Expansion of a Plane Wave into Gaussian Beams

Magnetic susceptibility measurements on topsoils have often been used during the last few years to detect anthropogenic pollution. In most cases, a Bartington susceptibility meter for field measurements was used. However, up to now, no standard procedure has been developed for carrying out such investigations. The purpose of our study was to test the compatibility of different set-ups of instruments used for this purpose and the possible influences of subjective (human) factors. Field magnetic susceptibility measurements, carried out with four different Bartington MS2D instruments in strictly defined positions, are very consistent both for low and high values. The correlation coefficient between the magnetic susceptibility values recorded with different Bartington MS2D probes reached 97–98%. A test area was mapped independently by two groups, without any restrictions concerning the choice and distribution of the measured points, but respecting a few standard conditions (e.g., measuring at a distance from tree trunks; on the flattest place possible; recording between 10–30 values per point). The resulting susceptibility maps show the same general features in both cases, suggesting that the measuring strategy applied is suitable for topsoil magnetic screening. The methodology proposed can be used to map magnetic susceptibility on a larger scale—for example Europe—providing large sets of representative data and eliminating border-transition biases and human errors.     

Decomposition of the wave field into optimized Gaussian packets

The decomposition of the wave field into optimized Gaussian packets represents a crucial step of the Gaussian packet prestack depth migration algorithm. The shape of optimized Gaussian packets, in the plane perpendicular to the central ray of the packet, depends not only on the frequency, but also on the coordinate of the intersection of the central ray of a Gaussian packet with the profile, on its arrival time, and on the component of the slowness vector along the profile. We express the amplitude of a Gaussian packet in the form of an integral transform similar to the forward coherentstate transform. Our method is suitable for a smooth distribution of the parameter determining the shape of a packet in the plane perpendicular to its central ray.     

Optimization of the shape of Gaussian beams

The applicability and accuracy of the Gaussian beam method depend on the proper choice of the shape of beams. Gaussian beams become inaccurate solutions of the elastodynamic equation if the velocity field changes considerably within the beam width. We present a procedure of determining the optimum initial shape of Gaussian beams based on minimizing the average squared widths of Gaussian beams and smoothing the distribution of the optimum parameters of Gaussian beams on the Hamiltonian hypersurface in the phase-space. The original method of smoothing represents an essential part of the algorithm, which is designed particularly for the optimization of the shape of Gaussian beams for Gaussian beam or packet migrations.     

Optimization of the shape of Gaussian beams of a fixed length

Summary The procedure of choosing the shape of Gaussian beams in order to minimize a given object function of a certain kind is proposed. The general form of the object function enables both the average square of the quadratic variation of the phase and the average square of the beamwidth to be minimized along the central ray. The error of the transformation of the Gaussian beams at the structural interfaces may also be taken into account. Most of the hitherto published suggestions of how to choose the shape of Gaussian beams are special cases of the described procedure. The aim of this paper is not to propose the object function to be minimized, but only to describe the minimization of a given object function. The minimization assumes the a priori known lengths of the central rays of the Gaussian beams (i.e. the lengths of the beams are not free parameters in the minimization procedure).

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A tale of two beams: an elementary overview of Gaussian beams and Bessel beams

An overview of two types of beam solutions is presented, Gaussian beams and Bessel beams. Gaussian beams are examples of non-localized or diffracting beam solutions, and Bessel beams are example of localized, non-diffracting beam solutions. Gaussian beams stay bounded over a certain propagation range after which they diverge. Bessel beams are among a class of solutions to the wave equation that are ideally diffraction-free and do not diverge when they propagate. They can be described by plane waves with normal vectors along a cone with a fixed angle from the beam propagation direction. X-waves are an example of pulsed beams that propagate in an undistorted fashion. For realizable localized beam solutions, Bessel beams must ultimately be windowed by an aperture, and for a Gaussian tapered window function this results in Bessel-Gauss beams. Bessel-Gauss beams can also be realized by a combination of Gaussian beams propagating along a cone with a fixed opening angle. Depending on the beam parameters, Bessel-Gauss beams can be used to describe a range of beams solutions with Gaussian beams and Bessel beams as end-members. Both Gaussian beams, as well as limited diffraction beams, can be used as building blocks for the modeling and synthesis of other types of wave fields. In seismology and geophysics, limited diffraction beams have the potential of providing improved controllability of the beam solutions and a large depth of focus in the subsurface for seismic imaging.     

地震数据的稀疏高斯束分解方法

本文研究了地震数据的稀疏分解问题.提出了一种用高斯束稀疏分解表示地震数据的方法.这是一个拟0范数约束优化问题.在求解拟0范数极小化问题的过程中,通过扫描同相轴的方法实现高斯束稀疏分解,数值实现上提出了使用一种快速单调下降的梯度优化方法.本文提出的稀疏优化方法同时具有去噪的功能,数据模拟试验表明了本方法的可行性和可靠性.     

Gaussian beams in inhomogeneous media: A review

The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich’s method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich’s method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich’s method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names. The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.     

