Computation of additional components of the first-order ray approximation in isotropic media

This study shows that the use of the first-order additional components of the ray method in the seismic wave field modeling is easy and that it can bring a substantial improvement of the standard ray results obtained with the zero-order ray approximation only. For the calculation of a first-order additional component, spatial derivatives of the parameters of the medium and spatial derivatives of the zero-order ray amplitude term are necessary. The evaluation of the former derivatives is straightforward; the latter derivatives can be calculated approximately from neighboring rays by substituting the derivatives by finite differences. This allows an effective calculation of the first-order additional terms in arbitrary laterally varying layered media.The importance of the first-order additional terms is demonstrated by the study of individual higher-order terms of the ray series representing elementaryP andS elastodynamic Green functions for a homogeneous isotropic medium. The study shows clearly that the consideration of the first-order additional terms leads to a more substantial decrease of the difference between approximate and exact elementary Green functions than any other higher-order term. With this in mind, effects of the first-order additional terms on the ray synthetic seismograms for aVSP configuration are studied. It is shown that the use of the additional terms leads to such phenomena, unknown in the zero-order approximation of the ray method, like quasi-elliptical and transverse polarization of a singleP wave or longitudinal polarization of a singleS wave.     

Comparison of Quasi-Isotropic Approximations of the Coupling Ray Theory with the Exact Solution in the 1-D Anisotropic “Oblique Twisted Crystal” Model

The coupling ray theory bridges the gap between the isotropic and anisotropic ray theories, and is considerably more accurate than the anisotropic ray theory. The coupling ray theory is often approximated by various quasi-isotropic approximations.Commonly used quasi-isotropic approximations of the coupling ray theory are discussed. The exact analytical solution for the plane S wave, propagating along the axis of spirality in the 1-D anisotropic oblique twisted crystal model, is then numerically compared with the coupling ray theory and its three quasi-isotropic approximations. The three quasi-isotropic approximations of the coupling ray theory are (a) the quasi-isotropic projection of the Green tensor, (b) the quasi-isotropic approximation of the Christoffel matrix, (c) the quasi-isotropic perturbation of travel times. The comparison is carried out numerically in the frequency domain, comparing the exact analytical solution with the results of the 3-D ray tracing and coupling ray theory software. In the oblique twisted crystal model, the three studied quasi-isotropic approximations considerably increase the error of the coupling ray theory. Since these three quasi-isotropic approximations do not noticeably simplify the numerical implementation of the coupling ray theory, they should deffinitely be avoided. The common ray approximations of the coupling ray theory do not affect the plane wave, propagating along the axis of spirality in the 1-D oblique twisted crystal model, and should be studied in more complex models.     

First-Order Perturbation Theory for Seismic Isochrons

A first-order perturbation theory for seismic isochrons is presented in a model independent form. Two ray concepts are fundamental in this theory, the isochron ray and the velocity ray, for which I obtain first-order approximations to position vectors and slowness vectors. Furthermore, isochron points are connected to a shot and receiver by conventional ray fields. Based on independent perturbation of the shot and receiver ray I obtain first-order approximations to velocity rays. The theory is applicable for 3D inhomogeneous anisotropic media, given that the shot and receiver rays, as well as their perturbations, can be generated with such model generality. The theory has applications in sensitivity analysis of prestack depth migration and in velocity model updating. Numerical examples of isochron and velocity rays are shown for a 2D homogeneous VTI model. The general impression is that the first-order approximation is, with some exceptions, sufficiently accurate for practical applications using an anisotropic velocity model.     

Interpolation of the coupling-ray-theory Green function within ray cells

The coupling–ray–theory tensor Green function for electromagnetic waves or elastic S waves is frequency dependent, and is usually calculated for many frequencies. This frequency dependence represents no problem in calculating the Green function, but may pose a significant challenge in storing the Green function at the nodes of dense grids, typical for applications such as the Born approximation or non–linear source determination. Storing the Green function at the nodes of dense grids for too many frequencies may be impractical or even unrealistic. We have already proposed the approximation of the coupling–ray–theory tensor Green function, in the vicinity of a given prevailing frequency, by two coupling–ray–theory dyadic Green functions described by their coupling–ray–theory travel times and their coupling–ray–theory amplitudes. The above mentioned prevailing–frequency approximation of the coupling ray theory enables us to interpolate the coupling–ray–theory dyadic Green functions within ray cells, and to calculate them at the nodes of dense grids. For the interpolation within ray cells, we need to separate the pairs of prevailing–frequency coupling–ray–theory dyadic Green functions so that both the first Green function and the second Green function are continuous along rays and within ray cells. We describe the current progress in this field and outline the basic algorithms. The proposed method is equally applicable to both electromagnetic waves and elastic S waves. We demonstrate the preliminary numerical results using the coupling–ray–theory travel times of elastic S waves.     

