sub-signals, evaluating the contribution of each sub-signal (i.e., each reciprocal ellipticity range) to the expected H/V result.For the study, single station seismic noise was recorded at 11 sites around the province of Alicante (southeastern Spain). The obtained results illustrate that high frequency peaks, above 1–2 Hz, are clearly represented by just using the sub-signals associated to the ellipticity range of [0–0.1], whereas low frequency peaks, down 0.5 Hz, need a wider ellipticity interval for being properly retrieved, i.e., [0–0.4] or even [0–0.6]. Therefore, linearly polarized waves’ contribution, range [0–0.1], is enough for retrieving the high frequency H/V peaks, but not for obtaining the complete H/V curve, especially the low frequency peaks. In these cases, the elliptical polarized Rayleigh waves’ contribution cannot be misestimated in HVSR computation.     

EFFECT OF SLAB FLEXIBILITY ON INTERACTION BETWEEN TWO CIRCULAR FOUNDATIONS

The paper is aimed at investigating the effect of foundation rigidity on dynamic stiffness for two circular foundations on a viscoelastic medium. To generate the dynamic stiffness, a substructure technique is employed. For the substructure of a viscoelastic medium, the solution for wave motion reported in Reference 11 is used. For the substructures of two flexible foundations, classical plate theory with the inertial force neglected is employed to find the displacement fields of the foundation plates subjected to the interaction stresses. Then, the continuity condition for all the substructures is imposed implicitly by using the variational principle; then with the help of the reciprocal theorem the dynamic stiffness for the two flexible foundations can be obtained. For the numerical study, the boundary condition at the rims of both foundation plates is assumed to be a hinge connection to superstructures. Some numerical investigations are performed and the effect of foundation rigidity on dynamic stiffness is examined. Some discussions and conclusions are also made.     

MORE ON THE LATEROLOG* DEVICE**

Basic Laterolog devices employ a measure-current focusing technique to provide well logs of formation resistivity that feature sharp vertical definition combined with a depth of investigation much improved over those of normals of comparable spacings. To develop the concept of measure-current focusing, a disk-electrode arrangement analogous to a guard-ring capacitor is considered first, leading finally to consideration of a real sonde in a cylindrical borehole. In the latter, auxiliary currents are adjusted to maintain substantially at zero the vertical potential gradients above and below the central measure-current electrode; this has the effect of focusing the measure-current beam. However, prediction of sonde responses on the basis of current patterns can in some cases be misleading. Indeed, for a given Laterolog array, one may use a (reciprocalmode) current pattern radically different from the one defined by the basic (direct-mode) measure-current focusing scheme and still obtain the same resistivity measurement. This feature is illustrated for Laterolog 3 by use of transfer impedances, and demonstrated for certain more general cases using the reciprocity theorem. Comparisons of Laterolog responses across typical thin beds, both non-invaded and invaded, with those of normal devices of a comparable range of spacings illustrate the improved vertical resolution obtained with Laterolog devices. The examples also show the different magnitudes of the auxiliary currents (thenceforth designated “bucking”- or “focusing” currents) used in the direct and reciprocal modes and illustrate how these currents must continuously vary in order to maintain the focused condition as the sonde moves through the bed. Finally, briefly considering possible applications of Laterolog-type arrays in surface prospecting, it is shown that a one-dimensional array laid out on the surface does not provide the same degree of measure-current focusing as is obtained in a borehole. Some improvement is obtained with a combination of two crossed linear arrays. Much more is obtained using a circular array.     

Dynamic stiffness matrices for two circular foundations

A systematic procedure is presented for generating dynamic stiffness matrices for two independent circular foundations on an elastic half-space medium. With the technique reported in References 1–3, the analytic solution of three-dimensional (3D) wave equations satisfying the prescribed traction due to the vibration of one circular foundation can be found. Since there are two analytic solutions for two prescribed tractions due to the vibrations of two circular foundations, the principle of superposition must be used to obtain the total solution. The interaction stresses (prescribed tractions) are assumed to be piecewise linear in the r-directions of both cylindrical co-ordinates for the two circular foundations. Then, the variational principle and the reciprocal theorem are employed to generate the dynamic stiffness matrices for the two foundations. In the process of employing the variational principle, a co-ordinate transformation matrix between two cylindrical co-ordinate systems is introduced. Some numerical results of dynamic stiffness matrices for the interaction of two identical rigid circular foundations are presented in order to show the effectiveness and efficiency of the present method, and some elaborations for its future extensions are also discussed.     