Diffraction of an arbitrary wave due to a half plane

Summary Diffraction problems of an arbitrary wave by a half plane is solved exactly with the help of the integral equation technic. The solution is made to depend on a simple real quadrature which readily evaluates in exact forms, for different complicated type of incident fields. The method is supposed to produce new results, some of which are placed in the paper. The problem of Sommer-feld's plane wave diffraction is also solved as a very simple special case.     

Interaction between parallel Gaussian electromagnetic beams in the ionosphere

In this paper, the propagation of two initially (z=0) parallel Gaussian electromagnetic beams, propagating in the z-direction in ionosphere, has been investigated. The nonlinearity in the dielectric function, responsible for the interaction between the beams arises from the redistribution of the electron density, caused by the nonuniform distribution of electron temperature determined by the Ohmic heating of the electrons and the energy loss of electrons on account of collisions. The wave frequencies have been assumed to be much larger than the electron collision frequency and gyrofrequency. A self-consistent solution of the electromagnetic wave equation and energy balance equation (considering the solar radiation) has been obtained in the paraxial approximation. Second-order coupled ordinary differential equations have been obtained for the distance between the centers of the beams and the beam widths in the x and y directions as a function of the distance of propagation along the z-axis. Using the available database for the mid-latitude daytime ionosphere, the equations have been solved numerically for a range of parameters and a discussion of the results has been presented. The physical basis for the fact that the beams move towards each other, when the resulting irradiance distribution of the two beams has a maximum in the space between the two beams, has been highlighted.     

The relation between Gaussian beams and Maslov asymptotic theory

Summary The application of Maslov asymptotic theory in a general 3-D mixed subspace of 6-D complex phase space is proposed to obtain the integral superpositions of Gaussian packets and beams. The ray method and the superposition of plane waves (Maslov method of Chapman and Drummond [7]) are special limiting cases of the above mentioned approach. The same high-frequency asymptotic expansion formulae for seismic body waves were derived previously in [8] using the Gaussian beam method.

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基于平面波照明的偏移成像补偿

受地下复杂构造和地震数据采集系统的影响,地震波对地下目标的照明出现不均匀,在地震数据的偏移成像中出现成像阴影.根据地震数据最小二乘偏移/反演理论,和把地震波场照明结果作为最小二乘偏移/反演中的Hessian矩阵的近似对偏移成像进行补偿的原理,提出一种应用平面波照明结果对平面波偏移成像结果进行补偿以消除偏移成像阴影的方法.这种基于平面波照明的偏移成像补偿方法相对于局部角度域的照明偏移成像补偿方法具有计算效率上的优势.     

Prestack shot-gather depth migration by a rigid flow of Gaussian wave packets

We introduce a new method for prestack depth migration of seismic common-shot gathers. The computational procedure follows standard steps of the reverse-time migration, i.e., downward continuation of the source and the receiver wavefields, followed by application of an imaging condition (e.g. zero-lag cross-correlation of these fields). In our method we first find a sparse data representation with a small number of Gaussian wave packets. We then approximate the downward wavefield propagation (for the source and the receiver fields) by a rigid flow of these wave packets along seismic rays. In this case, the wave packets are simply translated and rotated according to the ray geometry. One advantage of using Gaussian wave packets is that analytic formulas can be used for translation, rotation, and the application of the cross-correlation imaging condition. Moreover, they allow more sparse representations than competing methods. Finally we formulate a computationally and memory efficient migration procedure, as only few rays have to be traced, and since it is cheap to compute the cross-correlation for the intersecting rays.     

基于黏声介质的角度域高斯束逆时偏移方法研究

高斯束逆时偏移是一种兼具计算效率和成像精度的深度域成像方法,能够面向目标成像.地下介质中黏滞性普遍存在,利用传统各向同性或完全弹性的成像方法处理黏滞性探区的数据会降低分辨率,并导致成像位置不准确和振幅欠估计等问题.本文在高斯束逆时偏移的基础上,通过对震源点和检波点处的波场进行衰减补偿,并结合高斯束求解时的角度信息,实现了黏声介质角度域高斯束逆时偏移方法.最后通过模型和实际资料试算对本文方法的正确性和适用性进行了验证.试算结果表明:相比于声波高斯束逆时偏移,本文方法能够对黏滞性引起的吸收衰减进行有效补偿,同时提取的角度域共成像点道集(ADCIGs)不仅可以用于分角度叠加成像压制成像噪声,而且能够为后续的偏移速度分析提供支撑.     

Expansion of geoid heights into a spherical harmonic series

Summary An algorithm is derived to compute the coefficients of a spherical harmonic series for the geoid radius vector that are a solution to a system of linear equations, containing Stokes' constants of the geopotential.     

Seismic interferometry and estimation of the Green's function using Gaussian beams

This study investigates seismic interferometry in which the Green's function is estimated between two receivers by cross-correlation and integration over sources.For smoothly varying source strengths,the dominant contributions of the correlation integral come from the stationary phase directions in the forward and backward directions from the alignment of the two receivers.Gaussian beams can be used to evaluate the correlation integral and concentrate the amplitudes in a vicinity of the stationary phase regions instead of completely relying on phase interference.Several numerical examples are shown to illustrate how this process works.The use of Gaussian beams for the evaluation of the correlation integral results in stable estimates,and also provides physical insight into the estimation of the Green's function based on seismic interferometry.     