Two dimensional transient fundamental solution due to suddenly applied load in a half-space

A transient Green function due to suddenly applied line loads in an isotropic and homogeneous half-space is reported in this paper. The derivation of the half-space Green function in the Laplace and the Fourier transform spaces is first reviewed. Following an explicit inversion of the Fourier transform, the inverse Laplace transform is implemented along the contour integral on the p-complex plane in an integral form. The half-space Green function consists of full-space Green functions and a singularity-free complementary term. It can be easily incorporated into current transient boundary elements using the transient full-space Green function. Combined with finite elements, the half-space Green function can be used in a hybrid procedure to solve transient half-space problems without discretization of the free surface. Numerical results are presented to illustrate transient wave propagation in a half-space.     

A new approximation of the velocity-depth distribution and its application to the computation of seismic wave fields

Summary A new approximation of the velocity-depth distribution in a vertically inhomogeneous medium is suggested. This approximation guarantees the continuity of velocity and of its first and second derivatives and does not generate false low-velocity zones. It is very suitable for the computations of seismic wave fields in vertically inhomogeneous media by ray methods and its modifications, as it removes many false anomalies from the travel-time and amplitude-distance curves of seismic body waves. The ray integrals can be evaluated in a closed form; the resulting formulae for rays, travel times and geometrical spreading are very simple. They do not contain any transcendental functions (such asln (x) orsin ^–1, (x)) like other approximations; only the evaluation of one square root and of certain simple arithmetic expressions for each layer is required. From a computational point of view, the evaluation of ray integrals and of geometrical spreading is only slightly slower than for a system of homogeneous parallel layers and even faster than for a piece-wise linear approximation.     

PARABOLIC AND HYPERBOLIC PARAXIAL TWO-POINT TRAVELTIMES IN 3D MEDIA1

The 4 × 4 T -propagator matrix of a 3D central ray determines, among other important seismic quantities, second-order (parabolic or hyperbolic) two-point traveltime approximations of certain paraxial rays in the vicinity of the known central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two-component vector. Here, we provide parabolic and hyperbolic paraxial two-point traveltime approximations using the T -propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three-component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial-ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.     

Green–Ampt approximations

The solution to the Green and Ampt infiltration equation is expressible in terms of the Lambert W₋₁ function. Approximations for Green and Ampt infiltration are thus derivable from approximations for the W₋₁ function and vice versa. An infinite family of asymptotic expansions to W₋₁ is presented. Although these expansions do not converge near the branch point of the W function (corresponds to Green–Ampt infiltration with immediate ponding), a method is presented for approximating W₋₁ that is exact at the branch point and asymptotically, with interpolation between these limits. Some existing and several new simple and compact yet robust approximations applicable to Green–Ampt infiltration and flux are presented, the most accurate of which has a maximum relative error of 5 × 10⁻⁵%. This error is orders of magnitude lower than any existing analytical approximations.     

Spatial derivatives and perturbation derivatives of amplitude in isotropic and anisotropic media

Explicit equations for the spatial derivatives and perturbation derivatives of amplitude in both isotropic and anisotropic media are derived. The spatial and perturbation derivatives of the logarithm of amplitude can be calculated by numerical quadratures along the rays. The spatial derivatives of amplitude may be useful in calculating the higher-order terms in the ray series, in calculating the higher-order amplitude coefficients of Gaussian beams, in estimating the accuracy of zero-order approximations of both the ray method and Gaussian beams, in estimating the accuracy of the paraxial approximation of individual Gaussian beams, or in estimating the accuracy of the asymptotic summation of paraxial Gaussian beams. The perturbation derivatives of amplitude may be useful in perturbation expansions from elastic to viscoelastic media and in estimating the accuracy of the common-ray approximations of the amplitude in the coupling ray theory.     