地震动随机函数模型研究(Ⅰ)——模型建立

基于随机过程的随机函数解释,发展了一种随机地震动模型.模型包括两个部分,分别是基于理论物理关系的Fourier幅值谱模型和基于经验关系的Fourier相位谱模型.基于拟加速度谱概念建立的幅值谱模型能够反映强震记录的幅值特性,而Fourier相位谱模型则利用累积相位谱的单值连续特性,以线性多项式与三角级数组合的形式给出.模拟记录与实测记录对比结果显示两者不仅在波形结构上相似性良好,而且反应谱层次亦吻合良好.     

Vibration of surface foundations of arbitrary shapes

Presented is a systematic procedure for generating impedance (or compliance) matrices for foundations with arbitrary shapes, resting on an elastic half-space medium. A technique to decompose prescribed harmonic tractions on the half-space medium is employed to solve analytically the differential wave equations in cylindrical coordinates. However, the interaction stresses due to the vibration of a foundation with arbitrary shape are described in rectangular coordinates, and assumed to be piecewise constant in the region of the arbitrary shape. A coordinate transformation matrix is introduced for the piecewise constant tractions in order to use the solution of the differential wave equations in cylindrical coordinates. Finite element modelling is assumed in rectangular coordinates for the foundation itself. The impedance matrix is then obtained for the finite element model, using a variational principle and the reciprocal theorem. A simple example of a rigid square plate resting on a half-space medium and subjected to vertical excitation is used to demonstrate the efficiency and effectiveness of the procedure. Some numerical aspects are investigated and some possible extensions of the procedure are also discussed.     

Cyclic shearing and liquefaction of soil under irregular loading: an incremental model for the dynamic earthquake-induced deformation

The paper presents a mathematical model for the deformation of soil under irregular cyclic loading in the simple-shear conditions. The model includes the possible change in the effective pressure in saturated soil due to the cyclic shearing, the reciprocal influence of the effective pressure on the response of the soil to the shear loading, and the pore pressure dissipation due to the seepage of the pore fluid. The hysteresis curves for the strain–stress relationship are constructed in such a way that they produce both the required backbone curve and the required damping ratio as functions of the strain amplitude. At the same time, the approach enables the constitutive functions involved in the model to be specified in various ways depending on the soil under study. The constitutive functions can be calibrated independently of each other from the conventional cyclic shear tests. The constitutive model is incorporated in the boundary value problem for the dynamic site response analysis of level ground. A numerical solution is presented for the dynamic deformation and liquefaction of soil at the Port Island site during the 1995 Hyogoken-Nambu earthquake.     

Reflection and transmission of plane waves by a layer of compact inhomogeneities

Reflection and transmission of elastic wave motion by a layer of compact inhomogeneities has been analyzed. For identical inhomogeneities whose geometrical centers are periodically spaced, the problem has been formulated and solved rigorously. The reflected and transmitted longitudinal and transverse wave motions have been expressed as superpositions of wavemodes, where each wavemode has its own cut-off frequency. At its cut-off frequency a mode converts from a standing into a propagating wavemode. The standing wavemodes decay exponentially with distance to the plane of the centers of the inhomogeneities. At small frequencies only the lowest order modes of longitudinal and transverse wave motion are propagating. Reflection and transmission coefficients have been defined in terms of the coefficients of the zeroth-order scattered wavemodes. These coefficients have been computed by a novel application of the Betti-Rayleigh reciprocal theorem. They are expressed as integrals over the surface of a single inhomogeneity, in terms of the displacements and tractions on the surface of the inhomogeneity. The system of singular integral equations for the surface fields has been solved numerically by the boundary integral equation method. Curves show the reflection and transmission coefficients for the reflected and transmitted longitudinal and transverse waves as functions of the frequency. Some results are also presented for planar distributions of cracks whose spacing and size are random variables. Finally, dispersion relations are discussed for solids which are completely filled with periodically spaced inhomogeneities.     

Wavefield continuation datuming using a near surface model

When topography and low velocity zone differences vary greatly, conventional vertical static time shifts will cause wavefield distortion and influence wave equation seismic imaging for seismic data acquired on a complex near surface. In this paper, we propose an approach to datum correction that combines a joint tomography inversion with wavefield continuation to solve the static problem for seismic data on rugged acquisition topography. First, the near surface model is obtained by refracted wave tomography inversion. Second, the wavefield of sources and receivers are continued downward and upward to accomplish datum correction starting from a flat surface and locating the datum above topography. Based on the reciprocal theorem, Huygens＇ and Fresnel principles, the location of sources and receivers, and regarding the recorded data on the surface as a secondary emission, the sources and receivers are upward-continued to the datum above topography respectively. Thus, the datum correction using joint tomography inversion and wavefield continuation with the condition of a complex near surface is accomplished.     