Accuracy of kinematic wave and diffusion wave for spatial‐varying rainfall excess over a plane

The kinematic‐wave and diffusive‐wave approximations were investigated for unsteady overland flow resulting from spatially varying rainfall excess. Three types of boundary conditions were adopted: zero flow at the upstream end, and critical flow and zero depth‐gradient at the downstream end. Errors were derived by comparing the dimensionless profiles of the flow depth over the plane with those computed from the dynamic‐wave solution. It was found that the mean errors for both the approximations were independent of the type of rainfall excess distribution for KF₀² > 5, where K is the kinematic‐wave number and F₀ is the Froude number. Therefore, the regions (KF₀², F₀) where the kinematic‐wave and diffusive‐wave solutions would be fairly accurate and for any distribution of spatially varying rainfall, were characterized. The kinematic‐wave approximation was reasonably accurate, with a mean error of less than 5% and for the critical depth at the downstream end, for KF₀² ≥ 20 with F₀ ≤ 1; if the rainfall excess was concentrated in a portion of the plane, the field where the kinematic‐wave solution was found accurate, it was more limited and characterized for KF₀² > 35 with F₀ ≤ 1. The diffusive‐wave solution was in good agreement with the dynamic‐wave solution with a mean error of less than 5%, in the flow depth, for KF₀² ≥ 15 with F₀ ≤ 1; for rainfall excess concentrated in a portion of the plane, the accuracy of the diffusion wave solution was in a region more restricted and defined for KF₀² ≥ 30 with F₀ ≤ 1. For zero‐depth gradient at the downstream end, the accuracy field of the kinematic‐wave was found to be greater and characterized for KF₀² > 10 with F₀ ≤ 1; for rainfall excess concentrated in a portion of the plane, the region was smaller and defined for KF₀² > 15 with F₀ ≤ 1. The diffusive‐wave solution was found accurate in the region defined for KF₀² > 7·5, whereas for rainfall excess concentrated in a portion of the plane, the field of accuracy was for KF₀² > 12·5 with F₀ ≤ 1. The lower limits of the regions, defined on KF₀², can be considered generally valid for both approximations, but for F₀ < 1 smaller lower limits were also characterized. Finally, the accuracy of these approximations was influenced significantly by the downstream boundary condition. Copyright © 2002 John Wiley & Sons, Ltd.     

Theory and application of equivalent transformation relationships between plane wave and spherical wave

Based on the governing equations and the equivalent models,we propose an equivalent transformation relationships between a plane wave in a one-dimensional medium and a spherical wave in globular geometry with radially inhomogeneous properties.These equivalent relationships can help us to obtain the analytical solutions of the elastodynamic issues in an inhomogeneous medium.The physical essence of the presented equivalent transformations is the equivalent relationships between the geometry and the material properties.It indicates that the spherical wave problem in globular geometry can be transformed into the plane wave problem in the bar with variable property fields,and its inverse transformation is valid as well.Four different examples of wave motion problems in the inhomogeneous media are solved based on the presented equivalent relationships.We obtain two basic analytical solution forms in Examples Ⅰ and Ⅱ,investigate the reflection behavior of inhomogeneous half-space in Example Ⅲ,and exhibit a special inhomogeneity in Example Ⅳ,which can keep the traveling spherical wave in constant amplitude.This study implies that our idea makes solving the associated problem easier.     

Propagation of the radiofrequency ground wave transient over a finitely conducting plane earth

Summary The introduction of precision radio navigation systems employing pulse techniques and the ever increasing interest in spherics have stimulated considerable interest in the propagation of the ground wave transient over the surface of the earth. The theory of the propagation of a transient radio frequency ground wave over a finitely conducting plane earth is presented for the particular case of theNorton surface wave by a consideration of a wave, interrupted abruptly at one point in time (t=0), a wave interrupted abruptly at two points in time (t=0,T ₂) and a wave interrupted at one point in time followed by an exponential decay. The first case is illustrated by several numerical examples of a cosine current wave applied to a vertical electric dipole source. It is apparent that the method of the inverse aplace transform for the particular cases considered yields some simple mathematical formulas.     

TTI介质隐式有限差分平面波偏移

文章研究了TTI介质隐式有限差分(IFD)波场外推算子和TTI介质平面波偏移.与各向同性和VTI介质相比,TTI介质频散关系要复杂的多,很难得到频散关系的显性表达,也不能再只用对称的偶函数表示,需要加入一个奇函数.文中设计TTI介质IFD波场外推算子并用非线性优化方法求解算子系数.将各向同性介质下平面波偏移理论与TTI介质IFD外推算子相结合,实现了TTI介质下平面波偏移.将设计算子的频散关系曲线和偏移脉冲响应与理论解对比,验证了外推算子的精度;将平面波偏移算法用于全波TTI方程拟合并用极化分析得到的P波数据,验证了TTI介质平面波偏移的准确性和高效性.