区域面波群速度反演的球谐函数法

一个定义在球面局部区域的复杂的面波速度函数如果直接利用球谐函数拟合可能需要展开到很高阶的球谐系数．通过保角变换，把一个球面局部区域扩展到球面上更大的区域上，变换过程中面波速度保持不变，在变换后的球面域上用球谐函数来拟合速度函数，达到降低球谐系数阶数的目的，使面波群速度的反演变成了球谐系数的线性化反演.通过球谐系数分析，可得到反演的分辨率.该方法不仅适用于面波群速度反演，同样适用于各种球面区域场的分析.     

保角变换在区域面波群速度反演中的应用

通过保角变换，把一个球面局部区域扩展到球面上更大的球台域，再由球台域解析延拓到整个球面域.作为约束条件，变换过程中面波速度保持不变.在变换后的球面域上用球谐函数来拟合速度函数，达到降低球谐系数阶数的目的.面波群速度的反演变成了球谐系数的线性化反演，由球谐系数计算反演分辨核以及方差.该方法的优点是计算速度快，等值线光滑，构造界限清晰，该方法同样适用于各种局部球面区域的分析.     

波动方程反问题与广义RADON变换反演的一种关系

本文把波动方程反问题与广义Radon 变换的反演相联系。在假定弱散射条件下,把波动方程反问题转化成广义Radon 变换的反演问题,即如何从一系列关于目标函数在某类子流形上的积分值,去求出目标函数.这种转化提供了一种研究波动方程反问题的途径。     

用小波变换方法反演接收函数

提出了一种用小波变换方法反演接收函数的新方法. 通过对接收函数作离散小波变换，将接收函数展开到不同分辨尺度，从一给定的初始模型出发，分别在不同分辨尺度上用广义线性反演方法，对展开后的接收函数进行反演. 并将低阶接收函数的反演结果作为高阶接收函数的初始模型，在大尺度空间找到包含全局极小值的一个邻域，并逐步缩小该邻域.渐进地获取介质结构的跳变信息，从而保证反演结果稳定地收敛到全局极小点，降低接收函数波形反演对初始模型的依赖，尽可能克服波形反演的非唯一性，得到比较可靠的高分辨率的地壳上地幔速度结构．      

论如何根据某些流形上的积分以确定函数

如果两个变量x,y的函数(即平面上的点函数f(p)满足适当的正则条件,并沿着任意一条直线g积分,那么这些积分值F(g)就定义了一个线函数。本论文第一部分解决的问题是,这个泛函变换的反演,即给出下列问题的答案:我们通过这种方法能否得到满足适当正则条件的每一个线函数?若能,那么F是否唯一确定f?并且怎样求?     

基于广义模型约束的时间域航空电磁反演研究

由于航空电磁具有海量数据,因此快速有效的成像和反演手段至关重要.本文针对层状介质模型推导与实现了广义模型约束条件下时间域航空电磁一维反演.从正则化反演的目标函数出发,通过改变模型约束项构造Lp范数反演和聚焦反演,进而通过改变模型求解域构造出基于小波变换的稀疏约束反演.针对不同反演方法目标函数的构建方式,本文进一步从数学原理上分析不同反演方法的预期效果,并通过理论模型和实测数据进行验证.结果表明L0.8范数反演、聚焦反演和基于小波变换的稀疏约束反演可以得到更符合地下层状介质陡变界面的反演结果.     

关于区域性的地震面波层析反演方法的讨论

讨论了在地球表面局部区域进行面波层析反演时所用的各种反演方法，指出各种方法用于区域性层析反演时的优缺点，并就由面波资料对区域性的地壳和上地幔结构进行反演的方法──分块反演方法、本征函数展开法、球面雷当变换方法、Ｔａｒａｎｔｏｌａ概率方法和波形反演的方法进行了讨论。指出了各种方法的优缺点及适用范围，并对一些方法进行了数值模拟，给出了计算实例。关键词     

一类广义Radon变换的逆问题

由于CT技术的出现,Radon变换受到人们的重视,已被应用于许多领域,但是,还有一些问题(如地球物理中的层析成像问题),已知的数据往往是未知函数在曲线上的积分值,而不是古典Radon变换中的直线。这就使得人们不得不考虑所谓广义的Radon变换:即已知点函数在曲线上的积分值,要求恢复此函数。使我们感到欣慰的是,这方面的工作已取得了不少的进展。     

遗传算法反演地球等离子体层离子密度分布

本文介绍了采用一维遗传算法从地球等离子体层极紫外图像反演地球等离子体层He⁺密度的原理.首先采用通量管近似和磁偶极近似将三维问题转化为一维问题.通过引入权矩阵,将极紫外光强积分离散为求和函数,再采用一维实数编码遗传算法反演得到磁赤道面等离子体层He⁺密度,最后通过磁力线追迹得到三维密度分布.算法采用动态全球核心等离子体模式模拟的密度和光强分布作为初始输入参数,并通过遗传算法得到相应密度分布.反演结果表明,等离子体层密度相对误差在8%以内,光强相对误差趋于0,算法有效可行.本文研究为中国探月二期工程中月基极紫外图像反演奠定了基础.     