Comparison of Absolute and Relative Moment Tensor Solutions For The January 1997 West Bohemia Earthquake Swarm

Moment tensor solutions for 70 clustered events of the 1997 West Bohemia microearthquake swarm, as calculated by two different methods, are given. The first method is a single-event, absolute moment tensor inversion which inverts body-wave peak amplitudes using synthetic Green functions. The second method is a multiple-event, relative method for which Green functions are reduced to 2 geometrical angles of rays at the sources. Both methods yield similar moment tensors, which can be divided into at least two or three different classes of focal solutions, indicating that, during the swarm activity, different planes of weakness were active. The major source component of most events is a double couple. However, the deviations from the double-couple mechanisms seem to be systematic for some classes of solutions. Error analysis was based on transforming the estimate of the standard deviation of amplitudes extracting from the seismograms into confidence regions of the absolute moment tensor. They show that the non-DC components are significant at a fairly high confidence level.     

Parameter estimation in low order fractionally differenced ARMA processes

A class of regression type estimators of the parameterd in a fractionally differencedARMA (p, q) process is introduced. This class is an extension of the estimator considered by Geweke and Porter-Hudak. In a simulation study, we compared three estimators from this class together with two approximate maximum likelihood estimators which are based on two separate approximations to the likelihood. One approximation ignores the determinant term in the likelihood and the other includes a compensating factor for the determinant. When the determinant term is included, the estimate tends to be much less biased and is in general superior to the other estimate. The approximate maximum likelihood estimator out performed, by a large margin, the regression type estimators for pureARIMA (0,d,0) processes. However, forARIMA (1,d,1) processes, a regression type estimator turned out to be the best for realizations of length 400 in 3 out of the 5 cases we tried.     

Slowness-domain kinematical characteristics for horizontally layered orthorhombic media. Part II: Pre-critical slowness match

Part II of this paper is a direct continuation of Part I, where we consider the same types of orthorhombic layered media and the same types of pure-mode and converted waves. Like in Part I, the approximations for the slowness-domain kinematical characteristics are obtained by combining power series coefficients in the vicinity of both the normal-incidence ray and an additional wide-angle ray. In Part I, the wide-angle ray was set to be the critical ray (‘critical slowness match’), whereas in Part II we consider a finite long offset associated with a given pre-critical ray (‘pre-critical slowness match’). Unlike the critical slowness match, the approximations in the pre-critical slowness match are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. Moreover, for the pre-critical slowness match, there is no need to distinguish between the high-velocity layer and the other, low-velocity layers. The form of the approximations in both critical and pre-critical slowness matches is the same, where only the wide-angle power series coefficients are different. Comparing the approximated kinematical characteristics with those obtained by exact numerical ray tracing, we demonstrate high accuracy. Furthermore, we show that for all wave types, the accuracy of the pre-critical slowness match is essentially higher than that of the critical slowness match, even for matching slowness values close to the critical slowness. Both approaches can be valuable for implementation, depending on the target offset range and the nature of the subsurface model. The pre-critical slowness match is more accurate for simulating reflection data with conventional offsets. The critical slowness match can be attractive for models with a dominant high-velocity layer, for simulating, for example, refraction events with very long offsets.     

Analytical representation of Green’s functions of the Biot equations for elastic waves in a homogeneous two-phase medium

The system of Biot vector equations in the frequency space includes two elliptic-type vector partial differential equations with unknown displacement vectors in the solid and liquid phases. Considering the Biot equations, alongside with Pride’s equations, the key approaches to the theoretical study of the elastic waves in the two-phase fluid-saturated media, the author suggests an analytical solution for the inhomogeneous Biot equations in the frequency space, which is reduced to finding its fundamental solution (Green’s function). The solution of this problem consists of solutions for two systems of Biot equations. In the first system, only the first equation is inhomogeneous, while in the second system, only the second equation is inhomogeneous and, as it is shown, its right-hand side is exclusively a potential function. The fundamental solution of the full system of inhomogeneous Biot equations (in which both equations are inhomogeneous) is represented in the form of Green’s matrix-tensor, for the scalar elements of which the analytical relations are presented. The obtained formulas describing the elastic displacements of both the solid and liquid phases reflect three wave types, namely, compressional waves of the first and the second kind (the fast and the slow waves, respectively) and shear waves. Similar terms (those describing the same type of the elastic waves in the solid and liquid phases) in the expressions for Green’s functions are linked with each other through the coefficient that links the components of the displacement vectors of the solid and liquid phases corresponding to the given wave type.

