水平n层导电介质中时谐垂直电偶极子的电磁场

对于包含n层线性、均匀、各向同性的水平成层导电介质,本文应用汉克尔变换及其逆变换推导求得了位于顶层导电媒质中的时谐垂直电偶极子在该层产生的电磁场的表达通式．通过这些表达式能够很方便地求得包含任意多层导体半空间中的电磁场．文中最后计算了导电半体空间和二层导电介质中时谐垂直电偶极子产生的电磁场,验正了本文所得结果的正确性。     

谐变磁偶极激发下层状介质中球的电磁响应

本文在前人基础上,从理论计算角度研究了两层大地下球形导电体的偶极激发音频电磁场。探讨了电模式、磁模式场的耦合及其意义,并列出了剖面数据和频率响应数据,以适应移动源音频电磁法的实际需要。     

层状介质中水平磁偶极子电磁场空间分布特征

针对岩层中介电常数和磁导率随深度变化的情况,给出水平磁偶极子源电磁场的水平或垂直空间分布特征及相应的变化规律。采用Kong给出的汉克尔（Hankel）J0变换线性滤波器（241点）和汉克尔J1变换线性滤波器（241点）算法,选用均匀大地和两层地电模型,讨论介质的介电常数和磁导率变化时的水平磁偶极子电磁场空间分布特征。计算结果表明,磁场的峨分量幅值与磁导率反相关;磁场H2和电场Ey分量随着磁导率的增大而增大;低频电磁测深中介电常数变化时对电磁场基本没有影响。     

THE ABEL-FOURIER METHOD OF HANKEL TRANSFORMATION: APPLICATIONS TO SEISMIC DATA*

The formal equivalency between a Hankel transform and an Abel transform followed by a Fourier transform is used as a basis for the computation of synthetic seismograms and for performing plane-wave decomposition of both synthetic and field seismograms. The Abel-Fourier method performs better for near-offset or small ray-parameter, although at greater computational effort than calculations based upon the asymptotic Hankel transform.     

Vertical dynamic response of a disk on a saturated poroelastic half-space

This paper considers the vertical dynamic response of a disk on a saturated poroelastic half-space. Firstly the pressure-solid displacement form of the harmonic equations of motion for a poroelastic solid are developed from the form of the equations originally presented by Biot. These equations are solved by a new method. Then the mixed boundary value problem for the vertical harmonic vibration of a disk on a poroelastic half-space is studied. The two types of drainage conditions at the surface of the poroelastic half-space are considered: (a) the surface of the poroelastic half-space is assumed to be completely pervious both within and exterior to the plate; (b) The interface between the plate and the poroelastic half-space is assumed to be impervious and the exterior region is assumed to be pervious. By using the Hankel transform techniques, the paper develops the governing dual integral equations. These governing integral equations are further reduced to systems of standard Fredholm integral equations of the second kind by Abel transform.     

Time domain scattering of acoustic plane waves by vertical faults

Acoustic plane wave scattering at a vertical fault structure represents the simplest two-dimensional model of geophysical exploration that can be investigated by analytical techniques. The exact and complete solution, in the time domain, for the scattering of the pressure field of an acoustic plane wave normally incident on a vertical fault structure is determined adapting previous results given for the frequency domain. The wave form of the pressure field of the incident plane wave is expressed by a causal time function that decays exponentially with time at every point above the fault (z<0). The zero-order term of the scattered pressure field has been computed above the fault. This zero-order term consists of an inverse Fourier transform which reduces to a closed expression forx=0, and contains an integral of a Hankel function forx#0. The high frequency part of the inverse Fourier transform forx#0 is computed employing asymptotic expressions for the Hankel function. The integral of the asymptotic expression of the Hankel function reduces to: (i) a Fresnel integral which contains a plane wave term for |x||z|; and (ii) a stationary point plane wave term plus an upper limit term for |x|=O(|z|). For the latter case the plane wave term cancels, leaving a cylindrical wave emanated from the edge of the fault. The wave front is well defined in shape, in phase and in amplitude. The amplitude of the scattered field is discontinuous atx=0, presents a jump and is well defined for |x| small and is rather smooth for |x| large.     

A fundamental solution of a multilayered half-space due to an impulsive ring source

The fundamental solutions of axisymmetric elastodynamic problem for the multilayered half-space due to an impulsive ring source acting within a layered elastic media are derived in time domain with the aid of Laplace–Hankel mixed transform and transfer matrix techniques. In addition, an effective numerical procedure, which utilizes the fast Hankel transform algorithm, is also proposed to calculate these solutions. Illustrative examples have been given to demonstrate that the fundamental solutions can be readily evaluated and the numerical results are of high accuracy. The present solutions can be directly applied to determine the transient wave fields caused by a seismic source and show the potential application to the elastodynamic problems solved by the boundary element method.     