沿圆曲线的Radon变换数值解

研究具有紧支集且在支集内连续的二元函数沿上半圆曲线的Radon变换反演问题。基于对投影函数的Fourier变换，反演问题可以归结为具有弱奇性及震荡核的Abel积分方程的求解。我们证明了当圆曲线中心及半径在一定范围内变化时，在已知沿上半圆曲线的Radon变换情况下，这个积分方程的解具有唯一性，并给出了消除Abel积分方程弱奇性的数值方法。在考虑投影数据噪声的情况下，给出了多次加权改善系数矩阵条件数稳定的数值方法，并通过数值模拟验证所提出方法的有效性。      

单程波算子积分解的象征表示

单程波波场延拓算子在地震偏移成像中有重要应用.单程波波场延拓算子按其实现方式可分为Kirchhoff积分、空间隐式有限差分和Fourier变换方法,他们代表了算子的不同表示方法,当截断使用这些方法时会得到不同的精度.象征表示对这些方法的导出和精度分析有重要作用.算子作用于正弦波函数所得函数称为算子的象征.算子的象征是褶积算子Fourier变换的推广.Fourier变换方法则直接用象征函数的可分表示求出.空间隐式有限差分则可以用象征函数的Padè近似或部分分式导出.单程波算子在深度域的积分称为单程波算子积分解.本文推导了单程波算子积分解的象征表达式,给出了算子象征的代数运算的头几阶表达式,这些表达式还未在前人文献中发现.Kirchhoff积分所需格林函数可以通过象征函数和鞍点法导出.基于积分解的象征表达式给出了非对称走时公式,对改善Kirchhoff积分的聚焦性能有重要意义.     

A complex analysis approach to the motion of uniform vortices

A new mathematical approach to kinematics and dynamics of planar uniform vortices in an incompressible inviscid fluid is presented. It is based on an integral relation between Schwarz function of the vortex boundary and induced velocity. This relation is firstly used for investigating the kinematics of a vortex having its Schwarz function with two simple poles in a transformed plane. The vortex boundary is the image of the unit circle through the conformal map obtained by conjugating its Schwarz function. The resulting analysis is based on geometric and algebraic properties of that map. Moreover, it is shown that the steady configurations of a uniform vortex, possibly in presence of point vortices, can be also investigated by means of the integral relation. The vortex equilibria are divided in two classes, depending on the behavior of the velocity on the boundary, measured in a reference system rotating with this curve. If it vanishes, the analysis is rather simple. However, vortices having nonvanishing relative velocity are also investigated, in presence of a polygonal symmetry. In order to study the vortex dynamics, the definition of Schwarz function is then extended to a Lagrangian framework. This Lagrangian Schwarz function solves a nonlinear integrodifferential Cauchy problem, that is transformed in a singular integral equation. Its analytical solution is here approached in terms of successive approximations. The self-induced dynamics, as well as the interactions with a point vortex, or between two uniform vortices are analyzed.     

Numerical method for calculating the surface wave field in the presence of caustics

Due to the lateral heterogeneity of the upper layers of the Earth, paths of surface waves deviate from arcs of great circles. Because of the sphericity of the Earth, the paths intersect on a hemisphere opposite to the epicenter and form caustics consisting of two branches, with their tangent point being a cusp. For this reason, the field of surface waves cannot be analyzed in terms of the ray theory at distances larger than 90°. The asymptotic approach to the analysis of the field in the vicinity of such caustics is very ill-suited for numerical implementation. The difficulties of such an approach to the field calculation are aggravated by the fact that such caustics are superimposed in some regions. Therefore, it is suggested to use the theorem of representation, according to which the field within a certain contour is expressed as an integral whose integrand contains values of the function itself, its derivative along the normal to the contour, and Green’s function. The field on the contour (the circle bounding a hemisphere centered at the epicenter) is calculated by the ray method because rays do not intersect on this hemisphere. These data are used for the construction of the field on the opposite hemisphere assumed to be homogeneous, which enables the construction of Green’s function for this hemisphere. This limitation is not very stringent because the configuration of rays and caustics on this hemisphere is mainly determined by the field on the circle. The integral in the representation theorem is calculated numerically. Numerical examples are presented for models in which one caustic or two superimposed caustics form. These calculations yield constraints on variations in the amplitude and phase of the wave. Rayleigh wave fields are also calculated for a model of the real Earth. It is shown that, at some points, the Rayleigh wave spectrum can be strongly distorted because caustics corresponding to different periods differ in shape.     

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.     