     

Breakdown of boundary layers: (i) on moving surfaces; (ii) in semi-similar unsteady flow; (iii) in fully unsteady flow

Abstract

The breakdown and separation or reattachment of boundary layers adjoining a mainstream are studied in the three related situations (i)-(iii) of the title. For (i) the classical steady boundary layer generally admits a logarithmic singularity in the displacement when breakdown occurs on a downstream-moving surface whereas the corresponding singularity for an upstream-moving surface can be logarithmic or of minus-one-sixth form. Conversely, the breakdown can be delayed to the onset of zero mainstream flow, in which case the displacement singularity is again logarithmic. In certain flows these singularities prove to be removable locally, yielding a breakaway separation or reattachment and including the first known successes of a classical strategy in describing large-scale separation. Other flows, by contrast, require an interactive strategy. Again, even on a fixed surface a breakdown different from Goldstein's can be produced if there is a moving section of surface further upstream. The application to (ii), semi-similar unsteady boundary layers, e.g. near an impulsively started wedge-like trailing edge, then follows readily and predicts analogous forms of singularity. The corresponding singularity in displacement predicted for fully unsteady classical boundary layers, (iii), occurs within a finite time and, like (i) (usually) and (ii), a three-tiered breakdown is involved at first. Subsequently interaction comes into play. Comparisons with numerical and/or earlier work are noted. In all three situations (i)-(iii), although the dynamics involved near breakdown, separation or reattachment are predominantly inviscid, the presence of small viscosity is of significance in enforcing smoothness of the local velocity profiles.     

计算横向非均匀介质体波理论地震图的Maslov渐近法

本文运用Maslov渐近理论编写的二维横向非均匀介质中的理论地震图程序,与其它类型算法作了精确对比,结果表明,在层状介质模型中,本程序的结果无论振幅还是波形对比都与反射率法基本相同。对于二维横向非均匀模型,在射线理论的非奇点处,本算法与射线方法基本一致,在射线理论的奇异点处,Maslov方法消除了射线理论所固有奇点,提高了计算精度。     

计算横向非均匀介质体波理论地震图的Maslov渐近法

本文运用Maslov渐近理论编写的二维横向非均匀介质中的理论地震图程序,与其它类型算法作了精确对比,结果表明,在层状介质模型中,本程序的结果无论振幅还是波形对比都与反射率法基本相同。对于二维横向非均匀模型,在射线理论的非奇点处,本算法与射线方法基本一致,在射线理论的奇异点处,Maslov方法消除了射线理论所固有奇点,提高了计算精度。     

Asymptotic Elastodynamic Green Function in the Kiss Singularity in Homogeneous Anisotropic Solids

The far-field asymptotic formula is derived for the elastodynamic Green function in the kiss singularity in homogeneous anisotropic solids. In contrast to standard asymptotics in regular directions the derived formula is more complex and expressed in the form of a 1-D integral. This integral is specified for the kiss singularity along the symmetry axis in transverse isotropy and along the fourfold symmetry axes in tetragonal and cubic symmetries. The shape of the slowness surface in the singularity is regular in transverse isotropy and the amplitude of the Green function is expressed by means of the Gaussian curvature of this surface in the singularity. However, the shape of the slowness surface is irregular and the Gaussian curvature is not defined in the singularity in tetragonal or cubic symmetries. In this case, the amplitude of the Green function is expressed by means of the generalized Gaussian curvature.     

Errors Due to the Common Ray Approximations of the Coupling Ray Theory

The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented.     

Multifractal modelling and spectrum analysis: Methods and applications to gamma ray spectrometer data from southwestern Nova Scotia, Canada

Multifractal theory was developed for handling scale invariant fields instead of geometry only[1―4]. From a multifractal point of view, some fractal models, ordinary physical processes and relevant probability distribution types can be considered as special cases of multifractal models which provides new insight into the interrelationships between systems and subjects. For example, the low order moment exponents τ (0), τ (1), τ (2) or τ ″(1) obtained by means of the moment method determi…     

Multifocus moveout revisited: derivations and alternative expressions

The multifocus moveout of Gelchinsky et al. [Gelchinsky, B., Berkovitch, A., Keydar, S., 1997. Multifocusing homeomorphic imaging: Parts I and II: Course Notes, Special Course on Homeomorphic Imaging. Seeheim, Germany] is a powerful tool for stacking multicoverage data in arbitrary configurations. Based on general ray theoretical assumptions and on attractively simple geometrical considerations, the multifocus moveout is designed to express the traveltimes of neighbouring rays arbitrarily located around a fixed central, primary reflected or even diffracted, ray. In this work, the basic derivations and results concerning the multifocus approach are reviewed. A higher-order multifocus moveout expression that generalizes the corresponding one of Gelchinsky is obtained from slight modifications of the original derivation. An alternative form of the obtained multifocus expression that is best suited for numerical implementation is also provided. By means of a simple numerical experiment, we also comment on the accuracy of the multifocus traveltime approximations.