Torsional vibration of a finite cylindrical cavity in a two-layer transversely isotropic half-space

The aim of this paper is to present a rigorous investigation for a two-layered transversely isotropic linear elastic half-space containing a circular cylindrical cavity of length equal to the top layer undergoing mono-harmonic ring shape shear stress applied either on the vertical cylindrical surface or on the base of the cavity. To this end, a combination of Fourier cosine integral transform for depth and Hankel integral transform for radial distance are used, which translate the boundary value problem to a singular integral equation for the shear stress comes out from the continuity of two layers. The integral equation is solved for some collocation points with a smoothed variable of distance, which is adapted with the use of a free parameter. It is shown that, although the shear stress is highly singular, it does not highly depend on this free parameter. Both the analytical and numerical results are verified with both the static isotropic and dynamic transversely isotropic homogeneous cases. In addition, some new graphical results are presented for more understanding in engineering point of view.     

A GENERAL TECHNIQUE FOR THE DIRECT INTERPRETATION OF RESISTIVITY DATA OVER TWO-DIMENSIONAL STRUCTURES*

The indirect method of interpreting resistivity data is capable only of limited success because of the difficulty of calculating type curves for complex structures. Consequently a need arises for a direct method of interpretation for complex generalized structures. Such a technique for the direct interpretation of apparent resistivity data, obtained by electrical soundings carried out over two-dimensional structures, has been developed from an examination of the Hankel transform of such data. The method is based on the observation that the asymptotic expansion of the Hankel transform is critically dependent upon the minimum distance between the measuring device and the surface of discontinuity in resistivity. The variation in the across strike direction may be mapped by a sequence of depth soundings made parallel to each other and separated in the across strike direction.     

The fast Hankel transform1

An efficient algorithm is presented to compute the Hankel transform. The algorithm yields simultaneously all the required weights for a given order of the Bessel function using the fast Fourier transform. An additional shift is introduced to the filter abscissa besides Koefoed's shift to give a better filter performance.     

ON THE CONSTRUCTION OF THE 3D BAND-LIMITED EXTRAPOLATION OPERATOR IN THE SPACE-FREQUENCY DOMAIN1

Based on an expansion of the band-limited 3D extrapolation operator in terms of orthogonal Chebyshev polynomials, a closed form expression of the space-frequency response is presented. A key step is an evaluation of the (inverse) 2D Fourier transform of circularly symmetric functions, which is related to the (zero-order) Hankel transform. Hankel transforms of individual members of the orthogonal set of polynomials are available from tables and summation of series; hence, the real and the imaginary parts of the space-frequency response can be found in terms of cylindrical and spherical Bessel functions, respectively. The procedure permits an efficient and accurate evaluation of the space-frequency response.     

Analytical layer-element solution to axisymmetric dynamic response of transversely isotropic multilayered half-space

Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the corresponding algebraic operations and the Hankel transform, the analytical layer-elements of a finite layer and a half-space are obtained in the transformed domain. According to the continuity conditions between adjacent layers, the global stiffness matrix equation is obtained by assembling the analytical layer-element of each single layer. The solutions in the transformed domain are acquired by introducing the boundary conditions into the global stiffness matrix equation, and thus, the corresponding solutions in frequency domain are achieved by taking the inversion of Hankel transform. Finally, some numerical examples are given to illustrate the accuracy of the proposed method, and to study the influence of properties and the frequency of excitation on the dynamic response of the medium.     

New digital linear filters for Hankel J0 and J1 transforms

The numerical evaluation of certain integral transforms is required for the interpretation of some geophysical exploration data. Digital linear filter operators are widely used for carrying out such numerical integration. It is known that the method of Wiener–Hopf minimization of the error can be used to design very efficient, short digital linear filter operators for this purpose. We have found that, with appropriate modifications, this method can also be used to design longer filters. Two filters for the Hankel J₀ transform (61-point and 120-point operators), and two for the Hankel J₁ transform (47-point and 140-point operators) have been designed. For these transforms, the new filters give much lower errors compared to all other known filters of comparable, or somewhat longer, size. The new filter operators and some results of comparative performance tests with known integral transforms are presented. These filters would find widespread application in many numerical evaluation problems in geophysics.     