三维层状黏弹性半空间中球面SH、P和SV波源自由场

采用刚度矩阵方法结合Hankel积分变换,求解了层状黏弹性半空间中球面SH、P和SV波的自由波场.首先,在柱坐标系下建立层状黏弹性半空间的反轴对称(柱面SH波)和轴对称(柱面P-SV波)情况精确动力刚度矩阵.进而由Hankel变换将空间域内的球面波展开为波数域内柱面波的叠加,然后将球面波源所在层的上下端面固定,求得固定层内的动力响应和固定端面反力,将固端反力反向施加到层状黏弹性半空间上,采用直接刚度法求得固端反力的动力响应,叠加固定层内和固端反力动力响应,求得波数域内球面波源动力响应.最后由Hankel积分逆变换求得频率-空间域内球面波源自由场,时域结果由傅里叶逆变换求得.文中验证了方法的正确性,并以均匀半空间和基岩上单一土层中球面SH、P和SV波为例分别在频域和时域内进行了数值计算分析.研究表明基岩上单一土层中球面波自由场与均匀半空间情况有着本质差异;基岩上单一土层中球面波位移频谱峰值频率与场地固有频率相对应,基岩面的存在使得基岩上单一土层地表点的位移时程非常复杂,振动持续时间明显增长;阻尼的增大显著降低了动力响应的峰值,同时也显著减少了波在土层的往复次数.     

Transient electromagnetic response of a sphere in a layered medium

The integral equation for the electromagnetic response of a sphere in a layered medium may be solved as follows. First, the unknown time harmonic electric field in the sphere is expanded in spherical vector waves. Secondly, the coefficients for these wave functions are found by a set of equations. The equations are found by multiplying the integral equation throughout by each wave function and integrating over the spherical conductor.Once the unknown coefficients have been determined, then the transient response may be found by taking the inverse Fourier transform. In carrying out the Fourier transform one learns that for most of the time range used in prospecting, only the lowest order vector wave function is significant. A study of the singularities of the spectrum of the transient shows that, for the time range considered, only a single branch cut is significant. There are no pole type responses. That is, the field does not decay exponentially. Previous studies of a sphere in free space reported only pole type responses. That is, at the later stages, the field decays exponentially. This study shows that, in order to model satisfactorily the effect of the host rock on transient electromagnetic fields, the sphere must be placed in layered ground.     

Continuous wavelet-like filter for a spherical surface and its application to localized admittance function on Mars

We have developed a 2D isotropic continuous wavelet-like transform for a spherical surface. The transform is simply defined as the surface convolution between the original field and a kernel, based on the zeroth-order Bessel function with a spherical correction. This spherical correction violates the geometric similarity for the various scales of the kernels, which becomes more apparent at longer wavelengths. We found numerically that this transform is practically equivalent to a Gaussian bandpass filter in the spherical harmonic domain. We have applied this wavelet-like transform on the recently acquired Martian gravity and topography fields. Using a ratio constructed locally from these two fields, we have constructed a map describing the lateral variations of the localized admittance function on Mars.     

Spheroidal Integral Equations for Geodetic Inversion of Geopotential Gradients

The static Earth’s gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth’s surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet’s boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green’s kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth.     

Iterative Spherical Downward Continuation Applied to Magnetic and Gravitational Data from Satellite

In the last few decades, satellites have acquired various potential data sets hundreds of kilometers above the Earth’s surface. Conventionally, these global magnetic and gravitational data sets are approximated by using spherical harmonics that allow straightforward work with both fields outside the Earth’s mass. In this article, we present an alternative approach for working with potential data in mass-free space given over a regular coordinate grid on a spherical surface. The algorithm is based on an iterative scheme and the Poisson integral equation for the sphere. With help from the Fourier transform, global potential (magnetic or gravitational) data can efficiently be continued from a mean orbital sphere down to a reference surface without using the spherical harmonics. This is illustrated both with simulated magnetic field data and with real data from the satellite gradiometry mission GOCE. In the case of simulated magnetic data and the downward continuation for 450 km, we have achieved a root mean square at the level of 0.05 nT, while it was <1 E (eotvos) for real GOCE data continued for 250 km. The crucial point is to apply the algorithm twice as a large part of noise can be removed from the input data.     

地形表面反射强度的数学层析成像方法

若二维ＲＦ窄波反射函数是定义在地面一圆形区域内部的，并假定该圆域表面介质是弱反射介质，那么在给定时刻，接收发射器（如机载合成孔径雷达）所接收到的近回数据可视 过该圆域的一圆弧上的线积分值，Ｍｕｍｓｏｎ，ｅｔ，ａｌ。在假定圆域半径与接收发射器运动路径到该圆域中心的距离相比很小的情况，用直线上的线积分值来代替圆弧线上的线积分值，利用古典的Ｒａｄｏｎ变换方法，给出了该圆域反射函数的成像模式。本文将克服这