Parsimonious wave‐equation travel‐time inversion for refraction waves

We present a parsimonious wave‐equation travel‐time inversion technique for refraction waves. A dense virtual refraction dataset can be generated from just two reciprocal shot gathers for the sources at the endpoints of the survey line, with N geophones evenly deployed along the line. These two reciprocal shots contain approximately 2N refraction travel times, which can be spawned into refraction travel times by an interferometric transformation. Then, these virtual refraction travel times are used with a source wavelet to create N virtual refraction shot gathers, which are the input data for wave‐equation travel‐time inversion. Numerical results show that the parsimonious wave‐equation travel‐time tomogram has about the same accuracy as the tomogram computed by standard wave‐equation travel‐time inversion. The most significant benefit is that a reciprocal survey is far less time consuming than the standard refraction survey where a source is excited at each geophone location.     

Magnetohydrodynamic disturbances in the earth's core,III

Summary It is in the hope of establishing a theorem contributing to the origin of the Secular Variation of the geomagnetic field that this third paper under the same heading, is written, —In this paper it is supposed that amagnetic dipole is suddenly introduced in the earth's core to act as a source of disturbance to the exciting field taken as a poloidal one. The dipole axis is placed parallel to the original field and perpendicular to the mantle. Mathematical solutions have been obtained for the resulting fluid motion in the core as well as for the generated magnetohydrodynamic perturbations in the earth's core and in the mantle. It can be seen from the mathematical results obtained that although the disturbances in the core are so complicated, yet they are much less complicated in the mantle and specially at the plane boundary separating core and mantle.     

Bayesian probabilities of earthquake occurrences in Greece and surrounding areas

The Bayesian extreme-value distribution of earthquake occurrences has been adopted to estimate the seismic hazard in some seismogenic zones in Greece and surrounding regions. Seismic moment, slip rate, earthquake recurrence rate and magnitude were considered as basic parameters for computing the prior estimates of the seismicity. These estimates are then updated in terms of Bayes' theorem and historical estimates of seismicity associated with each zone.High probabilities for earthquakes withM6.0 have been obtained for the northwestern part of Greece as well as for the southwestern part of the Hellenic arc.     

Alternative theory for signal contribution sections and depth of investigation characteristics in electrical prospecting

We present an alternative to Roy's theorem for direct current regimes with the aim of validating the theoretical basis of signal contribution sections in electrical prospecting. Roy's theorem establishes that the electrical potential at a point can be expressed as an integral over all space of the electric field weighted by the gradient of inverse distance. The integrand is interpreted to represent elementary contributions to the potential that can be analyzed to compare different electrode arrays. Signal contribution sections and depth of investigation characteristics can be beautifully illustrated with important practical applications. However, the electric potential, being the solution of a boundary value problem, cannot be uniquely decomposed into elementary contributions. There is no guarantee for the integrand of a given integral to be meaningful in all situations. In the case of Roy's theorem, the concept has been severely criticized by respected scholars who challenged the scientific legitimacy of his approach. If we are going to keep the concept of elementary contributions alive, we need to go beyond Roy's theorem. In this paper, we develop an alternative theorem and show that it merges with the concept of sensitivity, which is unique and mathematically sound, and is also open to physical validation. This prevents any possible contradictions in the future and, equally important, eliminates the dichotomy between sensitivity and elementary contributions.     

Reciprocal experimental transplantations to assess effects of organic enrichment on the recolonization of benthic macrofauna in a subtropical estuary

Coastal benthic habitats are usually in a state of continuous recolonization as a consequence of natural disturbances or human activities. Recolonization patterns can be strongly affected by the quality of the sediment. We evaluated herein the macrobenthic recolonization of organically enriched sediments through a manipulative experiment involving reciprocal transplants between contaminated and non-contaminated intertidal areas. Regardless of the experimental treatments, the density of the polychaete Capitella sp. was extremely high in the contaminated area as well as the density of the gastropod Cylichna sp. in the non-contaminated area. We rejected the hypothesis that differences in sediment quality would determine macrofaunal recolonization at least in the considered scales of space in meters and time in weeks. The recolonization process in a subtropical estuarine environment was strongly dependent on the migration of adults present in the sediments adjacent to the experimental units.     

小波变换与信号瞬时特征分析

基于经典Hilbert变换计算信号瞬时参数（如瞬时频率等），当信号中噪声较强时计算结果不能很好地刻划有效信号特征．本文提出了用小波变换求能量有限实信号对应的解析信号的一个定理，在此基础上给出了用小波变换计算信号瞬时参数的算法．理论分析及模型算例结果表明，本文提出的方法计算精度高且有较强的抗噪声能力．对地震记录的褶积模型，深入地分析了不同尺度下地震记录小波变换结果及其对应的瞬时参数含义，这对实际应用有重要意义．