Axisymmetric transient waves in transversely isotropic half-space

A transversely isotropic material in the sense of Green is considered. Using a series of potential functions proposed in [Eskandari-Ghadi M. A complete solution of the wave equations for transversely isotropic media. J Elasticity 2005; 81:1–19], the solutions of the transient wave equations within a half-space under surface load are obtained in the Laplace–Hankel domain for axisymmetric problems. The solutions are investigated in detail in the special case of a surface point force pulse varying with time as Heaviside function. Using Cagniard–De Hoop method, the inverse Laplace transform and inverse Hankel transform of the solutions are then obtained in the form of integrals with finite limits. For validity of the analytical results, the final formulations for surface waves are degenerated for an isotropic material and compared with the existing formulation obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80], to show that they are exactly the same. The numerical evaluations of the integrals for some transversely isotropic materials as well as an isotropic one are obtained. The present approach is then numerically verified by comparing a particular case of displacements for the surface of an isotropic half-space subjected to a point load of Heaviside function with the solutions obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80]. In addition, the wave equations for the mentioned medium are obtained on the vertical line directly under the applied surface load. The final formulations are degenerated for an isotropic material and compared with the existing formulation given in Graff [Wave motion in elastic solids. New York: Dover Publications Inc; 1975 [New Ed edition, November 1991]], to show that they are also exactly the same. Then equations are presented in graphical forms using an appropriate numerical evaluation.     

热层大气潮汐微分算子与微分积分方程

本文进一步讨论了热层大气潮汐微分算子的问题,求得了有平均风场及风切变时的微分算子;从有限汉克尔（Hankel）变换的反演,求得了热层大气结构的微分积分方程,并讨论了这一方程的求解问题。     

Hankel transform filters for dipole antenna radiation in a conductive medium

We discuss the Hankel transforms related to a particular application, i.e. the dipole antenna radiation in conductive media, such as the antenna radiation in sea-bed electromagnetic applications. In this application, the electromagnetic wavefields decay very rapidly with distance. A good filter means that it can be used to evaluate weak fields. Exponential sampling transforms a Hankel transform into a convolution equation, which must be solved to obtain the filter coefficients. Here, we use a direct matrix inversion method to solve the convolution equation in the sample domain, instead of the Fourier transform method and the Wiener–Hopf method, previously used to solve the convolution equation. This direct method is conceptually simple and is suitable for our optimization process: by using the Sommerfeld identity, we search for the optimum sampling interval, which corresponds to the minimum wavefield, evaluated for a given length filter. The performances of the new filters obtained are compared with some well-known filters. We find that our filters perform better for our application; that is, for the same length filters, our filters are able to calculate weaker fields. For users working in similar applications, three sets of filters with lengths 61, 121, 241 are available from the author.     

The Green￠s function of the harmonic horizontal force applied at the interior of the saturated half-space soil

By using integral transform methods, the Green's functions ofhorizontal harmonic force applied at the interior of the saturated half-space soil are obtained in the paper. The general solutions of the Biot dynamic equations in frequency domain are established through the use of Hankel integral transforms technique. Utilizing the above- mentioned general solutions, and the boundary conditions of the surface of the half-space and the continuous con-ditions at the plane of the horizontal force, the solutions of the boundary value problem can be determined. By the numerical inverse Hankel transforms method, the Green's functions of the harmonic horizontal force are obtainable. The degenerate case of the results deduced from this paper agrees well with the known results. Two numerical examples are given in the paper.     

半空间饱和土在内部简谐水平力作用下的Ge函数

根据积分变换方法得出了半空间内部作用简谐水平力时的Gree函数．首先，利用Hankel积分变换方法，直接对频域内的Biot波动方程进行求解，得出Biot波动方程的通解；利用通解和半空间内部作用水平力时边界上的边界条件，以及力作用面上的连续性条件，可以得出上述边值问题的解；对于边值问题在变换域内的解进行相应的逆变换，就可以得出频域内的Gree函数．本文得到的线弹性退化解与文献中的结果吻合．最后，文中给出了两个算例．      

1-D and 2-D spatial frequency analysis of self-potential field anomalies,produced by a polarized sphere

This paper gives analytical expressions for the 1-D and 2-D frequency spectra of the self-potential field produced by a polarized sphere. In 1-D, the amplitude spectrum of the potential field leads to a criterion for determination of the depthh to the centre of the sphere. The polarization angle of the buried sphere can be calculated from the maximum point of the amplitude spectrum of the electric field. In 2-D, the depth to the centre of the polarized sphere can be calculated if the polarization is vertical.     

The Green''''s function of the harmonic horizontal force applied at the interior of the saturated half-space soil

IntroductionThe wave propagation problems in saturated soil are very important for the civil engineering, geophysics and seismology. Biot (1956,1962) established the theory of wave propagation in saturated soil firstly, and hereafter many researchers have used Biot theory to study wave propagation problems in saturated soil. By using integral transform and potential function method, Philippacopoulos (1988) studied the Lamb(s problem of a vertical point force applied to the surface of saturate…